### Absorbing phase transitions in coevolving networks

Author: Federico Vazquez, IFISC (CSIC-UIB).

Names and affiliation of other authors:
Federico Vazquez, Victor M. Eguiluz and Maxi San Miguel
IFISC (UIB)

Oral presentation

Abstract:
I will talk about a particular type of absorbing transitions that has recently been observed in models of adaptive networks. In these models, a node can change its state by interacting with its neighbors, and at the same time, links can be rewired depending on the state of the nodes at their ends. In this way, the dynamics of nodes and links are not independent, but they coevolve. It is found that when the rewiring is fast enough compare to the rate at which nodes update their states, the network breaks into disconnected components, each composed by nodes holding the same state. In order to
understand the mechanism of this fragmentation transition I will introduce a simple model, that possesses all the ingredients of related models, and has the advantage of being analytically tractable. A mean-field approximation
reveals an absorbing transition from an active to a frozen phase at a critical value of the rewiring probability $p_c=\frac{\mu-2}{\mu-1}$ that only depends on the average degree $\mu$ of the network. In finite-size systems, the active and frozen phases
correspond to a connected and a fragmented network respectively. The transition can be seen as the sudden change in the trajectory of an equivalent random walk at the critical point, resulting in an approach to the final frozen state whose time scale diverges as $\tau \sim |p_c-p|^{-1}$ near $p_c$.

Dynamics and evolution of biological and social networks. February 18-20th, 2008. Mallorca, Spain.