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1 .
Unit I
1
INTRODUCTION TO SOCIAL NETWORK
ANALYSIS (SNA)
Unit Structure
1.0 Objectives
1.1 Introduction
1.2 Introduction to networks and relations
1.2.1 Analyzing relationships to understand people and groups
1.2.2 Binary and valued relationships
1.2.3 Symmetric and asymmetric relationships
1.2.4 Multimode relationships
1.3 Using graph theory for social networks analysis
1.3.1 Adjacency matrices
1.3.2 Edge -lists
1.3.3 Adjacency lists
1.3.4 Graph traversals and di stances
1.3.5 Depth -first traversal
1.3.6 Breadth -first traversal paths and walks
1.3.7 Dijkstra’s algorithm
1.3.8 Graph distance and graph diameter
1.3.9 Social networks vs. link analysis
1.4 Ego -centric and s ocio-centric density.
1.5 Let us Sum Up
1.6 List of References
1.7 Bibliography
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2 1.0 OBJECTIVE
 To gain knowledge about Social Network Analysis
 To learn various relations
 To use graph theory for Social Network Analysis
 To find s hortest distance using Dijkstra’s algorithm
1.1 INTRODUCTION
Social network analysis (SNA) is the process of investigating social
structures through the use of networks and graph theory. It characterizes
networked structures in terms of nodes (individual a ctors, people, or things
within the network) and the ties, edges, or links (relationships or
interactions) that connect them.
The SNA structure is made up of node entities, such as humans, and ties,
such as relationships. The advent of modern thought and c omputing
facilitated a gradual evolution of the social networking concept in the form
of highly complex, graph -based networks with many types of nodes and
ties. These networks are the key to procedures and initiatives involving
problem solving, administrat ion and operations.
1.2 INTRODUCTION TO NETWORKS AND
RELATIONS
SNA can be used by a group as a process of “learning and understanding
the (formal and informal) networks that operate in a given
field” (Hovland, 2007). This extensive form of mind -mapping all ows the
group not only to identify networks, but also to highlight the patterns of
information exchange within the network. Although networks are often
created to pass information from one individual to another, and over time
the content is also shared wit h a wider network, it also gradually grows to
take in other outside contact and networks. An SNA focuses on the
structure of the relationships that weave between people and organisations
within a network.
Within every network there are starting points and branching points. Each
individual who forms a network, whether it us within a team or an
organisation, will exchange information with others who share the same or
common beliefs and ideas. They will be likely to share information with
friends, partners, an d relatives if they find the information interesting.
Maximising the appeal of this information can increase traffic to the sites
or pages of the distributing organisation.
What is a ‘social network’?
A social network is made up of what are called ‘nodes’ (points) and
‘links’, all of which are then identifiable categories of analysis. These
include people, groups, and organisations – which are usually the main
priority and concern for any type of social examination. Links in this type
of analysis which focu s on the ‘collective’ include social contacts and munotes.in

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3 exchangeable information. It has been argued for some time that
organisations are embedded in networks of larger social processes, which
they influence, and which also influence them (Hovland, p.10).
How do we use SNA?
“A range of methods can be used, including ethnography, participant
observation, key informant interviews, semi -structured interviews,
‘snowball’ sampling, focus groups, and content analysis of the media”
(Schelhas and Cerveny, 2002, Social N etwork Analysis for Collaboration
in Natural Resource Management).
“The aim is to construct a ‘map’ of the linkages that exist between people
in this field” (Hovland, 2005).
“Social network analysis is the mapping and measuring of relationships
and flows b etween people, groups, organisations, computers or other
information/knowledge processing entities” (Valdis Krebs, 2002). Social
Network Analysis (SNA) is a method for visualizing our people and
connection power, leading us to identify how we can best inte ract to share
knowledge.
1.2.1 Analyzing relationships to understand people and groups
The science of Social Network Analysis (SNA) boils down to one central
concept —our relationships, taken together, define who we are and how we
act. Our personality, educ ation, background, race, ethnicity —all interact
with our pattern of relationships and leave indelible marks on it. Thus, by
observing and studying these patterns we can answer many questions
about our sociality.
What is a relationship? In an interpersonal context, it can be friendship,
influence, affection, trust —or conversely, dislike, conflict, or many other
things.
1.2.2 Binary and valued relationships
Relationships can be binary or valued: “Max follows Alex on Twitter” is a
binary relationship while “Ma x retweeted 4 tweets from Alex” is valued.
In the Twitter world, such relationships are easily quantified, but in the
“softer” social world it’s very hard to determine and quantify the quality
of an interpersonal relationship.
A useful stand -in for strengt h of an interpersonal relationship is frequency
of communication. Besides being objectively measurable, frequency of
communication has been found by scientists to reflect accurately on the
emotional content, and amount of influence in a relationship. This would,
of course, not be true in many contexts (and you, my dear reader, are
probably busy coming up with counterexamples right now) —but in many
cases, for the lack of better data, frequency of communication works.
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4 1.2.3 Symmetric and asymmetric relations hips
It is easy to see that some relationships are asymmetric by nature.
Teacher/student or boss/employee roles presume a directionality of a
relationship, and do not allow for a symmetric tie back. Following on
Twitter and LiveJournal is directional by de finition —but a follow -back tie
can exist, thus symmetrizing the relationship
Other relationships are symmetric. Facebook friends and LinkedIn
connections require mutual confirmation —the software forces a symmetry
even when the real human relationship is as ymmetric.
In the real world, friendships and romantic relationships are asymmetric,
as much as we would like them not to be that way. Hence, we struggle
with unrequited love, one -sided friendships and other delusions of
popularity. Given good data, we can study these phenomena using SNA —
but such data would be very difficult to obtain and subject to self -reporting
and other biases.
1.2.4 Multimode relationships
Finally, we should mention that relationships can exist between actors of
different types —Corporat ions employ People, Investors buy stock in
Corporations, People possess Information and Resources, and so on. All of
these ties are described as bimodal or 2-mode
1.3 USING GRAPH THEORY FOR SOCIAL NETWORKS
ANALYSIS
What is graph theory?
The first thing you should know is that a graph is a mathematical structure
that allows you to represent everyday problems in a graphic way. In
addition, network theory allows you to represent only one type of
relationship (simple representation), but it also allows you to r epresent
more than one type (in that case, it would be called multiple).
Even eminent figures such as the founder of Facebook, Mark Zuckenberg,
have spoken of “social graphs” to represent the connections and
relationships that users of the social network h ave.
Graph theory is a branch of mathematics, the same branch that is also used
in computer science. It is based on both discrete and applied mathematics.
In this way, it manages to encompass different concepts.
Applying graph theory to social networks
Let’s think about the commercial strategy that any telecommunication
company seeking to know the composition of the links would carry out.
You would be interested in knowing which people we usually talk to and
thus adapt your commercial strategy to offer pers onalized offers and/or
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In addition to this, applying the networks to social networks can work to
adapt the products to the real needs, making them appear at the right time.
When we talk about networks applied to social networks the most
common is that they are used to “detect communities”. Thanks to the
algorithms we can see characteristics, attributes and relationships that
match within a group. When we analyze the subnetworks, we can see the
vertices that are most related to each other, and also how they relate to the
rest of the vertices.
If we look at the graph above, we can see that three different communities
have been detected, in which we can assume that all the members of the
same community have characteristics or attributes that coincide.
1.3.1 Adjacency matrices
Every network can be expressed mathematically in the form of an
adjacency matrix (Figure 4). In these matrices the rows and columns are
assigned to the nodes in the network and the presence of an edge is
symbolised by a numerical v alue. By using the matrix representation of
the network, we can calculate network properties such as degree, and other
centralities by applying basic concepts from linear algebra (see later in the
course).

Figure 4 Graphs by edge type and their adjacenc y matrices. munotes.in

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6 A network with undirected, unweighted edges will be represented by a
symmetric matrix containing only the values 1 and 0 to represent the
presence and absence of connections, respectively.
Directed and weighted networks can make use of differen t numerical
values in the matrix to express these more complex relationships. The sign
of the values, for example, is sometimes used to indicate stimulation or
inhibition.
1.3.2 Edge -lists
An edgelist is usually formatted as a table where the first two col umns
contain the IDs of a pair of nodes in the network that have a tie between
them. Optional additional columns may contain properties of the
relationship between the nodes (e.g., the value of a tie). Any pair of nodes
that does not have a tie between the m is usually not included in an
edgelist. This property is what makes edgelists a more efficient network
data storage format than sociomatrices (see below). Unobserved edges can
be encoded in edgelist format by including “NA” in the value column.
Here’s an example of a simple edgelist table with a value column:
Ego Alter Value
Harry Hermione 5
Harry Ron 6
Hermione Ron 5
The columns in an edgelist table are usually ordered “Ego” (often the
person who completed the interview or who was the subject of a fo cal
follow) followed by “Alter” (the person that the focal individual named or
interacted with). In the case of undirected network data, the ordering of
these columns does not matter, but in directed data, it
does. igraph and statnet software will encode d irected edgelist data with the
arrow pointing from the first to the second column, so if the ties you
recorded have reversed directionality from ego to alter (e.g., the alter gave
something to ego) you should flip the order of the columns before
converting the data to a network in order to get the edges properly directed
(if you want your network to show the direction that support flows in the
network). Most directed ties are straightforward to interpret but sometimes
it gets complicated, depending on your research design.
The final column in our imaginary edgelist contains a value for the edge. It
might be the number of years the pair have known each other, or some
measure of the strength or quality of their relationship.
1.3.3 Adjacency lists
An adjacency list is-- a hybrid between an adjacency matrix and an edge
list. An adjacency list is an array of linked lists that serves the purpose of
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7 easy to see which vertices are adjacent to any other vertices.
Each vertex in a graph can easily reference its neighbors through a linked
list.
Due to this, an adjacency list is the most common representation of
a graph. Another reason is that graph traversal problems often require us
to be able to easily figure out which nodes are the neighbors of another
node. In most graph traversal interview problems, we don't really need to
build the entire graph. Rather, it's important to know where we can travel
(or in other words, who the neighbors of a node are).
1.3.4 Graph traversals and distances
Graph traversal (also known as graph search) refers to the process of
visiting (checking and/or updating) each vertex in a graph. Such traversals
are classified by the order in which the vertices are visited. Tree
traversal is a special case of graph traversal . For any two locations in a
spatial network, their network distance is the length of the shortest path
between these two locations along the network. The shortest path is
computed based on the travel weight, such as travel distance or travel
time, of network edges
1.3.5 Depth -first traversal
Depth -first search (DFS) is an algorithm for traversing or searching tree or
graph data structures. The algorithm starts at the root node (selecting some
arbitrary node a s the root node in the case of a graph) and explores as far
as possible along each branch before backtracking.
1.3.6 Breadth -first traversal paths and walks
Breadth -first search (BFS) is an algorithm for searching a tree data
structure for a node that sati sfies a given property. It starts at the tree root
and explores all nodes at the present depth prior to moving on to the nodes
at the next depth level.
1.3.7 Dijkstra’s algorithm
Dijkstra’s algorithm is very similar to Prim’s algorithm for minimum
spanning tree. Like Prim’s MST, we generate a SPT (shortest path
tree) with a given source as a root. We maintain two sets, one set
contains vertices included in the shortest -path tree, other set includes
vertices not yet included in the shortest -path tree. At eve ry step of the
algorithm, we find a vertex that is in the other set (set of not yet
included) and has a minimum distance from thesource.
Below are the detailed steps used in Dijkstra’s algorithm to find the
shortest path from a single source vertex to all other vertices in the given
graph.
Algorithm
1) Create a set sptSet (shortest path tree set) that keeps track of vertices
included in the shortest -path tree, i.e., whose minimum distance from the
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8 2) Assign a distance value to all vertices in the input graph. Initialize all
distance values as INFINITE. Assign distance value as 0 for the source
vertex so that it is picked first.
3) While sptSet doesn’t include all vertices
a) Pick a vertex u which is not there in sptSet and has a minimum
distance value.
b) Include u to sptSet .
c) Update distance value of all adjacent vertices of u. To update the
distance values, iterate through all adjacent vertices. For every adjacent
vertex v, if the su m of distance value of u (from source) and weight of
edge u -v, is less than the distance value of v, then update the distance
value of v.
Let us understand with the following example:

The set sptSet is initially empty and distances assigned to vertice s are {0,
INF, INF, INF, INF, INF, INF, INF} where INF indicates infinite. Now
pick the vertex with a minimum distance value. The vertex 0 is picked,
include it in sptSet . So sptSet becomes {0}. After including 0 to sptSet ,
update distance values of its ad jacent vertices. Adjacent vertices of 0 are
1 and 7. The distance values of 1 and 7 are updated as 4 and 8. The
following subgraph shows vertices and their distance values, only the
vertices with finite distance values are shown. The vertices included in
SPT are shown in green colour.

Pick the vertex with minimum distance value and not already included in
SPT (not in sptSET). The vertex 1 is picked and added to sptSet. So
sptSet now becomes {0, 1}. Update the distance values of adjacent
vertices of 1. The distance value of vertex 2 becomes 12.
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Pick the vertex with minimum distance value and not already included in
SPT (not in sptSET). Vertex 7 is picked. So sptSet now becomes {0, 1,
7}. Update the distance values of adjacent vertices of 7. The distance
value of vertex 6 and 8 becomes finite (15 and 9 respectively).


Pick the vertex with minimum distance value and not already included in
SPT (not in sptSET). Vertex 6 is picked. So sptSet now becomes {0, 1, 7,
6}. Update the distance values of adjacen t vertices of 6. The distance
value of vertex 5 and 8 are updated.

We repeat the above steps until sptSet includes all vertices of the given
graph. Finally, we get the following Shortest Path Tree (SPT).



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10 1.3.8 Graph Distance and Graph Diameter
The distance between two vertices in a graph is the number of edges in a
shortest or minimal path. It gives the available minimum distance
between two edges. There can exist more than one shortest path between
two vertices.


Shortest Distance between 1 - 5 is 2 1 → 2 → 5
Diameter of graph –
The diameter of graph is the maximum distance between the pair of
vertices. It can also be defined as the maximal distance between the pair
of vertices. Way to solve it is to find all the paths and then find the
maximum of all .


