Depth-based hypergraph complexity traces from directed line graphs

Lu Bai, Francisco Escolano, Edwin R. Hancock

Abstract: Highlights • We aim to characterize hypergraphs in terms of structural complexity measures. • Straightforwardly measuring the complexity of a hypergraph tends to be elusive. • We transform a hypergraph into a line graph for measuring the complexities. • We compute depth-based complexity traces for the hypergraph on line graphs. • The complexity traces provide a flexible framework for hypergraphs and graphs. Abstract In this paper, we aim to characterize the structure of hypergraphs in terms of structural complexity measure. Measuring the complexity of a hypergraph in a straightforward way tends to be elusive since the hyperedges of a hypergraph may exhibit varying relational orders. We thus transform a hypergraph into a line graph which not only accurately reflects the multiple relationships exhibited by the hyperedges but is also easier to manipulate for complexity analysis. To locate the dominant substructure within a line graph, we identify a centroid vertex by computing the minimum variance of its shortest path lengths. A family of centroid expansion subgraphs of the line graph is then derived from the centroid vertex. We compute the depth-based complexity traces for the hypergraph by measuring either the directed or undirected entropies of its centroid expansion subgraphs. The resulting complexity traces provide a flexible framework that can be applied to both hypergraphs and graphs. We perform (hyper)graph classification in the principal component space of the complexity trace vectors. Experiments on (hyper)graph datasets abstracted from bioinformatics and computer vision data demonstrate the effectiveness and efficiency of the complexity traces.