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Despite of the extreme success of the spectral graph theory, there arerelatively few papers applying spectral analysis to hypergraphs. Chung firstintroduced Laplacians for regular hypergraphs and showed some usefulapplications. Other researchers treated hypergraphs as weighted graphs and thenstudied the Laplacians of the corresponding weighted graphs. In this paper, weaim to unify these very different versions of Laplacians for hypergraphs. Weintroduce a set of Laplacians for hypergraphs through studying high-orderedrandom walks on hypergraphs. We prove the eigenvalues of these Laplacians caneffectively control the mixing rate of high-ordered random walks, thegeneralized distances/diameters, and the edge expansions.

High-ordered Random Walks and Generalized Laplacians on Hypergraphs