Diameter: 3 BC → CF → FG
1.3.9 Social networks vs. link analysis
Another cousin to SNA is Link Analysis (LI). Some of you may have used
LI in business intelligence or law enforcement work or seen it on TV.
“Without a Trace” uses link analysis in ev ery episode; “Numbers” and
“Law and Order” resort to it on occasion.Link analysis is in many ways
similar to SNA —both talk about relationships in terms of nodes and edges
(Figure 1-1) and both try to derive the idea of who is more important in a
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However, LI allows for a mixing of differ ent node and edge types in the
same network —i.e, “ A gave $300 to B to procure drugs for C”. In this
example, bold words are nodes, or actors, and italic are actions, or edges.
The problem is understanding on a quantitative level whether the act of
giving m oney is different than the act of procuring drugs —and thus LI
relies on human -level understanding of language and is qualitative in its
pure form.
Most link analysis tools, including Analyst’s Notebook and Palantir,
include qualitative data gathering and t ools for qualitative decision -
making, and these are excellent and utilized widely in a number of
communities. However, the application of quantitative metrics centrality
measures is dangerous because mixing nodes and edges of different
meanings (e.g., mone y and telephone calls) produces a result that is
mathematically invalid. Unfortunately, this does not stop the software
from computing these metrics.
1.4 EGO -CENTRIC AND SOCIO -CENTRIC DENSITY
Ego-centric networks (or shortened to “ego” networks) are a part icular
type of network which specifically maps the connections of and from the
perspective of a single person (an “ego”). For example, if you were to ask
someone to name their friends (or any other type of “alter,” which is
defined as someone who is not th e ego), they will tell you who their
friends are. However, they will not tell you who the friends of other
people are. While in Figure 1.3 and 1.4 we had graphs of complete
networks, in this case, we are only getting a small part of the overall social
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Sociocentric network analysis involves the quantification of relationships
between people within a defined group – a classroom of children, a board
of directors, the residents of a village or town, the trading partners in a
bloc of nations.
1.5 LET US SUM UP
 Social network analysis (SNA) is the process of investigating social
structures through the use of networks and graph theory
 Graph theory is a branch of mathematics, the same branch that is
also used in computer science. It is based on both discr ete and
applied mathematics.
 The distance between two vertices in a graph is the number of
edges in a shortest or minimal path.
 Ego-centric networks are a particular type of network which specifically
maps the connections of and from the perspective of a s ingle person.
 Sociocentric network analysis involves the quantification of
relationships between people within a defined group.
 In Dijkstra’s algorithm to find the shortest path from a single
source vertex to all other vertices in the given graph.
1.6 LIS T OF REFERENCES
 Social Network Analysis: Methods and Applications by Katherine
Faust and Stanley Wasserman
 Social network analysis by John Scott
 The SAGE Handbook of Social Network Analysis
1.7 BIBLIOGRAPHY
 https://www.techopedia.com/definition/3205/social -network -analysis -
sna
 https://www.oreilly.com/library/view/social -network -
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13  https://www.ebi.ac.uk/training/online/courses/network -analysis -of-
protein -interaction -data-an-introduction/introduction -to-graph -
theory/graph -theory -adjacency -matrices/
 https://eehh -stanford.github.io/SNA -workshop/data -
import.html#edgelists
 https://algodaily.com/lessons/implementing -graphs -edge -list-
adjacency -list-adjacency -matrix
 https://www.geeksforgeeks.org/dijkstras -shortest -path-algorithm -
greedy -algo-7/
 https://www.geeksforgeeks.org/graph -measurements -length -distance -
diameter -eccentricity -radius -center/
 https://bookdown.org/omarlizardo/_main/2 -10-ego-centric -
networks.html
 Social network analysis by John Scott
 The SAGE Handbook of Social Network Analysis
1.8 UNIT END EXERCISE
1. Explain Social Network Analysis.
2. Briefly Explain
a) Adjacency matrices
b) Edge -lists
c) Adjacency lists
d) Graph traversals and distances
e) Depth -first traversal
f) Breadth -first traversal paths and walks
3. Explain Dijkstra’s algorithm
4. Short notes on: Graph distance and graph diameter
5. Explain Social networks vs. link analysis
6. Explain Ego -centric and socio -centric density.

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14 2
NETWORKS , CENTRA LITY AND
CENTRALIZATION IN SNA
Unit Structure
2.0 Objective
2.1 Understanding Networks
2.1.1 Density
2.1.2. Reachability
2.1.3. Connectivity
2.1.4. Reciprocity
2.1.5. Group -external and Group -internal ties in networks
2.1.6. Ego networks
2.1.7. Extracting and visualizing ego networks
2.1.8. Structural holes
2.2. Centrality
2.2.1. Degree of centrality
2.2.2. Closeness Centrality
2.2.3. Betweenness centrality
2.2.4. Local and global centrality
2.2.5. Centralization and graph centers
2.2.5. Notion of importance wi thin network
2.2.6. Google Pagerank Algorithm
2.3. Analyzing Network Structure
2.3.1. Bottom -up approaches
2.3.1.1. Cliques
2.3.1.2. N -cliques
2.3.1.3. N -clans
2.3.1.4. K -plexes
2.3.1.5. K -cores
2.3.1.6. F -groups
2.3.2. Top -down approaches
2.3.2.1. Compon ents
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15 2.3.2.3. Lambda sets and bridges
2.3.2.4. Factions.
2.4 Summary
2.5 Reference for further reading
2.6. Model Questions
2.0 OBJECTIVES
After going through this unit, you will be able to:
 Understand Network structure and i ts components
 Explain Connectivity components of networks
 Define Network centrality components like density, reachability
connectivity and reciprocity
 Describe the Structure holes in the networking structure
 Analyse of the network structure using the SNA structures cliques,
components
 Practice Google page ranking algorithm
2.1. UNDERSTANDING NETWORKS
Network consists of nodes and edges. Nodes are also called actors or
vertices. Edges are also called links or ties. In social Network Analysis
nodes represent people. The actors establish the relationship through edges
with each other. The figure 2.1.1. represents the structure of network.













Node D
Node A
Node A
A
Node E
Node C
Node B
Edge Edge Edge Edge Edge Edge
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2.1.1 Density
The proportions of all the possible ties or connections denote the density
of a network. For a complete valued network the density is calculated as


The ratio between the total number of ties with the total number of
possible ties gives the density of the network. The density drives the social
capita and social constraints among a network.
Knoke and Kuklinski (1982) selected a subset of 10 organizations with
two relationships from 95 organizations with 13 different types of
relationships. In this the Money exchange is recorded in KNOKM,
information exchange in KNOKI which is used for analysis h ere. In the
Matrix 1 represents no ties and 2 represents there exists a relationship.
When calculating density self -ties are ignored and cohesion is taking onto
consideration. The density of Matrix #1 is different from density of
Matrix #2 which shows the density depends on the relationship exists
within the actors based on the criteria selected.
Table 2 .1.1. .A part of KNOKI Data Set

Matrix#1 KNOKI
COUN COMM EDUC INDU MAYR WRO NEWS UWAY WELF WEST
COUN 0 1 0 0 1 0 1 0 1 0
COMM 1 0 1 1 1 0 1 1 1 0
EDUC 0 1 0 1 1 1 1 0 0 1
INDU 1 1 0 0 1 0 1 0 0 0
MAYR 1 1 1 1 0 0 1 1 1 1
WRO 0 0 1 0 0 0 1 0 1 0
NEWS 0 1 0 1 1 0 0 0 0 0
UWAY 1 1 0 1 1 0 1 0 1 0
WELF 0 1 0 0 1 0 1 0 0 0
WEST 1 1 1 0 1 0 1 0 0 0

Density 0.49
Standard Deviation 0.4999 Density =

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Table 2.1.2. A part of KNOKM Data Set

2.1.2. Reachability
If any actor is reached by another actor then there exists a reachability of
nodes. In this case we can trace a target actor/node from a given source
node. This does not consider the number of actors lies in the middle of the
path way. In directed, asymmetric data the source can reach the target but
the target actor cannot reach the source actor. For instance there is a
directed tie from actor A to B, now B can be reachable from A, but A
cannot be reachable from B.
Table 2.1.3 Table 2.1.4
Matrix
#3 Matrix
#3
A B C D E A B C D E
A 0 1 1 0 1 A 0 1 0 0 1
B 1 1 0 0 1 B 1 0 0 0 1
C 0 0 1 1 1 C 0 0 0 1 1
D 1 1 1 0 0 D 1 0 0 0 0
E 1 0 1 1 0 E 1 0 0 1 0

Matrix#2 KNOKM
COUN COMM EDUC INDU MAYR WRO NEWS UWAY WELF WEST
COUN 1 0 1 0 1 0 0 1 1 1
COMM 0 0 1 0 0 0 0 0 0 0
EDUC 0 0 0 0 0 0 0 1 0 0
INDU 0 1 1 0 0 0 1 1 1 0
MAYR 0 1 1 0 0 0 0 1 1 0
WRO 0 0 0 0 0 0 0 0 0 0
NEWS 0 1 0 0 0 0 0 1 0 0
UWAY 0 0 0 0 0 0 0 0 1 1
WELF 0 0 1 0 0 0 0 1 0 0
WEST 0 0 0 0 0 0 0 0 0 0

Density 0.23
Standard Deviation 0.42
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18 In the given Table 2.1.3. B and C are reachable in all cases if they are
undirected. In the Table 2.1.4 it is not possible B and C are not reachable
in some cases.
2.1.3. Connectivity
Adjacency refers the direct connection between tw o actors, it may be
directed or un -directed. If there are more path ways connect two actors
then there is a high connectivity to reach the actors from different paths.
Table 2.1.5 Point connectivity of KNOKE information exchange

From the Table 2.1.5 the Actor 6 is having weak connectivity. If any Actor
refuses to send message to one then it is difficult for Actor 6 to get most of
the information.
2.1.4. Reciprocity
SymmetricDyadic data: denotes that a pair of actors that may be connected
or not. With directed data there are four possible dyadic relationships. A
and B are two actors. In directed data there are four possible dyadic
relationship/ties. They are A sends to B, B sends to A, Both sends each
other, Both do not have any connection. A network which has more
number of null or reciprocated ties over asymmetric connections are more
equal or stable network.
Degree of Reciprocity:
In a population data there are two ways of inde xing the degree of
reciprocity. They are i) Dyad Method ii) Arc Method Matrix#5
1 2 3 4 5 6 7 8 9 10
COUN COMM EDUC INDU MAYR WRO NEWS UWAY WELF WEST
1 5 5 3 4 5 1 6 4 4 3
2 5 8 3 5 8 1 6 5 3 4
3 3 3 4 4 3 1 4 3 3 3
4 5 5 3 5 5 1 5 4 3 4
5 5 8 3 5 8 1 6 5 3 5
6 1 1 1 1 1 1 2 1 2 1
7 5 6 3 5 6 1 6 4 2 3
8 5 5 3 5 5 1 5 5 4 4
9 3 3 3 3 3 1 3 3 3 3
10 4 5 3 4 5 1 4 4 3 5
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19 i) Dyad Method:
This calculates the proportion of pairs of actors that are reciprocated. In
the Figure 2.1.2. the A actor and B actor are reciprocated. But the Actor B
and Actor C are not reciprocated. Possible reciprocations are AB, BC, AC
(3 Numbers). But available is only AB (1 Number). Therefore the degree
is 1/3 ie 0.3333.













Figure 2.1.2.
ii) Arc Method:
This method focuses on relations. This calculates the percentage of all
possible ties in the parts of available reciprocated structures. The degree
is defined as number of ties that are involved in reciprocal relations
relative to the total number of actual ties. In this ex ample it is 2/3 ie 0.667.
The number of ties involved are two. They are AB and BA. The number of
actual ties are three. They are AB, BA and BC.
2.1.5. Group -external and Group -internal ties in networks
The E -I (external - internal) index takes the number o f ties of group
members to outsiders, subtracts the number of ties to other group
members, and divides by the total number of ties. The resulting index
ranges from -1 to +1. Since this measure is concerned with any
connectionbetween members, the directions of ties are ignored.
The E -I index can be applied at three levels: the entire population, each
group, and each individual. That is, the network as a whole (all the groups)
can be characterized in terms of the bounded -ness and closure of its
subpopulation s. We can also examine variation across the groups in their
degree of closure; and, each individual can be seen as more or less
embedded in their group. The range of possible values of the E -I index is
restricted by the number of groups, relative group siz es, and totalnumber
of ties in a graph.
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20 Often this range restriction is quite severe, so it is important to re -scale the
coefficient torange between the maximum possible degree of "external -
ness" (+1) and the maximum possible degree of internalness.
2.1.6. Ego networks
Subnetworks that are centered on a certain node is called Ego Networks.
In Facebook and LinkedIn these are describes as “ Your Network”. But we
can access only our network. To compare Ego Networks of various people
a large dataset is requir ed. Ego networks are derived using Breath -First
Search and the depth is limited.
Network Distance : The distance between links/ties. For example a link
means Dev is the friend Sharma.Sastri is the friend of friend of Dev which
is distance of 2. Rao is the friend of friend of friend of Dev which is
having the network distance of 3.
Neighbourhood: It is the collection of ego and all nodes to whom ego has
a connection at some path length. In social networkanalysis, the
"neighbourhood" is almost always one -step; that is, it includes only ego
and actors that are directly adjacent.
N-step neighbourhood: It expands the definition of the size of ego's
neighbourhood by including all nodes to whom ego has aconnection at a
path length of N, and all the connections amo ng all of these actors.
Neighbourhoods of greater path length than 1 (i.e. egos adjacent nodes)
are rarely used in social network analysis.
An "out" neighbourhood include all the actors to whom ties are directed
from ego. An "in"neighbourhood include all the actors who sent ties
directly to ego.
2.1.7. Extracting and visualizing ego networks
Extraction of ego networks is simple as NetworkX network analyser
provides a ready -made function to do the job.
>>>net.ego_graph(cc,'justinbieber')

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21

Figure 2.1.3. Justin Bieber is in the Egypt Retweet dataset. His ego
network
.The ego_graph function returns a NetworkX graph object, and all the
usual metrics such as degree, betweenness can be computed on it .
Knowing the size of an ego network is important to understand the reach
of the information that a person can transmit.
Clustering coefficient: It is a metric that measures the proportion of
friends they are also friends with each other. This metric can b e applied to
entire networks In ego networks, the interpretation is dense ego networks
with a lot of mutual trust have a high clustering coefficient. Star networks
with a singlebroadcast node and passive listeners have a low clustering
coefficient.
Ego net works in the Egypt data:
## we need to convert the ego network from a Multi -graph to a simple
Graph
>>>bieb = net.Graph(net.ego_graph(cc,'justinbieber', radius=2))
>>>len(bieb)
22
>>>net.average_clustering(bieb)
0.0
>>>ghonim= net.Graph(net.ego_graph(cc,'G honim', radius=2))
>>>len(ghonim)
3450
>>>net.average_clustering(ghonim)
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22 Not only does WaelGhonim have a vastly larger retweet network (despite
having 100 times fewer followers then Bieber), his ego network is a
network of trust where p eople retweet messages from him and from each
other —a network where a revolutionary message can easily spread and be
sustained.
2.1.8. Structural holes
The term "structural holes" is coined by Ronald Buttto refer the positional
advantage/disadvantage of in dividuals that result from how they are
embedded in neighbourhoods. Imagine a network of three actors (A, B,
and C), in which each is connected to each of the others





Figure 2.1.4. Three actor network with no structural holes
In the figure 2.1.4. , Suppose that actor A in wanted to influence or
exchange with another actor. Assume that both B and C may have some
interest in interacting or exchanging, as well. Actor A will not be in a
strong bargaining position in this network, because both of A's po tential
exchange partners (B and C) have alternatives to treating with A; they
could isolate A, and exchange with one another.






