The service and application of a network is a behavioral process that is oriented toward its operations and tasks, whose metrics and evaluation are still somewhat of a rough comparison. This paper describes scenes of network behavior as differential manifolds. Using the homeomorphic transformation of smooth differential manifolds, we provide a mathematical definition of network behavior and propose a mathematical description of the network behavior path and behavior utility. Based on the principle of differential geometry, this paper puts forward the function of network behavior and a calculation method to determine behavior utility, and establishes the calculation principle of network behavior utility. We also provide a calculation framework for assessment of the network’s attack-defense confrontation on the strength of behavior utility. Therefore, this paper establishes a mathematical foundation for the objective measurement and precise evaluation of network behavior.