Random walks on digraphs, the generalized digraph laplacian and the degree of asymmetry

Yanhua Li, Zhi Li Zhang

Research output: Chapter in Book/Report/Conference proceedingConference contribution

17 Scopus citations

Abstract

In this paper we extend and generalize the standard random walk theory (or spectral graph theory) on undirected graphs to digraphs. In particular, we introduce and define a (normalized) digraph Laplacian matrix, and prove that 1) its Moore-Penrose pseudo-inverse is the (discrete) Green's function of the digraph Laplacian matrix (as an operator on digraphs), and 2) it is the normalized fundamental matrix of the Markov chain governing random walks on digraphs. Using these results, we derive new formula for computing hitting and commute times in terms of the Moore-Penrose pseudo-inverse of the digraph Laplacian, or equivalently, the singular values and vectors of the digraph Laplacian. Furthermore, we show that the Cheeger constant defined in [6] is intrinsically a quantity associated with undirected graphs. This motivates us to introduce a metric - the largest singular value of Δ := (ℒ̃ - ℒ̃T)/2 - to quantify and measure the degree of asymmetry in a digraph. Using this measure, we establish several new results, such as a tighter bound (than that of Fill's in [9] and Chung's in [6]) on the Markov chain mixing rate, and a bound on the second smallest singular value of ℒ̃.

Original languageEnglish (US)
Title of host publicationAlgorithms and Models for the Web Graph - 7th International Workshop, WAW 2010, Proceedings
Pages74-85
Number of pages12
DOIs
StatePublished - Dec 1 2010
Event7th International Workshop on Algorithms and Models for the Web Graph, WAW 2010 - Stanford, CA, United States
Duration: Dec 13 2010Dec 14 2010

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume6516 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other7th International Workshop on Algorithms and Models for the Web Graph, WAW 2010
CountryUnited States
CityStanford, CA
Period12/13/1012/14/10

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