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.

Keywords:

• Networks
• random graphs
• degree profile
• Wiener index
• asymptotic analysis

AMS Subject Classifications:

• Primary: 05C82
• Secondary: 05C12
• 60C05
• 60F05
• 90B15

Acknowledgments

The authors would like to thank Dominic Abela for developing and designing the initial prototype of an interactive application. The team from Catholic University would also like to thank the School of Arts and Sciences and the School of Engineering for their research support.

Data availability statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Maple™ is a trademark of Waterloo Maple Inc.

2 Note that this theorem does not require the regularity conditions needed in Theorem 3.1; the building sequence is entirely arbitrary.