On the Consequences of Power-Law Behavior in Partial Correlation Network Models

We study the cross-sectional dependence properties of a partial correlation network model with sparse power-law structure. We show that when the degree distribution of the network is power-law, the system exhibits a high degree of collinearity. More precisely, the largest eigenvalues of the inverse covariance matrix converge to an affine function of the degrees of the most interconnected vertices in the network. The result implies that the largest eigenvalues of the inverse covariance matrix are approximately power-law distributed, and that, as the system dimension increases, the eigenvalues diverge. As an empirical illustration we analyse a panel of stock returns of a large set of companies listed in the S&P500 and show that the covariance matrix of returns exhibits empirical features that are consistent with our power-law model