Title: Network Evolution under Delays.
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, in which individuals have only partial information on the state of the network when deciding on their actions. We aim to understand models incorporating network delay, where new individuals have information on a time-delayed snapshot of the system, in the following two settings:
Macroscopic delays: the ``information'' available to new individuals is the network structure at a past time, which scales proportionally with the current time, and vertices connect using standard preferential attachment type dynamics. Here, 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 has the 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''.
Mesoscopic delays: delays that are sublinear compared to the current time. Here, we develop probabilistic tools built on stochastic approximation to understand the asymptotics of local functionals, such as local neighborhoods and degree distributions.
We conclude by examining the impact of the delay distribution on the evolution of the degree of the network's initial founder. This work was done in collaboration with Shankar Bhamidi, Sayan Banerjee, and Akshay Sakanaveeti.