Node and Edge Eigenvector Centrality for Hypergraphs

Abstract

Network scientists have shown that there is great value in studying pairwise interactions between components in a system. From a linear algebra point of view, this involves defining and evaluating functions of the associated adjacency matrix. Recent work indicates that there are further benefits from accounting directly for higher order interactions, notably through a hypergraph representation where an edge may involve multiple nodes. Building on these ideas, we motivate, define and analyze a class of spectral centrality measures for identifying important nodes and hyperedges in hypergraphs, generalizing existing network science concepts. By exploiting the latest developments in nonlinear Perron-Frobenius theory, we show how the resulting constrained nonlinear eigenvalue problems have unique solutions that can be computed efficiently via a nonlinear power method iteration. We illustrate the measures on realistic data sets.

HAT: Hypergraph analysis toolbox

“…The incidence matrix is the primary representation of hypergraphs in HAT (Fig 1b) [10,23]. HAT targets the following hypergraph features and problems: (1) construction from data [24][25][26], (2) expansion and numeric representation [27][28][29], (3) characteristic structural properties (such as entropy [11], centrality [30], distance [13], and clustering coefficients [11]), (4) controllability [12], and (5) similarity measures [13]. The workflow for using HAT is outlined in Fig 1e.…”

“…The following structural properties of hypergraphs are computed: average distance between nodes is computed based on [13] (see Hypergraph Structural Properties in S1 File, Equation S1); the clustering coefficient is calculated based on [11] (Equation S2); hypergraph centrality is measured according to methods in [30,31], which employ a variety of techniques to solve the nonlinear eigenvalue problem. For a uniform hypergraph, entropy is computed according to [11], which is defined based on the higher-order singular values of the Laplacian tensor (Equation S3).…”

Important hyperedge sorting method based on grounded Laplacian matrix

“…In the development of hypernetwork theory in recent years, many theoretical and practical achievements have been obtained, and many scholars have proposed methods to identify important nodes in hypernetworks. Estrada et al 8 extended the definition of subgraph centrality and clustering coefficient in complex networks to hypernetworks, and used these two indicators to identify core nodes in three types of real-world hypernetworks; Tudisco et al 9 using the idea of HITS algorithm proposed a new class of centrality indicators based on eigenvectors; Battiston et al 10 summarized three different types of high-order node centrality indicators; Kovalenko et al 11 proposed to use a vector to represent the importance of nodes in a hypergraph, and accordingly reveal the different roles played by the same node in different orders of interaction;…”