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How big is the risk that a few initial failures of nodes in a network amplifyto large cascades that span a substantial share of all nodes? Predicting thefinal cascade size is critical to ensure the functioning of a system as awhole. Yet, this task is hampered by uncertain or changing parameters andmissing information. In infinitely large networks, the average cascade size canoften be well estimated by established approaches building on local treeapproximations and mean field approximations. Yet, as we demonstrate, in finitenetworks, this average does not even need to be a likely outcome. Instead, wefind broad and even bimodal cascade size distributions. This phenomenonpersists for system sizes up to $10^{7}$ and different cascade models, i.e. itis relevant for most real systems. To show this, we derive explicit closed-formsolutions for the full probability distribution of the final cascade size. Wefocus on two topological limit cases, the complete network representing a densenetwork with a very narrow degree distribution, and the star networkrepresenting a sparse network with a inhomogeneous degree distribution. Thosetopologies are of great interest, as they either minimize or maximize theaverage cascade size and are common motifs in many real world networks.

Explicit size distributions of failure cascades redefine systemic risk on finite networks