$0$ will also be referred to as society or the societal node. The full set of institutions, including the societal node, is denoted by ${\mathcal{N}}_0 := {\mathcal{N}}\cup \{0\}$. We refer to [@feinstein2014measures; @GY14] for further discussion of the meaning and concepts behind the societal node.

We will be extending the model from [@EN01] in this paper. In that work, any bank $i \in {\mathcal{N}}$ may have obligations $L_{ij} \geq 0$ to any other firm or society $j \in {\mathcal{N}}_0$. We will assume that no firm has any obligations to itself, i.e., $L_{ii} = 0$ for all firms $i \in {\mathcal{N}}$, and the society node has no liabilities at all, i.e., $L_{0j} = 0$ for all firms $j \in {\mathcal{N}}_0$. Thus the *total liabilities* for bank $i \in {\mathcal{N}}$ is given by $\bar p_i := \sum_{j \in {\mathcal{N}}_0} L_{ij} \geq 0$ and relative liabilities $\pi_{ij} := \frac{L_{ij}}{\bar p_i}$ if $\bar p_i > 0$ and arbitrary otherwise; for simplicity, in the case that $\bar p_i = 0$, we will let $\pi_{ij} = \frac{1}{n}$ for all $j \in {\mathcal{N}}_0 \backslash \{i\}$ and $\pi_{ii} = 0$ to retain the property that $\sum_{j \in {\mathcal{N}}_0} \pi_{ij} = 1$. On the other side of the balance 