Connectivity of inhomogeneous random graphs

TL;DR

This paper establishes conditions for the connectivity of inhomogeneous random graphs with intermediate density, extending classical results to more general distributions and kernel-based edge probabilities.

Contribution

It generalizes the classical connectivity threshold results for Erdős–Rényi graphs to inhomogeneous models with kernel-defined connection probabilities.

Findings

Connectivity threshold determined under weak assumptions

Generalizes classical G(n,p) results to inhomogeneous graphs

Provides conditions for connectivity in models with variable density

Abstract

We find conditions for the connectivity of inhomogeneous random graphs with intermediate density. Our results generalize the classical result for G(n, p), when p = c log n/n. We draw n independent points X_i from a general distribution on a separable metric space, and let their indices form the vertex set of a graph. An edge (i,j) is added with probability min(1, \K(X_i,X_j) log n/n), where \K \ge 0 is a fixed kernel. We show that, under reasonably weak assumptions, the connectivity threshold of the model can be determined.