We will discuss how one can endow a network with a landscape in a very simple and natural way. 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 (LAO protein binding network) are demonstrated. These examples show that the critical nodes play important roles in the structure and dynamics of such networks. This is a joint work with Weinan E and Jianfeng Lu.