@MISC{Chung_fourcheeger-type, author = {Fan Chung}, title = {Four Cheeger-type Inequalities for Graph Partitioning Algorithms ∗}, year = {} }

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Abstract

We will give proofs to four isoperimetric inequalities which are variations of the original Cheeger inequality relating eigenvalues of a graph with the Cheeger constant. The first is a simplified proof of the classical Cheeger inequality using eigenvectors. The second is based on a rapid mixing result for random walks by Lovász and Simonovits. The third uses PageRank, a quantitative ranking of the vertices introduced by Brin and Page. The fourth proof is by an improved notion, called the heat kernel pagerank. The four proofs lead to improved graph partitioning algorithms and, in particular, local partition algorithms with its computational complexity proportional to the size of its output instead of the total size of the graph.