Community Detection via Semidefinite Relaxation

Notes for lecture given by the author on November 7, 2014 as part of the special course: “Randomness, Matrices and High Dimensional Problems”, at IMPA, Rio de Janeiro, Brazil. The results presented in these notes are from [1]. 1. The problem we will focus on Let n be an even positive integer. Given two sets of n2 nodes consider the following random graph G: For each pair (i, j) of nodes, (i, j) is an edge of G with probability p if i and j are in the same set and q if they are in different sets. Each edge is drawn independently and p > q. (Think nodes as fans of Fluminense and Flamengo and edges representing friendships, in this model, fans of the same club are more likely to be friends) For which values of p and q can we recover the partition, with an efficient algorithm, from only looking at the graph G (with high probability)? 2. The interesting regime If p logn n then it is easy to see that each cluster will not be connected (with high probability) and so recovery is not possible. In fact, the interesting regime is when p = α log(n) n and q = β log(n) n , (2.1) for constants α > β. Let A be the adjacency matrix of G, meaning that Aij = { 1 if (i, j) ∈ E(G) 0 otherwise. (2.2) Let x ∈ R with xi = ±1 represent a partition (note there is an ambiguity in the sense that x and −x represent the same partition). Then, if we did not worry about efficiency then our guess (which corresponds to the Maximum Likelihood Estimator) would be the solution of