Taming complexity with Approximate Master Equations

Mean-field arguments are often used to model the dynamics of complex systems by tracking the average state of the system. In many systems, these approaches fail to capture the discreteness of available states, often counted in discrete number of agents or connections. For example, by assuming an infinite population, mean-field arguments ignore the possibility of a stochastic extinction of a dynamical process.

Master equations differ from mean-field approaches, even though they are also described through systems of differential equations. Instead of only tracking the average state of a system, master equations track the temporal evolution of the probability distribution of finding the system in a given state, over all possible states for that system.

Here are some references that might be of interest in combination with this tutorial:

• Modelling with the Master equation [Haag, 2017]

• Stochastic numerical methods: an introduction for students and scientists [Toral and Colet, 2014]. This is a general book on modeling methods for stochastic processes which features an extremely useful introduction to master equations as well as related numerical solution methods.

Our main goal is to provide a tutorial on the conceptual design of master equations and on computational to solve them. As applications, we will cover classic models from the study of complex systems: Lotka-Volterra predator-prey dynamics, or Susceptible-Infectious-Susceptible dynamics from disease modeling.

Table of content

Master Equations 101

• Definition and properties
• Numerical integration
• Method of Moments
• Explicit solution
• Approximate master equations

Applications

• Predator-prey
• Adaptive networks

References

Haa17

Günter Haag. Modelling with the master equation. Solution Methods, 2017.

HebertDNoelM+10

Laurent Hébert-Dufresne, Pierre-André Noël, Vincent Marceau, Antoine Allard, and Louis J Dubé. Propagation dynamics on networks featuring complex topologies. Physical Review E, 82(3):036115, 2010.

MNoelHebertD+10

Vincent Marceau, Pierre-André Noël, Laurent Hébert-Dufresne, Antoine Allard, and Louis J Dubé. Adaptive networks: coevolution of disease and topology. Physical Review E, 82(3):036116, 2010.

SAHebertD16

Samuel V. Scarpino, Antoine Allard, and Laurent Hébert-Dufresne. The effect of a prudent adaptive behaviour on disease transmission. Nature Physics, 12(11):1042–1046, 2016. URL: https://doi.org/10.1038/nphys3832, doi:10.1038/nphys3832.

SOTA+21

Guillaume St-Onge, Vincent Thibeault, Antoine Allard, Louis J Dubé, and Laurent Hébert-Dufresne. Master equation analysis of mesoscopic localization in contagion dynamics on higher-order networks. Physical Review E, 103(3):032301, 2021.

TC14

Raúl Toral and Pere Colet. Stochastic numerical methods: an introduction for students and scientists. John Wiley & Sons, 2014.

Wil05

Herbert S. Wilf. generatingfunctionology. CRC press, 2005.