(color online) The dependence of the order parameter on the time delay τ in excitable homogeneous random networks (EHRNs) without parameter-diversity (i.e., σ = 0.0). System parameters are chosen as a = 0.84, b = 0.07, ϵ = 0.04, D = 0.5, and N = 100. Here the order parameter is calculated by the formula , in which is the synchronization parameter of each oscillation in time-delayed EHRNs, and N_ALL=100 is the total number of independent samples performed for each set of parameters. Three distinct parameter regions, i.e., asynchronous region (domain I for ), transition region (domain II for , indicated by the blue rectangle), and synchronous region (domain III for ) are discovered. and represent the two critical values, which can be used to identify the asynchronous state, the weak synchronization, and the complete synchronization in EHRNs. The red rectangle indicates the parameter region of time delay (i.e., ), in which appropriate parameter-diversity can promote the synchronization performance in time-delayed EHRNs.