The document discusses social network analysis and community detection in networks. It first covers the concept of homophily, which is the tendency for similar people to associate with each other. It then introduces stochastic block models, which are generative models used to detect communities in networks. These models define parameters that represent how likely nodes within the same community are to connect. The maximum likelihood estimation approach is used to estimate the parameters and detect communities by finding the model that best explains the observed network connections.

1. Social and Economic
Network Analysis
UNIT – III
SOCIAL NETWORK ANALYSIS

2. Overview
Homophily
Community
Detection
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3. Homophily
NETWORKS, CROWDS AND MARKETS - CHAPTER 4
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4. Homophily
▪“Birds of feather flock together”
▪Similarity begets friendship – Plato
▪People loves those who are like
themselves - Aristotle
▪Homophily - we tend to be similar
to our friends
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5. Homophily
▪Links in a social network tend to connect people who are similar to one another
oImmutable characteristics: racial and ethnic groups; ages; etc.
oMutable characteristics: places living, occupations, levels of affluence, and interests, beliefs,
and opinions; etc
▪Friendship through common friend – Triadic closure (network structure)
▪Friendship through common school/college/job/interest – Contextual features
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6. Homophily Test
▪Consider a random network G = (V, E) where each node is assigned male with
probability p, and female with probability 1 - p
▪Consider any edge (i, j) Ꜫ E of this random network G
▪Both ends of edge will be male/female with probability p2 / (1-p)2
▪Let random variable Xij = 1 if it is a cross-edge, and Xij = 0 otherwise
▪Then Xij is a Bernoulli random variable such that P(Xij = 1) = 2p(1 - p)
Homophily Test: If the fraction of cross-gender edges is significantly less
than 2p(1 - p), then there is evidence for homophily
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7. Homophily Test
▪p = 2/3 and the fraction of cross-
gender edges is 5/18
▪On the other hand the fraction
of cross-gender edges in the
random network = 2p(1 - p) =
4/9 = 8/18
▪Note that 5/18 < 8/18 showing
evidence of homophily
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8. Homophily
SOCIAL INFLUENCE
▪Influenced by the people we are connected to
▪Socialization – existing links shapes people’s
interest
SELECTION
▪We select friends who are similar to us
▪Formation of new links
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9. Affiliation Networks
▪Affiliation networks are examples of a class of graphs called bipartite graphs
▪Affiliation networks represents the participation of a set of people in a set of foci
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10. Affiliation Networks
Triadic closure: The formation
of the link between B and C
with common friend A
Focal closure (selection): The
formation of the link between
B and C with common interest
A
Membership closure (social
influence): The formation of
the link between B and C
influenced by a friend A
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11. Affiliation Networks
Triadic Closure
Focal Closure
Membership
Closure
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12. Community Detection
SOCIAL AND ECONOMIC NETWORKS - CHAPTER 13
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13. Stochastic Block Models
▪Posterior Block modelling: By examining how network is generated from
underlying community structure one can detect communities in network
▪Goal: Define a model that can generate networks
oThe model will have a set of “parameters” that we will later used to detect communities
Given a set of nodes, how do communities “generate” edges of the network?
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14. Stochastic Block Models
1. Given the communities, generate the network using the model
2. Given a network, find the “best” community using the model developed
C
A
B
D E
H
F
G
C
A
B
D E
H
F
G
Generative model
C
A
B
D
H
C
A
B
D
H
Generative model
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15. Community-Affiliation Graph
Generative model B(V, C, M, {pc}) for graphs:
◦ Nodes V, Communities C, Memberships M
◦ Each community c has a single probability pc
◦ Later we fit the model to networks to detect communities
Model
Network
Communities, C
Nodes, V
Model
pA pB
Memberships, M
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16. AGM: Generative Process
◦ For each pair of nodes in community 𝑨, we connect them with prob. 𝒑𝑨
◦ The overall edge probability is:



−
−
=
v
u M
M
c
c
p
v
u
P )
1
(
1
)
,
( If 𝒖, 𝒗 share no communities:
𝑷 𝒖, 𝒗 = 𝜺
𝑴𝒖 … set of communities
node 𝒖 belongs to
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17. Detecting Communities
C
A
B
D E
H
F
G
Given a Graph 𝑮(𝑽, 𝑬), find the Model
1. Affiliation graph M
2. Number of communities C
3. Parameters pc
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18. Maximum Likelihood Estimation
Given: Data 𝑿
Assumption: Data is generated by some model 𝒇(𝚯)
𝒇 … model
𝚯 … model parameters
Want to estimate: 𝑷𝒇 𝑿 𝚯)
The probability that our model 𝒇 (with parameters 𝜣) generated the data 𝑿
We need to find the most likely model that could have generated the data:
arg max
Θ
𝑷𝒇 𝑿 𝚯)
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19. Example: MLE
Imagine we are given a set of coin flips
Task: Figure out the bias of a coin!
◦ Data: Sequence of coin flips: 𝑿 = [𝟏, 𝟎, 𝟎, 𝟎, 𝟏, 𝟎, 𝟎, 𝟏]
◦ Model: 𝒇 𝚯 = return 1 with prob. Θ, else return 0
◦ What is 𝑷𝒇 𝑿 𝚯 ? Assuming coin flips are independent
◦ So, 𝑷𝒇 𝑿 𝚯 = 𝑷𝒇 𝟏 𝚯 ∗ 𝑷𝒇 𝟎 𝚯 ∗ 𝑷𝒇 𝟎 𝚯 … ∗ 𝑷𝒇 𝟏 𝚯
◦ Then, 𝑷𝒇 𝑿 𝚯 = 𝚯𝟑
𝟏 − 𝚯 𝟓
◦ For example:
◦ 𝑷𝒇 𝑿 𝚯 = 𝟎. 𝟓 = 𝟎. 𝟎𝟎𝟑𝟗𝟎𝟔
◦ 𝑷𝒇 𝑿 𝚯 =
𝟑
𝟖
= 𝟎. 𝟎𝟎𝟓𝟎𝟐𝟗
◦ Data 𝑿 was most likely generated by coin with bias 𝚯 = 𝟑/𝟖
𝑷
𝒇
𝑿
𝚯
𝚯
𝚯∗
= 𝟑/𝟖
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20. Maximum Likelihood Estimation
Goal: Find 𝚯 = 𝑩(𝑽, 𝑪, 𝑴, 𝒑𝑪 ) such that:
How do we find 𝑩(𝑽, 𝑪, 𝑴, 𝒑𝑪 ) that maximizes the
likelihood?
Finding B means finding the bipartite affiliation network

P( | )
AGM
arg max

𝑮
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21. Maximum Likelihood Estimation
Given graph G(V,E) and Θ, we calculate likelihood that Θ generated G: P(G|Θ)
0 0.9 0.9 0
0.9 0 0.9 0
0.9 0.9 0 0.9
0 0 0.9 0
Θ=B(V, C, M, {pc})
0 1 1 0
1 0 1 0
1 1 0 1
0 0 1 0
G
P(G|Θ)
G
A B
arg max
𝐵(𝑽,𝑪,𝑴, 𝒑𝑪 )
ෑ
𝒖,𝒗∈𝑬
𝑷 𝒖, 𝒗 ෑ
𝒖𝒗∉𝑬
(𝟏 − 𝑷 𝒖, 𝒗 )
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