Figure 2.1.5. Three actor network with structural holes
Now imagine that we open a "structural hole" between actors B and C, as
in Figure 2.1.5. That is, a relation or tie is "absent" such thatB and C
cannot exchange (perhaps they are not aware of one another, or there are
very high transaction costs involved in forming atie).
In this situation, actor A has an advantaged posi tion as a direct result of
the "structural hole" between actors B and C. Actor A hastwo alternative
exchange partners; actors B and C have only one choice, if they choose to A B C
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23 (or must) enter into an exchange.Real networks, of course, usually have
more actor s.
2.2. CENTRALITY
Actors those who are connected with more number of actors in the
network has the maximum centrality. This type of actors are having
advantages as acting like bridges, third parties, getting more information,
transferring more informati on and having more alternative paths. The
effective measure of an actor's centrality and power potential is their
degree.






Figure 2.2.1. STAR network Figure 2.2.2. Line Network Figure 2.2.3. Ring Network
2.2.1 Degree of centrality
In the star netw ork given in Figure 2.2.1, actor A has more opportunities
and alternatives than other actors. If actor D chooses to not provide A with
a resource, the A has a number of other choices to get it; however, if D
does not prefer to exchange with A, then D canno t do any exchange.When
the number of ties of an actor is more then, the power of actors also will
be more. they have. In the star network, Actor A has degree six, all other
actors have degree one. This logic underlies measures of centrality and
power based on actor degree , which we will discuss below. Actors who
have more ties have greater opportunities because they have choices. This
autonomy makes them lessdependent on any specific other actor, and
hence more powerful.
In the line network given in Figure 2.2.2, matters are a bit more
complicated. The actors at the end of the line (A and G) are actually at a
structural disadvantage, but all others are apparently equal.
Now, consider the ring network given in Figure 2.2.3. in terms of degree.
Each actor has exactly the same number of alternative trading partners (or
degree), so all positions are equally advantaged or disadvantaged.
Generally, though, actors that are more central to the structure, in the
sense of having higher degree or more connections, ten d to have favoured
positions, and hence more power.

A B
GA C
FA E
D A B C D E F G
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24 2.2.2. Closeness and betweenness centrality
Actor A is closer to other actors than any other actor. Hence it is powerful
compared to all other actors. Power can be applied by direct bargaining
and exchan ge. Actors who are able to reach other actors at shorter path
lengths, or who are more reachable by other actors at shorter path lengths
have favoured positions. This structural advantage can be translated into
power. In the star network, actor A is at a g eodesic distance of one from
all other actors; each other actor is at a geodesic distance of two from all
other actors (but A). This logic of structural advantage underlies
approaches that emphasize the distribution of closeness and distance as a
source of power.
In actor closeness in ring network each actor lies at different path lengths
from the other actors, but all actors have identical distributions of
closeness, and again would appear to be equal in terms of their structural
positions.
In the line n etwork, the middle actor (D) is closer to all other actors than
are the set C, E, the set B, F, and the set A, G. Again, the actors at the ends
of the line, or at the periphery, are at a disadvantage.
Betweenness:
The reason that actor A is advantaged in the star network is because actor
A lies between each other pairs of actors, and no other actors lie between
A and other actors. If A wants to contact F, A may simply do so. If F
wants to contact B, they must do so by way of A. This gives actor A the
capac ity to broker contacts among other actors ie to extract "service
charges" and to isolate actors or prevent contacts.
The third aspect of a structurally advantaged position then is in being
between other actors. In the ring network, each actor lies between each
other pair of actors. Actually, there are two pathways connecting each pair
of actors, and each third actor lies on one, but not on the other of them.
Again, all actors are equally advantaged or disadvantaged. In the line
network, our end points (A,G) do not lie between any pairs, and have no
brokering power. Actors closer to the middle of the chain lie on more
pathways among pairs, and are again in an advantaged position.
2.2.3. Local and global centrality
A central point(Actor) was one which was 'at the centre' of a number of
connections, an actor with more direct contacts with other actors. The
point centrality is measured as by the degrees of the various points in the
graph. The degree centrality is the local centrality measure which counts
the num ber of ties connected by each node and points at individuals who
can quickly connect with the network. Since It is a local measure it does
not consider rest of the network and the importance of its value depends
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25 To calculate pop ular centrality measures – centrality degree, the
igraph package of CRAN in R is used.


















Figure 2.2.4. Undirected graph
From the random generated un -directed graph given in Figure 2.2.4 It is an
undirected network, a graph with b idirectional edges in contrast with a
directed graph in which the direction of an edge from one vertex to
another is considered, with 15 nodes and 30 edges.
The degree centrality of this graph would be calculated using
centr.degree() function. The 5 th n ode has the highest centrality.
Global centrality measures, considers the whole of the network. The most
widely used global centrality measures is closeness centrality. This
measure scores each node based on their closeness to all other nodes
within the ne twork.
# n = number of nodes/actors, m = the number of ties/edges > erdos.gr < - sample_gnm(n=10, m=25)
>
> plot(erdos.gr)
> erdos.gr < - sample_gnm(n=15, m=30)
> plot(erdos.gr)
> degree.cent < - centr_degree(erdos.gr, mode = "all")
>
>
> degree.cent$res
[1] 5 4 3 8 2 4 6 2 5 2 4 3 6 4 2

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26 It calculates the shortest paths between all nodes, then assigns each node a
score based on its sum of shortest paths and is useful for finding the
individuals who are best placed to influence the entire network most
quickly. closeness() function in igraph can be used to findout the global
centrality.
2.2.4. Centralization and graph centers
The whole graph centralized structure centralization point can be
examined. The concepts of density and centralization refer to differing
aspects of the overall ' compactness' of a graph. Density describes the
general level of cohesion in a graph; centralization describes the extent to
which this cohesion is organized around particular focal actors.
Centralization and density, therefore, are important complementary
measures. The result of centrality calculation will vary according to the
centrality type.














Figure 2.2.5 Centralities determined through R code
There are various methods suggested by the researchers to find out the
graph. In general graph c enter is calculated using I the closeness
centrality. To make more focused and accurate the degree and
betweenness centrality is calculated and the graph center actor is
determined. There are 49 centralities driven through the R code, it is given
in figure 2.2.5. Using sample centrality the result of 4 centralies is given in
the figure 2.2.6.This example uses Zachary dataset. > library(CINNA) > data("zachary")
> plot(zachary)
> pr_cent< -proper_centralities(zachary)
[1] "subgraph centrality scores"
[2] "Topological Coefficient"
[3] "Average Distance"
[4] "Barycenter Centrality"
[5] "BottleNeck Centrality"
[6] "Centroid value"
[7] "Closeness Centrality (Freeman )"
[8] "ClusterRank"
[9] "Decay Centrality"
[10] "Degree Centrality"
"…
…
[42] "Load Centrality"
[43] "Flow Betweenness Centrality"
[44] "Information Centrality"
[45] "Dangalchev Closeness Centrality"
[46] "Group Centrality"
[47] "Harmonic Centrality"
[48] "Local Bridging Centrality"
[49] "Wiener Index Centrality"
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27 >calculate_centralities(zachary, include = pr_cent[7:10])%>%
pca_centralities(scale.unit = TRUE)

2.2.5. Notion of importance within network
A degree -based measure of point centrality, therefore, corresponds to the
intuitive notion of how well connected a point is within its local
environment. Because this is calculated simply in terms of the number of
actors to which a particular point is adjacent, ignoring any indirect
connections it may have, the degree can be regarded as a measure of local
centrality. The simplest notion of closeness is, perhaps, that calculated
from the 'sum distance', the sum of the geod esic distances to all other
actors in the graph (Sabidussi, 1966). If the matrix of distances between
actors in an undirected graph is calculated, the sum distance of a point is
its column or row sum in this matrix (the two values are the same). A
point wi th a low sum distance is 'close' to a large number of other actors,
and so closeness can be seen as the reciprocal of the sum distance. In a
directed graph, of course, paths must be measured through lines which run
in the same direction, and, for this reas on, calculations based on row and
column sums will differ.
2.2.6. Google PageRank algorithm
PageRank centrality is determined through incoming links. PageRank was
originally developed for indexing webpages, but can be applied to social
networks as well, a s long as they are directed graphs
PageRank algorithm
PageRank is an iterative process, otherwise known as an anytime
algorithm. It is scaled between 0 and 1 and represents the likelihood that a
person following links (i.e., traversing the network, “surfi ng” the web, etc)
will arrive at a particular page or encounter a particular person. A 0.5
probability is commonly interpreted as a “50% chance” of an event.
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28 Hence, a PageRank of 0.5 means there is a 50% chance that a person
clicking on a random link will be directed to the document with the 0.5

PageRank.: Simplified version of PageRank Algorithm
1. Assume a small network of four nodes: (A)lice, (B)ob, (C)arol, and
(D)avid.
2. Initially, assign equal probability to A, B, C, and D:
$PR (A) =PR (B) =PC(C) =PR (D) =0.25$ .
3. If B, C, and D only link to A, A’s PageRank would be computed as
$PR(A)=PR(B)+PR(C)+PR(D)=0.25+0.25+0.25=0.75$ .
4. If a page has multiple outgoing links ( outdegree> 1 ) then its PageRank
contribution is equally divided b y all of the link targets.
5. Suppose that page B has a link to page C as well as to page A, while
page D has links to all three pages. The value of the link -votes is divided
among all the outbound links on a page. Thus, page B gives a vote worth
0.125 to page A and a vote worth 0.125 to page C. Only one third of D’s
PageRank is counted for A’s PageRank (approximately 0.083).
6. In the general case, PageRank can be computed as
$PR(N)=Sum_{i \in nodes} (PR(i)/out_degree(i))$ .
7. Repeat calculati on of PageRank for all nodes until the values stabilizes


2.3. ANALYSING NETWORK STRUCTURE
A network structure represents a group of nodes or actors or objects and
relationships or ties between them. Network is also termed as graph in
mathematics. The be haviour of nodes and ties are analysed using various
metrics. For example if we study about Twitter users then target users are
known as nodes and the followers or followings are considered as
relationships or ties. The analysis of this relationships and behavious play
vital in many applications. These analysis are carried through different
network structures like cliques, clans and factions.
2.3.1. Bottom -up approaches using cliques
Networks are composed of groups or sub -graphs. Two actors and a tie
which connects both actors form a "group." A clique extends the dyad by
adding members who are tied to all of the members in the group to it. The
size of cliques and clique -like subgroups build a map total network. In this
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29 how they are embedded in the web of overlapping groups in the larger
structure.
Example: The information on the strength, cost, or probability of relations
is available then bottom -up thinking to find maximal groups can be
applied.
Cliques
A clique is a sub -set of a network in where the actors of the network are
more closely connected to one another compared with the connection with
other actors of the network. Friendship network based on age, gender and
ideology among hu man groups are some example for cliques. Dyad is the
smallest group. A clique is the maximum number of actors who have all
possible ties present among themselves. Different type of cliques are given
in figure 2.3.1. The maximum possible actors are includ ed in a "Maximal
complete sub -graph".







Figure 2.3.1. Cliques


2.3.2. N -cliques
N-Cliques is defined as a clique with length of the path to connect two
sub-cliques N. Sometimes in sub -networks there may be some members
are not so tightly or closely connected may exist. This approach is used to
find long and complex groupings. Two methods are recommended to relax
the constraints of clique definition.to make it general.
Method 1 : Define an actor as a memb er of a clique if they are connected to
every other member of the group at a distance at least two. This takes the
concept friend of friend. In the given example A, B, C can be connected
with E through D. It takes two ties CD and DE to connect with E. Henc e
the sub -structure is called N - Clique. Here N represents the length of the
path to make connection which is two. Figure 2.3.2 shows the structure of
N-Clique.




2 Cliques 3 Cliques 4 Cliques B A A B
C C A B D
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30









Figure 2.3.2. N -Clique
Method 2 : Inner circle of actors for larger groupings also considered as N
Cliques. This follows clustering and co -membership concept. Figure 2.3.3
shows the N -cliques with clusters.














