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

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 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.