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

TL;DR

The paper studies a continuous-opinion model with compromise parameter $α$ and interaction-frequency parameter $β$ on structured networks, deriving a kinetic description and a Chapman–Enskog hydrodynamic framework to capture consensus formation. It shows that, on connected topologies, the system generically reaches consensus for $|α|<1$, with the approach governed by a Haff-type cooling of the opinion temperature and a scaling form $p_{ij}(s_1,s_2,t)=[2T(t)]^{-1}\phi_{ij}(c_1,c_2)$. Depending on topology, the system exhibits three regimes: (i) homogeneous scaling on all-to-all and certain lattices, (ii) weakly non-homogeneous behavior on ER/BA with degree-dependent temperatures, and (iii) spatially non-homogeneous states on regular lattices with phase-diagram–level transitions and pattern formation. The work links continuous opinion dynamics to granular-gas phenomenology, providing a unified framework to analyze consensus in networks and offering insights for complex social systems with sparse or heterogeneous interactions.

Abstract

A continuous-opinion model accounting for the social compromise propensity is theoretically 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, $α\in[-1,1]$ and $β\ge 0$. 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 $α\in(-1,1)$ and $β\ge 0$, 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 analyzed, by starting from the derivation of closed set of hydrodynamic-like equations using the Chapman-Enskog method ...

Figures (7)

• Figure 1: Simulations results of an all-to-all network with $N=100$ nodes. Top-left plot: the global temperature $T$ as a function of the time $t$. For $t/t_0=10^2$, from top to bottom, the lines correspond to $(\alpha,\beta)=(9/10,10),(7/10,10),(3/10,10),(9/10,1),(7/10,2),(3/10,1)$. The segments related to the decays $t^{-1/5}$ and $t^{-2}$ are the theoretical prediction of Eq. \ref{['eq:ttbet>0']} for $\beta=10$ and $\beta=1$, respectively. Top-right plot: the global temperature $T$ as a function of the time $\tau$ defined by Eq. \ref{['eq:tiempotau']}. From top to bottom: $(\alpha,\beta)=(9/10,1),(7/10,1),(3/10,1)$. Bottom-left plot: the scaled distribution $\phi$ for $(\alpha,\beta)=(9/10,1),(7/10,1),(3/10,1)$. Bottom-right plot: the scaled distribution $\phi$ for $(\alpha,\beta)=(9/10,10),(7/10,10),(3/10,10)$.
• Figure 2: Simulations results for an Erdös-Rényi network with $N=500$ agents and mean degree $4$. The top plots are analogous to Fig. \ref{['fig:1']}, with additional dashed lines that correspond to $-\alpha$. The bottom-left plot is as the corresponding one in Fig. \ref{['fig:1']}. The bottom-right plot shows the scaled distribution function $\phi$ for the sub-group of agents with degree $2$ (bimodal distribution), degree $4$, and $6$ (unimodal ones).
• Figure 3: Simulations results for a Barabási-Albert network with $N=500$ agents and mean degree $4$. The plots show the same data as in Fig. \ref{['fig:2']}.
• Figure 4: Sketch of the phase diagram with the five configurations described in the main text; see also next Figures. The vertical line at $\alpha_c$ is given by the critical condition \ref{['eq:crital']} of the linear stability analysis of Sec. \ref{['sec:lattice']}. The thin line, meeting the previous one, is given by the other condition \ref{['eq:critbe']}. The dashed lines indicate smooth/continuous transitions from one configurations to the others.
• Figure 5: Simulation results of a system of $N=900$ agents on a square 2D lattice with $\alpha=0.98$. The left column shows the time evolution of the kurtosis $e$, defined by Eq. \ref{['eq:kurt']}, and the right column the final state of the system. Bottom: $\beta=0$, in H$_\text{O}$. Top: $\beta=5$, in H$_\text{L}$.