  This systemic risk paper introduces inhomogeneous random financial networks
(IRFNs). Such models are intended to describe parts, or the entirety, of a
highly heterogeneous network of banks and their interconnections, in the global
financial system. Both the balance sheets and the stylized crisis behaviour of
banks are ingredients of the network model. A systemic crisis is pictured as
triggered by a shock to banks' balance sheets, which then leads to the
propagation of damaging shocks and the potential for amplification of the
crisis, ending with the system in a cascade equilibrium. Under some conditions
the model has ``locally tree-like independence (LTI)'', where a general
percolation theoretic argument leads to an analytic fixed point equation
describing the cascade equilibrium when the number of banks $N$ in the system
is taken to infinity. This paper focusses on mathematical properties of the
framework in the context of Eisenberg-Noe solvency cascades generalized to
account for fractional bankruptcy charges. New results including a definition
and proof of the ``LTI property'' of the Eisenberg-Noe solvency cascade
mechanism lead to explicit $N=\infty$ fixed point equations that arise under
very general model specifications. The essential formulas are shown to be
implementable via well-defined approximation schemes, but numerical exploration
of some of the wide range of potential applications of the method is left for
future work.
