## G(n,p)

We are familiar with the standard notion of random graphs where you fixed the number n of vertices and choose every edge to belong to the graph with probability 1/2 (or p) independently. This model is referred to as **G(n,1/2)** or more generally **G(n,p).**

## Random graphs with prescribed marginal behavior

Now, suppose that rather than prescribe the probability for every edge, you presecribe the marginal probability for every induced subgraph H on r vertices. (So r should be a small integer, 3,4,5, etc.) You make the additional assumptions that

a) the probability $p_H$ does not depend on the identity of the r vertices,

and

b) It depends only on the isomorphism type of $H$.

So, for example: for r=3 you can think about the case that $p_H = 1/9$ if H is a triangle or an empty graph and $p_H=7/54$ otherwise.

Once these $p_H$ are assigned you consider among all the probability distributions with these marginal behavior (if there are any) the one with maximal entropy. (But this choice **is negotiable**; if there is something different worth doing this is fine too.)

## My questions:

1) Are these models studied in the literature?

2) When are such $p_H$'s feasible?

3) Given such feasible $p_H$'s say on graphs with 4 vertices, is there a quick algorithm to sample from the maximal-Entropy distribution which will allow to experiment with this model?

### Background

This question is motivated by a recent talk by Nati Linial in our "basic notion" seminar on extremal graph theory. (Maybe these are well studied models that I simply forgot, but I don't recall it now.)