# Reducing Spreading Processes on Networks to Markov Population Models

## TL;DR

This paper introduces a novel lumping scheme that reduces complex network-based epidemic models to Markov Population Models, enabling more efficient analysis and approximation of spreading processes.

## Contribution

The authors propose a new node-partitioning lumping method that transforms complex network epidemic models into Markov Population Models, facilitating the use of existing approximation techniques.

## Key findings

- Lumping reduces the state space size significantly.
- Different counting abstractions affect approximation accuracy.
- Numerical examples demonstrate the method's effectiveness.

## Abstract

Stochastic processes on complex networks, where each node is in one of several compartments, and neighboring nodes interact with each other, can be used to describe a variety of real-world spreading phenomena. However, computational analysis of such processes is hindered by the enormous size of their underlying state space. In this work, we demonstrate that lumping can be used to reduce any epidemic model to a Markov Population Model (MPM). Therefore, we propose a novel lumping scheme based on a partitioning of the nodes. By imposing different types of counting abstractions, we obtain coarse-grained Markov models with a natural MPM representation that approximate the original systems. This makes it possible to transfer the rich pool of approximation techniques developed for MPMs to the computational analysis of complex networks' dynamics. We present numerical examples to investigate the relationship between the accuracy of the MPMs, the size of the lumped state space, and the type of counting abstraction.