In this work we analyze the coherence dynamics and estimate decoherence times of quantum states in a network where a central dissipative oscillator is coupled with N − 1 peripheral noninteracting dissipative oscillators. The results obtained here are compared with those in part I of this work where a symmetric network was considered. This comparison helps us to understand the influence of the topology of a network on the coherence dynamics of quantum superposition states. As in part I, master equations are derived for regimes of both weak and strong coupling between the oscillators. Decoherence times of particular states of the network are computed and the results are analyzed in the light of state swap and recurrence processes of reduced states of the network. The linear entropies of the joint and reduced systems are also analyzed.