Erdős–Rényi graphs

Let V = {1, 2, …, n}, and let (Xi,j : 1 ≤ i < j ≤ n) be independent Bernoulli random variables with parameter p. For each pair i < j, we place an edge 〈i, j〉 between vertices i and j if and only if Xi,j = 1. The resulting random graph is named after Erdős and Rényi, and it is commonly denoted Gn,p. The density p of edges may vary with n, for example, p = λ/n with λ ∈ (0, ∞), and one commonly considers the structure of Gn,p in the limit as n → ∞.

The original motivation for studying Gn,p was to understand the properties of ‘typical’ graphs. This is in contrast to the study of ‘extremal’ graphs, although it may be noted that random graphs have on occasion manifested properties more extreme than graphs obtained by more constructive means.

Random graphs have proved an important tool in the study of the ‘typical’ runtime of algorithms.