Maximal Biclique: A complete bipartite subgraph, that isn't a subgraph of another complete bipartite subgraph.

Given a bipartite graph $G=(V_{1}\cup V_{2}, E)$ where $|V_{1}|=|V_{2}|$ with probability $p$ of there being an edge from any $a\in V_{1}$ to any $b\in V_{2}$, what is the expected number of maximal bicliques?

What I have worked out is the upper and lower bounds:

Lower Bound: 1 or 2. If $|E|$ is divisible by $n$, then $\frac{|E|}{n}$ nodes can be connected to completely to $\frac{|E|}{n}$ other nodes, making one maximal biclique. Otherwise, connect $\frac{|E|}{n}$ nodes completely to $\frac{|E|}{n}$ nodes and connect one node to $|E|(mod\ n)$ nodes.

Upper Bound: There are $2^n$ unique subsets, the empty set and the entire set not included, leaves $2^n-2$ subsets. Therefore, there can be at most $2^n-2$ maximal bicliques. This upper bound is achievable when there are $n^2-n$ edges (I can prove this if anyone wants me to).

Both of these results are also easily extended to bipartite graphs where $|V_{1}|\neq |V_{2}|$.

The upper and lower bounds are both fairly trivial for the most part and it's the expected number of maximal bicliques that I've had the most trouble with. I've done a little work analyzing simple cases and brute forcing the expected number for small values of $n$ (I suppose I could write a program to brute force larger values of $n$), but it hasn't amounted to anything worth saying.

I'd appreciate any suggestions for methods of attacking the problem or references that I might find useful.