The distribution of minimum-weight cliques and other subgraphs in graphs with random edge weights

TL;DR

This paper analyzes the asymptotic distribution and mean of the minimum-weight subgraphs, such as cliques, in complete graphs with randomly weighted edges, providing bounds and applications for network structure inference.

Contribution

It offers the first asymptotic distribution results for minimum-weight subgraphs in weighted complete graphs, including explicit bounds and applicability to various distributions.

Findings

Asymptotic distribution of minimum-weight k-cliques determined

Explicit bounds on the distribution's CDF derived using Stein-Chen method

Applications to testing network edge weight independence

Abstract

We determine, asymptotically in $n$, the distribution and mean of the weight of a minimum-weight $k$-clique (or any strictly balanced graph $H$) in a complete graph $K_n$ whose edge weights are independent random values drawn from the uniform distribution or other continuous distributions. For the clique, we also provide explicit (non-asymptotic) bounds on the distribution's CDF in a form obtained directly from the Stein-Chen method, and in a looser but simpler form. The direct form extends to other subgraphs and other edge-weight distributions. We illustrate the clique results for various values of $k$ and $n$. The results may be applied to evaluate whether an observed minimum-weight copy of a graph $H$ in a network provides statistical evidence that the network's edge weights are not independently distributed but have some structure.