Emerging spectra characterization of catastrophic instabilities in complex systems

Abstract

Random matrix theory has been widely applied in physics, and even beyond physics. Here, we apply such tools to study catastrophic events, which occur rarely but cause devastating effects. It is important to understand the complexity of the underlying dynamics and signatures of catastrophic events in complex systems, such as the financial market or the environment. We choose the USA S&P-500 and Japanese Nikkei-225 financial markets, as well as the environmental ozone system in the USA. We study the evolution of the cross-correlation matrices and their eigen spectra over different short time-intervals or ‘epochs’. A slight non-linear distortion is applied to the correlation matrix computed for any epoch, leading to the emerging spectrum of eigenvalues, mainly around zero. The statistical properties of the emerging spectrum are intriguing—the smallest eigenvalues and the shape of the emerging spectrum (characterized by the spectral entropy) capture the system instability or criticality. Importantly, the smallest eigenvalue could also signal a precursor to a market catastrophe as well as a ‘market bubble’. We demonstrate in two paradigms the capacity of the emerging spectrum to understand the nature of instability; this is a new and robust feature that can be broadly applied to other physical or complex systems.

Network geometry and market instability

“…A major goal of this research is to evaluate different notions of discrete Ricci curvature for their ability to unravel the structure of complex financial networks and serve as indicators of market instabilities. Our study confirms that during a normal period the market is very modular and heterogeneous, whereas during an instability (crisis) the market is more homogeneous, highly connected and less modular [18,21,22,57]. Further, we find that the discrete Ricci curvature measures, especially Forman-Ricci curvature [43,48], capture well the system-level features of the market and hence we can distinguish between the normal or 'business-as-usual' periods and all the major market crises (bubbles and crashes).…”

“…Traditionally, the volatility of the market captures the ‘fear’ and the evaluated risk captures the ‘fragility’ of the market. Some of us showed in our earlier papers that the mean market correlation and the spectral properties of the cross-correlation matrices can be used to study the market states [ 20 ] and identify the precursors of market instabilities [ 22 ]. A goal of this study is to show that the state of the market can be continuously monitored with certain network-based measures.…”

“…Moreover, there are often multiple reasons leading to a market crash or a bubble burst. The study and characterization of such market events, including exogenous shocks, bubble bursts and anomalies, corresponding to such peaks has already been done in our earlier papers [ 20 – 22 , 30 ]. The findings of the present paper are in concordance with the earlier ones.…”

“…The study of empirical cross-correlations among stock prices goes back to more than two decades [11][12][13][14][15][16]. One of commonly adopted approaches for the modelling and analysis of complex financial systems has been correlation-based networks, and it has emerged as an important tool [11,12,[17][18][19][20][21][22].…”

Network-centric Indicators for Fragility in Global Financial Indices

“…Secondly, we have computed the eigenentropy [26] which involves the calculation of the Shannon entropy using the eigenvector centralities of the correlation matrix C τ (t) of market indices. Both mean correlation and eigenentropy have been shown to detect critical events in financial markets [26][27][28]. Thirdly, we have computed the risk corresponding to the Markowitz portfolio of the market indices, which is a proxy for the fragility or systemic risk of the global financial network [29].…”