Who’s Talking First? Consensus or Lack Thereof in Coevolving Opinion Formation Models

We investigate different opinion formation models on adaptive network topologies. Depending on the dynamical process, rewiring can either (i) lead to the elimination of interactions between agents in different states, and accelerate the convergence to a consensus state or break the network in noninteracting groups or (ii), counterintuitively, favor the existence of diverse interacting groups for exponentially long times. The mean-field analysis allows us to elucidate the mechanisms at play. Strikingly, allowing the interacting agents to bear more than one opinion at the same time drastically changes the model’s behavior and leads to fast consensus.

Figure 1

Convergence time for the $r$ VM vs $N$, for various rewiring probabilities. In all figures, the simulations are averaged over 100 realizations. Inset: $d$ VM.

Figure 2

Top: $k_{+}/⟨k⟩$ vs $m$ for the VM. Symbols: averages obtained from numerical simulations with $N=1000$, $⟨k⟩=10$. Continuous lines: numerical solution of the MF equations for the evolution of $\mathbf{x}=(m,l_{+-},l_{++})$, starting from initial conditions with $m$ close to 0. Bottom: $l_{++}$ and $l_{+-}$ vs $m$. The continuous black lines correspond to the numerical solution of the MF equations. The black symbols and the gray and brown lines correspond to single runs of the $d$ and $r$ VM, respectively, ($N=500$, $⟨k⟩=10$, $\Phi =0.4$). The inset shows the evolution of $m$ for the same runs (symbols for $d$ VM and dashed line for $r$ VM).

Figure 3

Convergence time for the direct (filled symbols) and reverse (open symbols) NG.

Figure 4

$⟨dm/dt⟩$ vs $m$ (symbols) for the $r$ NG. According to Eq. (3), changes in the magnetization come from $r$ VM-like interactions (dashed line) and those mediated by the 0-links, $l_{+0}$ and $l_{-0}$ (dash-dotted line). The upper inset gives $l_{+0}$ and $l_{-0}$, the other $k_{+}/⟨k⟩$, $k_{-}/⟨k⟩$. $\Phi =0.2$, $⟨k⟩=10$, $N={10}^4$.