Universal behavior of the linear threshold model on weighted networks

Abstract

The linear threshold model is widely adopted as a classic prototype for studying contagion processes on social networks, where nodes, representing individuals, are assumed to be in one of two states: inactive or active. Each inactive node can be activated via a threshold rule during evolution. Although both contagion mechanisms and network impacts have been well studied, very few studies paid attention on the effect of interacting strengths on the threshold rule. In this paper, a modified linear threshold model on weighted networks is proposed. On one hand, the weight of a link in the network is characterized by a power-law function of the product of endpoint degrees. On the other hand, peer influences on a node incorporate both the number of its active neighbors and associated link weights. The systematic dynamics is explored by the combination of the spin-glass theory and Monte-Carlo algorithm. In analogy to unweighted networks, a global cascade is not triggered in weighted networks when the average degree of nodes is either too small or too large, however, large cascades are realized within an intermediate range, which is referred to as the cascade window. Moreover, two regimes of the power exponent of the weight function are identified in which the system exhibits distinct behaviors: when networks are very sparse, there exist one extreme of the weight exponent making the system susceptible to large cascades; when the networks are relatively dense, on the contrary, there exists the other extreme of the weight exponent causing the system to maintain optimal robustness. All these results demonstrate the importance of both network connectivity and link weights, and offer a sophisticated description of social contagions.