Matrix-weighted networks for modeling multidimensional dynamics (2410.05188v1)

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.

• The paper introduces MWNs by replacing scalar weights with matrices to capture the multidimensional nature of network interactions.
• It develops mathematical constructs such as supra-Laplacian and coherence to explain how consistent transformational pathways yield stable consensus dynamics.
• Numerical validation using matrix-weighted stochastic block models confirms that coherent networks achieve non-trivial consensus and stable random walk behaviors.

Matrix-Weighted Networks for Modeling Multidimensional Dynamics

The paper of complex systems often involves examining interactions within a network framework, traditionally using scalar weights to represent the strength of connections. However, many real-world systems require a more sophisticated approach due to their inherently multidimensional nature. This paper introduces an advanced framework known as Matrix-Weighted Networks (MWNs) to model multidimensional interactions.

Matrix-Weighted Networks (MWNs)

In MWNs, edges are characterized by matrices instead of scalars, allowing for a richer representation of interactions between node states, which are vectors rather than simple values. These matrices can capture the multidimensional effects of interactions, such as in social networks where multiple dimensions of opinion dynamics intersect.

Key properties of MWNs include:

• Supra-Weight Matrix: A block matrix structure representing the network, where each block matrix encodes the multidimensional interaction between node pairs.
• Coherence: A central notion that defines whether different transformational pathways in the network lead to consistent outcomes. Coherence implies that the transformations along cycles in the network result in no net effect, ensuring the stability of derived patterns such as consensus.

Theoretical Results

The paper establishes the mathematical foundations of MWNs, identifying important constructs such as the supra-Laplacian and supra-transition matrices. These matrices help analyze key dynamics within the network, namely consensus dynamics and random walks, reshaped to account for the multidimensional nature of the network.

1. Consensus Dynamics: In coherent MWNs, consensus (or multi-consensus) is achieved where node states stabilize to non-trivial, consistent patterns across different sections of the network. In incoherent networks, initial patterns may emerge but will eventually settle to trivial states due to inherent inconsistencies.
2. Random Walks: For random walks on coherent MWNs, the mean direction of the state vectors stabilizes in line with the network's coherent structure. In contrast, incoherent networks either lack a steady state or resolve to the origin in terms of averaged node state vectors.

Numerical Validation

The theoretical claims are validated using simulations on matrix-weighted stochastic block models (MSBMs). Results consistently demonstrate that coherent networks exhibit expected behavior both in consensus dynamics and random acts, with observed time-scale separation in incoherent networks. These findings span various network configurations, reinforcing the robustness of the theoretical insights.

Implications and Future Directions

The introduction of MWNs has significant implications for complex systems modeling. It provides a broader framework allowing multidimensional analysis of networks where traditional scalar models fall short. This could be particularly transformative in fields like social network analysis, neuroscience, and multilayer network dynamics.

Additionally, MWNs pave the way for new methodologies in community detection, node centrality assessment, and other network-based analytical techniques that account for multidimensional interactions. Future research can explore relaxed notions of coherence to accommodate networks exhibiting partial coherence or evolving temporal coherence, broadening the applicability of MWNs in more dynamic contexts.

In summary, this paper lays a solid foundation for the use of MWNs in the analysis of complex systems with multidimensional interactions. It prompts a reevaluation of existing models and encourages the exploration of MWNs for enhanced analytical capacity in network science.