Collective variables: The key to understanding dynamics on networks
Networks of interacting agents are widely used to model dynamic social phenomena such as the spreading of a disease, the diffusion of a political opinion within a society, or the adoption of sustainable behaviors and technologies in response to environmental challenges. In such networks, nodes represent individual agents and edges stand for some type of social interaction. Each node has a state that evolves over time depending on the states of neighboring nodes. Often, stochastic effects are included to account for uncertainty in the dynamics and for the variability of agent behavior. These types of spreading processes lie at the heart of numerous open problems in a wide range of disciplines, such as understanding social collective behavior, assessing systemic risk in financial systems, or controlling modern power grids.
Although individual agent behavior may be intuitive and easy to understand, the emergent macroscopic dynamics resulting from their interactions are often complex and unpredictable. Small changes in the environment or agent behavior can lead to qualitatively different outcomes on a global scale, making it challenging to devise macroscopic models without prior knowledge of the system’s behavior. Even with simple interaction rules, analyzing emergent behavior analytically is often difficult. Hence, one usually resorts to numerical simulations for evaluating such systems, but these become computationally intensive or even infeasible for large networks. To balance detail with feasibility, reduced-order models are sought to capture central phenomena while enabling more nuanced insights and better computational efficiency.
At ZIB, we tackle this challenge by identifying low-dimensional representations that capture the most significant properties of a system’s dynamics. We develop constructive methods to find collective variables (CVs) that project the system into a lower-dimensional space, filtering out unnecessary detail while retaining essential information about the system’s behavior. Good CVs reduce dimensionality while also offering insights into macroscopic dynamics by highlighting the most relevant features.
Our research not only enables the identification of collective variables but also facilitates the derivation of a reduced macroscopic model – evolution equations for the macroscopic state defined by the CVs – that accurately approximates the low-dimensional projection of the original dynamics. While the reduced model often remains stochastic, our work demonstrates that, in certain cases, the random actions of many agents in the network effectively cancel out each other, leading to approximately deterministic macroscopic dynamics.
In collaboration with FU Berlin and the Potsdam Institute for Climate Impact Research (PIK), we derived conditions under which Markovian discrete-state systems on networks converge to a mean-field limit. This theory states that, for particular networks, the shares of nodes in certain classes align with the solution of a mean-field ordinary differential equation in the large population limit (Figure 1). Moreover, we proposed a method to algorithmically learn interpretable CVs for spreading processes on networks without requir-ing prior expert knowledge of a network’s topology or dynamics. The approach involves sampling net-work states, running multiple brief simulations from these states, and extracting optimal CVs by applying manifold learning techniques to the set of transition densities (Figure 2) – a method developed with input from ZIB researchers, known as the transition mani-fold approach. The learned CVs are then extended to unseen data using total-variation-regularized linear regression. These CVs are interpretable because the inferred parameters reveal the role and importance of specific network features.
The methods and results developed at ZIB have been successfully applied to spreading dynamics across various network types, including Erdős–Rényi random graphs, stochastic block models, random regular graphs, ring-shaped networks, and scale-free networks, demonstrating their flexibility and effectiveness. Our methods enable the exploration of spreading processes on networks, capturing high-impact phenomena like epidemic thresholds and network fragmentation. By incorporating real-world complexities, they offer valuable tools for analyzing and comprehending complex social dynamics.
Marvin Lücke, Stefanie Winkelmann
