Topology identification of stochastic complex networks is an important topic in network science. In modern identification techniques under a continuous framework, the controller has a negative dynamic gain (feedback gain), such that stochastic LaSalle's invariance principle (SLIP) is directly satisfied. In this article, the topology identification of stochastic complex networks is studied under aperiodic intermittent control (AIC). It is noteworthy that the AIC has a rest time, which indicates the SLIP is not valid since there is no negative feedback gained during this period. This motivates us to find other methods to obtain identification criteria. In this study, the graph-theoretic method and the stochastic analysis technique are integrated to obtain the almost surely exponential synchronization of drive-response networks. Furthermore, this integration enables the topology identification criteria of the drive network to be derived, which differs from previous work that directly utilized SLIP. It is worth mentioning that the topology identification criteria under the stochastic framework are first proposed based on the AIC in this work. The control strategy not only reduces the control cost but also makes it easier to operate. To enhance the application value of the network model, regime-switching diffusions, multiple weights, and nonlinear couplings are simultaneously considered. Finally, the proposed identification criteria are tested by using neural networks. At the same time, the validity of the theoretical results is further proved by numerical simulations.