Interlinkages across balance sheets of financial institutions may be modeled by a weighted directed graph G = (V, e) on the vertex set V = {1,…, n} = [n], whose elements represent financial institutions. The exposure matrix is given by e ∈ R^{n×n}, where the ij^{th} entry e(i, j) represents the exposure (in monetary units) of institution i to institution j. The interbank assets of an institution i are given by

A(i) := ∑_{j} e(i, j), which represents the interbank liabilities of i. In addition to these interbank assets and liabilities, a bank may hold other assets and liabilities (such as deposits).

The net worth of the bank, given by its capital c(i), represents its capacity for absorbing losses while remaining solvent. “Capital Ratio” of institution i, although technically, the ratio of capital to interbank assets and not total assets is given by

γ(i) := c(i)/A(i)

An institution is insolvent if its net worth is negative or zero, in which case, γ(i) is set to 0.

A financial network (e, γ) on the vertex set V = [n] is defined by

• a matrix of exposures {e(i, j)}_{1≤i,j≤n}

• a set of capital ratios {γ(i)}_{1≤i≤n}

In this network, the in-degree of a node i is given by

d^{−}(i) := #{j∈V | e(j, i)>0},

which represents the number of nodes exposed to i, while its out-degree

d^{+}(i) := #{j∈V | e(i, j)>0}

represents the number of institutions i is exposed to. The set of initially insolvent institutions is represented by

D_{0}(e, γ) = {i ∈ V | γ(i) = 0}

In a network (e, γ) of counterparties, the default of one or several nodes may lead to the insolvency of other nodes, generating a cascade of defaults. Starting from the set of initially insolvent institutions D_{0}(e, γ) which represent fundamental defaults, contagious process is defined as:

Denoting by R(j) the recovery rate on the assets of j at default, the default of j induces a loss equal to (1 − R(j))e(i, j) for its counterparty i. If this loss exceeds the capital of i, then i becomes in turn insolvent. From the formula for Capital Ration, we have c(i) = γ(i)A(i). The set of nodes which become insolvent due to their exposures to initial defaults is

D_{1}(e, γ) = {i ∈ V | γ(i)A(i) < ∑_{j∈D0} (1 − R(j)) e(i, j)}

This procedure may be iterated to define the default cascade initiated by a set of initial defaults.

So, when would a default cascade happen? Consider a financial network (e, γ) on the vertex set V = [n]. Set D_{0}(e, γ) = {i ∈ V | γ(i) = 0} of initially insolvent institutions. The increasing sequence (D_{k}(e, γ), k ≥ 1) of subsets of V defined by

D_{k}(e, γ) = {i ∈ V | γ(i)A(i) < ∑_{j∈Dk-1(e,γ)} (1−R(j)) e(i, j)}

is called the default cascade initiated by D_{0}(e, γ).

Thus D_{k}(e, γ) represents the set of institutions whose capital is insufficient to absorb losses due to defaults of institutions in D_{k-1}(e, γ).

Thus, in a network of size n, the cascade ends after at most n − 1 iterations. Hence, D_{n-1}(e, γ) represents the set of all nodes which become insolvent starting from the initial set of defaults D_{0}(e, γ).

Consider a financial network (e, γ) on the vertex set V = [n]. The fraction of defaults in the network (e, γ) (initiated by D_{0}(e, γ) is given by

α_{n}(e, γ) := |D_{n-1}(e, γ)|/n

The recovery rates R(i) may be exogenous or determined endogenously by redistributing assets of a defaulted entity among debtors, proportionally to their outstanding debt. The latter scenario is too optimistic since in practice liquidation takes time and assets may depreciate in value due to fire sales during liquidation. When examining the short term consequences of default, the most realistic assumption on recovery rates is zero: Assets held with a defaulted counterparty are frozen until liquidation takes place, a process which can in practice take a pretty long time to terminate.