Network trending; leadership, followership and neutrality among companies: A random matrix approach

In this article, we analyze the cross-correlation between returns of different stocks to answer the following important questions. The first one is: If there exists collective behavior in a financial market, how could we detect it? And the second question is: Is there a particular company among the companies of a market as the leader of the collective behavior? Or is there no specified leadership governing the system similar to some complex systems? We use the method of random matrix theory to answer the mentioned questions. Cross-correlation matrix of index returns of four different markets is analyzed. The participation ratio quantity related to each matrices' eigenvectors and the eigenvalue spectrum is calculated. We introduce shuffled-matrix created of cross correlation matrix in such a way that the elements of the later one are displaced randomly. Comparing the participation ratio quantities obtained from a correlation matrix of a market and its related shuffled-one, on the bulk distribution region of the eigenvalues, we detect a meaningful deviation between the mentioned quantities indicating the collective behavior of the companies forming the market. By calculating the relative deviation of participation ratios, we obtain a measure to compare the markets according to their collective behavior. Answering the second question, we show there are three groups of companies: The first group having higher impact on the market trend called leaders, the second group is followers and the third one is the companies who have not a considerable role in the trend. The results can be utilized in portfolio construction.