Many natural systems can be considered as a collection of interacting elements forming networks with non-trivial topologies. Nodes of the network can represent dynamical units (observables) whereas the weighted edges quantify correlations between the nodes. In many cases, the structure of such networks reveals high degree of complexity and some optimization procedures concentrating on key network properties are recommended. One example of such methodology is the minimal spanning tree (MST) which is a subgraph of a weighted network that minimizes sum of the edges weights spanning the network. Crucial in this context is determination of the correlations measure (edges weights) between the network nodes. In the standard MST approach weights are expressed by metric using the Pearson coefficient. This, however, imposes some limitations related to linearity and stationarity of the considered time series.   

 In this contribution we propose generalization of the MST methodology and call it the q-dependent minimum spanning tree (qMST). In the proposed approach the r_qcoefficient is applied to quantify the cross-correlation between time series with respect of the considered time scale and amplitude of the analysed signals. In effect, we are able to construct graphs reflecting variability of correlation structure with time and amplitude resolution selected by the r_q coefficient. Moreover, the methodology of estimation of the r_q based on the multifractal cross-correlation analysis (MFCCA) algorithm, makes qMST capable to deal with nonlinearity and nonstationarity of the time series. We demonstrate performance of the proposed methodology by applying it to the analysis of correlations between the American companies quoted on the New York Stock Exchange in the period 1998-1999. Our findings show that the topology of obtained qMST graphs strongly depends on the considered time scale and amplitude of the signals. In particular, analysis of the topological properties of the graphs estimated for different scale values show evolution of the correlation arrangement of the stocks from strongly centralized structure identified on the minute scales to sectorial organization on the monthly one. Moreover, amplitude filtering indicates that the strongest collective behaviour characterizes  intermediate-amplitude fluctuations whereas the largest one have more independent dynamics.   

In conclusion, the  qMST methodology offers a novel range of possibilities to study the hierarchical structure of the system with respect to time scales and amplitudes of the signals. The proposed technique can be of particular interest in financial engineering where it can be directly used to investment portfolio optimization.

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