Forest-fire model as a supercritical dynamic model in financial systems

Deokjae Lee, Jae-Young Kim, Jeho Lee, and B. Kahng

Phys. Rev. E 91, 022806 – Published 9 February 2015

Abstract

Recently large-scale cascading failures in complex systems have garnered substantial attention. Such extreme events have been treated as an integral part of self-organized criticality (SOC). Recent empirical work has suggested that some extreme events systematically deviate from the SOC paradigm, requiring a different theoretical framework. We shed additional theoretical light on this possibility by studying financial crisis. We build our model of financial crisis on the well-known forest fire model in scale-free networks. Our analysis shows a nontrivial scaling feature indicating supercritical behavior, which is independent of system size. Extreme events in the supercritical state result from bursting of a fat bubble, seeds of which are sown by a protracted period of a benign financial environment with few shocks. Our findings suggest that policymakers can control the magnitude of financial meltdowns by keeping the economy operating within reasonable duration of a benign environment.

Received 28 July 2014

DOI:https://doi.org/10.1103/PhysRevE.91.022806

Authors & Affiliations

Deokjae Lee¹, Jae-Young Kim², Jeho Lee^3,*, and B. Kahng^1,†

¹Center for Complex Systems Studies and CTP, Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea
²Department of Economics, Seoul National University, Seoul 151-747, Korea
³Graduate School of Business, Seoul National University, Seoul 151-747, Korea

^*jeho0405@snu.ac.kr
^†bkahng@snu.ac.kr

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Vol. 91, Iss. 2 — February 2015

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Images

Figure 1
(a) A schematic illustration of the model. (i) Empty (filled) nodes represent healthy (vulnerable) banks. (ii) A randomly chosen healthy bank becomes vulnerable by taking excessive risks, and a cluster develops composed of four connected vulnerable banks. (iii) One of the banks in the cluster is exposed to a random shock (represented as lightning), the exposed bank fails, and a cascade of bank failures is triggered throughout the entire cluster. (iv) Those failed nodes become healthy. The number of failed banks in (iii) is avalanche size. (b) A typical sequence of the avalanche sizes in a network with $10^{5}$ banks. Most avalanches are small, but a large-scale financial meltdown does occur as shown at time step 600. (c) The cluster size distribution of interconnected vulnerable banks at an onset of a large-scale financial meltdown. There exists a giant cluster (arrow), where complex transactional relationships among banks in vulnerable state will serve as a channel for financial crisis contagion.
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Figure 2
(a) Probability distributions of avalanche sizes for various $q$ . The tails of the distributions systematically deviate from a power-law behavior. (b) Probability distributions of avalanche sizes for various degree exponent values $γ$ . Simulations were run on scale-free networks with exponent $γ = 2.5$ for (a), containing $N = 10^{7}$ nodes and $L = 10 N$ links for (a) and (b).
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Figure 3
(a) Plot of data collapse in small-size avalanche region and (b) intermediate-size avalanche region for different system sizes $N$ . The parameter $q$ is fixed as $10^{- 4}$ . The underlying networks are scale-free networks with degree exponent $γ = 2.5$ . Panel (a) indicates the crossover point $s_{c 1}$ between the first and second region scales as $N^{0.53}$ , and panel (b) indicates the other crossover point $s_{c 2}$ between the second region and third region scales as $N$ .
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Figure 4
Average density of vulnerable nodes as a function of $q$ . The density is measured just before each shock. Degree exponent of the underlying network is taken as $γ = 2.5$ . The scaling behavior is evident in the range of $q$ where the plateau is observed in the average density.
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Figure 5
Data collapse of the avalanche size distributions on two dimensional square lattices of different sizes for a fixed $q = 0.7$ . The linear size of a lattice is denoted as $L$ . The collapse of the decreasing part indicates that the crossover point scales as $L^{1.9}$ . The inset shows the cutoff point of the increasing part scales along the order of the system size $L^{2}$ .
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