Large Deviation Properties of Minimum Spanning Trees for Random Graphs

TL;DR

This paper investigates the large deviation behavior of minimum spanning trees in random graphs, confirming theoretical predictions and revealing a phase transition at the percolation threshold.

Contribution

It provides the first detailed large deviation analysis of MST weights in complete and ER random graphs, including numerical validation.

Findings

Large deviation principle holds for both graph ensembles.

Distribution of MST weights changes notably at the percolation threshold c=1.

Numerical results match analytical predictions for complete graphs.

Abstract

We study the large-deviation properties of minimum spanning trees for two ensembles of random graphs with $N$ nodes. First, we consider complete graphs. Second, we study Erd\H{o}s-R\'{e}nyi (ER) random graphs with edge probability $p=c/N$ conditioned to be connected. By using large-deviation Markov chain sampling, we are able to obtain the distribution $P(W)$ of the spanning-tree weight $W$ down to probability densities as small as $10^{-300}$. For the complete graph, we confirm analytical predictions with respect to the expectation value. For both ensembles, the large deviation principle is fulfilled. For the connected ER graphs, we observe a remarkable change of the distributions at the value of $c=1$, which is the percolation threshold for the original ER ensemble.