Mixing trichotomy for random walks on directed stochastic block models

TL;DR

This work analyzes a directed stochastic block model with $m\ge 2$ communities and a tunable inter-community rewiring parameter $\alpha$, showing a three-way mixing behavior (subcritical, critical, supercritical) as $\alpha$ varies relative to the entropic time $t_{\text{ent}}$. The authors introduce the Directed Block Model ${\rm DBM}(n,m,p,\alpha)$, establish a robust entropic-time framework, and prove a mixing trichotomy with a cutoff in the subcritical regime, a two-timescale metastable relaxation in the strongly supercritical regime, and an interpolated profile at criticality. They develop a suite of tools—annealed random walks, local-tree approximations, quasi-stationary analysis, and coupling arguments—to demonstrate homogenization of the random environment and to quantify inter-community bottlenecks. The results extend the understanding of mixing on directed, multi-community graphs, highlighting how community structure can create new dynamical phases and how metastability and cutoff can coexist as complementary manifestations of the same underlying homogenization phenomenon. The methods and insights offer a blueprint for analyzing similar phase transitions in other non-reversible, networked Markov processes with bottlenecks.

Abstract

We consider a directed version of the classical Stochastic Block Model with $m\ge 2$ communities and a parameter $α$ controlling the inter-community connectivity. We show that, depending on the scaling of $α$, the mixing time of the random walk on this graph can exhibit three different behaviors, which we refer to as subcritical, critical and supercritical. In the subcritical regime, the total variation distance to equilibrium decays abruptly, providing the occurrence of the so-called cutoff phenomenon. In the supercritical regime, the mixing is governed by the inter-community jumps, and the random walk exhibits a metastable behavior: at first it collapses to a local equilibrium, then, on a larger timescale, it can be effectively described as a mean-field process on the $m$ communities, with a decay to equilibrium which is asymptotically smooth and exponential. Finally, for the critical regime, we show a sort of interpolation of the two above-mentioned behaviors. Although the metastable behavior shown in the supercritical regime appears natural from a heuristic standpoint, a substantial part of our analysis can be read as a control on the homogenization of the underlying random environment.

Figures (2)

• Figure 1: Plot of the (theoretical) limiting mixing profile in the subcritical case (left) and critical case (right) with $C=2$ and $m=2,3,4,5,6$.
• Figure 2: Plot of the (theoretical) limiting mixing profile in the supercritical case, with $m=2,3,4,5,6$, in the two timescales $t \asymp t_{\textup{ent}}$ (left) and $t \asymp \alpha^{-1}$ (right).

Theorems & Definitions (61)

• Theorem 1: BP
• Theorem 2
• Theorem 3: CF
• Proposition 4
• proof
• Corollary 5
• proof
• Lemma 6
• proof
• Lemma 7