We prove that the piecewise linear best-response dynamical systems of strategic interactions are asymptotically convergent to their set of equilibria on any weighted undirected graph. We study various features of these dynamical systems, including the uniqueness and abundance properties of the set of equilibria and the emergence of unstable equilibria. We also introduce the novel notions of social equivalence and social dominance on directed graphs, and demonstrate some of their interesting implications, including their correspondence to consensus and chromatic number of partite graphs. Examples illustrate our results.