Modeling Complex Networks

1. Modeling Complex Networks Troy Tassier University of Michigan and Fordham University ttassier@umich.edu www.umich.edu/~ttassier/

2. The Study of Social Interactions • Peer networks • Tipping Points • Participation in Riots • Voting Behavior • Drug use/ cigarette smoking • Crime/ Social Norms • Poverty traps

3. Outline • Today • Introduction to studying networks (definitions, concepts etc…) • General measures and properties of networks • Friday • An empirical example of network modeling • Weak Ties

4. The current state • Until recently most models of interaction ignored the actual network on which interactions occurred. • Random/ threshold models. I’ll participate in a riot if I see XX% also riot. • But your likelihood of rioting, smoking, doing drugs, and voting behavior probably depends more on your specific friends than on the macro-participation rate.

5. New Models of Social Interaction • Barabasi, Newman, Watts • How does the structure of interaction networks influence system behavior? • Crime (Kahan 1997, Picker 1997 Cooter 2000) • Job finding (Calvo, Jackson, Montgomery, Tassier)

6. Social Interaction Models (cont.) • Voting Behavior (Huckfeldt and Sprague) • Drugs, Education, Poverty (Arrow and Durlauf) • Spread of Rumors/ Word of Mouth Marketing (Farrell, Rosen) • Spread of Social Conventions (Morris, Young)

7. What is a network? • A collection of nodes (vertices) • Connected by edges (links) • Can be represented by a graph, a set of vectors, or an adjacency matrix.

8. Social Networks: • Friendships, Sexual Contacts, Marriage between Families, Boards of Directors, Committee Membership, Alumni Groups, Co-Authorships, Voting History, e-mail.

9. Technological and Information Networks: • Internet, Electric Power Grid, Roadways, Railways. • Academic Citations,World Wide Web.

10. Biological Networks • Food webs, Neural Networks, Metabolic Pathways.

11. Networks and the Eyeball Test • Examples: • Sexual contacts • Friendship networks • Food web

12. Type of Edges: • Directed: edges have a direction, only go one way (citations, one way streets) • Undirected: no direction (committee membership, two-way streets) • Weighted: Not all edges are equal. (Friendships)

13. Definitions: • Nodes, Edges, Graphs • Degree: the number of edges connected to a node. • In-degree: Number of incoming edges. • Out-degree: Number of outgoing edges. • In-degree=Out-degree for undirected graphs.

14. I=0 O=3 I=2 O=1

15. Definitions: • Component: The component to which a node belongs is the set of nodes that can be reached from it by traversing edges of the graph. • Geodesic path: Shortest path along the network from one node to another. • Diameter: Length of the longest geodesic path between any two nodes.

16. Characterizing Networks: • Characteristic Path Length • Clustering • Weak vs. Strong Ties • Degree Distribution • Mixing Patterns

17. Characteristic Path Length • Average geodesic distance between all nodes in the network.

18. Example: • D(ab)=1, D(ac)=1, D(ad)=2 • D(bc)=1, D(bd)=2 • D(cd)=1 • L=(1+1+2+1+2+1)/6 = 8/6 a d b c

19. Small World Phenomenon • Milgram Experiment • “Six Degrees” John Guarre • Six-degrees of… Kevin Bacon, Rogers Hornsby, Monica Lewinsky… • The Oracle of Bacon...

20. Clustering • For any set of K nodes there are connected to node i: • (K-1)K potential edges (directed graph) • (K-1)K/2 potential edges (undirected graph) • Clustering is the fraction of these edges that exist. • How many of my friends know each other?

21. Example: • Ka=2, Kb=2, Kc=3, Kd=1 • Ca=1 (because b and c connected) • Cb=1 (because a and b connected) • Cc=1/3 (a-b, not b-d, not a-d) • Cd=0 • Average clustering = (7/3) /4=7/12 a d b c

22. Weak Ties and Strong Ties • Close friends (strong ties) are more likely to know one’s other friends than are less-close friends (weak ties.) • Friendship networks with more weak ties tend to have less clustering. • Less clustering for a given number of connections implies a larger network span.

23. Importance of Weak Ties • Puts one nearer to more people. • Increases the size of one’s network for a given distance.

24. Me

25. Me

26. Implications of Weak Ties • With more weak ties: • Information diffuses faster. • Closer to everyone in the network. • Efficiency increases (?) • Less overlap of connections. • Networks more fragile (?)

27. Degree Distribution • Normal? • Exponential? • Scale Free (power law)? • How many people do you know on a first name basis with the following last name?

28. Mixing Patterns • Segregation in social networks. • Degree Correlation: • Assortative Mixing High degree nodes connect to other high degree nodes. • Dis-Assortative Mixing: High degree nodes connect to low degree nodes.

29. Models of Networks

30. Uniform Random Graphs • Given N nodes connect each pair with probabilty p<1. • Average degree: z=p(N-1) • Characteristic Path Length: L=log(N)/log(z) • Has the small world property.

31. Problems with the Model • Clustering: Probability two nodes are connected is p. • Any two nodes are equally likely to be connected. • The likelihood my two best friends know each other is equal to any other two people. • Degree distribution: Poisson. • No mixing structure, segregation, degree correlation etc...

32. One Dimensional Small World Model • Arrange N nodes in a ring. • Connect each node to k others in each direction. (The degree of each node is 2k.) • With probability p “re-wire” each connection from node i to a new node.

33. Properties of the Small World Model • If p=0 => regular lattice. • Large L, Large C • If p=1 => random network. • Small L, Small C • What about 0<p<1?

34. Growing random networks. • Menczer and Tassier Labor Market networks. • Agents make connections in order to find information about jobs. • Resulting networks are “Small World-ish”.

35. Preferential Attachment • Create a new node and connect it to existing nodes. • Connecting nodes chosen as a function of their degree. • Most connected nodes are the most likely to get new connections. • “Rich get richer” model.

36. Characteristics of P.A. • Small World. • High Clustering. • Power Law Degree Distribution. • Can fit data but does it make sense? • WWW, Citations (yes?) • Social Networks (probably not?)

37. An Exercise • Of the networks generated by the uniform random model, the small world model or the preferential attachment model rank which network is most robust to the failure of individual nodes. Why?

38. What is different about social networks and other networks concerning the transfer of information? • Compare the relative ease with which a virus would move across the networks generated by the three models. Why? • Compare the relative ease with which a new technological innovation would travel across the networks of the three models. • Compare the relative ease with which a rumor would travel across the networks of the three models.