Computing the p-Spectral Radii of Uniform Hypergraphs with Applications

Abstract

The p-spectral radius of a uniform hypergraph covers many important concepts, such as Lagrangian and spectral radius of the hypergraph, and is crucial for solving spectral extremal problems of hypergraphs. In this paper, we establish a spherically constrained maximization model and propose a first-order conjugate gradient algorithm to compute the p-spectral radius of a uniform hypergraph (CSRH). By the semialgebraic nature of the adjacency tensor of a uniform hypergraph, CSRH is globally convergent and obtains the global maximizer with a high probability. When computing the spectral radius of the adjacency tensor of a uniform hypergraph, CSRH stands out among existing approaches. Furthermore, CSRH is competent to calculate the p-spectral radius of a hypergraph with millions of vertices and to approximate the Lagrangian of a hypergraph. Finally, we show that the CSRH method is capable of ranking real-world data set based on solutions generated by the p-spectral radius model.

On the Effect of Data Dimensionality on Eigenvector Centrality

“…There have been great strides in the computation of the principal eigenvector of a hypergraph as a constrained optimization problem [11,12]. One can also consider the problem from a algebraic approach via the Lu-Man Method which was introduced in [13] and further developed in [14,15].…”

A refined bound for the Z1-spectral radius of tensors

“…The Z 1 -eigenvalue of tensors and its corresponding eigenvectors are useful for computing the limiting probability distribution in high order Markov chain [1,10] and the PageRank vector in multilinear PageRank models [7,11], and also have applications in image matching [5], best rank-one approximation of tensors [14,17], and hypergraph theory [2,8]. Definition 1.1.…”

The $α$-normal labeling method for computing the $p$-spectral radii of uniform hypergraphs

“…The p-spectral radius has been introduced by Keevash, Lenz and Mubayi [7] and subsequently studied by Nikiforov [11,12,5] and Chang et al [3]. Note that the pspectral radius λ (p) (G) shows remarkable connections with some hypergraph invariants.…”

“…This α-normal labeling method has been proved by many researches [6,9,8,1,13,17,18,16] to be a simple and effective method in the study of spectral radii of uniform hypergraphs. In this paper, we extend Lu and Man's method to the p-spectral radii of uniform hypergraphs for p = r. The α-normal labeling method (for p = r) is very different from the CSRH algorithm developed by Chang-Ding-Qi-Yan [3] to compute the p-spectrum radii of uniform hypergraphs numerically. Although our method can also be used to compute the p-spectral radius of a hypergraph G when G is highly symmetric or hypertree-like, this is not our main purpose.…”