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Information spreading on multiplex complex networks has been widely concerned. In reality, the constraints of inelastic resources, such as energy, time, money, and an individual's restrict contact capacities, can affect the information spreading process. In this paper, we propose an information spreading model with limited contact capacity on the two-layered multiplex network. In order to quantitatively study the information spreading, we adopt an edge-based compartmental theory. We find that increasing the contact capacity facilitates information spreading. If the multiplex networks with strong heterogeneous degree distributions, the information outbreak size grows continuously with the transmission probability. However, when the information spreads on multiplex networks with homogeneous or heterogeneous degree distributions, there exists a crossover phenomenon, in which the information outbreak size versus the transmission probability can grow continuously or discontinuously. Our theory agrees well with the numerical simulations.

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Many real-world phenomena can be described as complex contagions, which has attracted much attention in the field of network science. However, the effects of the heterogeneous adoption thresholds on complex contagions in weighted networks have not been systematically investigated. In this paper, we propose a heterogeneous complex contagion model on the weighted network, in which individuals have different adoption thresholds. For individuals with a relatively small adoption threshold, they are more likely to adopt the contagion and act as activists. An edge-weight based compartmental theory is developed to unveil spreading dynamics. Through extensive numerical simulations and theoretical analysis, we find that, for any weight distribution heterogeneity, with the increase of the activist fraction, the growth pattern of the final adoption size versus the information spreading probability changes from hybrid phase transition to a second-order continuous phase transition. Meanwhile, increasing the activist fraction can promote behavior spreading. Through bifurcation analysis, we discover that changing the heterogeneity of the weight distribution will not change the type of phase transition. Besides, reducing weight distribution heterogeneity can facilitate behavior spreading. Extensive numerical simulations verify that the theoretical solutions coincide with the numerical results very well. INDEX TERMS Complex contagions, heterogeneous adoption, weighted networks, threshold model, compartmental theory.