Commute times for a directed graph using an asymmetric Laplacian

Daniel Boley; Gyan Ranjan; Zhi Li Zhang

doi:10.1016/j.laa.2011.01.030

Commute times for a directed graph using an asymmetric Laplacian

Daniel Boley, Gyan Ranjan, Zhi Li Zhang

Computer Science and Engineering

Research output: Contribution to journal › Article › peer-review

50 Scopus citations

Abstract

The expected commute times for a strongly connected directed graph are related to an asymmetric Laplacian matrix as a direct extension to similar well known formulas for undirected graphs. We show the close relationships between the asymmetric Laplacian and the so-called Fundamental matrix. We give bounds for the commute times in terms of the stationary probabilities for a random walk over the graph together with the asymmetric Laplacian and show how this can be approximated by a symmetrized Laplacian derived from a related weighted undirected graph.

Original language	English (US)
Pages (from-to)	224-242
Number of pages	19
Journal	Linear Algebra and Its Applications
Volume	435
Issue number	2
DOIs	https://doi.org/10.1016/j.laa.2011.01.030
State	Published - Jul 15 2011

Bibliographical note

Funding Information:
Supported in part by NSF grants 0534286, 0626808, 0916750, DTRA grant HDTRA1-09-1-0050, and a Univ. of Minn. DTC DTI gr∗Corresponding author.ant. E-mail addresses: [email protected] (D. Boley), [email protected] (G. Ranjan), [email protected] (Z.-L. Zhang).

Keywords

Commute times
Directed graphs
Graph Laplacians
Wireless networks

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10.1016/j.laa.2011.01.030

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abstract = "The expected commute times for a strongly connected directed graph are related to an asymmetric Laplacian matrix as a direct extension to similar well known formulas for undirected graphs. We show the close relationships between the asymmetric Laplacian and the so-called Fundamental matrix. We give bounds for the commute times in terms of the stationary probabilities for a random walk over the graph together with the asymmetric Laplacian and show how this can be approximated by a symmetrized Laplacian derived from a related weighted undirected graph.",

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AB - The expected commute times for a strongly connected directed graph are related to an asymmetric Laplacian matrix as a direct extension to similar well known formulas for undirected graphs. We show the close relationships between the asymmetric Laplacian and the so-called Fundamental matrix. We give bounds for the commute times in terms of the stationary probabilities for a random walk over the graph together with the asymmetric Laplacian and show how this can be approximated by a symmetrized Laplacian derived from a related weighted undirected graph.

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KW - Wireless networks

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Commute times for a directed graph using an asymmetric Laplacian

Abstract

Bibliographical note

Keywords

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