The average number of distinct sites visited by a random walker on random graphs

TL;DR

This paper investigates how many unique sites a random walk visits on large random graphs, deriving a formula for the growth rate based on graph structure and validating it with simulations.

Contribution

It introduces a new method to calculate the average number of visited sites using message-passing equations linked to the graph's adjacency matrix.

Findings

Derived an expression for the growth rate prefactor B in S(n)=Bn.

Validated the theoretical predictions with numerical simulations.

Analyzed the scaling behavior with system size and connectivity.

Abstract

We study the linear large $n$ behavior of the average number of distinct sites $S(n)$ visited by a random walker after $n$ steps on a large random graph. An expression for the graph topology dependent prefactor $B$ in $S(n) = Bn$ is proposed. We use generating function techniques to relate this prefactor to the graph adjacency matrix and then devise message-passing equations to calculate its value. Numerical simulations are performed to evaluate the agreement between the message passing predictions and random walk simulations on random graphs. Scaling with system size and average graph connectivity are also analysed.