Temporal evolution of the degree distribution of alters in growing networks

The degree distribution of the neighbors of nodes in a network is a theoretically important tool that is invoked in diverse studies in network science, such as epidemics, network resilience, network search and observability, network synchronization, random walks, opinion dynamics, and other dynamical systems on networks. Many real networks grow, and their properties pertaining to the said phenomena evolve. There is a paucity of theoretical research on how the evolution of these properties depend upon time and upon the structure of the initial network. This paper addresses this problem by providing the first theoretical study of the temporal evolution of the nearest-neighbor degree distribution for arbitrary networks (with any size) in arbitrary times. The posited results enable the analysis of the structural properties of growing networks in the short-time and intermediary time regimes, which are typically ignored in favor of the steady state. We corroborate the solutions via Monte Carlo simulations on various topologies. As a byproduct of the obtained solutions, we also demonstrate that the existing result in the literature on the asymptotic behavior of the Pearson coefficient of growing networks under the preferential attachment mechanism is incorrect, and we present the correct solution.