English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The distribution of shortest path lengths (DSPL) of random networks providesuseful information on their large scale structure. In the special case ofrandom regular graphs (RRGs), which consist of $N$ nodes of degree $c \ge 3$,the DSPL, denoted by $P(L=\ell)$, follows a discrete Gompertz distribution.Using the discrete Laplace transform we derive a closed-form expression for themoment generating function of the DSPL of RRGs. From the moment generatingfunction we obtain closed-form expressions for the mean and variance of theDSPL. More specifically, we find that the mean distance between pairs ofdistinct nodes is given by $\langle L \rangle = \frac{\ln N}{\ln (c-1)} +\frac{1}{2} - \frac{ \ln c - \ln (c-2) +\gamma}{\ln (c-1)} + \mathcal{O} \left(\frac{\ln N}{N} \right)$, where $\gamma$ is the Euler-Mascheroni constant.While the leading term is known, this result includes a novel correction term,which yields very good agreement with the results obtained from directnumerical evaluation of $\langle L \rangle$ via the tail-sum formula and withthe results obtained from computer simulations. However, it does not accountfor an oscillatory behavior of $\langle L \rangle$ as a function of $c$ or $N$.These oscillations are negligible in sparse networks but detectable in densenetworks. We also derive an expression for the variance ${\rm Var}(L)$ of theDSPL, which captures the overall dependence of the variance on $c$ but does notaccount for the oscillations. The oscillations are due to the discrete natureof the shell structure around a random node. They reflect the profile of thefilling of new shells as $N$ is increased. The results for the mean andvariance are compared to the corresponding results obtained in other types ofrandom networks. The relation between the mean distance and the diameter isdiscussed.

The mean and variance of the distribution of shortest path lengths of random regular graphs