An interaction model on random networks: from social to airports   networks
  Social movements, neurons in the brain or even industrial suppliers are best described by agents evolving on networks with basic interaction rules. In these real systems, the connectivity between agents corresponds to the a critical state of the system related to the noise of the system. The new idea is that connectivity adjusts itself because of two opposite tendencies: on the one hand informations percolation is better when the network connectivity is small but all agents have rapidely the same state and the dynamics stops.   On the other hand, when agents have a large connectivity, the state of a node (opinion of a person, state of a neuron, ...) tends to freeze: agents find always a minority among their neighbours to support their state. The model introduced here captures this essential feature showing a clear transition between the two tendencies at some critical connectivity. Depending on the noise, the dynamics of the system can only take place at a precise critical connectivity since, away from this critical point, the system remains in a static phase. When the noise is very small, the critical connectivity becomes very large, and highly connected networks are obtained like the airports network and the Internet. This model may be used as a starting point for understanding the evolution of agents living on networks. 
Philippe Curty