Figures 2 .3.3. Clustering in Cliques

2.3.3. N -clans
The main issue of N -Cliques is there is a possibility of actors who are not
the members of the particular N Cilque may be within the group. In the
sociological applications it would create a problematic scenario
This issue of N -clique is resolve by introducing the additional constraint
like it should satisfy that the total span or path distance between any two
members of N -clique.
A
C B F
E D
Clique Cluster 1
Clique Cluster 3 Clique Cluster 2
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31 To overcome this problem, some analysts have suggested restricting N -
cliques by insi sting that the total span or path distance between any two
members of an N -clique also satisfy a condition. The additional restriction
has the effect of forcing all ties among members of an N -clique to occur
by way of other members of the n -clique
The n -clan approach is defined as that all the ties among actors occur
through other members of the group.
2.3.4. K -plexes
To relax the Clique constraints it is allowed to consider the actors who
have ties with members of cliques except some k members. This concep t
relaxes the strong assumptions of the "Maximal Complete Sub -Graph".
For example, if A has ties with B and C, but not D; while both B and C
have ties with D, all four actors could fall in clique under the K -Plex
approach. The figure 2.3.4 shows the struct ure of K -plex. This approach
says that a node is a member of a clique of size n if it has direct ties to n -k
members of that clique.











Figures 2.3.4. K -Plex
The k -plex approach would seem to have quite a bit in common with the
n-clique a pproach, but k -plex analysis often gives quite a different picture
of the sub -structures of a graph. Rather than the large and "stringy"
groupings sometimes produced by n -clique analysis, k -plex analysis tends
to find relatively large numbers of smaller gr oupings. This tends to focus
attention on overlaps and co -presence (centralization) more than solidarity
and reach.
The K -plex method of defining cliques tends to find "overlapping social
circles" when compared to the maximal or N -clique method. The k -plex
approach to defining sub -structures makes a good deal of sense for many
problems. It requires that members of a group have ties to (most) other
group members – ties by way of intermediaries (like the n -clique
approach) do not quality a node for membership . The picture of group
structure that emerges from k -plex approaches can be rather different from
that of n -clique analysis. Again, it is not that one is "right" and the other
"wrong." Depending on the goals of the analysis, both can yield valuable
insight s into the sub -structure of groups. A
B D C
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32 2.3.5. K -cores
K-cores contains a maximal group of actors who are all tied with a
number of say, k number of different individuals of the group. The k -core
approach is allowing actors to join the group if they are con nected to k
members, regardless of how many other members they may not be
connected to the actor. By varying the value of k (that is, how many
members of the group do you have to be connected to), different pictures
can emerge. K -cores can be (and usually are) more inclusive than k -
plexes. And, as k becomes smaller, group sizes will increase.
K-core analysis helps to identify the parts of a network that are more
connected than others. The number of immediate ties of a node is noted as
k. A k -core of 1 refe rs to all nodes that have a degree of 1 or more, ie. all
nodes are connected in the network. A k -core of 2 refers to the subset of
all nodes that have a degree of 2 or more, etc. Figure 2.3.5. shows three k -
cores with three different color nodes..


Figures 2.3.5 Three K.Cores
The k -core is used in the applications like, if an actor has ties to a
sufficient number of members of a group, they may feel tied to that group,
though they may not be knowing many members of the group..
2.3.6. F -groups
F-Groups
F-groups identifies maximal groups made up of "strongly transitive" and
"weakly transitive" triads. A strong tie triad is formed when, if there is a
tie XY and a tie YZ, there is also a tie XZ that is equal in value to the XY
and YZ ties. A weakly transiti ve triad is formed if the ties XY and YZ are
both stronger than the tie XZ, but the tie XZ is greater than some cut -off
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33











Figures 2.3.6. F -Groups
2.3.7 Top -down approaches
The overlaps and composition of components make the overa ll structure of
networks. In the bottom -up approach the actors build networks using
dynamic processes. Instead if the entire network is considered for the
analysis then it leads to top -down approach. Here rather than dyad, the
whole structure identifies th e substructures as parts that are locally denser
than the field as a whole. In a sense, this more macro lens is looking for
"holes" or "vulnerabilities" or "weak spots" in the overall structure or
solidarity of the network. These holes or weak spots define lines of
division or cleavage in the larger group, and point to how it might be de -
composed into smaller units. This top -down perspective leads us to think
of dynamics that operate at the level of group -selection, and to focus on
the constraints under whi ch actors construct networks.
Some methods are used to define the divisions and "weak spots" in a
network. The most common approaches are Components, Blocks and
cutpoints, Lambda sets and bridges and Factions
2.3.7.1. Components
Components of a graph are sub-graphs that are connected within, but
disconnected between sub -graphs. If a graph contains one or more
"isolates," these actors are components. More interesting components are
those which divide the network into separate parts, and where each part
has several actors who are connected to one another.
For directed graphs we can define two different kinds of components. A
weak component is a set of nodes that are connected, regardless of the
direction of ties. A strong component requires that there be a di rected path
from A to B in order for the two to be in the same component.
2.3.7.2. Blocks and cut -points
Removing the node from a graph is used to identify the weak point in the
graph. When a node is removed from a graph if the graph is divided into
separ ate parts that are dis -connected from each other, then that particular X
Z Y
X
Z Y
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34 node or actor is called cut -point. Theses cutpoints act as agents or brokers
among the disconnected graphs. The divisions into which cut -points divide
a graph are called blocks or bicom ponent. The maximal non -separable
blocks can be found through locating the cutpoints.










Figure 2.3.7. Blocks and cut -points
In this Figure 2.3.7. If actor 3 is removed then the graph will be separated
into 3 blocks with (1, 2), (4), and (5, 6, 7, 8). Here the cutpoint is node 3.
2.3.7.4. Lambda sets and bridges
Removing the connection from a graph is used to separate the graph into
blocks. The Lambda set approach ranks each of the relationships in the
network in terms of import ance by evaluating how much of the flows
among actors in the net go through each link. It then identifies sets of
relationships which, if disconnected, would most greatly disrupt the flow
among all of the actors. The math and computation is rather extreme,
though the idea is fairly simple. The lambda set idea has moved us quite
far away from the strict components idea. Rather than emphasizing the
"decomposition" or separation of the structure into un -connected
components, the lambda set idea is a more "cont inuous" one. It highlights
points at which the fabric of connection is most vulnerable to disruption.
2.3.7.5. Factions
Imagine a society in which each person was closely tied to all others in
their own subpopulation (that is, all sub -populations are cliqu es), and there
are no connections at all among sub -populations (that is, each sub -
population is a component). Most real populations do not look like this,
but the "ideal type" of complete connection within and complete
disconnection between sub -groups is a useful reference point for assessing
the degree of factionalization in a population. If we took all the members
of each "faction" in this ideal -typical society, and put their rows and
columns together in an adjacency matrix, we would see a distinctive
pattern of "1 -blocks" and "0 -blocks." All connections among actors within
a faction would be present, all connections between actors in different
factions would be absent.
3 1 5
2 6
4 7 8
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35 The "Final number of errors" can be used as a measure of the "goodness of
fit" of the "blocking" of the matrix. This count (27 in this case) is the sum
of the number of zeros within factions plus the number of ones in the non -
diagonal blocks (ties between members of different factions, which are
supposed to be absent in the ideal type). Si nce there are 49 total ties in our
data, being wrong on the locations of 27 is not a terribly good fit. It is,
however, the best we can do with four "factions." The four factions are
identified, and we note that two of them are individuals (10, 9), and one is
a dyad (3, 6).
The "blocked" or "grouped" adjacency matrix shows a picture of the
solution. We can see that there is quite a lot of density "off the main
diagonal" where there shouldn't be any. The final panel of the results
reports the "block densitie s" as the number of ties that are present in
blocks as proportions of all possible ties. This approach corresponds
nicely to the intuitive notion that the groups of a graph can be defined by a
combination of local high density, and the presence of "structu ral holes"
between some sets of actors and others. The picture then not only
identifies actual or potential factions, but also tells us about the relations
among the factions, potential allies and enemies, in some cases.
2.4 SUMMARY
This chapter discusses about understanding networks, Centrality and
analysing the network structure.
The understanding network structure explains how the density,
reachability, connectivity and reciprocity of a network are calculated.
More over the ego networks concept and benef its and demerits of
structured holes are given in this chapter. The role of centralization is
important in social network analysis. The different centralizations like
closeness, betweenness, degree of centralization are used to find the
importance of the nodes and the strength of the relationships. Alson with
structures the sub -structure in terms of groupings or cliques are used in
network analysis. The number, size, and connections among the sub -
groupings in a network can tell us a lot about the likely be havior of the
network as a whole. All the aspects of sub -group structure can be very
relevant to predicting the behavior of the network as a whole. Certain
individuals may act as "bridges" among groups, others may be isolates;
some actors may be cosmopolit ans, and others locals in terms of their
group affiliations. Such variation in the ways that individuals are
connected to groups or cliques can be quite consequential for their
behavior as individuals.
2.5 REFERENCE FOR FURTHER READING
1. Introduction to Social Network Methods: Robert A. Hanneman, Mark
Riddle, University of California, 2005, Published in digital form and
available at http://faculty.ucr.edu/~hanneman/nettext/index.html. munotes.in

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36 2. Social Network Analysis for Startups - Finding connections on the
social web: Maksim Tsvetovat, Alexander Kouznetsov, O'Reilly
Media, 2011.
3. Social Network Analysis - 3rd edition, John Scott, SAGE Publications,
2012.
4. Mark S. Handcock, David Hunter, Carter T. Butts, Steven M. Goodreau
and Martina Morris. 2003 statne t: An R package for the Statistical
Modeling of Social Networks http://www.csde.washington.edu/statnet
5. Vladimir Batagelj and Andrej Mrvar (2006), Pajek datasets
http://vlado.fmf.uni -lj.si/pub/networks/data/ .
6. Krackhardt and Stern (1988) developed a very simple and useful
measure of the group embedding based on comparing the numbers of
ties within groups and between groups
7. Getting Started in Social Network Analysis with NETDRAW, Bruce
Cronin University of Greenwich Business School, Occasional Paper
01/15, January 2015 brought to you by CORE View metadata, citation
and similar papers at core.ac.uk
8. Structural Holes, The Social Structure o f Competition, Ronald S. Burt
9. www.analytictech.com
10. https://www.datacamp.com/
2.6. MODEL QUESTIONS
1. How do you calculate Density an d reachability?
2. Compare Connectivity with reciprocity.
3. Explain Ego Networks with a real time application.
4. How do you calculate Centralization and graph centers? Explain.
5. Write the Google PageRAnk Algorithm.
6. How do N -cliques and N -clans "rela x" the definition of a clique?
7. Explain about K -plexes and K -cores.
8. How does the idea of a "block" relax the strict definition of a
component?
9. Explain the cutpoints with its advantages and disadvantages.
10. Discuss the bottom -up network structure s in detail.
 
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37 3
MEASURES OF SIMILARITY AND
STRUCTURAL EQUIVALENCE IN SNA
Unit Structure
3.1 Objectives
3.2 Introduction
3.3 Approaches to Network Positions and Social roles
3.3.1 Defining equivalence or similarity
3.3.2 Structural Equivalence
3.3.3 Automorphic Equivalen ce
3.3.4 Regular Equivalence
3.3.5 Finding Equivalence sets
3.3.6 Brute Force and Tabu Search
3.3.7 Equivalence of Distances
4 Maxsim
4.1 Measuring Similarity/Dissimilarity
1 Valued Relations
2.Pearson Correlations, Covariance and Cross -Products
3. Euclid ean, Manhattan and Squared Distance
5. Understanding clustering -agglomerative and divisive clusters
5.1 Binary Relations
5.2 Matches: Exact, Jaccard, Hamming
6. Summary
7. References
8. Miscellaneous Questions
3.1 OBJECTIVES
At the end of this uni t, student will able to munotes.in

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38  Describe network roles and positions, defining equivalence or
similarity
 Illustrate the concept of Brute Force and Tabu Search
 Explain the concept of similarity and dissimilarity measures
 Compare and contrast between different types of distances such as
Euclidean, Manhattan and squared distances.
3.2 INTRODUCTION
1. The ways that structural analysts look at network data. They look at
patterns in the overall structure (e.g. connectedness, density, etc.) and the
embeddedness of each ac tor (e.g. geodesic distances, centrality). A second
major way of going about examining network data by looking for "sub -
structures," or groupings of actors that are closer to one another than they
are to other groupings. For example, we looked at the meani ng of
"cliques" "blocks" and "bridges" as ways of thinking about and describing
how the actors in a network may be divided into sub -groups on the basis
of their patterns of relations with one another.
2. The central node of a "star" network is "closer" to all other members
than any other member A clique as a "maximal complete sub graph"
sounds tough, but, again, is easy to grasp. It is simply the biggest
collection of folks who all have connections with everyone else in the
group. Again, the idea is not dif ficult to grasp, because it is really quite
concrete: we can see and feel cliques.
3. The patterns of relations among social actors: the analysis of
"equivalence classes." Being able to define, theorize about, and analyze
data in terms of equivalence is im portant because we want to be able to
make generalizations about social behavior and social structure. That is,
we want to be able to state principles that hold for all groups, all
organizations, all societies, etc. To do this, we must think about actors n ot
as individual unique persons (which they are), but as examples of
categories -- sets of actors who are, in some defined way, "equivalent."
As an empirical task, we need to be able to group together actors who are
the most similar, and to describe what makes them similar; and, to
describe what makes them different, as a category, from members of other
categories.
4. Sociological thinking uses abstract categories routinely. "Working class,
middle class, upper class" are one such set of categories that de scribe
social positions. "Men and Women" are really labels for categories of
persons who are more similar within category than between category -- at
least for the purposes of understanding and predicting some aspects of
their social behavior. When categor ies like these are used as parts of
sociological theories, they are being used to describe the "social roles" or
"social positions" typical of members of the category.
5. Many of the category systems used by sociologists are based on
"attributes" of indivi dual actors that are in common across actors. If state
that "European -American males, ages 45 -64 are likely to have relatively munotes.in

