Greedy Random Walk

TL;DR

This paper introduces the Greedy Random Walk, a self-interacting process on graphs, analyzing its cover time and transience properties, with results showing linear cover time on certain graphs and transience in higher dimensions.

Contribution

It provides the first analysis of the Greedy Random Walk's cover time and transience, demonstrating linear cover time on specific graph families and transience in high-dimensional lattices.

Findings

Expected edge cover time is linear for certain graphs.

GRW is transient in $\\Z^d$ for all $d \geq 3$.

Examples include complete graphs, expanders, and hypercubes.

Abstract

We study a discrete time self interacting random process on graphs, which we call Greedy Random Walk. The walker is located initially at some vertex. As time evolves, each vertex maintains the set of adjacent edges touching it that have not been crossed yet by the walker. At each step, the walker being at some vertex, picks an adjacent edge among the edges that have not traversed thus far according to some (deterministic or randomized) rule. If all the adjacent edges have already been traversed, then an adjacent edge is chosen uniformly at random. After picking an edge the walk jumps along it to the neighboring vertex. We show that the expected edge cover time of the greedy random walk is linear in the number of edges for certain natural families of graphs. Examples of such graphs include the complete graph, even degree expanders of logarithmic girth, and the hypercube graph. We also show that GRW is transient in $\Z^d$ for all $d \geq 3$.