A simple generalization of the Erdos-Renyi family, the U|MAN distributions are uniform on the set of graphs, conditional on order (size) and the dyad census. As with the E-R case, there are two U|MAN variants. The first (corresponding to `method=="probability"`

) takes dyad states as independent multinomials with parameters \(m\) (for mutuals), \(a\) (for asymmetrics), and \(n\) (for nulls). The resulting pmf is then
$$
p(G=g|m,a,n) = \frac{(M+A+N)!}{M!A!N!} m^M a^A n^N,
$$
where \(M\), \(A\), and \(N\) are realized counts of mutual, asymmetric, and null dyads, respectively. (See `dyad.census`

for an explication of dyad types.)

The second U|MAN variant is selected by `method=="exact"`

, and places equal mass on all graphs having the specified (exact) dyad census. The corresponding pmf is
$$
p(G=g|M,A,N) = \frac{M!A!N!}{(M+A+N)!}.
$$

U|MAN graphs provide a natural baseline model for networks which are constrained by size, density, and reciprocity. In this way, they provide a bridge between edgewise models (e.g., the E-R family) and models with higher order dependence (e.g., the Markov graphs).