Random Walks and Evolving Sets: Faster Convergences and Limitations

Abstract

Analyzing the mixing time of random walks is a wellstudied problem with applications in random sampling and more recently in graph partitioning. In this work, we present new analysis of random walks and evolving sets using more combinatorial graph structures, and show some implications in approximating small-set expansion. On the other hand, we provide examples showing the limitations of using random walks and evolving sets in disproving the small-set expansion hypothesis.1. We define a combinatorial analog of the spectral gap, and use it to prove the convergence of nonlazy random walks. A corollary is a tight lower bound on the small-set expansion of graph powers for any graph.2. We prove that random walks converge faster when the robust vertex expansion of the graph is larger. This provides an improved analysis of the local graph partitioning algorithm using the evolving set process, and also derives an alternative proof of an improved Cheeger's inequality.3. We give an example showing that the evolving set process fails to disprove the small-set expansion hypothesis. This refutes a conjecture of Oveis Gharan and shows the limitations of all existing local graph partitioning algorithms in approximating small-set expansion.

Mean Field Analysis of Personalized PageRank with Implications for Local Graph Clustering

“…As a by-product of our analysis, in Subsection 6.3 we show that the natural cluster in our random graph model also does not really correspond to the problem of conductance minimization. This observation can be viewed as complementary to the results in [10] and [34].…”

“…At the same time this model appears to be a very natural benchmark for local graph clustering algorithms. Our concerns complement the limitations of PPR based clustering discussed in [10] and [34]. As in [10,34,23], we also note that the plain conductance might not be the best criterion for local graph clustering.…”

“…Our concerns complement the limitations of PPR based clustering discussed in [10] and [34]. As in [10,34,23], we also note that the plain conductance might not be the best criterion for local graph clustering.…”

“…where κ := m n . In [10] and [34] it has been observed that the graph conductance has significant limitations as a criterion for graph clustering. We show that in our random graph model the minimization of the graph conductance does not lead to the determination of the natural cluster.…”

“…The methods in [30,31,2,3,4] aim to find a local cut with target conductance. However, in [10] and [34] significant limitations of the random walks and PPR based local clustering methods are presented in terms of graph conductance and related quantities. As a by-product of our analysis, in Subsection 6.3 we show that the natural cluster in our random graph model also does not really correspond to the problem of conductance minimization.…”