A simple model of influence: Details and variants of dynamics

We consider a simple model of establishing influence in a network. Vertices (people) split into influence groups and follow the opinion of the leader – the influencer – of their group. Groups can merge, based on interactions between influencers (the ‘active vertices’ of the network, while the followers are the ’passive vertices’).
We study how the final number of influence groups depends on the way active vertices are chosen for interacting, considering two types of sparse graphs: the cycle Cn, which allows detailed analysis of various influencer algorithms, and random graphs G(n, p) where p = c/n for a constant c. We also introduce a simple dynamic Falling-Out model, which allows for rejection of opinion. In its most general form, as considered for G(n, p), one of the two interacting influencers can decide to follow the other influencer, or they both can reject the opinion of the other influencer and instead choose other influencers to follow.
Our analysis for the cycle is based on solving systems of recurrences using generating functions, and our analysis for the random G(n, p) graph uses the differential equation method.