cycles of length $k$.

Strategic Interactions in Networks\[Strategic Section\]
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In this section we present the game-theoretical model of strategic interactions considered in this paper and present interesting connections between the Nash equilibria and the eigenvalues of the adjacency matrix of the network.

The Model
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We represent the network of influences using a simple graph $\mathcal{G}$. Let $\mathcal{N}=\{1,\cdots ,n\}$ denote the set of $n$ players located at each node of the graph $\mathcal{G}$. We denote by $x_{i}\in \lbrack
0,\infty )$ the action chosen by agent $i$, and by $\mathbf{x}$ the vector that represents the joint actions for all agents. We denote by $\mathbf{x}%
_{-i}$ the vector of actions of all players excluding player $i$. As mentioned before, players interact according to a network of influences that we describe using its adjacency matrix $A_{\mathcal{G}}$. The interactions are assumed to be symmetric, $a_{ij}=a_{ji}$, and we do not allow self-loops, $a_{ii}=0$. The payoff function for agent $i$ is given by:$$U_{i}(x_{i},\mathbf{x}_{-i};\delta ,A_{\mathcal{G}})$$where $\delta \in \mathbb{R}$ is a parameter that can be tuned to change the influence of neighboring nodes on each player’s action.

\[Game Description\]Games with Linear Best Response Functions
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We study a class of games whose best response functions take a 