We consider a large random network, in which the performance of a node depends upon that of its neighbours and some external random influence factors. This results in random vector valued fixed-point (FP) equations in large dimensional spaces, and our aim is to study their almost-sure solutions. An underlying directed random graph defines the connections between various components of the FP equations. Existence of an edge between nodes $i,j$ implies the $i$-th FP equation depends on the $j$-th component. We consider a special case where any component of the FP equation depends upon an appropriate aggregate of that of the random `neighbour' components. We obtain finite-dimensional limit FP equations in a much smaller dimensional space, whose solutions aid to approximate the solution of FP equations for almost all realizations, as the number of nodes increases. We use Maximum theorem for non-compact sets to prove this convergence. We apply the results to study systemic risk in an example financial network with large number of heterogeneous entities. We utilized the simplified limit system to analyse trends of default probability (probability that an entity fails to clear its liabilities) and expected surplus (expected-revenue after clearing liabilities) with varying degrees of interconnections between two diverse groups. We illustrated the accuracy of the approximation using exhaustive Monte-Carlo simulations. Our approach can be utilized for a variety of financial networks (and others); the developed methodology provides approximate small-dimensional solutions to large-dimensional FP equations that represent the clearing vectors in case of financial networks.
Random fixed points, systemic risk and resilience of heterogeneous financial network