Competing dynamical processes on two interacting networks

We propose and study a model for the competition between two different dynamical processes, one for opinion formation and the other for decision making, on two interconnected networks. The networks represent two interacting social groups, the society and the Congress. An opinion formation process takes place on the society, where the opinion S of each individual can take one of four possible values (S=-2,-1,1,2), describing its level of agreement on a given issue, from totally against (S=-2) to totally in favor (S=2). The dynamics is controlled by a reinforcement parameter r, which measures the ratio between the likelihood to become an extremist or a moderate. The dynamics of the Congress is akin to that of the Abrams-Strogatz model, where congressmen can adopt one of two possible positions, to be either in favor (+) or against (-) the issue. The probability that a congressman changes his decision is proportional to the fraction of interacting neighbors that hold the opposite opinion raised to a power $\beta$. Starting from a polarized case scenario in which most people in the society are in favor (positive), but most congressmen are against (negative), we explore the conditions under which the society is able to influence and reverse the decision of the Congress. We find that, for a given $\beta=\beta*$, the entire system reaches a consensus in the positive state (the initial society's will) when the reinforcement overcomes a crossover value $r^*$ ($r>r^*$), while a negative consensus happens for $r \beta_c$. We develop an analytical mean-field approach that gives an insight of these regimes and shows that both dynamics are equivalent at the crossover line $(r^*,\beta^*)$.