Formation Social and Economic Networks Jafar Habibi MohammadAmin - - PowerPoint PPT Presentation

Correlation Patterns & Link Formation

Social and Economic Networks

Jafar Habibi MohammadAmin Fazli

Social and Economic Networks 1

ToC

• Correlation Patterns & Link Formation
• Correlation & Assortativity
• Homophily
• Affiliation
• Tracking Link Formation
• Spatial Segregation
• Readings:
• Chapter 3 from the Jackson book
• Chapter 4 from the Kleinberg book

Social and Economic Networks 2

Correlation & Assortativity

• Correlation patterns in SENs:
• What is the pattern of connections among nodes with property P and

property Q?

• Assortativity: Do high degree nodes connect to low degree nodes?

Social and Economic Networks 3

Correlation & Assortativity

• Important discoveries in economical studies:
• Serrano & Boguna: While large countries have a vast number of trades with each
• ther, there exist a negative correlation among degrees of countries in the trade

network.

• They show that in the trade network the average degree of a node with degree d,

can be modeled by 1

𝑒2

• They show that these networks can be described by a hub-and-spoke architecture.
• Important discoveries in social studies
• Core-Periphery patterns: There is a core of highly connected and interconnected

nodes and a periphery of less connected nodes

• Homophily and Segregation: Tendency to be connected to (structurally) similar nodes

Social and Economic Networks 4

Clustering Patterns

• Studying clustering behavior of SENs can help. Example:
• Correlation between clustering coefficient & degree:
• Goyal et al. show that in the network co-authorship among economic researchers has CC
• f 0.157 while the CC of high degree nodes is only 0.043!
• One way to study this is to compare cloverall and clavg
• Cloverall can be thought of as a weighted averaging of clustering across nodes with weights

proportional to the number of pairs of neighbors that the nodes have (exactly 𝑒𝑗

2 ).

Why? See on the black board.

• Clavg weights them equally.
• If Cloverall << ClavgClustering is significantly lower for high degree nodes and vice versa.

Social and Economic Networks 5

Network of Co-authorship Physics Mathematics Biology Cloverall/Clavg 0.45/0.56 0.15/0.34 0.09/0.60

Homophily

• People are more prone to maintain relationships with people who are

similar to themselves.

• Age
• Race
• Religion
• Many SENs are

segregated

• Shrum et al. : In middle

schools less than 10% of expected cross-race friendship exist.

Social and Economic Networks 6

Homophily

• Homophily can produce a division of a social network into densely-

connected, homogeneous parts that are weakly connected to each

• ther.
• Homophily provides natural basis for triadic closure: When B and C

has a common friend A, the principle of homophily suggests that B and C are each likely to be similar to A, and hence quite possible to be similar to each other

• There is an elevated chance that B-C will form even if neither of them is

aware that the other one knows A

Social and Economic Networks 7

Measuring Homophily

• Suppose that a p fraction of all

individuals are male and a q fraction are female.

• What is the probability that a given

edge has different gender endpoints? 2pq

• If the fraction of cross-gender edges

is significantly less than 2pq, then there is evidence for homophily

Social and Economic Networks 8

Inverse Homophily

• To have a fraction of cross-gender edges that is significantly more

than 2pq

Social and Economic Networks 9

Mechanisms Underlying Homophily

• Selection: the tendency of people to form friendships with others

who are like them.

• Explicit selection: choosing friends in a small group
• Implicit selection (opportunities to form links) : at more global levels such as

attending schools, living in neighborhoods or working for a company that are relatively homogenous compared to the population at large

• Socialization (Social Influence): people may modify their behaviors to

bring them more closely into alignment with the behaviors of their friends.

• Can be viewed as the reverse of selection

Social and Economic Networks 10

Selection & Socialization

• With selection, the individual characteristics drive the formation of

links, while with social influence, the existing links in the network serve to shape people’s (mutable) characteristics

• Have the people in the network adapted their behaviors to become

more like their friends, or have they sought out people who were already like them?

• Longitudinal study of social networks
• Example: Christakis & Fowler

Social and Economic Networks 11

Selection & Socialization

• Christakis & Fowler:
• The structure of a network between obese & non-obese people is observed for a

32-years period

• This network shows homophily patterns for the obesity feature.
• Three hypotheses are considered as a reason for this homophily:
• i) because of selection effects, in which people are choosing to form friendships with others
• f similar obesity status
• ii) because of the confounding effects of homophily according to other characteristics, in

which the network structure indicates existing patterns of similarity in other dimensions that correlate with obesity status

• iii) because changes in the obesity status of a person’s friends was exerting a (presumably

behavioral) influence that affected his or her future obesity status

• Statistical analysis show that iii is more significant that i and ii. Obesity is highly

contagion.

Social and Economic Networks 12

Selection & Socialization

• Crandel et al:
• Define similarity between

two Wikipedia editors as

• Baseline: the average of

similiarity for non- interacting editors

Social and Economic Networks 13

Affiliation

• The idea: we can consider link formation

enablers inside the network

• Foci or Focal point
• Affiliation network:
• Consider a node for each foci and a node for

each person

• If person A participate in foci X connect A to X
• Affiliation networks are bipartite
• but implicit relations among the nodes of both

sides exist

Social and Economic Networks 14

Selection & Socialization in Affiliation Networks

• We can inject focal

points into a SEN

• Social-Affiliation Network
• If two people participate in a shared focus, this provides them with an
• pportunity to become friends; and if two people are friends, they

can influence each other’s choice of foci.

Social and Economic Networks 15

Closures in Social-affiliation Networks

Social and Economic Networks 16

Link Formation in SENs

• The effect of triadic

closure (Kossinet & Watts)

• The baseline link

formation probability: 1 − 1 − 𝑞 𝑙

• Taking many

snapshots of a university email dataset and averaging T(k) over all the snapshots

Social and Economic Networks 17

Link Formation in SENs

• The effect of focal

closure (Kossinet & Watts)

• Considering class

schedule of students as focal points

Social and Economic Networks 18

Link Formation in SENs

• The effect of

membership closure (Backstrom et al.)

Social and Economic Networks 19

Spatial Segregation

• People live near
• thers like them

Social and Economic Networks 20

The Schelling Model

• Two types of agents: X & O
• An unsatisfied agent has less that t similar neighboring agents

Social and Economic Networks 21

The Schelling Model

• Sequentially, in each round an unsatisfied agent is chosen and is

moved to an unoccupied node where it will be satisfied

• Different movement can be considered (e.g. nearest good node)

Social and Economic Networks 22

The Schelling Model

• A larger example: (t = 3, 150 by 150 board with 2500 empty space,

random initial assignment of agents, move to random good places)

• Convergence After 50 steps

Social and Economic Networks 23

The Schelling Model

• A larger example: (t = 4, 150 by 150 board with 2500 empty space,

random initial assignment of agents, move to random good places)

Social and Economic Networks 24

The Schelling Model

• A larger example: (t = 4, 150 by 150 board with 2500 empty space,

random initial assignment of agents, move to random good places)

Social and Economic Networks 25

The Schelling Model

Social and Economic Networks 26