I am trying to understand a particular method of moments estimation algorithm. If you are interested in that, see this for more detailed description.
Long story short, I am trying to understand what it means for the neighbourhood matrix of a directed graph to have the following property: no element-wise product of each pair of rows results in a zero row. To put it another way, each pair of rows in the neighbourhood matrix has at least one common column full of ones. So e.g. (not necessarily a neighbourhood matrix) $\begin{pmatrix} 1&0&1\\1&1&1\end{pmatrix}$ has this property while $\begin{pmatrix} 1&0&0\\0&1&0\\0&0&1\end{pmatrix}$ does not.
So specifically:
1.) How can you represent this kind or property in a more graph theoretical/linear algebra way? I can only talk about it in a "programmatic" way, i.e. for all pairs of rows it holds that their dot product is not equal to zero.
2.) What can you tell about a graph whose neighbourhood matrix satisfies this property?
If neighborhood matrix is another name for adjacency matrix, the property means that every pair of nodes has a common out-neighbor. That is, for all $i,j\in N$ with $i<j$, there exists at least one $k\in N$ such that $(i,k) \in A$ and $(j,k) \in A$.