Cover and Hitting Times of Hyperbolic Random Graphs

TL;DR

This paper analyzes random walks on the giant component of Hyperbolic Random Graphs with power-law degree distributions, establishing asymptotic cover and hitting times, and providing insights into commute times using hyperbolic geometry and network flow techniques.

Contribution

It provides the first detailed asymptotic analysis of cover, hitting, and commute times for HRGs with degree exponent in (2,3), using hyperbolic geometry and flow methods.

Findings

Cover time is approximately n(log n)^2.

Maximum hitting time is about n log n.

Average hitting time is roughly n.

Abstract

We study random walks on the giant component of Hyperbolic Random Graphs (HRGs), in the regime when the degree distribution obeys a power law with exponent in the range $(2,3)$. In particular, we first focus on the expected time for a random walk to hit a given vertex or visit, i.e. cover, all vertices. We show that, a.a.s. (with respect to the HRG), and up to multiplicative constants: the cover time is $n(\log n)^2$, the maximum hitting time is $n\log n$, and the average hitting time is $n$. We then determine the expected time to commute between two given vertices a.a.s., up to a small factor polylogarithmic in $n$, and under some mild hypothesis on the pair of vertices involved. Our results are proved by controlling effective resistances using the energy dissipated by carefully designed network flows associated to a tiling of the hyperbolic plane, on which we overlay a forest-like structure.