Ihara coefficients: A flexible tool for higher order learning

Abstract

The aim of this paper is to seek a compact characterization of irregular unweighted hypergraphs for the purposes of clustering. To this end, we propose a novel hypergraph characterization method by using the Ihara coefficients, i.e. the characteristic polynomial coefficients extracted from the Ihara zeta function. We investigate the flexibility of the Ihara coefficients for learning relational structures with different relational orders. Furthermore, we introduce an efficient method for computing the coefficients. Our representation for hypergraphs takes into account not only the vertex connections but also the hyperedge cardinalities, and thus can distinguish different relational orders, which is prone to ambiguity in the hypergraph Laplacian. In experiments we demonstrate the effectiveness of the proposed characterization for clustering irregular unweighted hypergraphs and its advantages over the spectral characterization of the hypergraph Laplacian. © 2010 Springer-Verlag Berlin Heidelberg.