\in \mathbb{N}^*$ such as $A^k>0$.

\[def:communicates\] Let $A \in \mathcal{M}_N(\mathbb{R})$. Let $i$,$j$ be in $\{1,...,N\}$,

-   We say that $i$ *communicates* with $j$ (denoted $i \xrightarrow{A} j$) if there exists $n\geq0$ and $i_1,...,i_n \in \{1,...,N\}$ such that $$a_{ii_1}\prod_{l=1}^{n-1} (a_{i_{l}i_{l+1}}) a_{i_nj}>0.$$ If $i$ does not communicate with $j$ we write $i \nrightarrow j$.

-   We say that $i$ communicates with $j$ with $k$ *intermediates* if there exists $i_1,...,i_k \in \{1,...,N\}$ such that $$a_{ii_1}\prod_{l=1}^{k} (a_{i_{l}i_{l+1}}) a_{i_kj}>0,$$ and if there exists not $i_1,...,i_{k-1} \in \{1,...,N\}$ such that $$a_{ii_1}\prod_{l=1}^{k-1} (a_{i_{l}i_{l+1}}) a_{i_{k-1}j}>0,$$

-   Let $I,J \in \mathscr{P}(\{1,...,N\})$. We say that $I \xrightarrow{A} J$ if $$\exists i,j \in I \times J \text{ such that } i \xrightarrow{A} j.$$

If $A$ is a primitive matrix of size $N$, for each $i$, $j$ in $\{1,...,N\}, i\rightarrow j$. Moreover if there is $k$ such that $A^k >0$, then for each $i$, $j$ in $\{1,...,N\}$ $i$ communicates with $j$ with at most $k$ intermediates.

#### Examples.

Let us consider a matrix $A$ defined by $$A = \begin{pmatrix}
    1&1&0&0\\1&1&1&0\\0&1&1&1\\0&0&1&1
    \end{pmatrix}.$$ As we can see on the graph of the matrix $A$ (Fig. \[fig:graphprimitive matrix\]), every individual communicates with each other with at most 3 intermediates.

=\[ draw = black, thick, fill 