Spectral methods and cluster structure in correlation-based networks

We investigate how in complex systems the eigenpairs of the matrices derived from the correlations of multichannel observations reflect the cluster structure of the underlying networks. For this we use daily return data from the NYSE and focus specifically on the spectral properties of weight W_{ij} = |C|_{ij} - \delta_{ij} and diffusion matrices D_{ij} = W_{ij}/s_j- \delta_{ij}, where C_{ij} is the correlation matrix and s_i = \sum_j W_{ij} the strength of node j. The eigenvalues (and corresponding eigenvectors) of the weight matrix are ranked in descending order. In accord with the earlier observations the first eigenvector stands for a measure of the market correlations. Its components are to first approximation equal to the strengths of the nodes and there is a second order, roughly linear, correction. The high ranking eigenvectors, excluding the highest ranking one, are usually assigned to market sectors and industrial branches. Our study shows that both for weight and diffusion matrices the eigenpair analysis is not capable of easily deducing the cluster structure of the network without a priori knowledge. In addition we have studied the clustering of stocks using the asset graph approach with and without spectrum based noise filtering. It turns out that asset graphs are quite insensitive to noise and there is no sharp percolation transition as a function of the ratio of bonds included, thus no natural threshold value for that ratio seems to exist. We suggest that these observations can be of use for other correlation based networks as well.