Exact results in percolation theory on random graphs rely on a property known as the "tree ansatz'', which is known to be asymptotically true on the family on configuration graphs. More generally, the tree ansatz, also called mean field theory, can be used as the basis of approximations, which as numerous authors have remarked, can be surprisingly accurate. The question arises whether the tree ansatz can be useful for understanding financial systemic risk. In this talk, I will review the concepts underlying the tree ansatz, and explore how it can be embedded and used in models of financial contagion, such as the Eisenberg-Noe model and its alternatives. Along the way, I will propose definitions for "random financial network'' (RFN) and "locally treelike independence'' (LTI), and explore these definitions' mathematical consequences. In the end I will compare analytical approximations to Monte Carlo computations in some realistic network cascade examples, and show that there are indeed situations where the LTI approximation is "surprisingly" accurate. This provides some evidence that understanding of networks in other domains can help us in understanding financial networks.
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