Concepts and Algorithms for Higher Order Networks: Beyond Pairwise Interactions

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. Recently, there has been increased interest in the idea of accounting directly for higher order features. Such features may be built from the adjacency matrix---for example, a triangle involving nodes i, j and k arises when the three edges, i<->j, j<->k and k<->i are present. In other contexts, higher order information appears explicitly---for example, in a coauthorship network, a document involving three authors forms a natural triangle. I will discuss the use of tensor-based definitions and algorithms to exploit such higher order information. The algorithms also incorporate nonlinearities that increase flexibility. I will focus on spectral methods that extend classical concepts of node centrality and clustering coefficients. The underlying object of study will be a constrained nonlinear eigenvalue problem associated with a tensor. Using recent results from nonlinear Perron--Frobenius theory, we can establish existence and uniqueness under mild conditions, and show that such spectral measures can be computed efficiently and robustly with a nonlinear power method.

The talk is based on joint work with Francesca Arrigo (University of Strathclyde) and Francesco Tudisco (Gran Sasso Science Institute).

computational engineering, finance, and sciencenumerical analysis

Audience: researchers in the topic

E-NLA - Online seminar series on numerical linear algebra

Series comments: E-NLA is an online seminar series dedicated to topics in Numerical Linear Algebra. Talks take place on Wednesdays at 4pm (Central European Time) via Zoom and are initially scheduled on a weekly basis.

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