Figure 6: Relative density of cooperators against for different values of when and (see legend). For the lower occupation densities, for example, , all isolated cooperator clusters are able to survive for all the values of . But it is worth noting that the relative density of cooperators varies nonmonotonically with the increase of when , which indicates that moderately enlarging the gaps among the social ranks can facilitate cooperation, while exceedingly large or small gaps inhibit cooperation. Moreover, although cooperation is possible in a wide range of for , cooperation performs best when .