How Ideas grow: Critical Mass in the Linear Threshold Model

We study how ideas spread through a social network using the Linear Threshold Model. Each node i on the complete graph Kn is given a threshold Ɵi chosen uniformly at random from (0, 1]. This threshold indicates the fraction of the social network that must be active (or believe the idea) prior to node i becoming active. We start with an activated group of early adopters, called the seed set. Considering various scenarios, we use the probabilistic method to find lower bounds on size of a seed set which guarantees that all nodes become active with high probability. We characterize seed sets for both homogenous and heterogeneous influence by nodes. In the special case of a single seed node, we draw connections between the Linear Threshold Model and the Catalan numbers