We consider vector fixed point (FP) equations in large dimensional spaces involving random variables, and study their realization-wise solutions. We have an underlying directed random graph, that defines the connections between various components of the FP equations. Existence of an edge be- tween 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 neighbor components. We obtain finite dimensional limit FP equations (in a much smaller dimensional space), whose solutions approximate the solution of the random FP equations for almost all re- alizations, in the asymptotic limit (number of components increase). Our techniques are different from the traditional mean-field methods, which deal with stochastic FP equa- tions in the space of distributions to describe the station- ary distributions of the systems. In contrast our focus is on realization-wise FP solutions. We apply the results to study systemic risk in a large financial heterogeneous net- work with many small institutions and one big institution, and demonstrate some interesting phenomenon.
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