Exact Solution for Statics and Dynamics of Maximal Entropy Random Walk on Cayley Trees

TL;DR

This paper provides exact analytical solutions for the static and dynamic behavior of maximal entropy random walks on Cayley trees, revealing how tree structure influences relaxation times and stationary states.

Contribution

It offers the first exact solutions for MERW on Cayley trees, detailing the influence of tree structure on walk dynamics and stationary states.

Findings

MERW stationary state given by squared eigenvector elements

Shorter relaxation times for MERW compared to GRW

Three regimes of tree structure affecting statics and dynamics

Abstract

We provide analytical solutions for two types of random walk: generic random walk (GRW) and maximal entropy random walk (MERW) on a Cayley tree with arbitrary branching number, root degree, and number of generations. For MERW, we obtain the stationary state given by the squared elements of the eigenvector associated with the largest eigenvalue $\lambda_0$ of the adjacency matrix. We discuss the dynamics, depending on the second largest eigenvalue $\lambda_1$, of the probability distribution approaching to the stationary state. We find different scaling of the relaxation time with the system size, which is generically shorter for MERW than for GRW. We also signal that depending on the initial conditions there are relaxations associated with lower eigenvalues which are induced by symmetries of the tree. In general, we find that there are three regimes of a tree structure resulting in different statics and dynamics of MERW; these correspond to strongly, critically, and weakly branched roots.