Determining Exact Solutions for Structural Parameters on Hierarchical Networks With Density Feature

Abstract

The problem of determining closed-form solutions for some structural parameters of great interest on networked models is meaningful and intriguing. In this paper, we propose a family of networked models |$\mathcal{G}_{n}(t)$| with hierarchical structure where |$t$| represents time step and |$n$| is copy number. And then, we study some structural parameters on the proposed models |$\mathcal{G}_{n}(t)$| in more detail. The results show that (i) models |$\mathcal{G}_{n}(t)$| follow power-law distribution with exponent |$2$| and thus exhibit density feature; (ii) models |$\mathcal{G}_{n}(t)$| have both higher clustering coefficients and an ultra-small diameter and so display small-world property; and (iii) models |$\mathcal{G}_{n}(t)$| possess rich mixing structure because Pearson-correlated coefficients undergo phase transitions unseen in previously published networked models. In addition, we also consider trapping problem on networked models |$\mathcal{G}_{n}(t)$| and then precisely derive a solution for average trapping time |$ATT$|⁠. More importantly, the analytic value for |$ATT$| can be approximately equal to the theoretical lower bound in the large graph size limit, implying that models |$\mathcal{G}_{n}(t)$| are capable of having most optimal trapping efficiency. As a result, we also derive exact solution for another significant parameter, Kemeny’s constant. Furthermore, we conduct extensive simulations that are in perfect agreement with all the theoretical deductions.