Geometric Bounds on the Fastest Mixing Markov Chain

TL;DR

This work establishes vertex-based geometric barriers as the fundamental determinants of fast mixing for reversible Markov chains on graphs. It introduces vertex conductance Ψ^⋆ and a closely related matching conductance Υ^⋆, proving Ψ^⋆ governs the fastest possible mixing time via a Cheeger-type inequality and that Υ^⋆ ≍ Ψ^⋆ up to universal factors. To overcome the barriers of small Ψ^⋆, the authors construct an explicit, near-uniform stationary distribution Markov chain on a weighted BFS tree that achieves a mixing time scaling as roughly (diameter)^2 divided by the allowed deviation ε, with precise bounds in discrete and continuous time, and extend these ideas to time-inhomogeneous chains. The results include an exact continuous-time analogue and a time-inhomogeneous construction achieving perfect mixing in 2 diam(G) steps, highlighting the extra power gained by relaxing regularity constraints. The paper also presents rich examples, discusses connections to prior SDP and dual formulations, and outlines open questions about extending to general π, sparsification, and distributed implementations, marking a significant step in understanding the geometry of rapid mixing in networks.

Abstract

In the Fastest Mixing Markov Chain problem, we are given a graph $G = (V, E)$ and desire the discrete-time Markov chain with smallest mixing time $τ$ subject to having equilibrium distribution uniform on $V$ and non-zero transition probabilities only across edges of the graph. It is well-known that the mixing time $τ_\textsf{RW}$ of the lazy random walk on $G$ is characterised by the edge conductance $Φ$ of $G$ via Cheeger's inequality: $Φ^{-1} \lesssim τ_\textsf{RW} \lesssim Φ^{-2} \log |V|$. Analogously, we characterise the fastest mixing time $τ^\star$ via a Cheeger-type inequality but for a different geometric quantity, namely the vertex conductance $Ψ$ of $G$: $Ψ^{-1} \lesssim τ^\star \lesssim Ψ^{-2} (\log |V|)^2$. This characterisation forbids fast mixing for graphs with small vertex conductance. To bypass this fundamental barrier, we consider Markov chains on $G$ with equilibrium distribution which need not be uniform, but rather only $\varepsilon$-close to uniform in total variation. We show that it is always possible to construct such a chain with mixing time $τ\lesssim \varepsilon^{-1} (\operatorname{diam} G)^2 \log |V|$. Finally, we discuss analogous questions for continuous-time and time-inhomogeneous chains.

Figures (4)

• Figure 1: Dumbbell graph $D_\star$ with $n = 7$: two cliques connected to a single external vertex
• Figure 2: Star graph $G_\star$ with $n = 7$ a central vertex connected to leaves
• Figure 3: Matching graph $\mathcal{M}$ with $n = 7$: two cliques connected via a matching
• Figure 4: Source graph $\Sigma$ with $n = 7$ and $k = 3$: two cliques connected via a 'source'

Theorems & Definitions (52)

• Definition : Markov Chains on a Graph
• Definition : Spectral Gap, Relaxation Time and Mixing Time
• Definition : Fastest Mixing Markov Chain
• Definition A.1: Edge Conductance
• Definition A.2: Vertex Conductance
• Theorem A: Characterisation of Fast Mixing
• Definition C: Almost-Mixed Distributions
• Theorem B: Almost Mixing
• Definition D.1: RW on Weighted Graph
• Definition D.2: Normalisation