Predicting the emergence of spatial self-organization using dynamic networks*

Predicting and classifying the spontaneous emergence of order is of great importance to understanding biological systems. A famous example is the behavior of a flock of birds which can spontaneously change their direction of flight in unison. We are interested in the generic conditions under which order emerges in a large collection of such active interacting agents that can influence each other’s behavior over at long distances.
We recognize that in order to extract information on existence of interactions between agents, looking at complex spatial dynamics is not strictly necessary. Instead, we use the framework of dynamic networks to directly characterize the state of agents and their mutual influence. This approach allows us to concentrate on the dynamics of the order parameters instead of the dynamics of individuals. We can solve the resulting model analytically in a mean-field approximation, which gives us a second-order phase transition. Solving the full nonlinear model, we retain the critical temperature from the phase transition, and find a full phase diagram in terms of the key parameters, the orientational noise and the network rearrangement time.