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39 high incomes" I am talking about a group of people who are
demographically similar -- they share certain attributes (maleness,
European ancestry, biological age, and income). Structural analysis is not
particularly concerned with systems of categories (i.e. variables), that are
based on descriptions of similarity of individual attributes (some radical
structural analysts would even argue that such categories are not really
"sociological" at all). Structural analysts seek to define categories and
variables in terms of similarities of the patterns of relations among actors,
rather than attributes of actors. That is, the definition of a category, or a
"social role" or "social position" depends upon its relationship to another
category. Social roles and positions, structural analysts argue, are
inherently "relational."
6. What is a "worker?" We could mean a person who does labor (an
attribute, actually one shared by all humans). A more sociologically
interesting definition was given by Marx as a person who sells control of
their labor power to a capitalist. Note that the meaning of "worker"
depends upon a capitalist -- and vice versa. It i s the relation (in this case,
as Marx would say, a relation of exploitation) between occupants of the
two role that defines the meaning of the roles.
7. The point is: to the structural analyst, the building blocks of social
structure are "social roles" or "social positions." These social roles or
positions are defined by regularities in the patterns of relations among
actors, not attributes of the actors themselves. We identify and study social
roles and positions by studying relations among actors, not by studying
attributes of individual actors. Even things that appear to be "attributes of
individuals" such as race, religion, and age can be thought of as short -hand
labels for patterns of relations. For example, "white" as a social category is
really a shor t-hand way of referring to persons who typically have a
common form of relationships with members of another category -- "non -
whites." Things that might at first appear to be attributes of individuals are
really just ways of saying that an individual falls in a category that has
certain patterns of characteristic relationships with members of other
categories.
3.3 APPROACHES TO NETWORK POSITIONS AND
SOCIAL ROLES
1. Because "positions" or "roles" or "social categories" are defined by
"relations" among actors , we can identify and empirically define social
positions using network data. In an intuitive way, we would say that two
actors have the same "position" or "role" to the extent that their pattern of
relationships with other actors is the same. But, there a re a couple things
about this intuitive definition that are troublesome.
2. First, what relations to we take into account, among whom, in seeking
to identify which actors are similar and which are not? The relations that I
have with the university (as "Pro fessor") are similar in some ways to the
relations that my students have with the university: we are both governed
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40 with the university are very different from those of my students i n some
ways (e.g. the university pays me, students pay the university). Which
relations should count and which ones not, in trying to describe the roles
of "professor" and "student?" Indeed, why am I examining relations
among my students, me, and the unive rsity, instead of including, say,
members of the state legislature? There is no simple answer about what
the "right relations" are to examine; and, there is no simple answer about
who the relevant set of "actors" are. It all depends upon the purposes of
our investigation, the theoretical perspective we are using, and the
populations to which we would like to be able to generalize our findings.
Social network data analytic methods are of little use in answering these
conceptual questions.
3. The second prob lem with our intuitive definition of a "role" or
"position" is this: assuming that I have a set of actors and a set of relations
that make sense for studying a particular question, what do I mean that
actors who share the same position are similar in thei r pattern of
relationships or ties? The idea of "similarity" has to be rather precisely
defined. Again, there is no single and clear "right" answer for all purposes
of investigation. But, there are rigorous ways of thinking about what it
means to be "simil ar" and there are rigorous ways of actually examining
data to define social roles and social positions empirically. These are the
issues where there are some ways in which widely used methods can
provide guidance.
3.3.1 Defining equivalence or similarity
1. What do we mean when we say that two actors have "similar" patterns
of relations, and hence are both members of the same role or social
position? Network analysis most broadly defines two nodes (or other
more elaborate structures) as similar if they fal l in the same "equivalence
class." Frankly, that's no immediate help. But it does say that there is
something that would cause us to say two actors (or other structures) are
members of a "class" that is different from other "classes."
2. Now it becomes a question of what features of an actor's position place
them into a "class" with other actors? In what way are they "equivalent?"
3. There are many ways in which actors could be defined as "equivalent"
based on their relations with others. For example, w e could create two
"equivalence classes" of actors with out -degree of zero, and actors with
out-degree of more than zero. Indeed, a very large number of the
algorithms examined group sets of actors into categories based on some
commonality in their positi ons in graphs.
4. Three particular definitions of "equivalence" have been particularly
useful in applying graph theory to the understanding of "social roles" and
"structural positions." We will look at these in the next three chapters on
"structural equiv alence," "automorphic equivalence," and "regular
equivalence." Of these, "automorphic" has rarely been used in substantive
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41 5. The basic ideas of these three kinds of equivalence are easily explained
with three types of equivalence class as structura l, automorphic and
regular equivalence.
6. Figure given below depicts the Wasserman -Faust Network

Fig 1 Wasserman -Faust Network
3.3.2 Structural Equivalence
1. In this type of equivalence there if two nodes are said to be exactly
equivalent if they have the same relationships to all other nodes. It means
nodes follow same pattern as root node.
2. It is very specific as two actors must be exactly substitutable in order to
be structurally equivalent.
3. In fig 1 there are seven structural equivalence classe s as follows
3.1 There is no actor who has exactly the same set of ties as actor A (ties
to B, C, and D), so actor A is in a class by itself.
3.2 The same is true for actors B, C, and D. Each of these actors has a
unique set ties to others, so they form t hree classes, each with one
member.
3.3 E and F, however, fall in the same structural equivalence class. Each
has a single tie; and that tie is to actor B. Since E and F have exactly the
same pattern of ties with all other actors, they are structurally e quivalent.
3.4 Actor G, again, is in a class by itself. its profile of ties with the other
nodes in the diagram is unique.
3.5 Finally, actors H and I fall in the same structural equivalence class.
That is, they have exactly the same pattern of ties to a ll other actors.
4. Actors that are structurally equivalent are in identical "positions" in the
structure of the diagram. Whatever opportunities and constraints operate
on one member of a class are also present for the others. The nodes in a
structural e quivalence class are, in a sense, in the same position with
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42 5. Because exact structural equivalence is likely to be rare (particularly in
large networks), we often are interested in examining the degree of
structural equivalence , rather than the simple presence or absence of exact
equivalence.
6. Structural equivalence is the "strongest" form of that network analysts
usually consider. If we soften the requirements just a bit, we can often
find some interesting other patterns of equivalence.
3.3.3 Automorphic Equivalence
1. The idea of structural equivalence is powerful because it identifies
actors that have the same position, or who are completely substitutable.
But, even intuitively, you can probably imagine other "less strict "
definitions of what it means for two actors to be similar or equivalent.
2. Suppose that the graph in figure 1 described a franchise group of
hamburger restaurants. Actor A is the central headquarters, actors B, C,
and D are the managers of three differ ent stores. Actors E and F are
workers at one store; G is the lone worker at a second store; H and I are
workers at the third store.
3. Even though actor B and actor D are not structurally equivalent (they do
have the same boss, but not the same workers), they do seem to be
"equivalent" in a different sense. Both manager B and D report to a boss
(in this case, the same boss), and each has exactly two workers. These are
different people, but the two managers seem somehow equivalent. If we
swapped them, a nd also swapped the four workers, all of the distances
among all the actors in the graph would be exactly identical. In fact,
actors B and D form an "automorphic" equivalence class.
4. In above figure there are actually five automorphic equivalence class es:
{A}, {B, D}, {C}, {E, F, H, I}, and {G}. These classes are groupings
who's members would remain at the same distance from all other actors if
they were swapped, and, members of other classes were also swapped.
5. The idea of automorphic equivalence is that sets of actors can be
equivalent by being embedded in local structures that have the same
patterns of ties -- "parallel" structures. Large scale populations of social
actors (perhaps like hamburger restaurant chains) can display a great deal
of this sort of "structural replication." The faces are different, but the
structures are identical.
3.3.4 Regular Equivalence
1. Two nodes are said to be regularly equivalent if they have the same
profile of ties with members of other sets of actors that are al so regularly
equivalent. This is a complicated way of saying something that we
recognize intuitively.
2. Two mothers, for example, are "equivalent" because each has a certain
pattern of ties with a husband, children, and in -laws (for one example --
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43 to the same husband (usually) or the same children or in -laws. That is,
they are not "structurally equivalent." Because different mothers may have
different numbers of husbands, children, and in -laws, they will not be
automorphically equivalent. But they are similar because they have the
same relationships with some member or members of another set of actors
(who are themselves regarded as equivalent because of the similarity of
their tie s to a member of the set "mother").
3. In above figure there are three equivalence classes as first is actor A; the
second is composed of the three actors B, C, and D; the third is composed
of the remaining five actors E, F, G, H, and I.
4. The easiest cla ss to see is the five actors across the bottom of the
diagram (E, F, G, H, and I). These actors are regularly equivalent to one
another because a) they have no tie with any actor in the first class (that is,
with actor A, each has a tie with an actor in t he second class (either B or C
or D). Each of the five actors, then, has an identical pattern of ties with
actors in the other classes.
5. Actors B, C, and D form a class because they each have a tie with a
member of the first class (that is, with actor A) and b) they each have a tie
with a member of the third class. B and D actually have ties with two
members of the third class, whereas actor C has a tie to only one member
of the third class; this doesn't matter, as there is a tie to some member of
the third class.
6. Actor A is in a class by itself, defined by a) a tie to at least one member
of class two and b) no tie to any member of class three.
7. As with structural and automorphic equivalence, exact regular
equivalence may be rare in a large populati on with many equivalence
classes. Approximate regular equivalence can be very meaningful though,
because it gets at the notion of which actors fall in which social roles, and
how social roles (not role occupants) relate to one another.
3.3.5 Finding Equiv alence sets
1. The formal definition says that two actors are regularly equivalent if
they have similar patterns of ties to equivalent others. Consider two men.
Each has children (though they have different numbers of children, and,
obviously have differen t children). Each has a wife (though again, usually
different persons fill this role with respect to each man). Each wife, in turn
also has children and a husband (that is, they have ties with one or more
members of each of those sets). Each child has ties to one or more
members of the set of "husbands" and "wives."
2. What is important in identifying actors is that each “husband” have
atleast one tie to a person in the “wife” category and at least one person in
the “child” category. That is husband are eq uivalent to each other because
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44 3. But there would seem to be a problem with this fairly simple definition.
If the definition of each position depends on its relations with other
positi ons , where do we start?
4. There are a number of algorithms that are helpful in identifying regular
equivalence sets. UCINET provides some methods that are particularly
helpful for locating approximately regularly equivalent actors in valued,
multi -relati onal and directed graphs.
5. Consider the Wasserman -Faust example network. However that this is a
picture of order -giving in a simple hierarchy. That is all ties are directed
from the top of the diagram and moves towards downwards as shown in
below figure, where we will find a regular equivalence characterization of
this graph.

Fig 2 Directed Tie Version of the Wasserman - Faust Network
6. For a first step, characterize each node as either a "source" (an actor that
sends ties, but does not receive them), a "repeater" (an actor that both
repeats and sends), or a "sink" (an actor that receives ties, but does not
send). The source is A; repeaters are B, C, and D; and sinks are E, F, G, H,
and I. There is a fourth logical possibility. An "isolate" is a node tha t
neither sends nor receives ties. Isolates form a regular equivalence set in
any network, and should be excluded from the regular equivalence
analysis of the connected sub -graph.
7. Consider the three "repeaters" B, C, and D. In the neighborhood (that is,
adjacent to) actor B are both "sources" and "sinks." The same is true for
"repeaters" C and D, even though the three actors may have different
numbers of sources and sinks, and these may be different (or the same)
specific sources and sinks. We cannot def ine the "role" of the set {B, C,
D} any further, because we have exhausted their neighborhoods.
8. Now consider our "sinks" (i.e. actors E, F, G, H, and I). Each is
connected to a source (although the sources may be different). We have
already determined, in the current case, that all of these sources (actors B,
C, and D) are regularly equivalent. So, E through I are equivalently
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45 9. The result of {A} {B, C, D} {E, F, G, H, I} satisfies the c ondition that
each actor in each partition have the same pattern of connections to actors
in other partitions. The permuted adjacency matrix is shown in table 1
Table 1 Permuted Wasserman -Faust network to show regular equivalence
classes
A B C D E F G H I
A - 1 1 1 0 0 0 0 0
B 0 - 0 0 1 1 0 0 0
C 0 0 - 0 0 0 1 0 0
D 0 0 0 - 0 0 0 1 1
E 0 0 0 0 - 0 0 0 0
F 0 0 0 0 0 - 0 0 0
G 0 0 0 0 0 0 - 0 0
H 0 0 0 0 0 0 0 - 0
I 0 0 0 0 0 0 0 0 -

Here 1 Means there is tie between two nodes and zero means no
connection between two nodes and – indicates same node cannot have tie
between them
10 Table 2 presents Block image of regular equivalence classes in directed
Wasserman -Faust network
A B,C,D E,F,G,H,I
A - 1 0
B,C,D 0 - 1
E,F,G,H,I 0 0 -

Here {A} sends to one or more of {BCD} but to none of {EFGHI}.
{BCD} does not send to {A}, but each of {BCD} sends to atleast one of
{EFGHI}. None of {EFGHI} send to any of {A} or of {BCD}.
11. For directed binary graphs, the neighborhood search method applied in
wasserm an-Faust network works quite well. For binary graphs that are not
directed, usually the geodesic distance among actors is computed and used
instead of raw adjacency. For graphs with valued relations (Strength, cost,
probability) a method for identifying ap proximate regular equivalence was
developed by white and Reitz.
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46 1.With binary data, numerical algorithms are used to search for classes of
actors that satisfy the mathematical definitions of automorphic
equivalence. When t he new graph and the old graph have the same
distances among nodes, the graphs are isomorphic and the swapping that
done identifies the isomorphic sub -graphs.
2. One approach to binary data, “all permutations” (Netowrk ->Roles &
Positions>Automorphic>All Pe rmutations ) literally compares every
possible swapping of nodes to find isomorphic graphs with even a small
graph i.e method is nothing but a brute force method. An alternative
approach with the same intent (“optimization by tabu
search”)(Network>Roles & Positions>Exact>optimization) can much
more quickly sort nodes in to a user -defined number of partitions in such a
way as to maximize automorphic equivalence.
3. When we have measures of the strength, cost or probability of relations
among nodes(i.e value d data), exact automorphic equivalence is far less
likely. It is possible however, to identify classes of approximately
equivalent actors on the basis of their profile of distance to all other actors.
The “equivalence of distances” method (Network>Roles & Positions>
Automorphic>Maxsim) Produces measures of the degree of automorphic
equivalence for each pair of nodes, which can be examined by clustering
and scaling methods to identify approximate classes.
4. Brute Force - All Permutations
4.1 The automorphism s in a graph can be identified by the brute force
method of examining every possible permutation of the graph. With a
small graph, and a fast computer, this is a useful thing to do. Basically,
every possible permutation of the graph is examined to see if i t has the
same tie structure as the original graph. For graphs of more than a few
actors, the number of permutations that need to be compared becomes
extremely large.
4.2 Let’s use Networks>Roles& Positions>Automorphic>All
Permutations to search the Wasser man-Faust Network shown in figure
below

Figure 3 Wasserman -Faust Network for Automorphic Permutations munotes.in

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47 Here there are five types of orbits as 1) Orbit 1 consist of only one Node
that is A 2) Orbit 2 consist of two nodes as B and D as they has the same
pattern of distance i.e they comprising of two leaves node as E,F, H,I. 3)
Orbit 3 comprising of Node c 4) Orbit 4 Comprising of 4 nodes as E,F,H,I
5) Orbit 5 comprising of Node G . Note that automorphism classes
identify groups of actors who have the same p attern of distance from other
actors, rather than sub -structures as in case of Node B and D.
5. Tabu Search - Optimization
5.1 For larger graphs, direct search for all equivalences is impractical both
because it is computationally intensive and because exac tly equivalent
actors are likely to be rare.
5.2 Network>Roles & Positions >Exact>optimization provides a
numerical tool for finding the best approximations to a user -selected
number of automorphism classes. In using this method, it is important to
explore a range of possible number of partitions to determine how many
partitions are useful in order to re -run the algorithm a number of times to
insure that a global, rather than local minimum has been found.
5.3 The method begins by randomly allocating nodes t o partitions. A
measure of badness of fit is calculated as the sum of squares for each row
and each column within each block, along with calculating the variance of
these sums of squares. Then Sum of variance is calculated across the block
to construct a m easure of badness of fit. Search continues to find
allocation of actors to partitions that minimizes this badness of fit statistic.
5.4 Here we are using the Knoke bureaucracies information exchange
network data for calculations of automorphisms. In the Kn oke information
data there are no exact automorphisms.

