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The authors have declared that no competing interests exist.

Analyzed the data: MS BS. Contributed reagents/materials/analysis tools: MS BS. Wrote the paper: MS BS AG JSM. Reviewed the literature: AG JSM. Wrote the Supplementary Information: MS.

The financial crisis illustrated the need for a functional understanding of systemic risk in strongly interconnected financial structures. Dynamic processes on complex networks being intrinsically difficult to model analytically, most recent studies of this problem have relied on numerical simulations. Here we report analytical results in a network model of interbank lending based on directly relevant financial parameters, such as interest rates and leverage ratios. We obtain a closed-form formula for the “critical degree” (the number of creditors per bank below which an individual shock can propagate throughout the network), and relate failures distributions to network topologies, in particular scalefree ones. Our criterion for the onset of contagion turns out to be isomorphic to the condition for cooperation to evolve on graphs and social networks, as recently formulated in evolutionary game theory. This remarkable connection supports recent calls for a methodological rapprochement between finance and ecology.

In the financial sector, shock propagation mechanisms are at the core of systemic risk [

In recent years, random network theory [

An essential insight of Allen and Gale [

Our first goal in this paper is to sharpen these results by introducing a model of interbank lending that allows the “critical degree” separating these two regimes to be computed as an explicit function of a small number of relevant financial parameters: (interbank and external) interest rates, liquidity requirement, leverage ratio. As we shall see, this critical degree is pivotal in deriving an analytical estimate of the number of failures induced by a single shock given these parameters. Our results complement those of [

Our second goal is to analyze the role of degree heterogeneity in financial networks with regard to systemic risk. It has long been known [

The paper is organized as follows. We begin by describing our model of interbank lending networks, first in some generality and then under simplifying assumptions. Next we show how the number of failures induced by an individual shock can be estimated analytically by means of a mean-field-type approximation, in which Cayley trees (regular networks without loops) play an instrumental role. We then compare our results with numerical simulations of both homogenous and scalefree random networks. We close with a few remarks concerning the policy implications of our work, and point out an intriguing biological analogy.

We present a model of the structure of interbank lending as a random weighted directed network, in which a node _{ij} a loan of amount _{ij} made by _{i} = ∑_{j ← i} _{ij}, is therefore the total interbank exposure of bank _{i} = ∑_{j → i} _{ji}, is in turn the total liability of bank

In addition to its interbank liabilities _{i}, we assume that each bank _{i} (e.g. deposits). We assume that all of these (interbank and senior) liabilities will be available for reinvestment in external investment opportunities at a later time period. On the asset side, we further introduce liquid assets _{i} (e.g. bonds) as well as illiquid assets _{i} (e.g. buildings). The total assets _{i} and total liabilities _{i} of bank _{i} = _{i}+_{i}+_{i} and _{i} = _{i}+_{i}; the difference _{i} = _{i}−_{i} is the net worth of bank

assets _{i} |
liabilities _{i} |
---|---|

liquid assets _{i} |
senior liabilities _{i} |

illiquid assets _{i} |
interbank borrowings _{i} |

interbank loans _{i} |
net worth _{i} |

Basel III [_{i} = _{i}/_{i} (ratio of networth to total assets) and the _{i} = _{i}/_{i} (ratio of liquid assets to total assets). By definition, lowering the ratios Λ_{i} and _{i} increases the exposure of bank

We now introduce a discrete-time investment dynamic, through which a bank can either increase or decrease its net worth _{i}. We assume that the investment period is shorter than the time needed to liquidate illiquid assets. The period begins with the balance sheet introduced in _{i} to invest in some external opportunity, at some interest rate _{i}. (Successful investments correspond to _{i} > 1, hazardous ones correspond to _{i} < 1; in the worst case scenario, the investment is lost in full, viz. _{i} = 0.) We denote _{i} = (_{i} − 1)_{i} the profit made in this transaction. (If a bank only borrows and does not lend, _{i} > 0 and _{i} = 0, we take _{i} = (_{i} − 1)_{i}; equivalently, the profit is defined by _{i} = (_{i} − 1)max{_{i}, _{i}}.) In the second step, a bank uses this profit and its liquid assets _{i} to repay its interbank liabilities _{i} with an interest _{i}. When a bank

From a mathematical perspective, finding the interbank repayments _{i} (a.k.a. the clearing vector) amounts to solving the system of ^{+} = max{ ⋅, 0} and the sum ranges over _{ij} of bank _{ij} = _{ij} _{i}/_{i}, as first proposed by Eisenberg and Noe [

While the set of _{i}, _{i}, Λ_{i} and _{ij} have unit value, so that _{i} = _{i} = _{i}, where _{i} is the degree of node _{i} = 0), and (_{i} take the same value _{0}, call it the “shocked bank”), for which _{i0} = 0.

The reciprocity of the “core” of real financial networks is very high [

Our third assumption represents a worst case scenario in which all banks within a financial network are operating at the limit of regulatory caps, such as the Basel norms. The fourth assumption is due to the model set up, whereby the two periods we analyze are much shorter than the time in which illiquid assets could be liquidated. The assumption of uniform external interest rates is made based on an assumption that the banks are operating in a similar environment.

