[Paper Review] Generalized group-based epidemic model for spreading processes on networks: GgroupEM

Sifat Afroj Moon, Faryad Darabi Sahneh+1 more|arXiv (Cornell University)|Aug 16, 2019

Complex Network Analysis Techniques13 references4 citations

TL;DR

This paper introduces GgroupEM, a generalized group-based epidemic model framework for compartmental spreading processes on complex networks, enabling efficient simulation via state-space reduction through node grouping. By leveraging continuous-time Markov processes and mean-field approximations, GgroupEM achieves significant computational speedups with bounded aggregation error, while supporting multilayer networks and diverse epidemic models like SIS and SIR.

ABSTRACT

We develop a generalized group-based epidemic model (GgroupEM) framework for any compartmental epidemic model (for example; susceptible-infected-susceptible, susceptible-infected-recovered, susceptible-exposed-infected-recovered). Here, a group consists of a collection of individual nodes. This model can be used to understand the important dynamic characteristics of a stochastic epidemic spreading over very large complex networks, being informative about the state of groups. Aggregating nodes by groups, the state space becomes smaller than the individual-based approach at the cost of aggregation error, which is strongly bounded by the isoperimetric inequality. We also develop a mean-field approximation of this framework to further reduce the state-space size. Finally, we extend the GgroupEM to multilayer networks. Since the group-based framework is computationally less expensive and faster than an individual-based framework, then this framework is useful when the simulation time is important.

Motivation & Objective

• Address the computational infeasibility of individual-based epidemic modeling on very large networks, such as those in livestock, social, or communication systems.
• Reduce state space complexity by aggregating nodes into groups without sacrificing accuracy, using a framework that bounds aggregation error via the isoperimetric inequality.
• Generalize existing group-based models to support any compartmental epidemic model (e.g., SIS, SIR, SEIR), increasing flexibility and applicability.
• Develop a mean-field approximation to further reduce computational cost while preserving dynamic characteristics of the spreading process.
• Extend the framework to multilayer networks to model interdependent or overlapping network structures in real-world systems.

Proposed method

• Formulate a continuous-time Markov process for group-based dynamics, where each group represents a collection of nodes with aggregated states across compartments.
• Define transition indication matrices ($\Delta_{\delta}, \Delta_{\beta}$) and state transition indicators ($\mathcal{V}_{i,m}$) to model nodal and edge transitions between compartments.
• Construct the global transition rate matrix $\Theta$ using tensor operations ($\circ$) to combine group-level dynamics with network topology.
• Apply a mean-field approximation to reduce the state space size by assuming statistical independence of group states, enabling faster simulation.
• Extend the framework to multilayer networks by modeling inter-layer and intra-layer interactions through layered adjacency matrices $\mathcal{A}_g$.
• Use the isoperimetric inequality to theoretically bound the aggregation error introduced by grouping, ensuring model reliability.

Experimental results

Research questions

• RQ1 How can a group-based framework be generalized to support any compartmental epidemic model, including SIS, SIR, and SEIR?
• RQ2 What is the trade-off between computational efficiency and accuracy when aggregating nodes into groups, and how can the error be theoretically bounded?
• RQ3 To what extent does the mean-field approximation reduce simulation time while preserving the dynamic behavior of the original individual-based model?
• RQ4 How can the group-based framework be extended to model spreading processes on multilayer networks with interdependent or overlapping structures?
• RQ5 In what scenarios does the group-based approach outperform individual-based models in terms of speed and scalability without significant loss of predictive fidelity?

Key findings

• The GgroupEM framework reduces the state space size significantly compared to individual-based models, enabling faster simulation on large-scale networks.
• The aggregation error introduced by grouping is strongly bounded by the isoperimetric inequality, ensuring theoretical reliability of the model.
• For a two-group system with $\mathcal{N}_1=3$ and $\mathcal{N}_2=3$, the group-based model captures $3$ and $4$ distinct states respectively, demonstrating manageable state space growth.
• The mean-field approximation further reduces computational complexity by assuming statistical independence of group states, enabling scalable simulations.
• The framework successfully extends to multilayer networks by modeling inter-layer and intra-layer transitions through layered adjacency matrices.
• The $\Theta_{\delta_1}$ and $\Theta_{\beta_1}$ matrices derived for SIS processes demonstrate how group-level transition dynamics are encoded using matrix operations on group states and network structure.