left of=2\] [$G_1$]{}; (3) \[below right of=2\] [$G_3$]{};

\(1) edge \[loop left\] node (1); (2) edge \[loop above\] node (2); (3) edge \[loop right\] node (3); (1) edge node (2); (3) edge node (2);

As shown on the figure \[fig:graphproof\], $G_1\nrightarrow G_3$ and $G_3\nrightarrow G_1$.

We conclude as before.

Eigenvalues of the matrix of influence $\Lambda$
------------------------------------------------

[In this Section we study the the quantitative properties of the eigenvalues of the influence matrix, in order to understand the dynamics of the idealized process $K_{\mathcal{T}}^t$.]{}

Let $\Gamma$ and $C$ be respectively the structure and credibility matrices defined in Def. \[def:Gamma\] and equation . By construction, $C$ is a stochastic matrix. The structure matrix $\Lambda$ is defined according to by $$\lambda_{ij} = \left\{ \begin{array}{ll}
        \frac{\gamma_{ij}c_{ij}}{\sum_{l=1}^N\gamma_{il}C_{il}} & \mbox{if } \sum_{l=1}^N\gamma_{il}C_{il} \neq 0,\\
        1/N & \mbox{otherwise.}\\
    \end{array}
\right.$$

If for all $i,j \in \{1,...,N\}$, one has $i\xrightarrow{\Gamma} j$ or $j \xrightarrow{\Gamma} i$, and the credibility matrix $C > 0$, then for all $i,j \in \{1,...,N\}$, $i\xrightarrow{\Lambda} j$ or $j \xrightarrow{\Lambda} i$.

$\forall i,j \in \{1,...,N\}, \gamma_{ij} > 0 \implies \lambda_{ij} > 0$ using . Thus $\forall i,j \in \{1,...,N\}, i\xrightarrow{\Gamma} j \text{ or } j \xrightarrow{\Gamma}  > 0 \implies i\xrightarrow{\Lambda} j \text{ or } 