Structure of a financial cross-correlation matrix under attack

We investigate the structure of a perturbed stock market in terms of correlation matrices. For the purpose of perturbing a stock market, two distinct methods are used, namely local and global perturbation. The former involves replacing a correlation coefficient of the cross-correlation matrix with one calculated from two Gaussian-distributed time series while the latter reconstructs the cross-correlation matrix just after replacing the original return series with Gaussian-distributed time series. Concerning the local case, it is a technical study only and there is no attempt to model reality. The term ‘global’ means the overall effect of the replacement on other untouched returns. Through statistical analyses such as random matrix theory (RMT), network theory, and the correlation coefficient distributions, we show that the global structure of a stock market is vulnerable to perturbation. However, apart from in the analysis of inverse participation ratios (IPRs), the vulnerability becomes dull under a small-scale perturbation. This means that these analysis tools are inappropriate for monitoring the whole stock market due to the low sensitivity of a stock market to a small-scale perturbation. In contrast, when going down to the structure of business sectors, we confirm that correlation-based business sectors are regrouped in terms of IPRs. This result gives a clue about monitoring the effect of hidden intentions, which are revealed via portfolios taken mostly by large investors.