Cascades of Rumors and Information in Highly Connected Networks with Thresholds

Percolation or cascades on random networks, representing spread of information, propagation of rumors or adoption of innovations, are typically analyzed using generating functions. This approach requires that the network be assumed infinite and weakly connected. These assumptions are not obeyed by real or simulated networks on which this theory is often used. In this paper we offer a theory that assumes a finite network with arbitrary average nodal degree and apply it to the case where cascades follow a threshold rule, that is, that a node will change state (“flip”) only if a fraction, exceeding a given threshold, of its neighbors has changed state previously. The model is a dynamic Markov model whose state transition matrix, recalculated after each step, records the probability that a node of degree