TY - JOUR

T1 - The mean and variance of the distribution of shortest path lengths of random regular graphs

AU - Tishby, Ido

AU - Biham, Ofer

AU - Kühn, Reimer

AU - Katzav, Eytan

N1 - Funding Information: The authors wish to thank one of the anonymous referees for insightful comments and analysis that helped to clarify the nature of the oscillations of the mean and variance of the DSPL. This work was supported by the Israel Science Foundation Grant No. 1682/18. Publisher Copyright: © 2022 The Author(s).

PY - 2022/7/1

Y1 - 2022/7/1

N2 - The distribution of shortest path lengths (DSPL) of random networks provides useful information on their large scale structure. In the special case of random regular graphs (RRGs), which consist of N nodes of degree c 3/4 3, the DSPL, denoted by P(L = "), follows a discrete Gompertz distribution. Using the discrete Laplace transform we derive a closed-form (CF) expression for the moment generating function of the DSPL of RRGs. From the moment generating function we obtain CF expressions for the mean and variance of the DSPL. More specifically, we find that the mean distance between pairs of distinct nodes is given by lnNln(c-1)+12-lnc-ln(c-2)+3ln(c-1)+OlnNN, where 3 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 direct numerical evaluation of via the tail-sum formula and with the results obtained from computer simulations. However, it does not account for an oscillatory behavior of as a function of c or N. These oscillations are negligible in sparse networks but detectable in dense networks. We also derive an expression for the variance Var(L) of the DSPL, which captures the overall dependence of the variance on c but does not account for the oscillations. The oscillations are due to the discrete nature of the shell structure around a random node. They reflect the profile of the filling of new shells as N is increased. The results for the mean and variance are compared to the corresponding results obtained in other types of random networks. The relation between the mean distance and the diameter is discussed.

AB - The distribution of shortest path lengths (DSPL) of random networks provides useful information on their large scale structure. In the special case of random regular graphs (RRGs), which consist of N nodes of degree c 3/4 3, the DSPL, denoted by P(L = "), follows a discrete Gompertz distribution. Using the discrete Laplace transform we derive a closed-form (CF) expression for the moment generating function of the DSPL of RRGs. From the moment generating function we obtain CF expressions for the mean and variance of the DSPL. More specifically, we find that the mean distance between pairs of distinct nodes is given by lnNln(c-1)+12-lnc-ln(c-2)+3ln(c-1)+OlnNN, where 3 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 direct numerical evaluation of via the tail-sum formula and with the results obtained from computer simulations. However, it does not account for an oscillatory behavior of as a function of c or N. These oscillations are negligible in sparse networks but detectable in dense networks. We also derive an expression for the variance Var(L) of the DSPL, which captures the overall dependence of the variance on c but does not account for the oscillations. The oscillations are due to the discrete nature of the shell structure around a random node. They reflect the profile of the filling of new shells as N is increased. The results for the mean and variance are compared to the corresponding results obtained in other types of random networks. The relation between the mean distance and the diameter is discussed.

KW - distribution of shortest path lengths

KW - mean

KW - moments

KW - random network

KW - random regular graph

KW - variance

UR - http://www.scopus.com/inward/record.url?scp=85132342795&partnerID=8YFLogxK

U2 - https://doi.org/10.1088/1751-8121/ac6f9a

DO - https://doi.org/10.1088/1751-8121/ac6f9a

M3 - Article

SN - 1751-8113

VL - 55

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

IS - 26

M1 - 265005

ER -