Temporal Gillespie Algorithm: Fast Simulation of Contagion Processes on Time-Varying Networks.

Christian L Vestergaard,Mathieu Génois,Marcel Salathé

doi:10.1371/journal.pcbi.1004579

Abstract

Stochastic simulations are one of the cornerstones of the analysis of dynamical processes on complex networks, and are often the only accessible way to explore their behavior. The development of fast algorithms is paramount to allow large-scale simulations. The Gillespie algorithm can be used for fast simulation of stochastic processes, and variants of it have been applied to simulate dynamical processes on static networks. However, its adaptation to temporal networks remains non-trivial. We here present a temporal Gillespie algorithm that solves this problem. Our method is applicable to general Poisson (constant-rate) processes on temporal networks, stochastically exact, and up to multiple orders of magnitude faster than traditional simulation schemes based on rejection sampling. We also show how it can be extended to simulate non-Markovian processes. The algorithm is easily applicable in practice, and as an illustration we detail how to simulate both Poissonian and non-Markovian models of epidemic spreading. Namely, we provide pseudocode and its implementation in C++ for simulating the paradigmatic Susceptible-Infected-Susceptible and Susceptible-Infected-Recovered models and a Susceptible-Infected-Recovered model with non-constant recovery rates. For empirical networks, the temporal Gillespie algorithm is here typically from 10 to 100 times faster than rejection sampling.

Highlights

Networks have emerged as a natural description of complex systems and their dynamics [1], notably in the case of spreading phenomena, such as social contagion, rumor and information spreading, or epidemics [1,2,3]
When studying how e.g. diseases spread in a population, intermittent contacts taking place between individuals—through which the infection spreads—are best described by a timevarying network. This object captures both their complex structure and dynamics, which crucially affect spreading in the population
The dynamical process in question is usually studied by simulating it on the time-varying network representing the population

Summary

Introduction

Networks have emerged as a natural description of complex systems and their dynamics [1], notably in the case of spreading phenomena, such as social contagion, rumor and information spreading, or epidemics [1,2,3]. The dynamics of contagion processes occurring on a network are usually complex, and analytical results are attainable only in special cases [3, 4]. Such results almost systematically involve approximations [3, 4]. The development of fast algorithms is important for the characterization of contagion phenomena, and for large-scale applications such as simulations of world-wide epidemics [2, 5]. It is readily adapted to the simulation of Poisson processes on static networks [12,13,14,15,16] and can be generalized to non-Markovian processes [17]

Methods

Results

Discussion

Conclusion