Online social networks such as Twitter and Facebook have gained tremendouspopularity for information exchange. The availability of unprecedented amountsof digital data has accelerated research on information diffusion in onlinesocial networks. However, the mechanism of information spreading in onlinesocial networks remains elusive due to the complexity of social interactionsand rapid change of online social networks. Much of prior work on informationdiffusion over online social networks has based on empirical and statisticalapproaches. The majority of dynamical models arising from information diffusionover online social networks involve ordinary differential equations which onlydepend on time. In a number of recent papers, the authors propose to usepartial differential equations(PDEs) to characterize temporal and spatialpatterns of information diffusion over online social networks. Built onintuitive cyber-distances such as friendship hops in online social networks,the reaction-diffusion equations take into account influences from variousexternal out-of-network sources, such as the mainstream media, and provide anew analytic framework to study the interplay of structural and topicalinfluences on information diffusion over online social networks. In thissurvey, we discuss a number of PDE-based models that are validated with realdatasets collected from popular online social networks such as Digg andTwitter. Some new developments including the conservation law of informationflow in online social networks and information propagation speeds based ontraveling wave solutions are presented to solidify the foundation of the PDEmodels and highlight the new opportunities and challenges for mathematicians aswell as computer scientists and researchers in online social networks.
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