Nonequilibrium phase transition in negotiation dynamics

We introduce a model of negotiation dynamics whose aim is that of mimicking the mechanisms leading to opinion and convention formation in a population of individuals. The negotiation process, as opposed to “herdinglike” or “bounded confidence” driven processes, is based on a microscopic dynamics where memory and feedback play a central role. Our model displays a nonequilibrium phase transition from an absorbing state in which all agents reach a consensus to an active stationary state characterized either by polarization or fragmentation in clusters of agents with different opinions. We show the existence of at least two different universality classes, one for the case with two possible opinions and one for the case with an unlimited number of opinions. The phase transition is studied analytically and numerically for various topologies of the agents’ interaction network. In both cases the universality classes do not seem to depend on the specific interaction topology, the only relevant feature being the total number of different opinions ever present in the system.

Figure 1

(Color online) Convergence time $t_{\mathrm{conv}}$ of the model as a function of $\beta -{\beta }_c$ (with ${\beta }_c=1∕3$) in the case of a fully connected population of $N$ agents. We show data for the original model (circles) with unlimited number of opinions per agent, and for models with a finite number $m$ of different opinions. Increasing $m$, the power-law fits give exponents that differ considerably from the value $-1$.

Figure 2

(Color online) Time $t_m$ required to a population on a fully-connected graph to reach a (fragmented) active stationary state with $m$ different opinions. For every $m>2$, the time $t_m$ diverges at some critical value ${\beta }_c\left(m\right)<{\beta }_c$.

Figure 3

(Color online) Convergence time $t_{\mathrm{conv}}$ of the model as a function of $\beta -{\beta }_c$ on networks with different topological properties: the UCM networks with degree distributions $P\left(k\right)\sim k^{-\gamma }$, $\gamma =2.5$, and $\gamma =3$, and the ER homogeneous random graphs. Simulations are shown for networks of $N={10}^5$ nodes and average degree $⟨k⟩=10$, both for $m=2$ (“2w,” open symbols) and the original model with unlimited memory (filled symbols). The numerical integration of Eqs. (2) is in good agreement with the simulations.

Figure 4

(Color online) Behavior of the critical value ${\beta }_c$ as a function of the exponent $\gamma$ of the degree distribution $P\left(k\right)\sim k^{-\gamma }$, as obtained from the numerical solution of the evolution equations for $n_i^k$, for both direct (black circles) and reverse (red squares) strategies. Bottom: comparison between the values of ${\beta }_c\left(\gamma \right)$ obtained from the equations and from numerical simulations on UCM networks of $N={10}^3$ agents, for direct (open symbols) and reverse (full symbols) strategies.