combination with Condition (*i*) in Proposition \[Kranton condition\], will allow us to derive sufficient conditions for the existence of a Nash equilibrium in terms of structural properties of the network.

The Stability of Nash Equilibria
--------------------------------

We present conditions for stability of a Nash equilibrium in terms of $%
\lambda _{n}(A_{S})$. A Nash equilibrium $\mathbf{x}$ is asymptotically stable when the system of differential equations:$$\begin{array}{c}
\dot{x}_{1}=h_{1}(\mathbf{x})=f_{1}(\mathbf{x};\delta ,A_{\mathcal{G}%
})-x_{1}, \\ 
\vdots  \\ 
\dot{x}_{n}=h_{n}(\mathbf{x})=f_{n}(\mathbf{x};\delta ,A_{\mathcal{G}%
})-x_{n},%
\end{array}%$$is locally asymptotically stable around $\mathbf{x}$. One can prove the following necessary and sufficient condition for an equilibrium $\mathbf{x}$ to be asymptotically stable [@BKA10]:

An equilibrium profile $\mathbf{x}$ is asymptotically stable if and only if $%
\delta <-1/\lambda _{n}(A_{S})$ and $\delta \sum_{j=1}^{n}a_{ij}x_{j}>1,$ for all inactive agents $i\in \mathcal{N}\backslash S$.

From the above lemma and Remark \[Active Remark\], we conclude that if $%
\delta <-1/\lambda _{n}(A_{\mathcal{G}})$, there is a unique Nash equilibrium and it is asymptotically stable. These results show the close connection between the smallest and the largest eigenvalues of the adjacency matrix of interactions and the outcome of games with linear best response functions in a network. On the other hand, the results in this section are applicable if we are able to compute the eigenvalues 