• Open Access

Analytically Solvable Model of Spreading Dynamics with Non-Poissonian Processes

Hang-Hyun Jo, Juan I. Perotti, Kimmo Kaski, and János Kertész
Phys. Rev. X 4, 011041 – Published 17 March 2014

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 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 the 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 a 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.

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  • Received 22 November 2013

DOI:https://doi.org/10.1103/PhysRevX.4.011041

This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Authors & Affiliations

Hang-Hyun Jo1,*, Juan I. Perotti1, Kimmo Kaski1, and János Kertész2,1

  • 1BECS, Aalto University School of Science, FI-00076 Espoo, Finland
  • 2Center for Network Science, Central European University, H-1051 Budapest, Hungary

  • *hang-hyun.jo@aalto.fi

Popular Summary

In human societies, information, ideas, certain behavior, or diseases spread through interactions between individuals. What controls the speed of these spreading processes? It turns out that not only is the structure of the connections between individuals important as we naturally expect, but the timing of interaction events is also crucial. One of the major temporal patterns of human interactions is their “burstiness.” But there are very few analytic results that illuminate how burstiness influences the dynamics of spreading processes, important as this question is to developing strategies for halting epidemic outbreaks or promoting diffusion of innovations. In this paper, we present a number of concrete analytic results based on a model of “bursty” spreading dynamics, providing a fundamental reference for the interpretation of numerical simulations and empirical data.

The model of bursty spreading dynamics we have constructed is rather simple: Each node in the connectivity network is either “susceptible” or “infected” and can go through cycles of activation and deactivation, whose lengths follow a statistical distribution. An active infected node can infect a susceptible node, whether or not the latter is active. We have succeeded in obtaining analytic results on how the number of infected nodes depends on time for arbitrary distributions of the inter-event time. In particular, for distributions that stylize burstiness, our solutions reveal that burstiness speeds up the spreading process in the early stage but slows it down in the late time. This result clarifies some of the existing controversies related to numerical results only.

The simple but solvable model together with the analytic results should be of high interest to those who are working to understand dynamics of complex systems.

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Vol. 4, Iss. 1 — January - March 2014

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