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This paper gives a mathematical description of the directed network capacity definitions and theorems of computing capacities towards the network. Prove of the theorem correctness is also given. According to them, the paper proposes an algorithm of computing the directed network capacities based on spanning tree diagram and even an advanced optimization algorithm. In addition, it discusses network capacity expansion problems with constraints. And then, the optimization algorithm in the scenario of directed network with limited costs is given, whose feasibility and process are illustrated via examples.