Trajectory-based networks exhibit strong heterogeneous patterns amid human behaviors. We propose a notion of causal time-varying dynamic Bayesian network (cTVDBN) to efficiently discover such patterns. While asymmetric kernels are used to make the model better adherence to causal principles, the variations of network connectivities are addressed by an adaptive over-fitting control. Compact regularization paths are obtained by approximate homotopy to make the solution tractable. In our experiments, cTVDBN structure discovery has successfully revealed the evolution of time-varying relationships in a ring road system, and provided insights for plausible road structure improvements from a traffic flow dataset.