We study strategic games on weighted directed graphs, in which the payoff of a player is defined as the sum of the weights on the edges from players who chose the same strategy, augmented by a fixed non-negative integer bonus for picking a given strategy. These games capture the idea of coordination in the absence of globally common strategies. We identify natural classes of graphs for which finite improvement or coalition-improvement paths of polynomial length always exist, and, as a consequence, a (pure) Nash equlibrium or a strong equilibrium can be found in polynomial time. The considered classes of graphs are typical in network topologies: simple cycles correspond to the token ring local area networks, while open chains of simple cycles correspond to multiple independent rings topology from the recommendation G.8032v2 on the Ethernet ring protection switching. For simple cycles these results are optimal in the sense that without the imposed conditions on the weights and bonuses a Nash equilibrium may not even exist. Finally, we prove that the problem of determining the existence of a Nash equilibrium or of a strong equilibrium in these games is NP-complete already for unweighted graphs and with no bonuses assumed. This implies that the same problems for polymatrix games are strongly NP-hard.
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