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.
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)