Time-Biased Random Walks and Robustness of Expanders

TL;DR

This paper examines how time-homogeneous and time-adaptive random walks on expanders respond to edge-weight perturbations under a Lipschitz constraint. It proves a dichotomy: small Lipschitz perturbations preserve a constant spectral gap, while larger perturbations can induce polynomially small gaps, illustrating robustness limits. It then leverages a time-adaptive ε-biased random walk to achieve linear cover times on bounded-degree expanders, and provides a matching lower bound showing linearity fails for vanishing bias; the work relies on a trajectory-tree framework and refined potential arguments. The results advance understanding of robustness for mixing times and show that adaptive strategies can dramatically improve exploration efficiency on expander graphs, with implications for sampling, derandomization, and network algorithms.

Abstract

Random walks on expanders play a crucial role in Markov Chain Monte Carlo algorithms, derandomization, graph theory, and distributed computing. A desirable property is that they are rapidly mixing, which is equivalent to having a spectral gap $γ$ (asymptotically) bounded away from $0$. Our work has two main strands. First, we establish a dichotomy for the robustness of mixing times on edge-weighted $d$-regular graphs (i.e., reversible Markov chains) subject to a Lipschitz condition, which bounds the ratio of adjacent weights by $β\geq 1$. If $β\ge 1$ is sufficiently small, then $γ\asymp 1$ and the mixing time is logarithmic in $n$. On the other hand, if $β\geq 2d$, there is an edge-weighting such that $γ$ is polynomially small in $1/n$. Second, we apply our robustness result to a time-dependent version of the so-called $\varepsilon$-biased random walk, as introduced in Azar et al. [Combinatorica 1996]. We show that, for any constant $\varepsilon>0$, a bias strategy can be chosen adaptively so that the $\varepsilon$-biased random walk covers any bounded-degree regular expander in $Θ(n)$ expected time, improving the previous-best bound of $O(n \log \log n)$. We prove the first non-trivial lower bound on the cover time of the $\varepsilon$-biased random walk, showing that, on bounded-degree regular expanders, it is $ω(n)$ whenever $\varepsilon = o(1)$. We establish this by controlling how much the probability of arbitrary events can be ``boosted'' by using a time-dependent bias strategy.

Figures (2)

• Figure 3.1: Illustration of the construction of $R$ for a given $S \subseteq V$, where $\alpha=e^2$. A new sequence starts if the size of the $S_i$ is at least as large as the size of the previous set, and a sequence ends as soon as $S_i$ is not larger than $\frac{1}{2} \alpha$ times of the size of all previous sets in that sequence.
• Figure 5.1: Illustration of a (non-lazy) walk on a non-regular graph (shown on the lest) starting from $u$ with the objective of having visited $z$ by time $t=2$, this corresponds to a set of trajectories $S=\{uyz,uwz\}$. The conditional probabilities of achieving this, from the given node/trajectory, are given in blue (left) for the SRW, and in red (right) for the $\frac{1}{3}$-TBRW. The strategy used by the $\frac{1}{3}$-TBRW in this example is one that biases towards a vertex maximising the chance of reaching the target.

Theorems & Definitions (34)

• Theorem 1.1: Simplified Version of \ref{['res:robust']}
• Proposition 1.2: Corollary of \ref{['pro:lower']}
• Theorem 1.3: Corollary of \ref{['thm:cover']}
• Corollary 1.4
• Corollary 1.5
• Theorem 1.6
• Theorem 1.7: Simplified Version of \ref{['nonregbound']}
• Definition 2.1: Lipschitz Condition
• Lemma 2.2: cf. ETB
• Theorem 3.1