Unstable network fragmentation in co-evolution of Potts spins and system topology

• A fragmented network is unstable in quasi-ergodic coevolution.
• The system slowly evolves towards a single fully ordered component.
• Dynamics of clusters “surfaces” is responsible for the transition shape.

We investigate co-evolution of discrete qq-state Potts model and the underlying network topology, where spin changes and link re-wiring follow the same canonical ensemble dynamics in a constant temperature. It means that there are no absorbing, frozen states present in our model. Depending on the temperature TT and probability of link dynamics pp the system can exist in one of three states: ordered, disordered and ordered clusters (fragmented network), with the last being unstable and slowly relaxing into ordered state. The transition from ordered clusters to globally ordered system is characterized by non-exponential, slow growth of the order parameter. We investigate this process analytically and explain the transition characteristics as the result of the dominance of activity of “surface” nodes in each ordered cluster, as opposed to “bulk” nodes that are inactive.