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All of these systems can be modeled as complex networks [–] in which nodes are individuals and links are contacts between them. To investigate the diffusion mechanisms and to predict and control the dynamics of social contagions, many successful models, both Markovian [, ] and non-Markovian [–], have been proposed. Unlike such biological contagions as epidemic spreading, the social reinforcement effect is ubiquitous in social contagions [, –]. For example, if an individual has nine friends adopt a new behavior and then a tenth friend adopts it, the individual will take the actions of all ten friends into account when deciding whether to adopt the behavior, since multiple confirmations of the credibility and legitimacy of the behavior are needed. Researchers have found that the social reinforcement effect markedly alters both the type of phase transition that occurs and the resulting final state [, ]. The primary factor in these models is the number of individuals who adopt a behavior, which is an order parameter of a phase transition. Watts found that when social reinforcement is introduced into a steady state system the change in average degree causes both continuous and discontinuous phase transitions []. Wang et al found that the initial seed size (degree exponent) has a critical point below (above) which the order parameter increases discontinuously with the transmission probability, otherwise it increases continuously []. Time-delays are pervasive in such real-world complex networks as social contagions and biological systems [–]. When individuals seek to accomplish tasks they are limited by the available time and energy, and thus time-delays become an issue. Although previous studies have found that time-delays affect both the evolution and the steady state of biological contagions [, ] and synchronizations [–], we still lack systematic theoretical and numerical simulations that address the effect of time-delays on the dynamics of social contagions, and especially for the temporal transition of the contagion process. One theoretical challenge is dealing with the non-Markovian characteristic of the contagion process and its two factors: memory and time-delays. On the one hand, the social reinforcement induces the memory effect, since for a susceptible to adopt they must remember their accumulated information. On the other, time-delays also cause a non-Markovian effect. Retrieved 12 April 2015. • Rina Chandran (22 October 2002).. Namma kudumbam serial wiki. Business Line. ![]() To study the effects of time-delays we use a non-Markovian spreading threshold model in which there are time-delays in the adoption process. The probability that an individual has a time-delay correlates with the number of its contacts (its degree). The time-delay quantifies the exposure time needed after the individual receives enough information to adopt the behavior. We develop a generalized edge-based compartmental theory to study the contagion process. Using extensive numerical simulations and theoretical analyses, we find that a relatively long time-delay causes a microtransition in the temporal evolution of the fraction of individuals in the recovered state in which individuals no longer share the behavioral information with neighbors. This magnitude exhibits multiple phase transitions in the social contagion dynamics, but does not affect the final adoption size. The transition is sharper when high degree nodes (hubs) have a higher probability of experiencing time-delays. |
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