Matrix-weighted networks for modeling multidimensional dynamics

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 the different opinions of other individuals, which can be better characterized by matrices. In this talk, I will introduce a general framework for modeling such multidimensional interacting dynamics: matrix-weighted networks (MWNs). In particular, I will present the mathematical foundations of MWNs and discuss consensus dynamics, from classic cases to signed networks, then complex-weighted networks, and finally MWNs 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 previous analysis.