CSE 522 – Algorithmic and Economic Aspects of the Internet

1. CSE 522 – Algorithmic and Economic Aspects of the Internet Instructors: Nicole Immorlica Mohammad Mahdian

2. Previously in this class • Some common properties of social networks • Need for generative models • Several generative models for power law dist • Optimization • Multiplicative processes • Preferential growth • Power law graph models: • preferential attachment

3. This lecture • Analysis of the degree sequence of preferential attachment graphs • Other power law graph models • The copying model • Heuristically optimized tradeoffs • models for small world networks (time permitting)

4. Preferential attachment, recap. • Start with a graph with one node. • Vertices arrive one by one. • When a vertex arrives, it connects itself to one (m, in general) of the previous vertices, with probability proportional to their degrees.

5. Preferential attachment • Heuristic analysis (Barabasi-Albert): degree distribution follows a power law with exponent -3. • Theorem (Bollobas, Riordan, Spencer, Tusnady). For d < n1/16, the fraction of vertices that have degree d is almost surely around

6. Copying models • Kleinberg et al. 1999 and Kumar et al. 2000 • Vertices join one by one, and each new vertex connects to m old vertices (picked as follows). • A new vertex picks an old vertex uniformly at random as its prototype. • For each link on the prototype, the new vertex copies the link with probability p, or replaces the link by a link to a randomly selected vertex with probability 1-p. • Captures the power law, as well as the “locally dense, globally sparse” features of the web.

7. Heuristically Optimized Tradeoff • Fabrikant, Koutsoupias, Papadimitriou, 2002 • Each node is a point in the unit square • Nodes arrive one by one • Upon arrival, node i connects to a node j that minimizes .dij + hj, where dij is the Euclidean distance between i and j, and hj is the graph distance between j and node 1 (the center).

8. The FKP model

9. Small World Networks • Low average distance L • Definition: The average distanceL of a network is the number of edges in the shortest path between two vertices, averaged over all pairs of vertices. • High clustering coefficient C • Definition: The clustering coefficientC of a network is the probability that two neighbors of a random vertex are connected by a single edge.

10. Small World Networks • Many examples • Film actors: edge means actors appeared in a film together • Power grid: edge represents high-voltage transmission lines between generators, transformers, or substations • Neural network of worm C. elegans: two neurons joined by an edge if connected by synapse or gap junction Data from Watts-Strogatz

11. Models • Regular network, e.g. Cnk • High clustering coefficient (C ¼ ¾) • High average distance (L ¼ n/2k) • Random network, e.g. G(n,k/n) • Low average distance (L ¼ ln(n)/ln(k)) • Low clustering coefficient (C ¼ k/n)

12. Watts & Strogatz Model • Add a small amount of random noise • Start with regular graph, e.g. Cnk • Randomly “rewire” each edge with probability p