Network exploration by random walks: A large deviation perspective

TL;DR

This paper analyzes the exploration behavior of random walks on networks, using large deviation theory and continuous time formalism to understand the distribution of visited nodes, independent of network topology at short times.

Contribution

It introduces a continuous time random walk framework to study large deviations in node exploration, extending beyond idealized fully connected networks.

Findings

Distribution of visited nodes at small times is topology-independent.

Waiting time distribution determines exploration properties at short times.

Mapping to coupon collector problem aids in estimating visitation probabilities.

Abstract

We study exploration properties of a random walk on a network. For a fully connected network we find that the problem can be mapped to the well known coupon collector problem, thus allowing us to estimate form of $P(S,t)$: the distribution of number of distinct nodes $S$ visited by the random walk upto time $t$. From a practical point of view, however, both the fully connected network and hops taking place after fixed intervals are an idealization. We solve this problem by introducing the formalism of continuous time random walks wherein the random walk spends a random amount of time a node before hopping to its neighboring node. The formalism allows us to study the large deviation limit of $P(S,t)$ under very mild conditions that the distribution of waiting times $\psi(\tau)$ exhibits analyticity at small times. Furthermore, we find that at small times, the properties of $P(S,t)$ are largely independent of the network topology, and are governed solely by the waiting time characteristics.