Eigen-entropy measure to study phase separation in market behavior

One of the spectacular examples of a complex system is the financial market, which displays rich correlation structures among price returns of different assets. The eigenvalue decomposition of a correlation matrix into partial correlations - market, group and random modes, enables identification of dominant stocks or "influential leaders" and sectors or "communities". The correlation-based network of leaders and communities changes with time, especially during market events like crashes, bubbles, etc. Using a novel entropy measure - eigen-entropy, computed from the eigen-centralities (ranks) of different stocks in the correlation-network, we extract information about the "disorder" (or randomness) in the market and its modes. The relative-entropy measures computed for these modes enable us to construct a "phase space", where the different market events undergo "phase-separation" and display "order-disorder" transitions, as observed in critical phenomena in physics. We choose the US S&P-500 and Japanese Nikkei-225 financial markets, over a 32-year period, and study the evolution of the cross-correlation matrices computed over different short time-intervals or "epochs", and their corresponding eigen-entropies. We compare and contrast the empirical results against the numerical results for Wishart orthogonal ensemble (WOE), which has the maximum disorder (randomness) and hence, the highest eigen-entropy. This new methodology helps us to better understand market dynamics, and characterize the events in different phases as anomalies, bubbles, crashes, etc. This can be easily adapted and broadly applied to the studies of other complex systems such as in brain science or environment.