Laplacian Spectral Properties of Graphs from Random Local Samples

Abstract

The Laplacian eigenvalues of a network play an important role in the analysis of many structural and dynamical network problems. In this paper, we study the relationship between the eigenvalue spectrum of the normalized Laplacian matrix and the structure of 'local' subgraphs of the network. We call a subgraph local when it is induced by the set of nodes obtained from a breath-first search (BFS) of radius r around a node. In this paper, we propose techniques to estimate spectral properties of the normalized Laplacian matrix from a random collection of induced local subgraphs. In particular, we provide an algorithm to estimate the spectral moments of the normalized Laplacian matrix (the power-sums of its eigenvalues). Moreover, we propose a technique, based on convex optimization, to compute upper and lower bounds on the spectral radius of the normalized Laplacian matrix from local subgraphs. We illustrate our results studying the normalized Laplacian spectrum of a large-scale e-mail network. IntroductionUnderstanding the relationship between the structure of a network and its eigenvalues is of great relevance in the field of network science (see [3], [16] and references therein). The growing availability of massive databases, computing facilities, and reliable data analysis tools has provided a powerful framework to explore this relationship for many real-world networks. On the other hand, in many cases of practical interest, one cannot efficiently retrieve and/or store the exact full topology of a large-scale network. Alternatively, it is usually easy to retrieve local samples of the network structure. In this paper, we focus our attention on local sample of the network structure given in the form of a subgraph induced by the set of nodes obtained from a breath-first search (BFS) of small radius r around a particular node.We study the relationship between the normalized Laplacian spectrum of a graph and a random collection of local subgraphs. We show how local structural information contained in these localized subgraphs can be efficiently aggregated to infer global properties of the

Half Sampling on Bipartite Graphs

“…We refer the reader to [1,[3][4][5][6][7][8][14][15][16][17][18][19][20]25] for just a sampling of such results. There are also some related works on filter banks and wavelets on graphs, for example [12,13,23,24].…”