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I want to populate a 0-1 matrix, which is an adjacency matrix, corresponding to a directed graph, with weights on the elements that are 1. In other words, I want to generate an $N\times N$ matrix $A$ so as to target an $N$ vector of row sums(in expectation) and simultaneously all column sums should sum to 1. In addition to this, I have a prefixed number of elements which are set to zero. For example, beginning with: $$ \left[\begin{array}{rrr} 0 & 1 & 1 \\ 1 & 0 & 0 \\ 1 & 1 & 0 \end{array}\right] $$ and the row sum vector $[1.5, 0.25, 1]^{T}$, I want to end up with $$ \left[\begin{array}{rrr} 0 & a_{12} & a_{13} \\ a_{21} & 0 & 0 \\ a_{31} & a_{32} & 0 \end{array}\right] $$ under the following conditions:

$\mathrm{E}[a_{12} + a_{13}] = 1.5$

$ \mathrm{E}[a_{21}] = 0.25$

$\mathrm{E}[a_{31}+a_{32}] = 1$

$a_{21}+a_{31} = 1$

$a_{12}+a_{32} = 1$

$a_{13} = 1$

I realise that each column can be drawn from the Dirichlet distribution, for example, columns 1 and 2 from $\mathrm{R}^{2}$(Dirichlet distribution of order 2) and column 3 from $\mathrm{R}$(Dirichlet distribution of order 1, or simply a beta distribution), but is there any standard procedure to target the row sums as well?

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