Network evolution with Macroscopic Delays: asymptotics and condensation

Abstract

Driven by the explosion of data and the impact of real-world networks, a wide array of mathematical models have been proposed to understand the structure and evolution of such systems, especially in the temporal context. Recent advances in areas such as distributed cyber-security and social networks have motivated the development of probabilistic models of evolution where individuals have only partial information on the state of the network when deciding on their actions. This paper aims to understand models incorporating \emph{network delay}, where new individuals have information on a time-delayed snapshot of the system. We consider the setting where one has macroscopic delays, that is, the ``information'' available to new individuals is the structure of the network at a past time, which scales proportionally with the current time and vertices connect using standard preferential attachment type dynamics. We obtain the local weak limit for the network as its size grows and connect it to a novel continuous-time branching process where the associated point process of reproductions \emph{has memory} of the entire past. A more tractable `dual description' of this branching process using an `edge copying mechanism' is used to obtain degree distribution asymptotics as well as necessary and sufficient conditions for condensation, where the mass of the degree distribution ``escapes to infinity''. We conclude by studying the impact of the delay distribution on macroscopic functionals such as the root degree.