Slowdown for the geodesic-biased random walk

TL;DR

This paper demonstrates that in certain graphs, a geodesic-biased random walk can significantly slow down the process of reaching a target, with expected hitting times growing exponentially.

Contribution

It introduces the counterintuitive phenomenon that geodesic bias can exponentially increase hitting times in bounded-degree graphs.

Findings

Expected hitting time can grow exponentially with graph size.

Geodesic bias may slow down random walks significantly.

Counterintuitive effects of bias on random walk efficiency.

Abstract

Given a connected graph $G$ with some subset of its vertices excited and a fixed target vertex, in the geodesic-biased random walk on $G$, a random walker moves as follows: from an unexcited vertex, she moves to a uniformly random neighbour, whereas from an excited vertex, she takes one step along some fixed shortest path towards the target vertex. We show, perhaps counterintuitively, that the geodesic-bias can slow the random walker down exponentially: there exist connected, bounded-degree $n$-vertex graphs with excitations where the expected hitting time of a fixed target is at least $\exp (\sqrt[4]{n} / 100)$.