This project described an approach to define critical nodes in networks, which sheds light on crucial nodes on pathways connecting different regions of networks. Critical point analysis is introduced for functions defined on networks. The concept of local minima/maxima and saddle points of different indices are defined, by extending the notion of gradient flows and minimum energy path to the network setting. Persistent homology is used to design efficient numerical algorithms for performing such analysis. Applications to some examples of social and biological networks are demonstrated. These examples show that the critical nodes play important roles in the structure and dynamics of such networks.