Matrix-weighted networks for modeling multidimensional dynamics

TL;DR

This paper introduces matrix-weighted networks (MWNs) as a new framework to model complex systems with multidimensional interactions, extending traditional scalar-weighted networks to better capture real-world dynamics.

Contribution

The paper develops the mathematical foundation of MWNs and explores their consensus dynamics and random walks, revealing new steady states that generalize community and balance concepts.

Findings

MWNs exhibit non-trivial steady states.

Coherence in MWNs influences community structures.

Generalizes traditional network notions.

Abstract

Networks are powerful tools for modeling interactions in complex systems. While traditional networks use scalar edge weights, many real-world systems involve multidimensional interactions. For example, in social networks, individuals often have multiple interconnected opinions that can affect different opinions of other individuals, which can be better characterized by matrices. We propose a novel, general framework for modeling such multidimensional interacting dynamics: matrix-weighted networks (MWNs). We present the mathematical foundations of MWNs and examine consensus dynamics and random walks within this context. Our results reveal that the coherence of MWNs gives rise to non-trivial steady states that generalize the notions of communities and structural balance in traditional networks.