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We study the network replicator equation and characterize its fixed points onarbitrary graph structures for $2 \times 2$ symmetric games. We show arelationship between the asymptotic behavior of the network replicator and theexistence of an independent vertex set in the graph and also show that complexbehavior cannot emerge in $2 \times 2$ games. This links a property of thedynamical system with a combinatorial graph property. We contrast this byshowing that ordinary rock-paper-scissors (RPS) exhibits chaos on the 3-cycleand that on general graphs with $\geq 3$ vertices the network replicator withRPS is a generalized Hamiltonian system. This stands in stark contrast to theestablished fact that RPS does not exhibit chaos in the standard replicatordynamics or the bimatrix replicator dynamics, which is equivalent to thenetwork replicator on a graph with one edge and two vertices ($K_2$).

Generalized Hamiltonian Dynamics and Chaos in Evolutionary Games on Networks