Locally computable approximations for spectral clustering and absorption times of random walks

电路游走与Graver游走及线性规划与整数规划

Circuit and Graver Walks and Linear and Integer Programming

本文证明线性规划的电路游走可仅用线性代数运算在多项式时间内计算，而整数规划的Graver游走在有预言机时多项式可计算，无预言机时即使已知最优解也难以计算，并推广到有界树深和子行列式矩阵。

Towards Classifying the Polynomial-Time Solvability of Temporal Betweenness Centrality

In static graphs, the betweenness centrality of a graph vertex measures how many times this vertex is part of a shortest path between any two graph vertices. Betweenness centrality is efficiently computable and it is a fundamental tool in network science. Continuing and extending previous work, we study the efficient computability of betweenness centrality in temporal graphs (graphs with fixed vertex set but time-varying arc sets). Unlike in the static case, there are numerous natural notions of being a "shortest" temporal path (walk). Depending on which notion is used, it was already observed…

关于简单折线行走数量和等价规范化的图识别

On Numbers of Simplicial Walks and Equivalent Canonizations for Graph Recognition

通过简单折线行走数量定义的SW细化过程，用于识别有限路径宽图的属性，其计算复杂度为$O(kn^{3k})$。

Hearing the clusters in a graph: A distributed algorithm

We propose a novel distributed algorithm to cluster graphs. The algorithm recovers the solution obtained from spectral clustering without the need for expensive eigenvalue/vector computations. We prove that, by propagating waves through the graph, a local fast Fourier transform yields the local component of every eigenvector of the Laplacian matrix, thus providing clustering information. For large graphs, the proposed algorithm is orders of magnitude faster than random walk based approaches. We prove the equivalence of the proposed algorithm to spectral clustering and derive convergence rates.…

Sharp Bounds on Random Walk Eigenvalues via Spectral Embedding

Spectral embedding of graphs uses the top k non-trivial eigenvectors of the random walk matrix to embed the graph into R^k. The primary use of this embedding has been for practical spectral clustering algorithms [SM00,NJW02]. Recently, spectral embedding was studied from a theoretical perspective to prove higher order variants of Cheeger's inequality [LOT12,LRTV12]. We use spectral embedding to provide a unifying framework for bounding all the eigenvalues of graphs. For example, we show that for any finite graph with n vertices and all k >= 2, the k-th largest eigenvalue is at most 1-Omega(k^3…

Spectral clustering algorithms for the detection of clusters in block-cyclic and block-acyclic graphs

We propose two spectral algorithms for partitioning nodes in directed graphs respectively with a cyclic and an acyclic pattern of connection between groups of nodes. Our methods are based on the computation of extremal eigenvalues of the transition matrix associated to the directed graph. The two algorithms outperform state-of-the art methods for directed graph clustering on synthetic datasets, including methods based on blockmodels, bibliometric symmetrization and random walks. Our algorithms have the same space complexity as classical spectral clustering algorithms for undirected graphs and …