Dynamic networks and directed percolation

We introduce a model for dynamic networks, where the links or the strengths of the links change over time. We solve the model by mapping dynamic networks to the problem of directed percolation, where the direction corresponds to the time evolution of the network. We show that the dynamic network undergoes a percolation phase transition at a critical concentration pc, that decreases with the rate r at which the network links are changed. The behavior near criticality is universal and independent of r. We find that for dynamic random networks fundamental laws are changed. i) The size of the giant component at criticality scales with the network size N for all values of r, rather than as N2/3 in static networks. ii) In the presence of a broad distribution of disorder, the optimal path length between two nodes in a dynamic network scales as N1/2, compared to N1/3 in a static network.

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