BioND | Early fragmentation in the adaptive voter model on directed networks

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zschaler2012early

Early fragmentation in the adaptive voter model on directed networks

Gerd Zschaler, Gesa A. Böhme, Michael Seißinger, Cristián Huepe and Thilo Gross

Phys. Rev. E 85, 046107, 2012

We consider voter dynamics on a directed adaptive network with fixed out-degree distribution. A transition between an active phase and a fragmented phase is observed. This transition is similar to the undirected case if the networks are sufficiently dense and have a narrow out-degree distribution. However, if a significant number of nodes with low out degree is present, then fragmentation can occur even far below the estimated critical point due to the formation of self-stabilizing structures that nucleate fragmentation. This process may be relevant for fragmentation in current political opinion formation processes.

$Typical trajectories from network simulations. The state of the network is characterized by the density of active links $\rho$ and the magnetization $m$. The trajectories shown correspond to networks with Poissonian out-degree distributions with $\langle k \rangle = 4 $ (red/light gray), $ \langle k\rangle = 8 $ (black), and an out-degree distribution following $ P_{\rm out\}(k) \sim k^{-2} $ (blue/dark gray). The trajectories initially drift along a parabola of active states (dotted red, dashed black, and dash-dotted blue lines, denoting analytical results from Eq. (10)). However, only the black trajectory reaches a consensus state, whereas the others eventually collapse to a fragmented state. The inset shows a time series of ρ from the scale-free network shortlybefore fragmentation.$

Figure 1: Typical trajectories from network simulations. The state of the network is characterized by the density of active links $\rho$ and the magnetization $m$. The trajectories shown correspond to networks with Poissonian out-degree distributions with $\langle k \rangle = 4 $ (red/light gray), $ \langle k\rangle = 8 $ (black), and an out-degree distribution following $ P_{\rm out\}(k) \sim k^{-2} $ (blue/dark gray). The trajectories initially drift along a parabola of active states (dotted red, dashed black, and dash-dotted blue lines, denoting analytical results from Eq. (10)). However, only the black trajectory reaches a consensus state, whereas the others eventually collapse to a fragmented state. The inset shows a time series of ρ from the scale-free network shortlybefore fragmentation.