Persistence in Random and Disordered Networks.

To better understand the lifetime and temporal dynamics of activities and trends in social networks, we initiated investigations of diffusive persistence in various graphs. Persistence is defined as the probability that the diffusive field at a given node has not changed sign up to a certain time (or in general, that node remained inactive/active). We investigated disordered networks (characterized by the fraction of removed edges) and found that the behavior of the persistence probability depended on the topology of the network. In 2D networks we have found that above the percolation threshold diffusive persistence scale scales similarly to that of the original two-dimensional regular lattice, i.e., a power law with an exponent of 0.18. At the percolation threshold, the scaling changes to one with 0.12. This new exponent is the result of the interplay of diffusive persistence and the underlying structural transition in the disordered lattice at the percolation threshold. In contrast, we found that in random networks without a regular structure, such as Erdös-Rényi networks, no simple power-law scaling behavior exists above the percolation threshold.