The Moran forest

TL;DR

This paper studies a Markov chain on graphs that converges to a distribution over forests, revealing detailed properties of the resulting forest structures, including degree distributions, component sizes, and asymptotic behaviors.

Contribution

It provides a complete characterization of the stationary distribution of the Moran forest process and analyzes its structural properties and asymptotic behaviors.

Findings

Largest tree size is asymptotically α log n, with α ≈ 2.18.

Maximum vertex degree is asymptotically log n / log log n.

Distribution of the number of trees and degree distribution are fully characterized.

Abstract

Starting from any graph on $\{1, \ldots, n\}$, consider the Markov chain where at each time-step a uniformly chosen vertex is disconnected from all of its neighbors and reconnected to another uniformly chosen vertex. This Markov chain has a stationary distribution whose support is the set of non-empty forests on $\{1, \ldots, n\}$. The random forest corresponding to this stationary distribution has interesting connections with the uniform rooted labeled tree and the uniform attachment tree. We fully characterize its degree distribution, the distribution of its number of trees, and the limit distribution of the size of a tree sampled uniformly. We also show that the size of the largest tree is asymptotically $\alpha \log n$, where $\alpha = (1 - \log(e - 1))^{-1} \approx 2.18$, and that the degree of the most connected vertex is asymptotically $\log n / \log\log n$.

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