## Abstract

Normalized cut is a widely used measure of separation between clusters in a graph. In this paper we provide a novel probabilistic perspective on this measure. We show that for a partition of a graph into two weakly connected sets, V = A {multiset union} B, the multiway normalized cut is approximately MN cut ≈ 1/τ_{A→B} + 1/τ_{B→A}, where τ_{A→B} is the unidirectional characteristic exit time of a random walk from subset A to subset B. Using matrix perturbation theory, we show that this approximation is exact to first order in the connection strength between the two subsets A and B, and we derive an explicit bound for the approximation error. Our result implies that for a Markov chain composed of a reversible subset A that is weakly connected to an absorbing subset B, the easy-to-compute normalized cut measure is a reliable proxy for the chain's spectral gap.

Original language | American English |
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Pages (from-to) | 757-772 |

Number of pages | 16 |

Journal | SIAM Journal on Matrix Analysis and Applications |

Volume | 34 |

Issue number | 2 |

DOIs | |

State | Published - 2013 |

Externally published | Yes |

## Keywords

- Characteristic exit time
- Generalized eigenvalues
- Graph partitioning
- Markov chain
- Matrix perturbation theory
- Normalized cut
- Perturbation bounds