Analytically Solvable Model of Spreading Dynamics with Non-Poissonian Processes

TL;DR

This paper introduces an analytically solvable model for spreading dynamics with non-Poissonian processes, revealing how burstiness influences the speed and convergence of spreading in different time regimes.

Contribution

It provides the first exact solution for SI spreading with arbitrary inter-event time distributions, explicitly considering lower bounds and contrasting early and late time behaviors.

Findings

Burstiness accelerates early and intermediate spreading phases.

Power-law inter-event times slow down late-time convergence.

Algebraic decay characterizes the approach to full infection in finite systems.

Abstract

Non-Poissonian bursty processes are ubiquitous in natural and social phenomena, yet little is known about their effects on the large-scale spreading dynamics. In order to characterize these effects we devise an analytically solvable model of Susceptible-Infected (SI) spreading dynamics in infinite systems for arbitrary inter-event time distributions and for the whole time range. Our model is stationary from the beginning, and the role of lower bound of inter-event times is explicitly considered. The exact solution shows that for early and intermediate times the burstiness accelerates the spreading as compared to a Poisson-like process with the same mean and same lower bound of inter-event times. Such behavior is opposite for late time dynamics in finite systems, where the power-law distribution of inter-event times results in a slower and algebraic convergence to fully infected state in contrast to the exponential decay of the Poisson-like process. We also provide an intuitive argument for the exponent characterizing algebraic convergence.