Summary: We consider the path-counting problem that asks how many paths exist between any two different nodes in a network after deleting an arbitrary number of edges or nodes from the original network. Using path-counting methods, we propose a quantitative method for measuring the robustness of the network-structured system. First we define the connectivity function and attempt to obtain the expected connectivity function for the network. Applying the Monte Carlo method, we estimate expected edge deletion and node deletion connectivity functions when an arbitrary number of edges or nodes are deleted from the original network. We attempt to approximate the expected edge deletion connectivity function using an appropriate nonlinear function with two parameters. We also show the numerical results of applying the path-counting method with their analysis in order to quantitatively evaluate the connectivity for some special types of networks.