The Moran forest

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 $\log n / \log\log n$.