Matrix-weighted networks for modeling multidimensional dynamics

TL;DR

This work presents the mathematical foundations of MWNs and reveals that the coherence of MWNs gives rise to non-trivial steady states that generalize the notions of communities and structural balance in traditional networks.

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 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.

Figures (8)

• Figure 1: Illustration of a MWN, where each node is equipped with 3 state variables. In an opinion dynamics setting, each variable can be interpreted as an opinion on a specific topic. Here, we illustrate how the state vector of node $v_i$ affects the state vector of $v_j$ and vice versa.
• Figure 2: Example of the block structure of a coherent MWN, where the label indicates the orthogonal matrix weights and $\mathbf{O}_{13} = \mathbf{O}_{12}\mathbf{O}_{23}$
• Figure 3: Consensus dynamics in a three-block MSBM with $120$ nodes and $10$ dimensional state space. Red, yellow, and blue represent nodes in blocks 1, 2, and 3, respectively. Stars indicate theoretical steady states. We set $n_d = 10$, $p_{\text{in}} = 0.3$, $p_{\text{out}} = \{0.1, 0.3\}$. A, B: Coherent case, no topological community structure. C, D: Coherent case, strong topological community structure. E, F: Incoherent case, no topological community structure. G, H: Incoherent case, strong topological community structure. Panels A, C, E, G show PCA projections of state vector trajectories. In panels B, D, F, H, each blue line represents the distance to the steady state of a node, and the red line represents the average distance. The colorband indicates the 95% confidence interval estimated by $10^3$ bootstrap sampling.
• Figure 4: Random walk dynamics in a three-block MSBM with $120$ nodes and $10$ dimensional state space. Red, yellow, and blue represent nodes in blocks 1, 2, and 3, respectively. The directions of the theoretical steady states are indicated by solid lines. We set $n_d = 10$, $p_{\text{in}} = 0.3$, $p_{\text{out}} = \{0.1, 0.3\}$. A--C: Coherent case, no topological community structure. D--F: Coherent case, strong topological community structure. G--I: Incoherent case, no topological community structure. J--L: Incoherent case, strong topological community structure. Panels A, D, G, J show PCA projections of state vector trajectories. Panels B, E, H, K show angular distances to steady state (blue: individual nodes, red: average). Panels C, F, I, L show distances to origin (blue: individual nodes, red: average). The colorband indicates the 95% confidence interval estimated by $10^3$ bootstrap sampling.
• Figure 5: Consensus dynamics in a three-block MSBM with 120 nodes and 3-dimensional state space. Red, yellow, and blue represent nodes in blocks 1, 2, and 3, respectively. The stars indicate the theoretical steady states. We set $n_d = 3$, $p_{\text{in}} = 0.3$, $p_{\text{out}} = \{0.1, 0.3\}$. A--C: Coherent case, no topological community structure. D--F: Coherent case, strong topological community structure. G--I: Incoherent case, no topological community structure. J--L: Incoherent case, strong topological community structure. Panels A, D, G, J show PCA projections of state vector trajectories. Panels B, E, H, K show distances to steady state (blue: individual nodes, red: average). Panels C, F, I, L show distances to origin (blue: individual nodes, red: average). The colorband indicates the 95% confidence interval estimated by $10^3$ bootstrap sampling.

Theorems & Definitions (13)

• Proposition 1
• proof : Proof of Proposition \ref{['pro:L-semi-definite']}
• Definition 2
• Theorem 3: structural theorem for balance
• proof : Proof of Theorem \ref{['the:balance-part']}
• Theorem 4
• proof : Proof of Theorem \ref{['the:balance-L']}
• Proposition 5
• proof
• Proposition 6