Distribution of the number of cycles in directed and undirected random regular graphs of degree 2

Ido Tishby, Ofer Biham, Eytan Katzav, Reimer Kühn

Research output: Contribution to journalArticlepeer-review

Abstract

We present analytical results for the distribution of the number of cycles in directed and undirected random 2-regular graphs (2-RRGs) consisting of N nodes. In directed 2-RRGs each node has one inbound link and one outbound link, while in undirected 2-RRGs each node has two undirected links. Since all the nodes are of degree k=2, the resulting networks consist of cycles. These cycles exhibit a broad spectrum of lengths, where the average length of the shortest cycle in a random network instance scales with lnN, while the length of the longest cycle scales with N. The number of cycles varies between different network instances in the ensemble, where the mean number of cycles (S) scales with lnN. Here we present exact analytical results for the distribution PN(S=s) of the number of cycles s in ensembles of directed and undirected 2-RRGs, expressed in terms of the Stirling numbers of the first kind. In both cases the distributions converge to a Poisson distribution in the large N limit. The moments and cumulants of PN(S=s) are also calculated. The statistical properties of directed 2-RRGs are equivalent to the combinatorics of cycles in random permutations of N objects. In this context our results recover and extend known results. In contrast, the statistical properties of cycles in undirected 2-RRGs have not been studied before.

Original languageAmerican English
Article number024308
JournalPhysical Review E
Volume107
Issue number2
DOIs
StatePublished - Feb 2023
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2023 American Physical Society.

Fingerprint

Dive into the research topics of 'Distribution of the number of cycles in directed and undirected random regular graphs of degree 2'. Together they form a unique fingerprint.

Cite this