Fig 4 Knoke Information Data Network
Here for this fit of automorphic equivalence models is given in terms of
mean value as
Partition Fit
2 4.366
3 4.054
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48 5 3.504
6 3.328

So in automor phic equivalence between nodes is actually operates on the
profile of distance between nodes or actors, Here in above table Partition 2
and 3 are equidistant as they are having mean value as 4.366 and 4.054.
3.3.7 Equivalence of Distances
1 Maxsim
1. When we have information on the strength, cost, or probability of
relations (i.e. valued data), exact automorphic equivalence could be
expected to be extremely rare. But, since automorphic equivalence
emphasizes the similarity in the profile of distances of actor s from others,
the idea of approximate equivalence can be applied to valued data.

2. Network>Roles & Positions>Automorphic>MaxSim generates a
matrix of "similarity" between shape of the distributions of ties of actors
that can be grouped by clustering and sc aling into approximate classes.
The approach can also be applied to binary data, if we first convert the
adjacency matrix into a matrix of geodesic near -ness (which can be treated
as a valued measure of the strength of ties).

3. The algorithm begins with a (reciprocal of) distance or strength of
tie matrix. The distances of each actor re -organized into a sorted list from
low to high, and the Euclidean distance is used to calculate the
dissimilarity between the distance profiles of each pair of actors.

4. The a lgorithm scores actors who have similar distance profiles as
more automorphically equivalent. Again, the focus is on whether actor u
has a similar set of distances, regardless of which distances, to actor v.
Again, dimensional scaling or clustering of the distances can be used to
identify sets of approximately automorphically equivalent actors.

5. Example -Line Network


Fig 5 Line Network Graph

Here automorphic equivalence of geodesic distances in the line network is
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49

Fig 6 Binary Adjacency Matrix converted to reciprocals of geodesic
distances

Here in above first the matrix is converted in to a geodesic distance matrix
where distance between actors is given. So according to matrix first the
actor 3 and actor 5 has same geode sic distance of 5.13, second the actor 2
and 6 has of 4.56, third 1 and 7 has same geodesic distance and cluster out
actor is 4 whose distance is not matching with any of these actors. Then at
last step Euclidean distance between these lists is calculated as a measure
of the non -automorphic -equivalence and hierarchical clustering is applied.

6. Example 2 - Maxsim method has applied on donors data of
California political campaigns, where strength of ties is measured among
the actors with the number of positions in campaigns they have in
common when either contributed.
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50


Fig 6 Truncated California automorphic equivalence

Here the above the figure represents only small part of large data set
which shows that a number of non -Indian casinos and race -tracks cluster
together and separately from some other donors who are primarily
concerned with education and ecological issues.

7. The identification of approximate equivalence classes in valued data
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51 the structure of the graph as a whole. By emphasizing distance profiles,
however, it is possible to finds classes of actors that include nodes that are
quite distant from one another, but at a similar distance to all the other
actors. That is, actors that have similar positions in the network as a
whole.

3.4.1 Valued Relations
1 Pearson Correlations, Covariance and Cross -Products
1. Valued Relations - a)A common approach for indexing the similarity
of two valued variables is the degree of linear association bet ween the
two. Exactly the same approach can be applied to the vectors that describe
the relationship strengths of two actors to all other actors.b)As with any
measures of linear association, linearity is a key assumption. It is often
wise, even when data are at the interval level (e.g. volume of trade from
one nation to all others) to consider measures with weaker assumptions
(like measures of association designed for ordinal variables).

2. Pearson, Correlation

2.1 The correlation measure of similarity is part icularly useful when the
data on ties are "valued," that is, tell us about the strength and direction of
association, rather than simple presence or absence.

2.2 Pearson correlations range from -1.00 (meaning that the two actors
have exactly the opposite ties to each other actor), through zero (meaning
that knowing one actor's tie to a third party doesn't help us at all in
guessing what the other actor's tie to the third party might be), to +1.00
(meaning that the two actors always have exactly the same tie to other
actors - perfect structural equivalence).

2.3 Pearson correlations are often used to summarize pair -wise
structural equivalence because the statistic (called "little r") is widely used
in social statistics. If the data on ties are truly nominal, or if density is very
high or very low, correlations can sometimes be a little troublesome, and
matches (see below) should also be examined.

2.4 Figure shown below the the correlations of the ten Knoke
organization's profiles of in and out information ties. We are applying
correlation, even though the Knoke data are binary. The UCINET
algorithm Tools >Similarities will calculate correlations for rows or
columns.

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52 2.5 We can see, for example, that node 1 and node 9 have identical
patterns of ties; there is a moderately strong tendency for actor 6 to have
ties to actors that actor 7 does not, and vice versa.

2.6 The Pearson correlation measure does not pay attention to the
overall prevalence of ties (the mean of the row or column), and it does not
pay attention to differences between actors in the variances of their ties.
Often this is desirable - to focus only on the pattern, rather than the mean
and variance as aspects of similarity between actors.

2.7 Covari ance Matrix - we might want our measure of similarity to
reflect not only the pattern of ties, but also differences among actors in
their overall tie density. Tools>similarities will also calculate the
covariance matrix.

2.8 Cross product - If we want to includ e differences in variances across
actors as aspects of (dis)similarity, as well as means, the c ross-product
ratio calculated in Tools>Similarities might be used.

2.Euclidean, Manhattan and Squared Distances

1. An alternative approach to linear correlatio n(and its relatives) is to
measure the “distance” or “dissimilarity” between the tie profiles of each
pair of actors. Several “distance” measures are fairly commonly used in
network analysis particularly the Euclidean distance or squared Euclidean
distance .

2. These measures are not sensitive to the linearity of association and can
be used with either valued or binary data.

3. Figure below shows the Euclidean distances among the Knoke
organizations calculated using Tools>Dissimilarities and Distances>Std
Vector Dissimilarities/distances


Fig 8 Euclidean distance in sending for Knoke information network

4. The Euclidean distance between two vectors is equal to the square root
of the sum of the squared differences between them. That is the distance
betwe en Actor A and Actor C is subtracted from the distance of actor B to
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53 the other actors (D,E,F etc) and summed. The square root of the sum is
then taken.

5. A closely related measur e is the “Manhattan” or block distance between
the two vectors. This distance is simply the sum of the absolute difference
between the actors ties to each alter, summed across the alters.
3.4.2 Understanding clustering -agglomerative and divisive clusters
1. Agglomerative hierarchical clustering of nodes on the basis of the
similarity of their profiles of ties to other cases provides a "joining tree" or
"dendogram" that visualizes the degree of similarity among cases - and
can be used to find approximate equ ivalence classes.

2.Tools>Cluster>Hierarchical proceeds proceeds by initially placing each
case in its own cluster. The two most similar cases (those with the highest
measured similarity index) are then combined into a class. The similarity
of this new class to all others is then computed on the basis of one of three
methods.

2.On the basis of the newly computed similarity matrix, the
joining/recalculation process is repeated until all cases are "agglomerated"
into a single cluster. The "hierarchical" part of the method's name refers
to the fact that once a case has been joined into a cluster, it is never re -
classified. This results in clusters of increasing size that always enclose
smaller clusters.

3. The "Average" method computes the similarity of the average scores in
the newly formed cluster to all other clusters; the "Single -Link" method
(a.k.a. "nearest neighbor") computes the similarities on the basis of the
similarity of the member of the new cluster that is most similar to each
other case no t in the cluster.

4. The "Complete -Link" method (a.k.a. "farthest neighbor") computes
similarities between the member of the new cluster that is least similar to
each other case not in the cluster. The default method is to use the cluster
average; single -link methods will tend to give long -stringy joining
diagrams; complete -link methods will tend to give highly separated
joining diagrams.

5.The hamming distance in information sending in the Knoke network was
computed and the results were stored as a file . This file was then input to
Tools>cluster>Hierarchical. The “Average” method was to be used, and
that the data were “dissimilarities”. The results are shown in figure below munotes.in

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54


Fig 9 Clustering of Hamming distances of information sending in the
Knoke Netw ork

6. The first graphic shows that nodes 1 and 9 were the most similar and
joined first. The graphic by the way can be rendered as a more polished
dendrogram using Tools>dendrogram>draw on data saved from the
cluster tool. At the next step, there are thr ee clusters(cases 2 and5, 4 and 7
and 1 and 9).

7. The joining continues until (at the 8th step) all cases are agglomerated in
to a single cluster. This gives a clear picture of the similarity of cases and
the groupings or classes of cases. But there are really eight pictures here
(one for each step of the joining). Which is the “right” Solution?.

8. Again, there is no single answer. Theory and a substantive knowledge
of the processes giving rise to the data are the best guide. The second
panel "Measur es of cluster adequacy" can be of some assistance. There
are a number of indexes here, and most will (usually) give the similar
answers.

9. As we move from the right (higher steps or amounts of agglomeration)
to the left (more clusters, less agglomeratio n) fit improves. The E -I index
is often most helpful, as it measures the ratio of the numbers of ties within
the clusters to ties between clusters. Generally, the goal is to achieve
classes that are highly similar within, and quite distinct without. Here, one
might be most tempted by the solution of the 5th step of the process
(clusters of 2+5, 4+7+1+9, and the others being single -item clusters).
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55 10. To be meaningful, clusters should also contain a reasonable percentage
of the cases. The last panel sho ws information on the relative sizes of the
clusters at each stage.
3.4.3 Binary Relations
1 Matches: Exact, Jaccard, Hamming
1. If the information that we have about the ties among our actors is
binary, correlation and distance measures can be used, but may not be
optimal. For data that are binary, it is more common to look at the
vectors of two actor's ties, and see how closely the entries in one "match"
the entries in the other.

2. Matches: Exact

2.1 Figure below shows the result for the columns relation of th e Knoke
bureaucracies


Fig 10 Proportion of Matches for Knoke Information receiving

2.2 These results show similarity in a way that is quite easy to interpret.
The number 0.625 in the cell 2,1 means that in comparing actor no 1 and
actor no 2, they have the same tie(present or absent) to other actors 62.5%
of the time. The measure is particularly useful with multi -category
nominal measures of ties, it also provides a nice scaling for binary data.

2.3 In some networks connections are very sparse. Indeed, if one were
looking at ties of personal acquaintance in very large organizations, the
data might have very low density. Where density is very low, the
"matches" "correlation" and "distance" measures can all show relatively
little variation among the actors, and m ay cause difficulty in discerning
structural equivalence sets (of course, in very large, low density networks,
there may really be very low levels of structural equivalence).
3. Jaccard
3.1 One approach to solve problem of matches and coefficient is to use
jaccard method which states that to calculate the number of times that both
actors report a tie (or the same type of tie) to the same third actors as a
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56 where neither X or Y are tied to Z, and ask, of the total ties that are
present, what percentage are in common.
Fig 11 Jaccard Coefficient for information receiving profiles in Knoke
network
3.2 Again the same basic picture emerges. The uniqueness of actor no 6,
though is emphasize d. Actor six is more unique by this measure because
of the relatively small number of total ties that it has -- this results in a
lower level of similarity when "joint absence" of ties are ignored. Where
data are sparse, and where there are very substanti al differences in the
degrees of points, the positive match coefficient is a good choice for
binary or nominal data.
4.Hamming Distance
4.1 The hamming distance is the number of entries in the vector for one
actor that would need to be changed in order to make it identical to the
vector of the other actor. These differences could be either adding or
dropping a tie, so the Hamming distance treats joint absence as similarity.

Fig 12 Hamming distance of information receiving in Knoke Network
Summary
In this section we studied about various methods described above that are
used in social network analysis to find out the strength between two ties or
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57 equivalence, automorphic equivalence, Valued relations and Binary
relations and how the distance between them is measured using pearson
correlation, covariance, agglomerative clustering, exact, jaccard and
hamming distances.
References
[1] “Introduction to Social Network Methods” by Robert A. Hannem an
University of California
Questions
Q1.How are network roles and social roles different from network "sub -
structures" as ways of describing social networks?
Q2. Explain the differences among structural, automorphic, and regular
equivalence.
Q3. Actors wh o are structurally equivalent have the same patterns of ties
to the same other actors. How do correlation, distance, and match
measures index this kind of equivalence or similarity?
Q4. If the adjacency matrix for a network can be blocked into perfect sets
of structurally equivalent actors, all blocks will be filled with zeros or with
ones. Why is this?
Q5. If two actors have identical geodesic distances to all other actors, they
are (probably) automorphically equivalent. Why does having identical
distances to all other actors make actors "substitutable" but not necessarily
structurally equivalent?
Q6. Regularly equivalent actors have the same pattern of ties to the same
kinds of other actors -- but not necessarily the same distances to all other
actors, or ties to the same other actors. Why is this kind of equivalence
particularly important in sociological analysis?

