  In this paper we present matrix game-theoretic models for joint routing,
network coding, and scheduling problem. First routing and network coding are
modeled by using a new approach based on compressed topology matrix that takes
into account the inherent multicast gain of the network. The 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, 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,
path, and network components. The outputs of this game model are mixed strategy
vectors of the second and the third players at equilibrium. Strategy vector of
the second player specifies optimum multi-path routing and network coding
solution while mixed strategy vector of the third players indicates optimum
switching rate among different network components or membership probabilities
for 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 fictitious playing method. Numerical and simulation
results prove the superior performance of the proposed techniques compared to
the conventional schemes using hard graph coloring.
