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While multilinear algebra appears natural for studying the multiwayinteractions modeled by hypergraphs, tensor methods for general hypergraphshave been stymied by theoretical and practical barriers. A recently proposedadjacency tensor is applicable to nonuniform hypergraphs, but is prohibitivelycostly to form and analyze in practice. We develop tensor times same vector(TTSV) algorithms for this tensor which improve complexity from $O(n^r)$ to alow-degree polynomial in $r$, where $n$ is the number of vertices and $r$ isthe maximum hyperedge size. Our algorithms are implicit, avoiding formation ofthe order $r$ adjacency tensor. We demonstrate the flexibility and utility ofour approach in practice by developing tensor-based hypergraph centrality andclustering algorithms. We also show these tensor measures offer complementaryinformation to analogous graph-reduction approaches on data, and are also ableto detect higher-order structure that many existing matrix-based approachesprovably cannot.

Scalable tensor methods for nonuniform hypergraphs