Scale-Free Network in Stock Markets

We study the cross-correlations in stock price changes among the S&P 500 companies by introducing a weighted random graph, where all vertices (companies) are fully connected via weighted edges. The weight of each edge is distributed in the range of [−1, 1] and is given by the normalized covariance of the two modified returns connected, where the modified return means the return minus the mean over all companies. We define an influence-strength at each vertex as the sum of the weights on the edges incident upon that vertex. Then we find that the influence-strength distribution in its absolute magnitude |q| follows a power-law, P (|q|) ∼ |q|−δ, with exponent δ ≈ 1.8(1).