On the Limiting Spectral Distributions of Stochastic Block Models

Abstract

Erdős–Rényi graph is a random graph in which the probability of a connection between two nodes follows a Bernoulli distribution independently. The stochastic block models (SBM) are an extension of the Erdős–Rényi graph by dividing nodes into $K$ subsets, known as blocks or communities. Let $\widetilde{A}_{N} = (\widetilde{A}_{ij}^{(N)})$ be an $N \times N$ normalized adjacency matrix of the SBM with $K$ blocks of any sizes, and let $\mu_{\widetilde{A}_{N}}$ be the empirical spectral density of $\widetilde{A}_{N}$.

In this paper, we first showed that if the connecting probabilities between nodes of different blocks are zero, then $\lim_{N \to \infty} \mu_{\widetilde{A}_{N}} = \mu$ exists almost surely, and we gave the explicit formulas for $\mu$ and its Stieltjes transform, respectively. Second, we showed under a suitable condition on the maximum of connecting probability between nodes in different blocks, say by $\zeta_{0}$, $\mu_{\widetilde{A}_{N}}$ converges both in probability and expectation as first $N \to \infty$ and then $\zeta_{0} \to 0$.