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

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

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)

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.

Preferential attachment Heuristic analysis (Barabasi-Albert): degree distribution follows a power law with exponent -3. Theorem (Bollobas, Riordan, Spencer, Tusnady). For d < n 1/16, the fraction of vertices that have degree d is almost surely around

Copying models Kleinberg et al and Kumar et al  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.

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 .d ij + h j, where d ij is the Euclidean distance between i and j, and h j is the graph distance between j and node 1 (the center).

The FKP model

Small World Networks Low average distance L  Definition: The average distance L 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 coefficient C of a network is the probability that two neighbors of a random vertex are connected by a single edge.

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 L actual L random C actual C random Film actors Power grid C. Elegans Data from Watts-Strogatz

Models Regular network, e.g. C n k  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)

Watts & Strogatz Model Add a small amount of random noise  Start with regular graph, e.g. C n k  Randomly “rewire” each edge with probability p