Question:
What are, provided their existence, examples of functions $f$ with the following properties:
\begin{align}f:& \ \mathbb{N}\times\mathbb{N}\ni(i,j)&\mapsto\ \beta\in\lbrace 0,1\rbrace\\ &f(i,i)&=\quad\quad\quad\quad0\\ &f(i,j)&=\quad\quad f(j,i)\\ &\sum_{i=1}^{\infty}{f(i,j)}&=\quad\quad k\in\mathbb{N}\\ &\sum_{j=1}^{\infty}{f(i,j)}&=\quad\quad k\in\mathbb{N}\\ &F\in\lbrace0,1\rbrace^{n\times n},\ 1\le i,j \le n,\ F_{ij}=f(i,j)&\implies \ |F|\ne 0\end{align}\begin{align}f:& \ \mathbb{N}\times\mathbb{N}\ni(i,j)&\mapsto\ \quad\quad\quad\quad\quad\ \beta\in\lbrace 0,1\rbrace\\ &f(i,i)&=\quad\quad\quad\quad\quad\quad\quad\quad\quad\ 0\\ &f(i,j)&=\quad\quad\quad\quad\quad\quad\quad\ f(j,i)\\ &\sum_{i=1}^{\infty}{f(i,j)}&=\quad k\in\mathbb{N}\quad \forall n\ge n_0\ge k\\ &\sum_{j=1}^{\infty}{f(i,j)}&=\quad k\in\mathbb{N}\quad \forall n\ge n_0\ge k\\ &F\in\lbrace0,1\rbrace^{n\times n},\ 1\le i,j \le n,\ F_{ij}=f(i,j)&\implies\quad\quad\quad\quad\quad\quad \ |F|\ne 0\end{align}
The calculation of $f(i,j)$ must not depend on $n$ but would ideally be parameterized by $k\in\mathbb{N}$
In reply to @PuckRombach's comment: Another restriction is that, given $k$ the $f(i,j)$ must yield matrices with the described properties for all $n\ge n_0\ge k$