Interacting Social Processes on Interconnected Networks

We propose and study a model for the interplay between two different dynamical processes -one for opinion formation and the other for decision making- on two interconnected networks A and B. The opinion dynamics on network A corresponds to that of the M-model, where the state of each agent can take one of four possible values (S = -2, -1, 1, 2), describing its level of agreement on a given issue. The likelihood to become an extremist (S = +/- 2) or a moderate (S = +/- 1) is controlled by a reinforcement parameter r >= 0. The decision making dynamics on network B is akin to that of the Abrams-Strogatz model, where agents can be either in favor (S = +1) or against (S = -1) the issue. The probability that an agent changes its state is proportional to the fraction of neighbors that hold the opposite state raised to a power beta. Starting from a polarized case scenario in which all agents of network A hold positive orientations while all agents of network B have a negative orientation, we explore the conditions under which one of the dynamics prevails over the other, imposing its initial orientation. We find that, for a given value of beta, the two-network system reaches a consensus in the positive state (initial state of network A) when the reinforcement overcomes a crossover value r*(beta), while a negative consensus happens for r < r*(beta). In the r - beta phase space, the system displays a transition at a critical threshold beta(c), from a coexistence of both orientations for beta < beta(c) to a dominance of one orientation for beta > beta(c). We develop an analytical mean-field approach that gives an insight into these regimes and shows that both dynamics are equivalent along the crossover line (r*, beta*).