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This article studies multiagent differential graphical games in linear dynamical networks. We point out that in existing graphical game formulations, being global Nash and being distributed are two contradicting properties. In particular, we prove that the best response strategy, most widely used in solving graphical games, leads to Nash but does not provide distributed solutions. On the other hand, the minmax strategy, recently developed in solving graphical games, provides distributed solutions but prevents the agents from reaching a global Nash equilibrium. In this article, we propose a novel differential graphical game formulation, which promises the existence of solutions that 1) lead to global Nash equilibrium and 2) are distributed in the sense that each agent only uses the state information of its own and its direct neighbors. Stability and Nash equilibrium properties of the proposed graphical game are proven, respectively. Simulation studies are conducted to illustrate the theoretical results.