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arXiv · 1606.04925

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

Alan Frieze·Wesley Pegden·Gregory Sorkin
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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.

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Alan Frieze, Wesley Pegden, Gregory Sorkin. 2016-06-15. The distribution of minimum-weight cliques and other subgraphs in graphs with random edge weights. https://arxiv.org/abs/1606.04925

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