Some empirical and theoretical attributes of random multi-hooking networks

AbstractMulti-hooking networks are a broad class of random hooking networks introduced in [8] 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: Networksrandom graphsdegree profileWiener indexrecurrenceasymptotic analysisAMS subject classifications:: Primary: 05C82Secondary: 05C1260C0560F0590B15DisclaimerAs a service to authors and researchers we are providing this version of an accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proofs will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to these versions also. AcknowledgmentsThe authors would like to thank Dominic Abela for developing and designing the initial prototype of an interactive application.Data availability statementThe data that support the findings of this study are available from the corresponding author upon reasonable request.Notes1 MapleTM is a trademark of Waterloo Maple Inc.2 Note that this theorem does not require the regularity conditions needed in theorem 1; the building sequence is entirely arbitrary.