The dynamics of polarized beliefs in networks governed by viral diffusion and media influence

This paper studies the evolution of polarized beliefs governed by the intertwined dynamics of viral diffusion and media influence in influence networks. It addresses the question of how different forms of influence interact with each other. First, we propose a Markov chain to model the dynamics of individuals as they transition between three belief states (neutral, positive and negative) based on the states of their neighbors. This stochastic system assumes that individuals are influenced via the links of the network or through the global effect of mass media. For exponential and scale-free networks, we approximate this stochastic system by deterministic differential equations and define the homogeneous mean-field system and heterogeneous mean-field system, respectively. Studying stability conditions for these deterministic dynamical systems, we analyze the fraction of neutral, positive and negative individuals in the population. Also, we determine the conditions under which desired dynamical transitions happen for the targeted population. These conditions allow us to predict macroscopic measures of dynamics of adoption in influence networks. Finally, the derived analytical results are validated using simulations of four synthetic networks: Watts–Strogatz, random regular, Barabasi–Albert and small-world forest-fire, as well as five real-world networks: ego-Facebook, Deezer, Livemocha, a Facebook interaction network and Douban. Also, we demonstrate how the proposed model can be leveraged by marketing campaigns for optimal resource allocations between viral marketing and media marketing to minimize the number of final negative individuals in different network settings.