  This paper studies $n$-person simultaneous-move games with linear best
response function, where individuals interact within a given network structure.
This class of games have been used to model various settings, such as, public
goods, belief formation, peer effects, and oligopoly. The purpose of this paper
is to study the effect of the network structure on Nash equilibrium outcomes of
this class of games. Bramoull\'{e} et al. derived conditions for uniqueness and
stability of a Nash equilibrium in terms of the smallest eigenvalue of the
adjacency matrix representing the network of interactions. Motivated by this
result, we study how local structural properties of the network of interactions
affect this eigenvalue, influencing game equilibria. In particular, we use
algebraic graph theory and convex optimization to derive new bounds on the
smallest eigenvalue in terms of the distribution of degrees, cycles, and other
relevant substructures. We illustrate our results with numerical simulations
involving online social networks.
