Memory and burstiness in dynamic networks

A discrete-time random process is described which can generate bursty sequences of events. A Bernoulli process, where the probability of an event occurring at time $t$ is given by a fixed probability $x$, is modified to include a memory effect where the event probability is increased proportionally to the number of events which occurred within a given amount of time preceding $t$. For small values of $x$ the inter-event time distribution follows a power-law with exponent $-2-x$. We consider a dynamic network where each node forms, and breaks connections according to this process. The value of $x$ for each node depends on the fitness distribution, $\rho(x)$, from which it is drawn; we find exact solutions for the expectation of the degree distribution for a variety of possible fitness distributions, and for both cases where the memory effect either is, or is not present. This work can potentially lead to methods to uncover hidden fitness distributions from fast changing, temporal network data such as online social communications and fMRI scans.