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

This lecture Probabilistic generative models for social networks (in particular web graph)

Why look for generative models? Designing and testing algorithms for the web  E.g.: Compressing the web graph Designing crawling strategies Search algorithms on P2P networks … Explaining why web has certain properties  For example, the central limit theorem tells us why we often see the Gaussian distribution in practice.  Is there a similar explanation for the power law distribution? Predicting what “might” happen in the future  E.g.: An AIDS epidemic? An Internet black out? A residential segregation?

Characteristics of a good model Simple Plausible Exihibits the observed properties  Power law  Small world  Locally dense, globally sparse

Power law distribution From last lecture: power laws everywhere!  Income distribution (Pareto 1896)  Word frequencies (Estoup 1916, Zipf 1932)  City population (Auerbach 1913, Zipf 1949)  Scientific productivity (Lotka 1926)  Internet graph degree dist (FFF 1999)  Web graph degree dist (BKMRRSTW 2000)  Dist. of file sizes  … Why?

Models and explanations for power law Optimization (“power law is the best design”)  Mandelbrot 1953: Zipf’s law is the most efficient design.  Carlson & Doyle 1999, Fabrikant et al (HOT) Monkeys typing randomly  Miller 1957: even a monkey typing randomly can generate a power law. Multiplicative processes & Log-normal dist.  Gibrat 1930, Champernowne 1955, Gabaix 1999 Preferential growth (“the rich get richer”)  Simon 1955, Yule 1925

Log-normal distribution Central limit Thm: Product of many indep. distributions is approximately log-normal.

Multiplicative process and power law Multiplicative processes can sometimes generate power law instead of log-normal:  Multiplicative process with a minimum Chambernowne 1953, Gabaix 1999  Random stopping time Montroll and Schlesinger 1982,1983

Preferential growth The system “grows”. The probability of a new member joining a group is proportional to its current size. Simon 1955, Yule 1925 (for biological systems) Barabasi and Albert 1999: preferential attachment for web graph

Random graph models Erdos-Renyi random graphs G(n,p)  n vertices, there is an edge between each pair independently with probability p. G(n,p) at a glance:  Average degree np. Binomial degree dist.  p < 1/n: union of small simple connected comp.  p > 1/n: a “giant” complex component emerges (still many small connected components)  p > ln(n)/n: connected.

The ACL model Proposed by Aiello, Chung, and Lu, Fix a degree sequence d (e.g., power law). Put d i copies of the i’th vertex. Pick a random matching. Contract the d i copies of the i’th vertex  Essentially a variant of G(n,p), with the degree distribution explicitly enforced.

Preferential attachment 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