In this study, we consider users' online communication rhythms (online social rhythms) as coupled oscillators in a complex social network. Users' rhythms may be entrained onto those of their friends, and macro-scale pattern of such rhythms can emerge. We investigated the entrainment in online social rhythms and long-range correlations of the rhythms using an avatar communication dataset. We indicated entrainment in online social rhythms to emerge if the strength of a new connection reaches a threshold. This entrainment spread via densely-connected clusters. Consequently, long-range correlations of online social rhythms extended to about 36% of the network, although offline social life naturally restricts online social rhythms. This research supports an understanding of human social dynamics in terms of systems of coupled oscillators.