Date Published: February 23, 2017
Publisher: Public Library of Science
Author(s): Thomas R. Hurd, James P. Gleeson, Sergey Melnik, Renaud Lambiotte.
We introduce a probabilistic framework that represents stylized banking networks with the aim of predicting the size of contagion events. Most previous work on random financial networks assumes independent connections between banks, whereas our framework explicitly allows for (dis)assortative edge probabilities (i.e., a tendency for small banks to link to large banks). We analyze default cascades triggered by shocking the network and find that the cascade can be understood as an explicit iterated mapping on a set of edge probabilities that converges to a fixed point. We derive a cascade condition, analogous to the basic reproduction number R0 in epidemic modelling, that characterizes whether or not a single initially defaulted bank can trigger a cascade that extends to a finite fraction of the infinite network. This cascade condition is an easily computed measure of the systemic risk inherent in a given banking network topology. We use percolation theory for random networks to derive a formula for the frequency of global cascades. These analytical results are shown to provide limited quantitative agreement with Monte Carlo simulation studies of finite-sized networks. We show that edge-assortativity, the propensity of nodes to connect to similar nodes, can have a strong effect on the level of systemic risk as measured by the cascade condition. However, the effect of assortativity on systemic risk is subtle, and we propose a simple graph theoretic quantity, which we call the graph-assortativity coefficient, that can be used to assess systemic risk.
The study of contagion in financial systems is topical in light of the recent global credit crisis and the resultant damage inflicted on financial institutions. Contagion  refers to the spread of dangerous shocks through a system of financial institutions, with each successive shock causing increasing pressure on the remaining components of the system. The term systemic risk refers to the contagion-induced threat to the financial system as a whole, due to the failure of one (or more) of its component institutions.
In this section we specify the two constituent parts of our interbank model: network structure and dynamics. The structure or skeleton of the network is modelled as a random directed graph. The dynamics is determined by the bank balance sheets and the rules for the propagation of defaults through the interbank network.
In this section, we consider two examples of stylized interbank networks and show that the analytical results obtained above match well to the Monte Carlo simulations when N, the number of nodes in the network, is sufficiently large. Unless specified otherwise, we adopt the choice of parameters made for the model of Ref. :
In summary, we have described here an analytical framework which can predict the systemic risk of a networked system of financial institutions. The qualitative type of networks one can address has been extended compared to most existing work, in particular by the inclusion of the non-independent connections between nodes. In this more general setting we find the cascade is described by a vector-valued fixed point problem that reduces to well-understood scalar problems in special cases. We also observed that graph assortativity can strongly affect the course of contagion cascades, and hence showed the importance of incorporating assortativity in numerical and analytical treatments of banking network models. Our analytic framework will enable extensive studies of alternative network topologies. In such studies the cascade condition and cascade frequency provide two easily computed and useful measures of systemic risk by which to compare different network topologies. However, the daunting range of network variables means that both analytical and numerical studies must be carefully framed to address specific issues, for example, to uncover other key determinants of systemic risk, and to deal with finite-size deviations from the infinite-N theory. Finally, we anticipate that future work can show how the approach described here may be further extended to include partial recovery models (such as Ref. ) and stochastic balance sheets.