Low prevalence, quasi-stationarity and power-law behavior in a model of contagion spreading

While contagion (information, infection, etc.) spreading has been extensively studied recently, the role of active local agents has not been fully considered. Here, we propose and study a model of spreading which takes into account the strength or quality of contagions as well as the local probabilistic dynamics occurring at various nodes. Transmission occurs only after the quality-based fitness of the contagion has been evaluated by the local agent. We study such spreading dynamics on Erdös-Rényi as well as scale free networks. The model exhibits qualitydependent exponential time scales at early times leading to a slowly evolving quasi-stationary state. Low prevalence is seen for a wide range of contagion quality for arbitrary large networks. We also investigate the activity of nodes and find a power-law distribution with a robust exponent independent of network topology. These properties, while absent in standard theoretical models, are observed in recent empirical observations. Copyright c © EPLA, 2012 Introduction. – Spreading, defined broadly, as transmission of contagions (e.g., information or virus) from one agent to the next is a fundamental process with application in a wide range of disciplines including physics, epidemiology and social sciences. In many cases of spreading phenomena, transmission only occurs after some local conditions are met. From a theoretical point of view, this requires construction of models of contagion dynamics (spreading) which take into account the content of the contagion as well as the role of the individual agents receiving/evaluating/transmitting contagion. It is therefore surprising that such models have not been proposed and studied, despite the fact that much attention has been given to spreading phenomena, typically in a mean-field approximation, in the past decade [1–4]. In this letter, we propose and study a model of contagion spreading which takes into account the “quality” of contagion as well as the role of the individual local agents in the spreading phenomenon. We find many realistic features in our model, including quality-dependent fast initial spreading followed by extremely slow dynamics, as well as robust power-law behavior in activities of individual agents. Low prevalence (a)E-mail: [email protected] is also observed as a generic but limiting behavior of our model, e.g., as system size diverges. It is hardly possible to overemphasize the importance of contagion dynamics. Modern telecommunications including emails and text-messages, and more recently social networks like Facebook and Twitter have revolutionized the process of information spreading. Network and/or viral marketing [5], opinion formation and rumor/innovation spreading [6,7], recruiting and talent searches [8], disaster response and relief efforts, mobilizing masses [9], as well as epidemic spreading [1,2,10] are a few examples of how contagion spreading through complex networks is of vital importance in our modern way of life. Scarcity of reliable empirical results, as well as the ease to capture the general features of spreading, have led many authors to study spreading along the lines of epidemiology [1,11–13]. In such approaches, one typically uses SI, SIR, or SIS models on a complex network where the letters in the acronyms refer to the state (Susceptible, Infected, or Recovered) of the agents on the network. Assuming a mean-field transmission rate λ, one typically finds a low threshold for spreading where a large part of the network is “infected” in a relatively short time, with some dependence on network topology [2,13–15]. However, such results are in contrast