Dynamic networks and directed percolation

Abstract

We introduce a model for dynamic networks, where the links or the strengths\nof the links change over time. We solve the model by mapping dynamic networks\nto the problem of directed percolation, where the direction corresponds to the\nevolution of the network in time. We show that the dynamic network undergoes a\npercolation phase transition at a critical concentration $p_c$, which decreases\nwith the rate $r$ at which the network links are changed. The behavior near\ncriticality is universal and independent of $r$. We find fundamental network\nlaws are changed. (i) For Erd\\H{o}s-R\\'{e}nyi networks we find that the size of\nthe giant component at criticality scales with the network size $N$ for all\nvalues of $r$, rather than as $N^{2/3}$. (ii) In the presence of a broad\ndistribution of disorder, the optimal path length between two nodes in a\ndynamic network scales as $N^{1/2}$, compared to $N^{1/3}$ in a static network.\n