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58 4
TWO -MODE NETWORKS FOR SNA
Unit Structure
4.0. Objectives
4.1. Understanding Two -mode networks
4.1.1 Bi-partite data structures
4.1.2. Visualizing two -mode data
4.1.3. Q uantitative analysis
4.1.3.1. Two-mode Singular val ue decomposition (SVD) analysis
4.1.3.2. Two-mode factor analysis
4.1.3.3. T wo-mode correspondence analysis
4.1.4. Q ualitative analysis
4.1.4.1. Two -mode core -periphery analysis
4.1.4.2. Two -mode factions analysis
4.1.5. Affiliation Networks
4.1.6. Attribute N etworks
4.2. Summary
4.3. R eferences
4.4. Model Questions
4.0. OBJECTIVES
After going through this unit, you will be able to:
 Explicate Two -mode Network s and its applications
 Comprehend Bi -partite data structure
 Compare the applications of SVD , factor and correspondence
analysis
 Analyse the methods of qualitative analysis
 Describe the importance of affiliation and attribute networks
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59 4.1. UNDERSTANDING TWO -MODE NETWORKS
Nowadays, more data in the network are in 2 -mode structure. This means
that it represents two different types of actors and ties to define the
connections between the one group of actors with other group of actors.
This two -mode data network analyses the Macro -Micro relationships
between the actors.















Figure 4.1.1 .1.
In fi gure 4.1.1.1. two types of actors, one set of actors represented by
circle with red colour and another set of actors by rectangle with blue
colour are connected through the ties. In this the red circles belong to one
group of actors and the blue rectangles belong to another group of actors.
Among the two types of actors one is macro actor, who plays major role in
the society and having relationships with themselves. The other is micro
actors, who plays the roles with the macro actors and in certain occasio ns
they are as well interconnected with themselves. These macro and micro
actors establish the ties between them. This structure is termed as two -
mode network.
The table 4.1.1.1. represents the matrix form of Davis data (Davis et al.,
Homans 1950 ). This da ta is collected by the author over nine -month
period by closely watching and observing the social activities of 18
women in Southern women’s club.
During that period, various subsets of these women had met in a series of
14 informal social events . This da ta shows the list of events E1..E14
attended by the women given in the data table. The women attended the
various activities like going to a store, attending a meeting of a club, a
church s upper, a party, a meeting of an association etc .
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60 Table 4.1.1.1. D avis Southern Women’s’ Matrix data
E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12 E13 E14
EVELYN 1 1 1 1 1 1 0 1 1 0 0 0 0 0
LAURA 1 1 1 0 1 1 1 1 0 0 0 0 0 0
THERESA 0 1 1 1 1 1 1 1 1 0 0 0 0 0
BRENDA 1 0 1 1 1 1 1 1 0 0 0 0 0 0
CHARLOTTE 0 0 1 1 1 0 1 0 0 0 0 0 0 0
FRANCES 0 0 1 0 1 1 0 1 0 0 0 0 0 0
ELEANOR 0 0 0 0 1 1 1 1 0 0 0 0 0 0
PEARL 0 0 0 0 0 1 0 1 1 0 0 0 0 0
RUTH 0 0 0 0 1 0 1 1 1 0 0 0 0 0
VERNE 0 0 0 0 0 0 1 1 1 0 0 1 0 0
MYRNA 0 0 0 0 0 0 0 1 1 1 0 1 0 0
KATHERINE 0 0 0 0 0 0 0 1 1 1 0 1 1 1
SYLVIA 0 0 0 0 0 0 1 1 1 1 0 1 1 1
NORA 0 0 0 0 0 1 1 0 1 1 1 1 1 1
HELEN 0 0 0 0 0 0 1 1 0 1 1 1 1 1
DOROTHY 0 0 0 0 0 0 0 1 1 1 0 1 0 0
OLIVIA 0 0 0 0 0 0 0 0 1 0 1 0 0 0
FLORA 0 0 0 0 0 0 0 0 1 0 1 0 0 0

From the data choices of parties attended by the women as macro
structures may affect the choices of the individual women. These types of
data are two -mode data. The women with one set and the activities with
another set, how the women are tied up with activities can be derived from
the data. This type of activity is also known as macro -micro activity.
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61 In this chapter, the concepts are explained through data which describe the
contributions of a small number of large donors to campaigns supporting
and opposing ballot initiatives in Cali fornia during the period 2000 to
2004. There are 44 members are taken for data analysis. There are 44
initiatives for the donors. Hence t he data set has towo modes such as i)
Donors ii) Initiatives
Two different forms of the data are used.
i) Valued Data
The relations between donors and initiatives using a simple ordinal scale
are described by this type of data. The following table shows the details
about the actor code.
Action Actor code
Contribution towards opposing a particular
initiative -1
No Contribut ion 0
Contribution towards in support of the
initiative +1

ii) Binary Data
The binary data specifies the contribution in the campaign on each
initiative with binary values as given in the table.
Donor Contribution Actor code
Contributed +1
Not Contributed 0

4.1.1 . BI-PARTITE DATA STRUCTURES
The rectangular data matrix is used for storing 2 mode data. The actors
represented in rows and events represented in columns. Figure 4.1.1.2.
shows a portion of the valued data set use d for the analysis.



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62 Table 4.1.1.2. Rectangular data array of California political
donations data

From the given table 4.1.1.2.:
Donors contributed donations in opposition to:
The ballot initiatives 7, 9, and 10 ( (Having more minus -1 values)
Donors contributed a donat ion supporting of:
The ballot initiative 8.(Having more +1 values)
Common approach to two -mode data : It is to convert ed into two one -
mode data sets, and examine relations within each mode separately. For
example, create a data set of actor -by-actor ties , measuring the strength of
the tie between each pair of actors by the number of times that they
contributed on the same side of initiatives, summed across the 40 -some
initiatives. A one-mode data set o f initiative -by-initiative ties can be
created and the coding the strength of the relation as the number of donors
that each pair of initiatives had in common .
Using the suitable tool one-mode data sets are created from a two-mode
rectangular data array . A retrieval technique is used to convert the two
mode dataset into affiliations of one mode valued data set.
The row mode (actors) is selected . The cross -product method for binary
data (Table 4.1.1.3) takes each entry of the row for actor A, and multiplies
it times the same entry for actor B, and then sums t he result. With binary
data, each product is 1 only if both actors were "present" at the event, and
the sum across events yields the number of events in common - a valued
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63 Table 4.1.1.3. Actor -by-actor tie strengths



However in thi s case, the cross -product method to valued data is used to
convert two -mode into one -mode network.
Actors involved status Initiative Result
Neither Actors donated 0,0 (0 * 0) 0
One donated , One not
donated a) 1,0 (+1 * 0)
b) (-1,0) ( -1 * 0) 0 & No tie
Both donated
(in the same direction) a) (1,1) or ( -1,-1)
b) (+1 * +1) or ( -1 *
-1) +1 & Positive
tie
Both donated
(in opposite direction) a) (1, -1) (+1 * -
1)
b) (-1,1) ( -1,+1) -1 & Negative
tie

The minimums method examines the entrie s for the two actors at each
event, and selects the minimum value. For binary data, the result is the
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64 For va lued data, the minimums method the tie between the two actors is
equal to the weaker of the ties of the two actors to the event. This
approach is commonly used when the original data are measured as
valued .
Illustration: The teachers association participated in 16 campaigns. The
association took the same position on issues as the Democratic party (actor
7) ten more ti mes than taking opposite (or no) position. The restaurant
association (node 10) took an opposite position to Mr. Bing (node 9) more
frequently than supporting (or no) position.
Resulting one -mode matrices of actors -by-actors and events -by-events are :
value d matrices . This indicating the strength of the tie based on co -
occurrence .
Two-mode data are sometimes stored in the "bipartite" matrix. A bipartite
matrix is formed by adding the rows as additional columns, and columns
as additional rows. For example, a bipartite matrix of the donors data
would have 68 rows (the 23 actors followed by the 45 initiatives) by 68
columns (the 23 actors followed by the 45 initiatives). The two actor -by-
event blocks of the matrix are identical to the original matrix; the two n ew
blocks (actors by actors and events by events) are usually coded as zeros.
The tool converts two -mode rectangular matrices to two -mode bipartite
matrices . In the tool the data to be entered are
i) Two-mode dataset
ii) Value to fill with in mode ties
iii) Make the result symmetric or not
iv) Output dataset
The value to fill within -mode ties usually zero in the developed tool , so
that actors are connected only by co -presence at events, and events are
connected only by having actors in common. Algorithm for one -mode data
is applied to get the result.
4.1.2. VISUALIZING TWO -MODE DATA
Graphs can be used to visualize 2 -mode data. Both actors and events are
treated as nodes, and lines are used to show the connections of actors to
events (there will be no lines from actors to a ctors directly, or from events
to events).. Figure 4.1.2.1 shows one rendering of the California donors
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65


Figure .4.1.2.1 . Two -mode valued network of California donors and
initiatives
Findings from map:
i) Actors that are close t ogether, are connected because they have similar
profiles of events . For example, the Cahualla and Morongo I ndians in
the lower left corner.
ii) The two tribes were jointly involved in initiatives about gambling
(P70) and environment (P40).
Numeric Methods C aptures
i) Clustering of actors based on events
ii) Co-presence of Actors bring the events together
Final result is Bundles of (Clusters of) actors or events
4.1.3. QUANTITATIVE ANALYSIS
This is an approach that emphasizes statistical and mathematical analysis
to hep to find out the real dimension of the problem. This method mainly
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66 approach is mainly used to find out the various types of relationships
between the actors. In addition this a pproach helps to determine the
solutions for the various issues and used to make valuable decisions. In
this chapter two types of Quantitative analysis i) SVD ii) Factor analysis
are illustrated.
4.1.3.1. T wo-mode Singular valu e decomposition (SVD) analysi s
Factor and Component Analysis : The approach of locating, or scoring,
individual cases in terms of their scores on factors of the common variance
among multiple indi cators.
Scale or Index: Done in terms of participation of the actors in the events.
It is applied either to actors or to events. T he events can be scaled i n terms
of the patterns of co -participation of actors , but weight the actors
according to their frequency of co -occurrence.
Joint variance dimension can be determined and the actors and even ts are
mapped into the same space. This gives information about the
a) Actors those are similar in terms of their participation in events .
b) Events that are similar in terms of what actors participate in them .
c) Actors and E vents that are located near .
Clusters of actors and events that are similarly located may form
meaningful types or domains of social action.
Interpretation of the fundamental factors or dimensions would result in
why the actors and events are having the ties.
Two-mode SVD analysis
Singular Value Decomposition (SVD) : It is a method of identifying t he
factors underlying two -mode valued data. The method of extracting
factors (singular values) differs somewhat from conventional factor and
component s analysis, so for factoring results both SVD and 2 -mode
should be examined.
Example for SVD: Input a matrix of 23 major dono r by 44 California
ballot initiatives. Each actor is scored as -1 if they contributed in
opposition to the initiative, +1 if they contributed in favo ur of the
initiativ e, or 0 if they did not contribute. The resulting matrix is valued
data that can be examined with SVD and factor analysis; however, the low
number of contributors to many initiatives, and the very restricted variance
of the scale are not ideal.


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67 Table 4 .1.2.2 Two -mode scaling of California donors and initiatives
by Single Value decomposition: Singular valu es



Above table shows the “ singular values" extracted from the rectangular
donor -by-initiative matrix usin g the standard tool. The "singular valu es"
are similar to "eigenvalues" in the more common factor and components
scaling techniques.
Result: The joint "space" of the variance among donors and initiatives is
not well captured by a simple characterization.
Issue: If we could easily make sense o f the patterns with ideas like "left/
right" and "financial/moral" as underlying dimensions, there would be
only a few singular values that explained substantial portions of the joint
variance. This result tells us that the ways that actors and events "go
together" is not clean, simple, and easy in this case.
Solution: To solve the issue how the events and donors are "scaled" or
located on the underlying dimensions , the ballot initiatives in Table
4.1.2.2.. shows the location, or scale scores of each of the ballot
proposition on the first six underlying dimensions of this highly multi -
dimensional space.


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68 Table 4.1 .2.2. SVD of California donors and initiatives: Scaling of
initiatives



First dimension : Locate initiatives supporting public expendi ture for
education and social welfare toward one pole, and initiatives supporting
limitation of legis lative power toward the other though interpretations like
this are entirely subjective.
Second & Higher Dimensions : This specifies that initiatives can als o be
seen as differing from one another in other ways. But, the results locate or
scale the donors along the same underlying dimensions. These loadings
are shown in table 4.1.2.3. .
Table 4.1 .2.3. SVD of California donors and initiatives: Scaling of
donor s
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69 Result Analysis : In the positive end of dimension one , the Democratic
party, public employees and teachers unions are found ; at the opposite
pole, Republicans and some b usiness and professional groups are found.
Map: The locations of the actors and ev ents in a scatterplot are visualized
and defined by scale scores on the various dimensions. The map in Figure
4.2.1.1. shows the results for the first two dimensions of this space.


Figure 4.2.1.1 SVD of California donors and initiatives: Two -
dimensiona l map
Result Discussion: First dimension :(left -right in the figure) seems to
have its poles "anchored" by differences among the initiatives;
Second dimension (top -bottom) seems to be defined more by differences
among groups (with the exception of proposi tion 56).
The result produce s some interesting clusters that show groups of actors
along with the issues that are central to their patterns of participation. The
Democrats and unions cluster (upper right) along with a number of
particular propositions in which they were highly active (e.g. 46, 63).
Corporate, building, and venture capitalist cluster (more loosely) in the
lower right, along with core issues that formed their primary agenda in the
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70 4.1.3.2. Two -mode factor analysis
Factor analysis provides an alternative method to SVD to the same goals
such as identifying underlying dimensions of the joint space of actor -by-
event variance, and locating or scaling actor s and events in that space. The
method used by factor an alysis to identify the dimensions differs from
SVD. Table 4.1.3.1. shows the eigenvalues (by prin ciple components)
calculated using the tool.
Table. 4.1.3. 1. Eigen values of two -mode factoring of California
donors and initiatives




Solution: It is different from SVD, and considerable dimensional
complexities are given in the joint variance of actors and events.
Simple characterizations of the underlying dimensions (e.g. "left/right") do
not provide very accurate predictions about the locations of individual
actors or events. The factor analysis method does produce lower
complexity than SVD.
The scaling of actors on the first three factors given in the following table
4.1.3.2. . The first factor, by this method, produces a similar pattern to
SVD. At one pole are Democrats and unions, at the other lie many
capitalist groups. There are, however, some notable differences (e.g.
AFSCME).




