I'm confronted with the following problem:

Let $G=(V,E)$ be a directed graph with edge costs $c:E\rightarrow \mathbb{R}$ (Negative cycles do not matter). Let $V=\{v\_1,\dots,v\_n\}$.
For Matrices $A$ and $B$ $\in \mathbb{R}^{n \times n}$, we define a matrix product $\odot$ as follows: $A \odot B = C$ with
$$c\_{i,j} = \min\left\{a\_{i,l}+b\_{l,j}|1\leq l \leq n\right\}.$$
We write $A=A^{\odot 1}, A \odot A = A^{\odot 2}$, etc.
Let $M \in \mathbb{R}^{n \times n}$ be given by
$$m\_{i,j} = c(v\_i,v\_j)$$
with $$c(v\_i,v\_j)=\infty \text{ if }(v\_i,v\_j) \not\in E$$
Interpret the values of the matrix $M^{\odot k}$ for $K\in \mathbb{N}$, $k\geq 1$.

**Does anyone know this function? For what purposes it can be used?**

I wrote a script to study the behaviour of this matrix $M^{\odot k}$ and tested some instances. It seems that:

If there is a negative cycle the entries $m\_{i,j}$ tend to $-\infty$ $ \forall i,j$for large $k$.

If there is a no negative cycle the entries $m\_{i,j}=\infty$ $ \forall i,j$ for large $k$

If there are negative edges but no negative cycles some $m\_{i,j}=\infty$ and some $-\infty<m\_{i,j}<\infty$ for large $k$.