A Cheeger Inequality for Size-Specific Conductance

Abstract

The $\hat{I}¼$-conductance measure proposed by Lovász and Simonovits is a size-specific conductance score that identifies the set with smallest conductance while disregarding those sets with volume smaller than a $\hat{I}¼$ fraction of the whole graph. Using $\hat{I}¼$-conductance enables us to study the network structures in new ways. In this manuscript we study a modified spectral cut for $\hat{I}¼$-conductance that is a natural relaxation of the integer program of $\hat{I}¼$-conductance and show that the optimum of this program has a two-sided Cheeger inequality with $\hat{I}¼$-conductance.

Accepted by Discrete Mathematics

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