Controllability of Hypergraphs

TL;DR

This paper introduces a tensor algebra-based framework for analyzing the controllability of hypergraphs, providing a new measure called the minimum number of control nodes (MCN) and demonstrating its applications in biological networks.

Contribution

It develops a novel tensor-based dynamical system model for hypergraphs and derives a Kalman-rank-like condition for controllability, along with an efficient heuristic for MCN.

Findings

MCN relates to hypergraph degree distribution in simulations

The proposed method effectively assesses robustness in biological networks

A new tensor-based approach advances hypergraph controllability analysis

Abstract

In this paper, we develop a notion of controllability for hypergraphs via tensor algebra and polynomial control theory. Inspired by uniform hypergraphs, we propose a new tensor-based multilinear dynamical system representation, and derive a Kalman-rank-like condition to determine the minimum number of control nodes (MCN) needed to achieve controllability of even uniform hypergraphs. We present an efficient heuristic to obtain the MCN. MCN can be used as a measure of robustness, and we show that it is related to the hypergraph degree distribution in simulated examples. Finally, we use MCN to examine robustness in real biological networks.