Our model then creates a single shock to the system, in the form of one bank losing its external investement. Due to interbank lending, all other banks in the network can in principle feel the effects of this shock, either directly (first neighbors, direct creditors) or indirectly (higher order neighbors, indirect creditors). Within this setting, our objective is to estimate the understand the onset of contagion, in particular through the _{0} such that

We begin our investigation of the model by considering the simplest network topology, namely a network with uniform degree

When _{0} inherits only a small fraction of _{0}’s losses—and none fails. For networks with incrementally decreasing degree _{0} gradually increases, until at some point shocked bank _{0}’s weakest neighbor also fails. If degree _{0} also approach criticality, and start failing as well.

This sequence of transitions, involving higher and higher order neighbors of the shocked bank, defines an ordered sequence of “critical degrees” _{(p)}(^{p−1} is the number of nodes at distance _{0}. The values of these critical degrees provide a measure of the robustness of the network with respect to a shock: the higher the critical degrees, the more fragile the financial structure.

The expression for each _{0} are critical. This gives in particular

Real-world financial networks being anything but regular, the usefulness of the exact solution above would seem to be extremely limited. It turns out to be the opposite. In the regime where failures are unlikely to extend beyond the first neighbors of the shocked bank—which is the case for most realistic values of the financial parameters, as illustrated by the low values of

We will assume that, on a general random network, _{i} is smaller than the critical degree _{i}. While this approximation clearly cannot capture all the dynamics of a single network, it does provide a tractable starting point to study the statistics of failure contagion in a given ensemble of random networks. Within this approximation, we obtain the following results (see

First, we have an explicit lower bound on the expected number of failures
_{0} has a subcritical degree. Here _{k ≥ 1}

Second, we show that, whether the network is Poisson-distributed (^{k}/^{−γ}), the failures distributions _{0} to have large degree _{0} to have subcritical degree

To test the validity of these findings, we analyzed two additional types of random networks for which the conditional probability distribution ^{−3}, see

Here

For both network types, we generated 10^{4} random networks for each value of the mean degree

The dashed line indicates the value of the critical degree

Observe the “robust-yet-fragile” nature of scalefree networks: while the maximum expected number of failures is lower than for ER networks, the probability of catastrophic failures is much higher.

The close agreement for these values of the financial parameters (and any other values such that

We have considered the effect of financial variables such as interest rates, leverage ratio and financial exposure on the robustness of interbank systems vis-à-vis individual shocks. Focusing first on regular networks, we obtained an explicit formula for the critical degree, below which failures begin to propagate through the network. From this, we then showed how to derive a simple but reliable lower bound on the expected number of failures and failures distribution in random (and possibly strongly heterogeneous) networks. Besides an in-depth study of cascades beyond first neighbors of the shocked bank, interesting extensions of our work could include a non-linear relation between interbank exposure and network degree, overlapping portfolios, multiple or probabilistic shocks, multiple-period dynamics, and amplifications of failures. Additionally, our assumptions, particularly reciprocal loans, could be relaxed to more realistically represent real-world networks.

The highly stylized character of our model notwithstanding, our results shed new light on important aspects of systemic risks, such as the association between contagion and interest rate policy [

We observed that our

What is more, this interpretation establishes a direct link with a seemingly unrelated problem: the condition for the evolution of cooperation, famously investigated by Hamilton [

Denoting _{ij} the amount repaid by bank

_{i}+_{i}−_{i}+∑_{j ≠ i} _{ji} ≥ _{i}, bank _{i} in full, hence for each

_{i}+_{i}−_{i}+∑_{j ≠ i} _{ji} < _{i}, bank

_{i}+_{i}−_{i}+∑_{j ≠ i} _{ji} ≤ 0, bank _{ij} = 0 for each

We call

In this paper we considered three classes of networks: Cayley trees, ER networks and BA networks. They are defined as follows.

Cayley trees are graphs without loops in which each node is connected to a fixed number of neighbors _{0}, the number of nodes at distance _{0} is ^{d−1}.

ER networks are the simplest random networks: given

BA networks are obtained by means of a stochastic growth process. Starting from a complete graph over (say)

Here we estimate the probability _{1}(_{1}) that _{1} first neighbors of the shocked bank fail. Clearly, the probability that _{0} itself) is larger than the probability that _{0} fail, i.e. _{1}(_{1}(

Consider a random network with degree distribution _{0} has degree _{0} has a subcritical degree. According to our “mean-field” assumption, the probability that _{1} neighbors of _{0} fail is given by the probability _{1} neighbors have subcritical degree, times the probability _{1} first neighbors have supercritical degree, times the number of choices of _{1} failing neighbors among _{0} has _{1}⟩ = ∑_{F1 ≥ 1} _{1} _{1}) (see

Let us now consider the limit of _{1} ≫ 1 (hence for shocked banks with degree

^{−z} ^{k}/

Thus, in Poisson distribued networks, the failures distribution is Poissonian with mean

_{γ} = ^{γ} in the degree distribution ^{γ}. This allows to perform the _{2}𝓕_{1} is the Gauss hypergeometric function, whose asymptotics for large parameters is given in [_{1}(_{1}) is scalefree with exponent

In both cases, the tail of _{1}(_{1}) has the same nature (Poisson or power-law) as the degree distribution itself.

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We thank the participants and staff of the 2013 Santa Fe Institute Complex Systems Summer School (where this project was initiated) for a very stimulating experience. We are especially indebted to T. J. Carter, R. Martinez and M. M. King and for their help in the early stages of this research, and to B. Vaitla for drawing our attention to Ref. [