## 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 - ℒ̃^{T})/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 language | English (US) |
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Title of host publication | Algorithms and Models for the Web Graph - 7th International Workshop, WAW 2010, Proceedings |

Pages | 74-85 |

Number of pages | 12 |

DOIs | |

State | Published - 2010 |

Event | 7th International Workshop on Algorithms and Models for the Web Graph, WAW 2010 - Stanford, CA, United States Duration: Dec 13 2010 → Dec 14 2010 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 6516 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 7th International Workshop on Algorithms and Models for the Web Graph, WAW 2010 |
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Country/Territory | United States |

City | Stanford, CA |

Period | 12/13/10 → 12/14/10 |

### Bibliographical note

Funding Information:The work was supported in part by the National Science Foundation grants CNS-0905037 and CNS-1017647, the DTRA grant HDTRA1-09-1-0050, and a University of Minnesota DTC DTI grant.

Funding Information:

★ The work was supported in part by the National Science Foundation grants CNS-0905037 and CNS-1017647, the DTRA grant HDTRA1-09-1-0050, and a University of Minnesota DTC DTI grant.