In this lecture, guided by the applications to higher-order network, we will discuss some topological approaches to graphs and hypergraphs beyond topological data analysis (TDA), including GLMY homology introduced by S. T. Yau et al, and it generalizations. We will start to explain the motivations of mathematical explorations on the subject from the views of scientific challenges in high-order complex network. After reviewing some topological approaches on the subject, we will report some of our recent progress on the applications of GLMY theory in biology and materials. Finally, we will discuss the homotopy theory of digraphs and its connections to the fundamental higher-order structures of complex network.