The structures of molecules and materials determine their functions. Understanding the structure and function relationship is the holy grail of molecular and materials sciences. However, the rational design of molecules and materials with desirable functions remains a grand challenge despite decades of efforts. A major obstacle is the lack of an intrinsic mathematical characteristic that attributes to a specific function. This work introduces persistent path topology (PPT) to effectively characterize directed networks extracted from functional units, such as constitutional isomers, cis-trans isomers, chiral molecules, Jahn-Teller isomerism, and high-entropy alloy catalysts. Path homology (PH) theory is utilized to decipher the role of mirror-symmetric sublattices that hinder the formation of periodic unit cells in amorphous solids. Topological perturbation analysis (TPA) is proposed to reveal the critical target in the blood coagulation system. The proposed topological tools can be directly applied to systems biology, omics sciences, topological materials, and machine learning study of molecular and materials sciences.