This master thesis focuses on the particular problem about the cut-size property of different networks. The scope of networks is from trivial network models (e.g., random graph) to real networks (e.g., Power grid network and Facebook network). The Susceptible-Infected-Susceptible (SIS) epidemic model can describe the spreading processes of information or diseases on networks. Within this thesis, we explore the cut-size property of networks and seek the relations between the cut-size property and the spreading behaviors. Our result deduces the cut-size property and the relevant physical meanings of the real networks. In such a way, a deeper understanding of the cut-size would help researchers to obtain insights of real networks. Our results may also contribute to the study of control of dynamic processes on networks.