Let $G \sim G(n,p)$ be a directed Erdős–Rényi random graph
with $n$ vertices and the probability $p$ that
there is a directed edge between any two ordered pairs of vertices.
What is the expected number of cycles in $G$?
Is there an exact formula or an upper bound and lower bound
on the expected number of simple cycles in $G$?
Best Answer
Let $X$ be the total number of cycles, $X_k$ the number of cycles of length $k$ in $G$, and $Y_k$ the number of cycles of length $k$ in a random graph of size $k$.
Note that:
The last one is because a cycle of length $k$ is a permutation of numbers from $1$ to $k$ but the first vertex does not matter (a cycle does not have a first vertex). The probability of each edge being present is $p$, so the probability of $k$ edges being present is $p^k$.
We have:
\begin{align*} {\sf E}[X] &= \sum_{k=2}^n E[X_k] \\ &= \sum_{k=2}^n {n \choose k} E[Y_k] \\ &= \sum_{k=2}^n {n \choose k} (k-1)! p^k \\ &= \sum_{k=2}^n \frac{n!}{k!(n-k)!} (k-1)! p^k \\ &= \sum_{k=2}^n \frac{n!}{(n-k)!k} p^k \\ &= \sum_{k=2}^n \frac{n\ldots (n-k+1)}{k} p^k \\ \end{align*}
For a simple lower bound we can drop the first half of the sum: \begin{align*} &\geq \sum_{k=n/2}^{n} \frac{n\ldots (n-k+1)}{k} p^k \\ &\geq \sum_{k=n/2}^{n} \frac{(n/2)^{k}}{n} p^k \\ &\geq \frac{1}{n}\sum_{k=n/2}^{n} (np/2)^{k} \\ &= \frac{(\frac{np}{2})^{n+1}-(\frac{np}{2})^{\frac{n}{2}}}{\frac{np}{2}-1} \end{align*}