Pavel Kolev

A Note On Spectral Clustering

Spectral clustering is a popular and successful approach for partitioning the nodes of a graph into clusters for which the ratio of outside connections compared to the volume (sum of degrees) is small. In order to partition into k clusters, one first computes an approximation of the bottom k eigenvectors of the (normalized) Laplacian of G, uses it to embed the vertices of G into k-dimensional Euclidean space R^k, and then partitions the resulting points via a k-means clustering algorithm. It is an important task for theory to explain the success of spectral clustering. Peng et al. (COLT, 2015) made an important step in this direction. They showed that spectral clustering provably works if the gap between the (k+1)-th and the k-th eigenvalue of the normalized Laplacian is sufficiently large. They proved a structural and an algorithmic result. The algorithmic result needs a considerably stronger gap assumption and does not analyze the standard spectral clustering paradigm; it replaces spectral embedding by heat kernel embedding and k-means clustering by locality sensitive hashing. We extend their work in two directions. Structurally, we improve the quality guarantee for spectral clustering by a factor of k and simultaneously weaken the gap assumption. Algorithmically, we show that the standard paradigm for spectral clustering works. Moreover, it even works with the same gap assumption as required for the structural result.

Dirichlet Eigenvalues, Local Random Walks, and Analyzing Clusters in Graphs

A cluster \(S\) in a massive graph \(G\) is characterised by the property that its corresponding vertices are better connected with each other, in comparison with the other vertices of the graph. Modeling, finding and analyzing clusters in massive graphs is an important topic in various disciplines. In this work we study local random walks that always stay in a cluster \(S\). Moreover, we initiate the study of the local mixing time and the almost stable distribution, by analyzing Dirichlet eigenvalues in graphs. We prove that the Dirichlet eigenvalues of any connected subset \(S\) can be used to bound the \(\epsilon \)-uniform mixing time, which improves the previous best-known result. We further present two applications of our results. The first is a polynomial-time algorithm for finding clusters with an improved approximation guarantee, while the second is the significance ordering of vertices in a cluster.

Two Results on Slime Mold Computations

Abstract We present two results on slime mold computations. In wet-lab experiments by Nakagaki et al. (2000) [1] the slime mold Physarum polycephalum demonstrated its ability to solve shortest path problems. Biologists proposed a mathematical model, a system of differential equations, for the slimes adaption process (Tero et al., 2007) [3] . It was shown that the process convergences to the shortest path (Bonifaci et al., 2012) [5] for all graphs. We show that the dynamics actually converges for a much wider class of problems, namely undirected linear programs with a non-negative cost vector. Combinatorial optimization researchers took the dynamics describing slime behavior as an inspiration for an optimization method and showed that its discretization can e-approximately solve linear programs with positive cost vector (Straszak and Vishnoi, 2016) [14] . Their analysis requires a feasible starting point, a step size depending linearly on e, and a number of steps with quartic dependence on opt / ( e Φ ) , where Φ is the difference between the smallest cost of a non-optimal basic feasible solution and the optimal cost (opt). We give a refined analysis showing that the dynamics initialized with any strongly dominating point converges to the set of optimal solutions. Moreover, we strengthen the convergence rate bounds and prove that the step size is independent of e, and the number of steps depends logarithmically on 1 / e and quadratically on opt / Φ .