Graphical Nash Equilibria and Replicator Dynamics on Complex Networks

Pairwise-interaction graphical games have been widely used in the study and design of strategic interaction in multiagent systems. With regard to this issue, one entitative problem is actually to understand how the interaction structure of agents affects the strategy configuration of Nash equilibria. This paper intends to study the effect of interaction networks on Nash equilibria in pairwise-interaction graphical games. We first show that interaction networks may induce new strategy equilibria in pairwise-interaction graphical games and then provide graphical conditions for the existence of these network-induced equilibria. Furthermore, to determine Nash equilibria of pairwise-interaction graphical games, a graphical replicator dynamics model is formulated, and its connection with graphical games is established. In detail, it is shown that every Nash equilibrium of the graphical games corresponds to a fixed point of the graphical replicator dynamics and that every asymptotically stable fixed point of the graphical replicator dynamics corresponds to a strict pure Nash equilibrium of the graphical games. The obtained results are applied in understanding coordination in complex networks and determination of structural conflicts in signed graphs. This work may provide new insights into understanding and designing strategy equilibria and dynamics in games on networks.