Number of autoregulation and FFL motifs in a network
The average number of auto-regulatory motifs (self-edges), is equal to the number of edges E times the probability that an edge is a self-edge which is $p_{self}=1/N$, with N being the total number of nodes. Therefore, $&lt;N_{self}&gt;_{rand} \approx E/N$
&amp; 3. According to U.Alon's book ("Introduction to Systems Biology"), the mean number of times that a subgraph G occurs in a random network is given by the following formula: $&lt;N_G&gt;\approx a^{-1}N^n p^g$.


Explaining what each term of the formula means:


α: is a number that includes combinatorial factors related to the structure and symmetry of each subgraph, equal to 1 for feed-forward loop (FFL) and equal to 3 for feed-back loop (FBL)
$N^n$: is the number of ways of choosing a set of n nodes out of N. Because there are N ways of choosing the first one, times N-1$\approx$N ways of choosing the second one, and so on..(this approximation is true for large networks)
$p^g$ : is the probability to get the g edges in the appropriate places (each with probability p)