Some empirical and theoretical attributes of random multi-hooking networks

Abstract

Multi-hooking networks are a broad class of random hooking networks introduced in [H. Mahmoud, Local and global degree profiles of randomly grown self-similar hooking networks under uniform and preferential attachment, Adv. Appl. Math. 111 (2019), p. 101930.] wherein multiple copies of a seed are hooked at each step, and the number of copies follows a predetermined building sequence of numbers. For motivation, we provide two examples: one from chemistry and one from electrical engineering. We explore the empirical and theoretical local degree distribution of a specific node during its temporal evolution. We ask what will happen to the degree of a specific node at step n that first appeared in the network at step j. We conducted an experimental study to identify some cases with Gaussian asymptotic distributions, which we then proved. Additionally, we investigate the distance in the network through the lens of the average Wiener index for which we obtain a theoretical result for any building sequence and explore its empirical distribution for certain classes of building sequences that have systematic growth.