Consensus as cooling: A granular gas model for continuous opinions on structured networks

Abstract

and numerically analysed. An agent’s opinion is represented by a real number that can be changed through social interactions with her neighbours. The proposed dynamics depends on two fundamental parameters, 𝛼 ∈ [−1, 1] and 𝛽 ≥ 0. If an interaction takes place between two agents, their relative opinions decreases an amount given by 𝛼. The probability of two neighbours to interact is proportional to the 𝛽-power of their relative opinions. We unveil the behaviour of the system for all physical relevant values of the parameters and several representative interaction networks. When 𝛼 ∈ (−1, 1) and 𝛽 ≥ 0, the system always reaches consensus, with all agents having the mean initial opinion, provided the interaction network is connected. The approach to consensus can be characterized by means of the mean opinion and the temperature (or opinion dispersion) of each agent. Three scenarios have been identified. When the agents are well mixed, as with all-to-all interactions, a pre-consensus regime is seen, with all agents having zero mean opinion and the same temperature, following the Haff’s law of granular gases. A similar regime is observed with Erdös–Rényi and Barabási–Albert networks: mean opinions are zero but agents with different degrees have different temperatures, though still following the Haff’s law. Finally, the case of a square 2D lattice has been carefully analysed, by starting from the derivation of closed set of hydrodynamic-like equations using the Chapman–Enskog method. For 𝛼 larger than a critical value, that depends on the system size, the system keeps spatially homogeneous, with zero mean opinions and equal temperatures, as they approach consensus. Below this critical line, the system explores states with spatially non-homogeneous configurations that evolve in time. Numerically, it is found that the main role of 𝛽 is to change the local structure of the spacial opinion dispersion: while for 𝛽 small enough the system keeps locally isotropic, as 𝛽 increases, neighbouring agents with similar opinions tend to form local lineal structures.