  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.
