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In this paper, we present matrix game-theoretic models for joint routing, network coding, and scheduling problems. First, routing and network coding are modeled by using a new approach based on a compressed topology matrix that takes into account the inherent multicast gain of the network. Scheduling is optimized by a new approach called network graph soft coloring. Soft graph coloring is designed by switching between different components of a wireless network graph, which we refer to as graph fractals, with appropriate usage rates. The network components, which are represented by graph fractals, are a new paradigm in network graph partitioning that enables modeling of the network optimization problem by using the matrix game framework. In the proposed game, which is a nonlinear cubic game, the strategy sets of the players are links, paths, and network components. The outputs of this game model are mixed strategy vectors of the second and third players at equilibrium. The strategy vector of the second player specifies the optimum multipath routing and network coding solution, whereas the mixed strategy vector of the third player indicates the optimum switching rate among different network components or membership probabilities for an optimal soft-scheduling approach. Optimum throughput is the value of the proposed nonlinear cubic game at equilibrium. The proposed nonlinear cubic game is solved by extending a fictitious playing method. Numerical and simulation results prove the superior performance of the proposed techniques compared with the conventional schemes using hard graph coloring.

Joint Optimization of Scheduling and Routing in Multicast Wireless Ad Hoc Networks Using Soft Graph Coloring and Nonlinear Cubic Games