Effect of indirect social ties on cascading diffusion of information

Diffusion of commercial products, ideas or innovations are all examples of information cascade, where a small initial shock may trigger a global cascade. One assumption usually adopted by researchers is the threshold mechanism, which assumes that a person will convert its state (from inactive to active) if a certain fraction of its neighbors has done so. In the original threshold model, only the direct neighbors of a node participate in the decision making of that person. However, increasingly empirical studies show that the indirect social ties are also of importance in many aspects in real life. Here we study how these indirect social ties affect the information cascading process. Motivated by the real situation, we assume a fraction r of nodes is in immune state, which can not be influenced by others. We perform simulations on both random ER and community networks, and find that without immune nodes the indirect social ties play an inhibitory role in the information cascading process. However, as r increases, the existence of indirect social ties may also facilitate the spreading process to some extent.