Collective behaviour induced by network volatility

Many systems exhibit patterns of interaction that are largely sparse and volatile at the same time. Sparsity is a common trait in networks where links are costly, or the nodes involved have some kind of limited capacity. Volatility refers to the fact that edges tend to have very low persistence (compared to the observation period of the network evolution): the patterns of interaction are therefore characterised by a decay time after which the network topology is largely decorrelated with the previous time-step. Here, we introduce a simple model for temporal networks compatible with an arbitrary time-aggregated network, whose volatility can be adjusted. When volatility is too large, the instantaneous network experiences a percolation transition, to a largely disconnected structure. Interestingly, we show a non-trivial relationship between network volatility and the properties of dynamical processes taking place in the nodes of the system. We show that a phase transition towards between non-trivial dynamical states (like synchronisation, or infection propagation) is not-related to the topological transition to percolation, having different critical points. Moreover, we show that long range correlations emerge in the limit of very large network volatility.