# Matrix-weighted networks for modeling multidimensional dynamics

arXiv: https://arxiv.org/abs/2410.05188 · HTML: https://mlsanity.com/paper/2410.05188/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 56 references that are in the map.
- First posted on arXiv 7 Oct 2024.

## Builds on (most cited first, 21 of 21)

- [Tensorizing Neural Networks](https://mlsanity.com/paper/1509.06569/) (2015)
- [Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs](https://mlsanity.com/paper/2202.04579/) (2022)
- [Multilayer Networks](https://mlsanity.com/paper/1309.7233/) (2013)
- [Sheaf Neural Networks with Connection Laplacians](https://mlsanity.com/paper/2206.08702/) (2022)
- [PageRank beyond the Web](https://mlsanity.com/paper/1407.5107/) (2014)
- [Random walks and diffusion on networks](https://mlsanity.com/paper/1612.03281/) (2016)
- [Random Walks, Markov Processes and the Multiscale Modular Organization of Complex Networks](https://mlsanity.com/paper/1502.04381/) (2015)
- [Novel Multidimensional Models of Opinion Dynamics in Social Networks](https://mlsanity.com/paper/1505.04920/) (2015)
- [Heterogeneous Sheaf Neural Networks](https://mlsanity.com/paper/2409.08036/) (2024)
- [Complex networks with complex weights](https://mlsanity.com/paper/2212.06257/) (2022)
- [Dynamical Systems on Networks: A Tutorial](https://mlsanity.com/paper/1403.7663/) (2014)
- [Spreading and Structural Balance on Signed Networks](https://mlsanity.com/paper/2212.10158/) (2022)
- [Estimation of Laplacian spectra of direct and strong product graphs](https://mlsanity.com/paper/1507.03030/) (2015)
- [Graph-Theoretic Analysis of Belief System Dynamics under Logic Constraints](https://mlsanity.com/paper/1810.02456/) (2018)
- [Structural Balance and Random Walks on Complex Networks with Complex Weights](https://mlsanity.com/paper/2307.01813/) (2023)
- [Temporal networks: slowing down diffusion by long lasting interactions](https://mlsanity.com/paper/1305.2938/) (2013)
- [Continuous-time Opinion Dynamics on Multiple Interdependent Topics](https://mlsanity.com/paper/1805.02836/) (2018)
- [Consensus and Disagreement of Heterogeneous Belief Systems in Influence Networks](https://mlsanity.com/paper/1812.05138/) (2018)
- [Graph Convolutional Networks from the Perspective of Sheaves and the Neural Tangent Kernel](https://mlsanity.com/paper/2208.09309/) (2022)
- [Cooperative opinion dynamics on multiple interdependent topics: Modeling and analysis](https://mlsanity.com/paper/1807.04406/) (2018)
- [Multi-topic belief formation through bifurcations over signed social networks](https://mlsanity.com/paper/2308.02755/) (2023)

## Abstract (arXiv)

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.