From Agent-Based Markov Dynamics to Hierarchical Closures on Networks: Emergent Complexity and Epidemic Applications

TL;DR

The paper develops a rigorous, probabilistic framework for agent-based SIR dynamics on networks by casting the process as a continuous-time Markov chain and deriving a BBGKY-like hierarchy of marginal distributions. It introduces four closures—first-order, ergodic, second-order direct decoupling, and second-order conditional—then benchmarks them against exact solutions and Monte Carlo simulations on simple graphs and scale-free networks. The results show that the second-order conditional closure provides the best balance between tractability and fidelity, effectively capturing the influence of network topology, clustering, and interventions such as lockdowns. This hierarchical, network-aware approach offers a general methodology for modeling diffusion and cascading processes in complex systems beyond epidemics, with potential extensions to adaptive networks and multi-state agents.

Abstract

We explore a rigorous formulation of agent-based SIR epidemic dynamics as a discrete-state Markov process, capturing the stochastic propagation of infection or an invading agent on networks. Using indicator functions and corresponding marginal probabilities, we derive a hierarchy of evolution equations that resembles the classical BBGKY hierarchy in statistical mechanics. The structure of these equations clarifies the challenges of closure and highlights the principal problem of systemic complexity arising from stochastic but generally not fully chaotic interactions. Monte Carlo simulations are used to validate simplified closures and approximations, offering a unified perspective on the interplay between network topology, stochasticity, and infection dynamics. We also explore the impact of lockdown measures within a networked agent framework, illustrating how SIR dynamics and structural complexity of the network shape epidemic with propagation of the COVID-19 pandemic in Northern Italy taken as an example.

Figures (10)

• Figure 1: One-dimensional connected graph with initial infection of the first node $i=1$.
• Figure 2: Modelling epidemic in one-dimensional case: total infected (top figure) and recovered (bottom figure). Lines: $\bullet ~\bullet ~\bullet ~\bullet$ the first-order closure; -- -- -- -- the second-order direct decoupling closure; --------- the second-order conditional closure; o---o---o Monte-Carlo, ensemble averaging over 100 realisations; $\cdot\cdot\cdot\cdot\cdot$ Monte-Carlo, ensemble averaging over 1000 realisations. Simulation parameters: $\tilde{p}=p\Delta t=5\times 10^{-3},$$\tilde{q}=q\Delta t=8\times 10^{-4}$.
• Figure 3: Simulations of SIR epidemic on a tree (a), randomly generated graph (b) with a fixed degree of $d_i=4$ for every node Lines: $\bullet ~\bullet ~\bullet ~\bullet$ the first-order closure; -- -- -- -- the second-order direct decoupling closure; --------- the second-order conditional closure; o---o---o Monte-Carlo, ensemble averaging over 100 realisations. Simulation parameters: $\tilde{p}=0.005$, $\tilde{q}=0.003$.
• Figure 4: Node degrees versus nodes (ordered by their degrees) for Erdős-Rényi (-- -- --), Barabasi-Albert (------) and random with a fixed degree ($\bullet ~\bullet ~\bullet ~\bullet$) graphs used in simulations.
• Figure 5: Simulations of SIR epidemic on the Erdős-Rényi (a), fixed node degrees (b) and Barabasi-Albert (c,d) graphs with peripheral (c) and central (d) initial conditions. Lines: $\bullet ~\bullet ~\bullet ~\bullet$ the first-order closure; -- -- -- -- the second-order direct decoupling closure; --------- the second-order conditional closure; o---o---o Monte-Carlo, ensemble averaging over 100 realisations. Simulation parameters: $\tilde{p}=0.005$, $\tilde{q}=0.003$.