The q-dependent detrended cross-correlation analysis of stock market

The properties of q-dependent cross-correlation matrices of stock market have been analyzed by using the random matrix theory and complex network. The correlation structures of the fluctuations at different magnitudes have unique properties. The cross-correlations among small fluctuations are much stronger than those among large fluctuations. The large and small fluctuations are dominated by different groups of stocks. We use complex network representation to study these q-dependent matrices and discover some new identities. By utilizing those q-dependent correlation-based networks, we are able to construct some portfolio by those most independent stocks which consistently perform the best. The optimal multifractal order for portfolio optimization is approximately $q=2$. These results have deepened our understanding about the collective behaviors of the complex financial system.