STAT 206: Random Graphs and Complex Networks

Spring 2003 Le ture 7: Criti al behavior of the Erd} os-Renyi model Le turer: David Aldous S ribe: Andrej Bogdanov In this le ture we investigate the size of the giant omponent in the Erd} os-Renyi random graph model with edge probability p 1=n. In this model, the presen e of ea h edge in the graph is de ided by an independent oin ip with su ess probability p. We show that for p slightly bigger than 1=n, the giant omponent has size of order n2=3. There are several proofs of this result, and we will opt for an intuitive, ba k-of-envelope heuristi argument. This argument has the advantage of showing o some sophisti ated on epts from probability like the Central Limit Theorem and Brownian motion. At the heart of this argument is a random pro ess that, starting from an arbitrary vertex v, exposes the neighbors of v, their neighbors, and so on, until the whole omponent of v is revealed. The sizes of the omponents are found by analyzing the dynami s of this pro ess. For the Erd} os-Renyi model, the relevant statisti s an be omputed exa tly, giving a detailed pi ture of the behavior of omponent sizes near p = 1=n. 7.1 The breadth rst spanning forest of a graph Let G be a (non-random) graph on vertex set [n℄. The breadth rst spanning forest of G is the spanning forest generated by the breadth rst sear h algorithm: Until all verti es of G have been visited, 1. Pi k a vertex v that has not been visited yet. Put v in a queue Q. 2. While Q is nonempty, pull a vertex v from the head of Q, draw edges to all its neighbors that have not been previously visited, and put these hildren at the tail of Q. Here is an example: