CSV886: Social, Economics and Business Networks. Lecture 2: Affiliation and Balance. R Ravi ravi+iitd@andrew.cmu.edu

Transcription

1 CSV886: Social, Economics and Business Networks Lecture 2: Affiliation and Balance R Ravi ravi+iitd@andrew.cmu.edu

2 Granovetter s Puzzle Resolved Strong Triadic Closure holds in most nodes in social networks Local bridges span across communities and are likely to provide new leads in job searches Under these circumstances, these local bridge links (job generating leads) will be weak ties (acquaintances) 2

3 Evidence: Clustering coefficient Fraction of pairs of neighbors that are friends Compute CC for A 3

4 Large Dataset Version Cellphone call network [Onnela+] Edge if conversations both ways in observed period (18 weeks) Tie Strength of an edge number of minutes spent in conversations 4

5 Neighborhood Overlap For an edge (A,B) # of common neighbors of A and B # of neighbors of A or B Q: What is the neighborhood overlap of a local bridge? 5

6 Tie Strength and Neighborhood Overlap [Onnela +] 6

7 Problem Break Does STC Hold? 7

8 Problem Break Does STC Hold? 8

9 Practice Identify the local bridges in this network 9

10 Practice Which one obeys STC? 10

11 Cross, Borgatti & Parker (2002) 11

12 Closure Nodes at the center Embeddedness Structural: Number of mutual neighbors Relational: Based on relations and other (nonnetwork) interactions High embeddedness implies high trust and integrity in the relationship (Coleman) Nodes with high trust can enable better group mobilization, coordination and performance 12

13 Brokerage - Nodes at the boundaries Span structural holes in organization Creativity amplifiers (by synthesizing ideas) Social gate-keeping across communities (Burt) Nodes connecting tightly interlinked communities otherwise isolated from each other can profit 13

14 Types of Social Capital Closure (cf. Coleman) vs Brokerage (cf. Burt) Bonding capital vs Bridging capital Mobilization vs Ideation Exploitation vs Exploration 14

15 Kongisberg Bridges over River Pregel: Euler Is it possible to take a scenic walking tour crossing every bridge exactly once? 15

16 Solve the puzzle Is it possible to take a scenic walking tour crossing every bridge exactly once? 16

17 Schedule 17

18 Homophily Birds of a feather flock together [Moody+] 18

19 A test for homophily Fraction p of boys, q of girls Independent links have Both ends boys Both ends girls Cross-gender Homophily test: If fraction of cross-gender links is significantly less than, there is evidence for homophily 19

20 Selection vs Social Influence Selection: immutable characteristics leading to friendship Social influence: friends influence your behavior producing alignment 20

21 Example longitudinal studies Networks of delinquent children Significant selection rather than just social influence [Cohen & Kandel] Networks of obese people Significant social influence rather than just selection [Christakis & Fowler] 21

22 Implications for Marketing Selection: Use the clustering to infer from one person s tastes, candidates for marketing to other (market research) Influence: Choose influencers in the cluster to promote the product (campaign) 22

23 Affiliation Networks Bipartite graphs relating people and their affiliated activities or foci The ruling class in the US 23

24 Network Forensics Find smallest affiliation network explaining these connections 24

25 Network Forensics Find smallest affiliation network explaining these connections 25

26 Network Forensics Find smallest affiliation network explaining these connections 26

27 Find the smallest affiliation network 27

28 Social Affiliation Networks Affiliation and social links together 28

29 Closure in social affiliation networks Recall closure creates a third edge in a triangle Three types based on the type of triangle closed 29

30 Three types of closure 30

31 at work 31

32 Evidence of triadic closure Chance of forming a link as function of number of mutual friends in a US university [Kossinets & Watts] 32

33 Evidence of focal closure Chance per day of forming links as function shared foci 33

34 Evidence of membership closure Probability of editing a wikipedia article as function of number of friends who have done so 34

35 Structural Balance Consider all pairs of interactions in a community Add + or labels to each one depending on friendly or hostile interaction What configurations are stable over time? Check all configurations of three entities 35

36 Which triangles are balanced? 36

37 Balance of the whole network A complete network is balanced if for every set of three nodes, the edges connecting them are labeled in a balanced configuration OR For every triangle, the number of edges is even (zero or two) 37

38 Examples of 4-node networks 38

39 Cartwright-Harary Theorem If a labeled complete graph is balanced, then either all pairs of nodes are friends or there are two groups of friends with every pair drawn from both groups being enemies [local definition of balance to global property of balanced networks] 39

40 Details In a balanced network that is not completely labeled +, nodes fall into two groups X and Y such that Every pair in X are friends (+) Every pair in Y are friends (+) Everyone in X is an enemy of everyone in Y (-) 40

41 Proof Assume there is a single edge Consider neighbor around a node A If nodes do really fall into two groups, friends of A must be in X and all enemies of A must be in Y 41

42 Proof Verify if the X and Y defined this way works 42

43 Applications Evolution of international alliances Europe leading up to WW 1 Separation of Bangladesh from Pakistan Trust in on-line communities 43

44 Recall WWI Alliances 44

45 Problem Break: Ex Add a new node and label links from it to all others to keep the network balanced 45

46 Problem Break: Ex Add a new node and label links from it to all others to keep the network balanced 46

47 Problem Break: Ex Add a new node and label links from it to all others to keep the network balanced 47

48 Problem Break: Ex When can you add a new node and label links from it to all others to keep the network balanced? 48