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.

翻译：网络是建模复杂系统中相互作用的强大工具。传统网络使用标量边权，而许多现实系统涉及多维相互作用。例如，在社交网络中，个体通常具有多个相互关联的观点，这些观点可能影响其他个体的不同观点，通过矩阵可以更好地刻画这种特性。我们提出了一种新颖的通用框架来建模此类多维相互作用动态：矩阵加权网络（MWNs）。我们阐述了MWNs的数学基础，并在此框架下研究了共识动态和随机游走过程。研究结果表明，MWNs的相干性会产生非平凡稳态，这些稳态推广了传统网络中社区和结构平衡的概念。