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- W4315705819 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 $ln N$, 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 $langle S rangle$ scales with $ln N$. Here we present exact analytical results for the distribution $P_N(S=s)$ 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 $P_N(S=s)$ 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." @default.
- W4315705819 created "2023-01-12" @default.
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- W4315705819 date "2023-01-09" @default.
- W4315705819 modified "2023-09-30" @default.
- W4315705819 title "The distribution of the number of cycles in directed and undirected random 2-regular graphs" @default.
- W4315705819 doi "https://doi.org/10.48550/arxiv.2301.03686" @default.
- W4315705819 hasPublicationYear "2023" @default.
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