Scale-Dependent Inverse Temperature Features Associated With Crashes in the US and Japanese Stock Markets

Some complex systems (e.g., an ecosystem) in direct contact with an environment can be assigned the temperature of the environment. Other complex systems, such as human beings, can maintain a core temperature of 36.5°C in environments with different temperatures, at least for a short period of time. Finally, for complex systems such as financial markets, whose environments we understand very little of, is there even a reasonable way to define a temperature? It is clear that human beings are almost never in thermal equilibrium with their surroundings, but can financial markets achieve detailed balance independently at all scales, or is information flow in such systems different at different scales? If we combine the information-theoretic picture with the thermodynamics picture of entropy, temperature is the driving force for changes in information content of a system. From an interactions point of view, the information content of a financial market can be computed from the cross correlations between its stocks. In their 2015 paper, Ye et al. (2015) constructed the normalized graph Laplacians in different time periods based on strong cross correlations between stocks listed on the New York Stock Exchange. By writing the partition function in terms of polynomials of the normalized graph Laplacian, Ye et al. computed the average energy E, entropy S, and inverse temperature β = 1/k_BT. This led us to an information-based definition of the inverse temperature. In this work, we investigated the inverse temperature β(ϵ,  n) at different times n and scales ϵ for two mature financial markets, using the S&P 500 and Nikkei 225 cross sections of stocks from January 2007 to May 2023. In the dynamics of β, the most prominent features are peaks at various times. We identified five esoteric and seven characteristic peaks and studied how they change with scale ϵ. The latter consists of a negative power-law dip followed by a positive power-law rise, with exponents narrowly distributed between 0.3–0.4. In addition, we constructed heat maps of β that reveal positive-, negative-, and infinite-slope cascades that hint at their possible exogenous and endogenous origins. Notably, the heat map of β confirmed that the 2007−2009 Global Financial Crisis was an endogenous crash in the US market, which in turn caused an exogenous crash in the Japanese stock market. To better understand the evolution of β, we analyzed ΔJ (the difference in the number of links) and ΔQ (the difference in the number of triangles) and found they oscillate in time. Occasionally, very intense swings of ΔQemerge over all scales, suggesting significant market-level reconstructions at these times.