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Speeding up random walk mixing by starting from a uniform vertex

Alberto Espuny Díaz, Patrick Morris, Guillem Perarnau, Oriol Serra

TL;DR

This paper provides a general framework to show logarithmic average mixing time for random walks on graphs with small bottlenecks and demonstrates its applicability on two random models for which the mixing time was known to be of order $(\log n)^2), speeding up the mixing to order $\log n$.

Abstract

The theory of rapid mixing random walks plays a fundamental role in the study of modern randomised algorithms. Usually, the mixing time is measured with respect to the worst initial position. It is well known that the presence of bottlenecks in a graph hampers mixing and, in particular, starting inside a small bottleneck significantly slows down the diffusion of the walk in the first steps of the process. The average mixing time is defined to be the mixing time starting at a uniformly random vertex and hence is not sensitive to the slow diffusion caused by these bottlenecks. In this paper we provide a general framework to show logarithmic average mixing time for random walks on graphs with small bottlenecks. The framework is especially effective on certain families of random graphs with heterogeneous properties. We demonstrate its applicability on two random models for which the mixing time was known to be of order $(\log n)^2$, speeding up the mixing to order $\log n$. First, in the context of smoothed analysis on connected graphs, we show logarithmic average mixing time for randomly perturbed graphs of bounded degeneracy. A particular instance is the Newman-Watts small-world model. Second, we show logarithmic average mixing time for supercritically percolated expander graphs. When the host graph is complete, this application gives an alternative proof that the average mixing time of the giant component in the supercritical Erdős-Rényi graph is logarithmic.

Speeding up random walk mixing by starting from a uniform vertex

TL;DR

This paper provides a general framework to show logarithmic average mixing time for random walks on graphs with small bottlenecks and demonstrates its applicability on two random models for which the mixing time was known to be of order \log n$.

Abstract

The theory of rapid mixing random walks plays a fundamental role in the study of modern randomised algorithms. Usually, the mixing time is measured with respect to the worst initial position. It is well known that the presence of bottlenecks in a graph hampers mixing and, in particular, starting inside a small bottleneck significantly slows down the diffusion of the walk in the first steps of the process. The average mixing time is defined to be the mixing time starting at a uniformly random vertex and hence is not sensitive to the slow diffusion caused by these bottlenecks. In this paper we provide a general framework to show logarithmic average mixing time for random walks on graphs with small bottlenecks. The framework is especially effective on certain families of random graphs with heterogeneous properties. We demonstrate its applicability on two random models for which the mixing time was known to be of order , speeding up the mixing to order . First, in the context of smoothed analysis on connected graphs, we show logarithmic average mixing time for randomly perturbed graphs of bounded degeneracy. A particular instance is the Newman-Watts small-world model. Second, we show logarithmic average mixing time for supercritically percolated expander graphs. When the host graph is complete, this application gives an alternative proof that the average mixing time of the giant component in the supercritical Erdős-Rényi graph is logarithmic.
Paper Structure (19 sections, 14 theorems, 79 equations, 1 figure)

This paper contains 19 sections, 14 theorems, 79 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Average mixing times
  3. Our results
  4. Application 1: Smoothed analysis on connected graphs
  5. Application 2: Giant components in random subgraphs of expanders
  6. Organisation
  7. Preliminaries
  8. Basic notation
  9. Random walks
  10. A general approach to average mixing times
  11. Proof overview
  12. Badly connected sets
  13. Stationary distributions
  14. Mixing time after contractions
  15. Hitting time of bad vertices
  16. ...and 4 more sections

Key Result

Theorem 1.5

For all $\epsilon>0$, $D\geq 4$ and $0<\alpha<1/D^2$, there exists a $C>0$ such that the following holds for all $n$ sufficiently large. Suppose $G$ is an $n$-vertex connected $(\alpha,D)$-spreader graph. Then,

Figures (1)

  • Figure 1: Schematic plot of the total variation distance starting at different vertices and the two average mixing times for $\epsilon= 0.05$. In red, the function $\frac{1}{n}\sum_{u\in V(G)} d_{\mathrm{TV}}(\mu_0^u P_G^t,\pi_G)$ and the dot representing $\bar{t}_{\mathrm{mix}}(G,\epsilon)$. In blue, the average of mixing times at different thresholds and the dot representing $\mathbb{E}(t_{\mathrm{mix}}^{(U_n)}(G,\epsilon))$.

Theorems & Definitions (39)

  • Remark 1.1
  • Remark 1.2
  • Definition 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Corollary 1.8
  • Theorem 1.9
  • Corollary 1.10
  • ...and 29 more