(color online) (a) The dependence of the synchronization parameter R on the diversity σ for time delay τ = 2.3 in EHRNs. The blue and the green dashed lines represent the two critical synchronization parameters and , which can be used to identify the asynchronous state, the weak synchronization, and the complete synchronization in time-delayed EHRNs. (b) The dependence of the maximum synchronization parameter (indicated by the left axis and shown by the red dots) and the proportion of weak synchronization p (indicated by the right axis and shown by the blue squares) on the diversity σ in time-delayed EHRNs. The proportion of weak synchronization p is defined as , where N_WS is the number of weak synchronization counted in the N_ALL=100 independent numerical samples performed for each set of parameters.