  We describe a general purpose algorithm for counting simple cycles and simple
paths of any length $\ell$ on a (weighted di)graph on $N$ vertices and $M$
edges, achieving a time complexity of
$O\left(N+M+\big(\ell^\omega+\ell\Delta\big) |S_\ell|\right)$. In this
expression, $|S_\ell|$ is the number of (weakly) connected induced subgraphs of
$G$ on at most $\ell$ vertices, $\Delta$ is the maximum degree of any vertex
and $\omega$ is the exponent of matrix multiplication. We compare the algorithm
complexity both theoretically and experimentally with most of the existing
algorithms for the same task. These comparisons show that the algorithm
described here is the best general purpose algorithm for the class of graphs
where $(\ell^{\omega-1}\Delta^{-1}+1) |S_\ell|\leq |\text{Cycle}_\ell|$, with
$|\text{Cycle}_\ell|$ the total number of simple cycles of length at most
$\ell$, including backtracks and self-loops. On Erd\H{o}s-R\'enyi random
graphs, we find empirically that this happens when the edge probability is
larger than circa $4/N$. In addition, we show that some real-world networks
also belong to this class. Finally, the algorithm permits the enumeration of
simple cycles and simple paths on networks where vertices are labeled from an
alphabet on $n$ letters with a time complexity of
$O\left(N+M+\big(n^\ell\ell^\omega+\ell\Delta\big) |S_\ell|\right)$. A Matlab
implementation of the algorithm proposed here is available for download.
