Cover times, blanket times, and majorizing measures

Abstract: We exhibit a strong connection between cover times of graphs, Gaussian processes, and Talagrand's theory of majorizing measures. In particular, we show that the cover time of any graph $G$ is equivalent, up to universal constants, to the square of the expected maximum of the Gaussian free field on $G$, scaled by the number of edges in $G$. This allows us to resolve a number of open questions. We give a deterministic polynomial-time algorithm that computes the cover time to within an O(1) factor for any graph, answering a question of Aldous and Fill (1994). We also positively resolve the blanket time conjectures of Winkler and Zuckerman (1996), showing that for any graph, the blanket and cover times are within an O(1) factor. The best previous approximation factor for both these problems was $O((\log \log n)^2)$ for $n$-vertex graphs, due to Kahn, Kim, Lovasz, and Vu (2000).

• The paper establishes that graph cover times are proportional to the square of the expected maximum of the Gaussian free field multiplied by the number of edges.
• It proves that the δ-blanket time is asymptotically equivalent to the cover time with a constant approximation factor, thereby resolving the blanket time conjecture.
• It introduces a deterministic polynomial-time algorithm to approximate cover times, effectively bridging discrete graph theory with continuous stochastic processes.

Cover Times, Blanket Times, and Majorizing Measures

Introduction

The paper "Cover times, blanket times, and majorizing measures" explores the interplay between cover times in graphs, Gaussian processes, and Talagrand's majorizing measures theory. It resolves longstanding open questions concerning graph cover times by establishing their equivalence, up to universal constants, with the square of the expected maximum of the Gaussian free field on a graph, scaled by the number of its edges.

Key Concepts and Results

Cover Times and Gaussian Processes

The cover time $t_{\mathrm{cov}(G)$ for a graph $G$ is the first time a random walk visits every vertex. The paper demonstrates that this cover time is fundamentally connected to the Gaussian free field (GFF)—a centered Gaussian process indexed by the graph vertices. Specifically, it shows that:

$t_{\mathrm{cov}(G) \asymp |E| \left( \max_{v \in V} \eta_v\right)^2,$

where $|E|$ is the number of edges and $\{\eta_v\}_{v \in V}$ represents the GFF on $G$. This equivalence is a significant result, linking discrete structures like graphs to continuous stochastic processes.

Blanket Times

The blanket time conjecture speculates that the $\delta$-blanket time $t_{\mathrm{bl}^{\circ}(G,\delta)$, which ensures that every vertex has been visited at least a $\delta$ fraction of the expected visit count at stationarity, is closely tied to the cover time. The paper proves for any graph $G$ and $\delta \in (0,1)$:

$t_{\mathrm{bl}^{\circ}(G,\delta) \asymp t_{\mathrm{cov}(G),$

thus resolving this conjecture with an $O(1)$ approximation factor, improving upon the previous $O((\log \log n)^2)$ factor.

Deterministic Algorithm for Cover Times

A deterministic polynomial-time algorithm calculates cover times within constant factors. This directly addresses a question posed by Aldous and Fill (1994) about whether such approximation is feasible without relying on simulation or probabilistic methods.

Implementation Approach

Gaussian Free Field and Effective Resistance

Practically, implementing the concepts of GFF involves calculating the effective resistance for a network graph, which characterizes potential differences akin to an electrical network. The GFF for a graph yields insights into vertex variance that can be leveraged to compute expected maxima, essential in estimating cover times.

Algorithmic Details

For efficient computation:

• Form the normalized graph Laplacian.
• Compute its pseudoinverse for commute times using techniques like matrix decomposition or iterative solvers.
• Use Gaussian sampling methods to estimate the supremum of the GFF accurately.

Randomized Algorithms

The paper gives rise to a randomized process where by simulating Gaussian fields and leveraging their expected supremum properties, one can estimate cover times in nearly linear time relative to the size of the graph—a significant improvement for large-scale network analysis.

Conclusion

This paper synthesizes discrete graph theory and continuous stochastic models, offering solutions to classical problems about random walks on networks. The connections made open pathways for more efficient graph algorithms and deeper understanding of diffusion processes over complex structures. Future challenges include exploring the dynamic behavior in evolving graphs and networks where edge weights or structures change over time, potentially expanding this framework's applicability.