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71 Table. 4.1.3. 2.. Loadings of donors




Table. 4.1.3. 3. Loadings of events





Table. 4.1.3.3. shows the loadings of the events. The patterns here
also have some similarity to the SVD results, but do differ considerably in
the specifics.
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72 4.1.3.3. Two -mode correspondenc e analysis
For binary data, the use of factor analysis and SVD is not recommended.
Factoring methods operate on the variance/covariance or correlation
matrices among actors and events. When the connections of actors to
events is measured at the binary leve l (which is very often the case in
network analysis) correlations may seriously understate covariance and
make patterns difficult to separate.
As an alternative for binary actor -by-event scaling, the method of
correspondence analysis can be used.
Correspo ndence analysis
i) It operates on multi -variate binary cross -tabulations
ii) It's distributional assumptions are better suited to binary data.
Example:
The political donor and initiatives data are dichotomized by assigning a
value of
i) 1 if an actor gav e a donation either in favour or against an initiative,
ii) Assigning a zero if they did not participate in the campaign on a
particular initiative.
The partisanship has been given more attention rather than simple
participation. Two data sets - one base d on opposition or not, one based on
support or not are created and two separate correspondence analy ses are
carriedout.
Table. 4.1.3.3. shows the location of events (initiatives) along three
dimensions of the joint actor -event space identified by the corr espondence
analysis method.
Table. 4.1.3.3. Event coordinates for co -participation of donors in California
initiative campaigns



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73 Result: Since these data do not reflect partisanship, only participation, it
reflects. H owever, that this method can be used to locate the initiatives
along multiple underlying dimensions that capture variance in both actors
and events.
Table 4.1.3.4. shows the scaling of the actors.
Table 4.1.3.4. Actor coordinates for co -participation of donors in California
initiative campaigns


The first dimension has some similarity to the Democrat/union versus
capitalist poles. But this difference means that the two groupings tend to
participate in different groups of initiatives. . Visualization is the best
approach to finding mean ingful patterns .
Figure 4.1.3.1. show s the plot of the actors and events in the first two
dimensions of the joint correspondence analysis space.
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74

Figure 4.1.3.1. Correspondence analysis two -dimensional map
Result : In the lower right there are some prop ositions regarding Indian
casino gambling represented by 68 and 70 . The other two propositions
regarding ecological/conservation issues are represented by 40 and 50.
Two of the major Native American Nations (the Cahualla and Morongo
band of Mission Indians ) are mapped together. The result is showing that
there is a cluster of issues that "co -occur" with a cluster of donors - actors
defining events, and events defining actors.
4.1.4. QUALITATIVE ANALYSIS
Actors and events are co-presence with each other . In the case of either an
actor was, or wasn't present, and the incidence matrix is binary , there will
be some issue when data are parsed in correspondence analysis.
Block Modeling : This is an alternative method for correspondence
analysis. It works directly on the binary incidence matrix by trying to
permute rows and columns to fit, as closely as possible, ideali zed images.
This method does n ot involve any of the distributional assumptions that
are made in scaling analysis.
4.1.4.1. Two-mode core -periphery analysis
The core -periphery structure is a typical pattern that divides both the rows
and the columns into two classes. One of the blocks on the main diagonal
is a high -density block which is known as core block ; the other block on
the main diagonal is a l ow-density block. The core -periphery model is
indifferent to the density of ties in the off -diagonal blocks.
Core: When the core -periphery model is applied to actor -by-actor data the
model identif ies a set of actors with high density of ties among themsel ves
known as core. This model shares many events in common . The "core"
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75 events in an event partition; and simultaneously a partition of events that
are closely connected to the acto rs in the core partition.
Periphery: In another set of actors who have very low density of ties
among themselves known as periphery by having few events in common.
Comparison: Between Core and Periphery
S.No. Core Periphery
1 Actors are able to coordina te
their actions Actors cannot coordinate their
actions.

2 Actor are at a structural
advantage in exchange relations
with actors in the periphery. No structural advantage with
core
3 It is a cluster of frequently co -
occurring actors and events. It consi sts of a partition of
actors who are not co -incident
to the same events; and a
partition of events that are
disjoint because they have no
actors in common.

Numerical methods using tools are used to search for the partition of
actors and of events that co mes as close as possibl e to the idealized image.
Table 4.1.4.1. shows a portion of the results of applying this method to
participation (not partisanship) in the California donors and initiatives
data.











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76 Table 4.1.4.1. Results of participation in the California donors and
initiatives data

Genetic algorithm is the numerical search method used by core and
periphery. T he measure of goodness of fit is stated in terms of a "fitness"
score where 0 means bad fit, 1 means excellent fit. The goodness of the
result by examining the density matrix is at the end of the output. If the
block model was completely successful, the 1,1, block should have a
density of one, and the 2, 2 block should have a density of zero.
Result Discussion: The blocked matrix show s a "core" composed of the
Democratic Party, a number of major unions, and the building industry
association who are all very likely to participate in a considerable number
of initiatives (proposition 23 through proposition 18). The remainder of
the actors are grouped into the periphery as both participating less
frequently, and having few issues in common. A considerable number of
issues are also grouped as "peripheral" in the sense that they attract few
donors, and these donors have little in common. The upper right) that core
actors in upper right do participate to some degree (.179) in peripheral
issues. In the lower left the peripheral actors participate somewhat more
heavily (.260) in core issues.

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77 4.1.4.2 . Two -mode Factions Analysis
Factions : Groupi ngs that have high density within the group, and low
density of ties between groups. This method is an alternative block model.
The subgroups factions choice in the tool f its this block model to one
mode data .for any number of specified factions. The two -mode choice fits
the same type of model to two -mode data for only two factions.
Factions model applied to one -mode actor data : Identifies t wo clusters
of actors who are closely tied to one another by attending all of the same
events, but very loosely conn ected to members of other factions and the
events that tie them together
. Factions model applied to one -mode event data Identif ies events that
are closely tied by having exactly the same participants.
The two -mode option in the tool applies the same app roach to the
rectangular actor -by-event matrix. This locate s joint groupings of actors
and events that are as mutually exclusive as possible. Figure 4.1.4.2.
shows the results of the two mode factions block model to the
participation of top donors in poli tical initiatives.
Table 4.1.4.2. Two mode factions model of California $1M donors and ballot
initiatives (truncated)

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78 Two measures of goodness -of-fit are available.
i) Fitness score : It is the correlation between the observed scores such as
0 or 1.The scores that should be present in each block.
ii)Densities in the blocks : It gives goodness of fit. For a factions ana lysis,
an ideal pattern is dense 1 - blocks along the diagonal and zero -blocks off
the diagonal .
Result Discussion: The fit of the two fa ctions model is not as impressive
as the fit of the core -periphery model. This suggests that an image of
California politics as one of two separate and largely disjoint issue -actor
spaces is not as useful as an image of a high intensity core of actors and
issues coupled with an otherwise disjoint set of issues and participants.
The blocking itself also is not very appealing, placing most of the actors in
one faction (with modest density of .401). The second faction is small, and
has a density (.299) that i s not very different from the off -diagonal blocks.
4.1.6 . AFFILIATION NETWORKS
Persons A and B are both members of a club. They can form an open triad,
or a structural hole , but it is infer ed that if A and B are members of the
same club (Figure 4.1.6.1) , they may know each other; and the triad is
closed. This is a weak inference. T o make a more concrete case, it should
be considered that if they were members of the club at the same time, or if
the club has multiple chapters in differen t cities, etc.







Figure 4.1.6.1 . Triadic closure and co -membership
Consider the same people are members of more then one club as shown in
the top of Figure 4.1.6.2. where nodes E, F, and H are co -members in 2
clubs . This presents a stronger association between the people , having a
common group identity. The co-memberships can be accumulated until
the connections are real, and weigh the inferred links accordingly.


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79

























Figure . 4.1. 6.2 Creating an affiliation network from a 2 -mode
network

Figure . 4.1.6 .2. shows two resulting projected networks i) A network of
people where links were determined through co -membership in groups ii)
A network of groups where links were determined by comembership of
people. To create these networks, count the come mberships for every one
of the people or for every one of the clubs.
These networks can be used for all so cial network analysis, but are
particularly for analysis with the island method and clustering techniques .
The reason is these networks are essentiall y networks of similarities or
correlations .
4.1.7 ATTRIBUTE NETWORKS
This an application of 2 -mode network analysis is based on the idea of
homophily meaning that similar, the idea that people who share interests
or attributes are more likely to talk to e ach other and form ties than people
who are very different.







1 2 3 A B C A G E B F C 2 1 3 H G F E D 1 1 1 1 1 2 2 1 2 H 1 1 1 2 3 Bipartite Affiliation Network – People Organizations
People OrganizatioUnipartite Representations
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80










Figure 4.1.7.1. Common Network (Election Hashtags and People )








Figure 4.1.7.2. Attribute network
(People Network, Hash tag Network)
When people become more tightly connected, they become more similar
in their views , but up to a certain limit. However, if one wants to build a
“suggest a friend” mechanism for their online social network, treating an
attribute or interest matrix as a 2 -mode net work can be a useful
mechanism. Every one of the pieces of information (tags, keywords, etc.)
could be treated as a node in a 2 -mode network, compute a person -to-
person affiliation network from it, and apply the island method or
clustering to find potentia l groupings of people. Then, to suggest friends,
pick the top links in the affiliation network.
An inverse affiliation network , attributes through people , could provide
very interesting insights as well. For example in mapping political
discourse on Twitt er, the tweets from several thousand people containing
the hashtag#election can be extracted, build a 2 -mode network from
people to hashtags, and compute a #hashtag through -people affiliation
network. From the figure 4.1.7.1 the election related tweets ar e given. The
tweets are given as blue circle for# hashtags and the people nodes are
specified in brown circles. If they are taken in attribute network the
resultant Figure (4.7.1.2) contains the people with election #hashtags and
others separately. In such networks, clusters will act as proxies for entire A BC D E F G A B C D
D E
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81 areas of discourse and will separate people with election hash tag and not.
The exploration of clusters may yield an idea of divisions inside
supporting campaign in the tweets.
4.3. SUMMARY
Two-mode data gives interesting possibilities for gaining insights into
macro -micro or agent -structure relations. With two -mode data, the macro -
structures (events) pattern the inte ractions among agents (or not) can be
determined. T he actors define and create macro struc tures by their patterns
of affiliation with them also explained through the illustration. . In
addition, to describe patterns of relations between actors and structures
simultaneously.
In this chapter some of the typical ways in which two -mode data arise i n
social network analysis, and the data structures that are used to record and
manipulate two -mode data are examined . The utility of two -mode graphs
(bi-partite graphs) in visualizing the "social space" defined by both actors
and events also exhibited in t his chapter.
The methods for trying to identify patterns in two -mode data that might
better help us describe and understand why actors and events "fit together"
in the ways they do. One class of methods derives from factor a nalysis and
related approaches. They can also be useful to identify groups of actors
and the events that "gotogether" when viewed through the lens of latent
abstract dimensions.
Another class of methods is based on block modeling. The goal of these
methods is to assess how well the obser ved patterns of actor -event
affiliations fit some prior notions of the nature of the "joint space" .To the
extent that the actor -event affiliations can be usefully thought of in these
ways, block models also then allow us to classify types or groups of act ors
along with the events that are characteristic of them.
Another important topic discussed in this chapter is affiliation and
attribute networks. The affiliation networks are used to examine the
comembership relations whereas the attribute networks are u sed to cluster
the networks based on the attributes.
For illustration the Davis data and California Teachers Association data is
used in this chapter.
4.4. REFERENCE FOR FURTHER READING
1. Introduction to Social Network Methods: Robert A. Hanneman, Mark
Riddle, University of California, 2005, Published in digital form and
available at http://faculty.ucr.edu/~hanneman/nettext/index.html.
2. Social Network Analysis for Startups - Finding connections on the social
web: Maksim Tsvetovat, Alexander Kouznetsov, O'Reilly Media, 2011. munotes.in

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82 3. Social Network Analysis - 3rd edition, John Scott, SAGE Publications,
2012.
4. Mark S. Handcock, David Hunter, Carter T. Butts, Steven M. Goodreau
and Martina Morris. 2003 statnet: An R package for the Statistical
Modeling of Soc ial Networks http://www.csde.washington.edu/statnet
5. Vladimir Batagelj and Andrej Mrvar (2006), Pajek datasets
 http://vlado.fmf.uni lj.si/pub/networks/data/.
6. Krackhardt and Stern (1988) developed a very simple and useful
measure of the group embeddin g based on comparing the numbers of
ties within groups and between groups
7. Getting Started in Social Network Analysis with NETDRAW, Bruce
Cronin University of Greenwich Business School, Occasional Paper
01/15, January 2015 brought to you by CORE View me tadata, citation
and similar papers at core.ac.uk
8. Structural Holes, The Social Structure of Competition, Ronald S. Burt
9. www.analytictech.com
10. https://www.datacamp.com/
11. https://networkdata.ics.uci.edu/netdata/html/davis.html
12. Finding Social Groups: A Meta -Analysis of the Southern Women
Data1 Linton C. Freeman University of California, Irvine
4.5. MODEL QUESTIONS
1. What is two -mode network? Explain with example.
2. How the bipartite network is managed? Give example.
3. What are the two alter native methods used in bipartite network?
4. State the purpose of visualising data.
5. Describe the quantitative analysis in social network using suitable
example.
6. Compare SVD with two -mode factor analysis
7. Illustrate how the results of correspondence analysis can be interpreted
8. Write short note on quality analysis.
9. Explain the affiliation networks briefly.
10. Summarize the purpose of attribute networks.
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