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  • Open Access

Matrix-Weighted Networks for Modeling Multidimensional Dynamics: Theoretical Foundations and Applications to Network Coherence

Yu Tian1,2,*, Sadamori Kojaku3,†, Hiroki Sayama3,4,‡, and Renaud Lambiotte5,§

  • *Contact author: yu.tian.researcher@gmail.com
  • †Contact author: skojaku@binghamton.edu
  • ‡Contact author: sayama@binghamton.edu
  • §Contact author: renaud.lambiotte@maths.ox.ac.uk
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Phys. Rev. Lett. 134, 237401 – Published 13 June, 2025

DOI: https://doi.org/10.1103/jw1k-6s7w

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 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 nontrivial steady states that generalize the notions of communities and structural balance in traditional networks.

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Physics Subject Headings (PhySH)

Corrections

6 November, 2025

Correction: An error in Eq. (1) has been fixed. The corresponding proofs in the Supplemental Material have been updated as well.

Article Text

Supplemental Material

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Outline

Information

Phys. Rev. Lett. 134, 237401– Published 13 June, 2025
  • Received 7 October 2024
  • Revised 16 December 2024
  • Accepted 21 May 2025
  • Corrected 6 November 2025
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