j \xrightarrow{\Lambda} i$.

If $\Gamma$ is primitive and $c_{\min}>0$, then $\Lambda$ is a primitive matrix. \[lemma:primitive\]

$\Gamma$ is primitive so there exists $k \in \mathbb{N}$ such that $\Gamma^k > 0$. In particular, $$\begin{aligned}
   \forall i,j \in \{1,...,N\}, \gamma^k_{ij} > 0 \iff&  \forall i,j \in \{1,...,N\}, \\
   &\sum_{0\leq l_1,...,l_{k-1}\leq N}\gamma_{il_1}\gamma_{l_1l_2}...\gamma_{l_{k-1}j} > 0 ,\\
   \iff& \forall i,j \in \{1,...,N\},  \exists l_1,...,l_{k-1} \in \{1,...,N\}, \\&\gamma_{il_1}\gamma_{l_1l_2}...\gamma_{l_{k-1}j} > 0.\end{aligned}$$ Moreover $C>0$ implies that $\forall i,j \in \{1,...,N\}, \gamma_{ij} > 0$. One can then conclude that $\lambda_{ij} > 0$.

Let us assume that $\Gamma > 0$. If $c_{\min}>0$, the social learning matrix $\Lambda$ is *bounded by below*: there exists $\underline m_\lambda > 0 $ such that $\forall i, j \in \{1, ..., N\}$, $\lambda_{ij} \geq \underline m_\lambda$. \[lemma:lambda&gt;eps\]

Let us set $\underline{m}_\gamma = \min_{i,j} \gamma_{ij}$. Since $\gamma_{ij} > \underline m_\gamma$ for all $i,j \in \{1,...,N\}$, one has that $$\lambda_{ij} > \mathds{1}_{\sum_{l=0}^N\gamma_{il}C_{il} = 0} \frac{1}{N} + \mathds{1}_{\sum_{l=0}^N\gamma_{il}C_{il} \neq 0}     \frac{\underline m_\gamma c_{\min}}{N(1 - N\underline m_\gamma)}.$$ It is then enough to choose $$\underline m_\lambda= \min \left(\frac{1}{N},\frac{\underline m_\gamma c_{\min}}{N(1 - N\underline m_\gamma)}\right).$$

\[lemma:upperBound\] Let $\Lambda$ be a stochastic matrix of size N that is bounded by below by $\underline m_\lambda$ as in Lemma \[lemma:lambda&gt;eps\]. Let us consider the sorted 