Number of autoregulation and FFL motifs in a network

Question

Suppose we have a network with N* nodes (N* is the number of internal nodes). Every directed link in this network exists with probability p. What would be the number of:

1. motifs that are auto-regulatory?
2. feed-forward loop motifs
3. feed-back loop motifs

Answer 1

1. 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, $<N_{self}>_{rand} \approx E/N$
2. & 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: $<N_G>\approx a^{-1}N^n p^g$.

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)