Abstract：The rising of online networks have provided a new ecosystem of individual interactions. Of particular interest are social contagions on duplex networks. Previous studies assume that each network is homogeneous, however, empirical analysis shows that nearly the online networks are heterogeneous. Motivated by this, we studied the threshold model on duplex networks with different online-offline coupling. Compared with uncorrelated coupling, the maximum positive coupling weakens systematic robustness for the sparse network, while enhances the robustness for the dense network. The maximum negative coupling, however, plays an opposite role. These effects are strengthened when the online network is heterogeneous.
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