Structural Entropy: Monitoring Correlation-Based Networks Over Time With Application To Financial Markets.

The concept of "Structural Diversity" of a network refers to the level of dissimilarity between the various agents acting in the system, and it is typically interpreted as the number of connected components in the network. This key property of networks has been studied in multiple settings, including diffusion of ideas in social networks and functional diversity of regions in brain networks. Here, we propose a new measure, "Structural Entropy", as a revised interpretation to "Structural Diversity". The proposed measure relies on the finer-grained network communities (in contrast to the network's connected components), and takes into consideration both the number of communities and their sizes, generating a single representative value. We then propose an approach for monitoring the structure of correlation-based networks over time, which relies on the newly suggested measure. Finally, we illustrate the usefulness of the new approach, by applying it to the particular case of emergent organization of financial markets. This provides us a way to explore their underlying structural changes, revealing a remarkably high linear correlation between the new measure and the volatility of the assets' prices over time.