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

Yu Tian1,2, Sadamori Kojaku3, Hiroki Sayama3,4

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This study introduces matrix-weighted networks (MWNs) to model complex systems with multidimensional interactions. MWNs reveal new insights into network structures, generalizing concepts like communities and structural balance.

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Area of Science:

  • Complex Systems Science
  • Network Theory
  • Mathematical Modeling

Background:

  • Traditional networks use scalar edge weights, limiting their ability to represent multidimensional interactions.
  • Real-world systems, such as social networks with interconnected opinions, often exhibit complex, multidimensional relationships.

Purpose of the Study:

  • To propose a general framework, matrix-weighted networks (MWNs), for modeling multidimensional interacting dynamics.
  • To establish the mathematical foundations for MWNs.
  • To analyze consensus dynamics and random walks on MWNs.

Main Methods:

  • Development of the mathematical framework for matrix-weighted networks.
  • Analysis of consensus dynamics within the MWN framework.
  • Examination of random walk processes on MWNs.

Main Results:

  • The mathematical framework for matrix-weighted networks (MWNs) is established.
  • Consensus dynamics and random walks are analyzed within the MWN context.
  • The coherence of MWNs leads to nontrivial steady states.

Conclusions:

  • Matrix-weighted networks provide a powerful generalization for modeling multidimensional interactions.
  • MWNs extend traditional network concepts like communities and structural balance.
  • The study lays the groundwork for analyzing complex systems with richer interaction structures.