Normalized cuts are approximately inverse exit times

Matan Gavish, Boaz Nadler

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


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/τA→B + 1/τB→A, where τA→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 languageAmerican English
Pages (from-to)757-772
Number of pages16
JournalSIAM Journal on Matrix Analysis and Applications
Issue number2
StatePublished - 2013
Externally publishedYes


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


Dive into the research topics of 'Normalized cuts are approximately inverse exit times'. Together they form a unique fingerprint.

Cite this