The majority of chemical and biological processes can be viewed as complex networks of states connected by dynamic transitions. It is fundamentally important to determine the structure of these networks in order to fully understand the mechanisms of underlying processes. A new theoretical method of obtaining topologies and dynamic properties of complex networks, which utilizes a first-passage analysis, is developed. Our approach is based on a hypothesis that full temporal distributions of events between two arbitrary states contain full information on number of intermediate states, pathways, and transitions that lie between initial and final states. Several types of network systems are analyzed analytically and numerically. It is found that the approach is successful in determining structural and dynamic properties, providing a direct way of getting topology and mechanisms of general chemical network systems. The application of the method is illustrated on two examples of experimental studies of motor protein systems.