Funnel theorems for spreading on networks

We derive the funnel theorems for the Bass and susceptible-infected models on networks that describe the spreading of innovations and epidemics. Let j be a node and divide the remaining nodes into L≥2 disjoint sets {Al}l=1L. The funnel theorems provide lower and upper bounds for the difference between the susceptibility probability of j and the product of its susceptibility probability on the L modified networks in which j can only be influenced by incoming edges from Al. In particular, one can let L be equal to the indegree of j, so that in the modified networks, j is only influenced by one incoming edge. We illustrate how the funnel theorems can be used to obtain exact explicit expressions for the adoption/infection probabilities of nodes and for the expected adoption/infection level in various types of networks, both with and without cycles.