Commute times for a directed graph using an asymmetric Laplacian

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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 languageEnglish (US)
Pages (from-to)224-242
Number of pages19
JournalLinear Algebra and Its Applications
Issue number2
StatePublished - 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: (D. Boley), (G. Ranjan), (Z.-L. Zhang).


  • Commute times
  • Directed graphs
  • Graph Laplacians
  • Wireless networks


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