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 other, 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 𝑒 2 • 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 of 0.157 while the CC of high degree nodes is only 0.043! • One way to study this is to compare cl overall and cl avg • Cl overall 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 𝑒 𝑗 2 ). Why? See on the black board. • Cl avg weights them equally. • If Cl overall << Cl avg Clustering is significantly lower for high degree nodes and vice versa. Network of Co-authorship Physics Mathematics Biology Cl overall /Cl avg 0.45/0.56 0.15/0.34 0.09/0.60 Social and Economic Networks 5
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 other. • 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 of 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 opportunity 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 others 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
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