Despite its importance for rumors or innovations propagation, peer-to-peer collaboration, social networking or Marketing, the dynamics of information spreading is not well understood. Since the diffusion depends on the heterogeneous patterns of human behavior and is driven by the participants' decisions, its propagation dynamics shows surprising properties not explained by traditional epidemic or contagion models. Here we present a detailed analysis of our study of real Viral Marketing campaigns where tracking the propagation of a controlled message allowed us to analyze the structure and dynamics of a diffusion graph involving over 31,000 individuals. We found that information spreading displays a non-Markovian branching dynamics that can be modeled by a two-step Bellman-Harris Branching Process that generalizes the static models known in the literature and incorporates the high variability of human behavior. It explains accurately all the features of information propagation under the "tipping-point" and can be used for prediction and management of viral information spreading processes.
Each day, millions of conversations, emails, SMS, blog posts and comments, instant messages, tweets or web pages containing various types of information are exchanged between people. Humans natural inclination to share information with others in a "viral" fashion stems from the need of socializing and seeks to gain reputation, influence, trustworthiness or popularity [1]. Such viral dissemination of information through social networks, commonly known as "Word-of-Mouth" (WOM), is of paramount importance in our everyday life. In fact, it is known to influence purchasing decisions to the extent that 2/3 of the United States economy is driven by those kind of personal recommendations [2]. WOM is also important to understand sales and customer value [3,4], opinion formation or rumor spreading in social networks [5,6] or to determine the influence of each person in its social neighborhood [7,8]. Despite its importance and due to the difficulty (or inability) to capture this phenomenon, detailed empirical data on how humans disseminate information are scarce [9], population aggregated [10] or indirect [11,12]. Moreover, most studies have concentrated on asymptotical stationary properties of information difussion [13][14][15][16]. This has hampered the study of the dynamics of information diffusion and indeed most of its understanding comes from theoretical propagation models running on empirical or synthetic social networks in an approach borrowed from epidemiology [17][18][19]. In those models, information diffusion equates to the propagation of virus or diseases that spontaneously pass to others by contagion through the active social connections of the infected (i.e. informed) agents. However, information diffusion mechanisms are fundamentally different from those operating in disease spreading. In fact, passing a message along has a perceived transmission cost, its targets are consciously selected among potentially interested individuals [20,21], depends on human volition and, ultimately, is executed on the individuals' activity schedule. An obvious implication of those peculiarities is that information spreading is bound to depend on the large variability observed both on the volume and frequency of human activities and on the perceived value/cost of transmitting the information. For example, the number of emails sent by individuals per day [22], the number of telephone calls placed by users [23], the number of blog entries by user [24,25], the number of web page clicks per user [26], and the number of a person's social relationships [27] or sexual contacts [28] show large demographic stochasticity. In fact these numbers are distributed according to a powerlaw (or Pareto) distribution, inconsistent with the mild Gaussian or Poissonian stochasticity around populationaveraged values traditionally assumed in epidemiological models [29]. The same large variability pattern applies to the human activities time dynamics: for example, email response delays, market trading frequencies or inter-event time of web page visits, telephone calls, etc. are well described by power-law or log-normal distributions [22,30,31]. Recent research has shown that such high variability in human behavior alters substantially the temporal dynamics of information diffusion and does not merely introduce some stochasticity in populationaveraged models [9,32,33]. Thus, it is important to incorporate this human behavior into the models.
Besides, information diffusion travels through social arXiv:1110.1884v1 [physics.soc-ph] 9 Oct 2011 connections thereby depending on the properties of the social networks where it spreads. For example, simulations on synthetic scale-free networks showed that if information flowed through every social connection the epidemic threshold would be significantly lowered to the extent that it could disappear [13,34], so that any rumor, virus or innovation might reach a large fraction of individuals in the population no matter how small the probability of being infected. Given the fact that social networks are scale-free [35] those results predict that there is a strong interplay between network structure and the spreading process. However such is not the case for information spreading processes. Our daily experience indicates that most rumors, innovations or marketing messages do not reach a significant part of the population [36]. As mentioned earlier, the information transmission perceived cost prevents it from traveling inexpensively through all possible network paths. Therefore when participants assess the value of the information being passed, the impact of their social network structure on the diffusion process might be diminished. Unfortunately the true extent of such influence remains unknown in general. Moreover, the reach of information can be affected by the dynamics of human communication [33] and thus it is important to understand the interplay between the static and dynamical properties of information d
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