Stability and analysis of multiagent network systems with state-dependent switching typologies have been a fundamental and longstanding challenge in control, social sciences, and many other related fields. These already complex systems become further complicated once one accounts for asymmetry or heterogeneity of the underlying agents/dynamics. Despite extensive progress in analysis of conventional networked decision systems where the network evolution and state dynamics are driven by independent or weakly coupled processes, most of the existing results fail to address multiagent systems where the network and state dynamics are highly coupled and evolve based on status of heterogeneous agents. Motivated by numerous applications of such dynamics in social sciences, in this paper we provide a new direction toward analysis of dynamic networks of heterogeneous agents under complex time-varying environments. As a result we show Lyapunov stability and convergence of several challenging problems from opinion dynamics using a simple application of our framework. In particular, we introduce a new class of asymmetric opinion dynamics, namely, nearest neighbor dynamics, and show how our framework can be used to analyze their behavior. Finally, we extend our results to game-theoretic settings and provide new insights toward analysis of complex networked multiagent systems using exciting field of sequential optimization.