graph expansion analysis

This work addresses rapid mixing of Markov Chain Monte Carlo (MCMC) algorithms on high-dimensional discrete distributions—such as vertex colorings of graphs and bases of matroids. We introduce a novel analytical framework centered on **spectral independence**, unifying spectral independence, the local-to-global theorem, and Oppenheim’s trickle-down theorem for the first time. This yields a cohesive spectral analysis paradigm applicable to both graphical models and matroid structures. We prove that spectral independence implies polynomial mixing time; under natural conditions, it yields the optimal $O(n log n)$ mixing time for Glauber dynamics and tight spectral gap bounds for base-exchange walks. Our results systematically extend Anari et al.’s matroid sampling framework, providing universal, tight convergence guarantees for probabilistic inference and randomized algorithms over combinatorial structures.

This work extends the classical KKL theorem to the setting of high-dimensional expanders (HDX), circumventing the strong expansion assumptions required by prior results. By introducing a general local-to-global analytic framework that integrates the link structure of simplicial complexes with spectral techniques, the authors establish the first dimension-dependent global KKL theorem on non-trivial two-sided expanding complexes under merely local KKL-type conditions. This approach dispenses with strong expansion hypotheses and applies broadly to dense clique complexes and Ramanujan complexes. As corollaries, the paper derives the first KKL-type inequality for combinatorial HDX and a Kruskal–Katona-type small-set expansion result in this context.

This work addresses the mixing time of Glauber dynamics for the hard-core and Ising models. Methodologically, it introduces— for the first time in spin-system sampling analysis—the *local connectivity constant* of a graph, integrating spectral independence theory, *k*-nonbacktracking matrices, and high-dimensional expander tools to transcend conventional analyses relying solely on maximum degree or spectral radius. The main contributions are: (i) tight, unified upper bounds on mixing time that strictly improve upon Sinclair et al. (2017) and Hayes (2006); (ii) significantly broadened applicability across graph families; and (iii) refined convergence characterizations for numerous classical graph structures—including expanders, bounded-treewidth graphs, and random regular graphs—thereby establishing a locally structure-aware, optimal sampling analysis framework.

This study investigates the theoretical bounds on edge expansion and modularity in preferential attachment random graphs where each new node introduces $h \geq 2$ edges, aiming to elucidate their connectivity and community structure properties. Employing probabilistic graph theory and combinatorial analysis, the work establishes novel probabilistic bounds on the edge expansion of small vertex subsets for small values of $h$, and leverages these results to derive a tighter global upper bound on modularity. These findings advance the theoretical understanding of structural evolution in dynamic networks and provide new analytical foundations for assessing network robustness and community detection.

The computational complexity of graph isomorphism testing remains unresolved, particularly for highly symmetric graphs whose adjacency matrices possess repeated eigenvalues—causing ambiguity in the solution space for conventional methods. This paper introduces a novel continuous optimization framework: it reformulates the discrete matching problem via orthogonal and doubly stochastic relaxations, and—crucially—systematically characterizes how eigenvalue multiplicity governs the geometric structure of the feasible solution space. Building on this insight, we propose a subspace constraint strategy that effectively suppresses spurious solutions induced by symmetry. Our algorithm employs the Frank–Wolfe method, integrating spectral matrix analysis with iterative projection-based optimization. Extensive evaluation on highly symmetric benchmarks—including strongly regular graphs, complete graphs, and the Petersen graph—demonstrates substantial improvements in both efficiency and robustness of isomorphism detection.

High-dimensional expanders must simultaneously satisfy spectral expansion and coboundary expansion, and prior constructions relied on sophisticated tools from algebraic number theory. This work proposes a remarkably simple combinatorial method based on projections of flag complexes—chains of subspaces—to construct high-dimensional expander complexes with subpolynomial degree and nearly linear size, without invoking deep algebraic machinery. For the first time, this construction achieves local spectral expansion, coboundary expansion, and commuting coboundary expansion concurrently. As a consequence, it yields the first nearly linear-sized combinatorial hypergraph suitable for “1%” agreement testing protocols and further leads to a streamlined, nearly linear-sized PCP construction.

This study addresses the long-standing open problem of analyzing the mixing time of switch chains for graph degree sequence realizations. The proposed approach introduces high-dimensional expansion theory, modeling the realization space as faces of a simplicial complex. By analyzing the simplicial switch chain, the authors bound the mixing time of the classical chain. Specifically, they prove that high-codimension links are strong spectral expanders and compare Dirichlet energies across different update strategies. This work establishes an O(Δ²m log m) upper bound on the mixing time, confirming convergence within O(n log n) steps under bounded maximum degree and thereby resolving a longstanding conjecture in the field.

This work addresses the challenge of efficiently sampling from the hardcore model on random regular bipartite graphs when the fugacity parameter λ exceeds the uniqueness threshold. The authors propose a novel Markov chain that integrates two complementary mechanisms and introduce an analytical framework based on spectral expansion of simplicial complexes. By combining the trickle-down theorem with structural properties of the underlying graph, they establish rapid mixing for λ ≲ 1/√Δ—surpassing the uniqueness threshold for the first time in this setting. This breakthrough yields an efficient approximate sampling algorithm for hardcore configurations and leads to a fully polynomial randomized approximation scheme (FPRAS) for the partition function, significantly extending the known theoretical limits for this model.

This work addresses the problem of characterizing graph structures with constant mixing time from a purely combinatorial perspective—a longstanding gap, as classical spectral graph theory only yields logarithmic mixing-time bounds. We propose a novel combinatorial criterion based on *bipartite density of small sets*, which is strictly weaker than near-optimal spectral radius constraints yet strictly stronger than small-set vertex expansion. Our method integrates spectral analysis, random walk theory, and extremal combinatorics to establish a necessary and sufficient condition linking small-set bipartite density directly to constant mixing time. We rigorously prove this equivalence and validate its effectiveness across several canonical graph families, including expanders, random regular graphs, and certain product graphs. This result provides the first combinatorial characterization equivalent to constant mixing time, bridging a fundamental theoretical gap in mixing-time analysis. It establishes a new structural paradigm for both analyzing and constructing rapidly mixing graphs, with implications for algorithm design, sampling, and distributed computation.

本文通过集成Bochner方法简化并强化了分析，解决了单纯复形的全局下-上行走谱隙问题，利用维度和谱影响量的关系改进了现有结果。