Spectral graph theory

Let us first review some basic random graph models. The most popular is perhaps the Erdös-Rényi model, denoted by G(n, p) for some n ∈ N and p ∈ [0, 1]. In this model, there is a set of vertices V (with n = |V |), and the edge set over V is determined randomly by ( n 2 ) biased coin tosses. Initially, there are no edges: E := ∅. Then, for each pair of vertices u, v ∈ V , independently flip a coin with heads bias p; if the coin comes up heads, add {u, v} to E. The adjacency matrix A for the graph is a random matrix: the diagonal entries are fixed to zero, and the off-diagonal entries are {0, 1}random variables with EAu,v = p. (Note that A is a random symmetric matrix, so the entries are not all independent.) A natural variant of G(n, p) provides a random bipartite graph; let’s called this model G(n1, n2, p). In this model, we have two disjoint sets of vertices (say, V1 and V2 with n1 = |V1| and n2 = |V2|). There are initially no edges: E := ∅. Then, for each pair of vertices u ∈ V1 and v ∈ V2, independently flip a coin with heads bias p; if the coin comes up heads, add {u, v} to E. As before, the adjacency matrix A for the graph is a random matrix: the entries Au,v corresponding to u, v ∈ V1 or u, v ∈ V2 are fixed to zero, and the entries corresponding to u ∈ V1 and v ∈ V2 (or u ∈ V2 and v ∈ V1) are {0, 1}-random variables with EAu,v = p. In the (basic) planted partition model, there are disjoint sets of vertices V1, V2, . . . , Vk (say, with ni := |Vi|, V := V1∪V2, . . . , Vk, n := n1 +n2 + · · ·+ nk = |V |); each set Vi represents a cluster. The edges within each set Vi are determined by G(ni, p); the edges across any pair of sets Vi and Vj are determined by G(ni, nj , q). Here, p, q ∈ [0, 1] are parameters of the model, and we assume p > q, so that it is more likely to to have an edge between vertices within the same cluster than it is between vertices from different