We consider a class of adaptive network models where links can only be created or deleted between nodes in different states. These models provide an approximate description of a set of systems where nodes represent agents moving in physical or abstract space, the state of each node represents the agent's heading direction, and links indicate mutual awareness. We show analytically that the adaptive network description captures the phase transition to collective motion in swarming systems and that the properties of this transition are determined by the number of states (discrete heading directions) that can be accessed by each agent.
Figure 1: Phase diagrams of adaptive network systems with \(M = 3\) possible movement directions. The system transitions from unordered to ordered collective motion as the noise level \(w_0\) is decreased. At \(M\geq 3\) this transition occurs in a discontinuously, creating a zone of bistability and hysteresis.