New collectivity measures for financial covariances and correlations

Complex systems are usually non-stationary and their dynamics is often dominated by collective effects. Collectivity, defined as coherent motion of the whole system or some its parts, manifests itself in time-dependent structures covariance correlation matrices. The largest eigenvalue corresponds to a whole, while other large, isolated, eigenvalues indicate collectivity parts system. In case finance, these industrial sectors. By removing latter effects much better revealed. We measure remaining which we refer average sector collectivity. identify signals around Lehman Brothers crash after dot-com bubble burst. For crash, find potential precursor. analyze 213 US stocks over period more than 30 years from 1990 2021. plot versus corresponding study market trajectory two dimensional space spanned both collectivities. Therefore, capture precise way. Additionally, observe that larger values accompanied trend shifts mean covariances correlations. As 2015/2016 stock markets changed fundamentally.