The structure of redundancy in multivariate time series

Redundancy network is proposed to display the structure of dependence between the variables in a complex system. It is realized by three steps. First, from a multivariate time series, one extracts the principal components whose accumulative contribution reaches a specified percent. Second, the combination coefficients whose absolute values are larger (smaller) than a specified threshold are preserved (replaced with zeros). Third, the variables corresponding to the preserved combination coefficients within every principal component are linked together, resulting into a redundancy network. As a typical example we investigate the multi-variate series of prices for component stocks of the Hang Seng index for Hongkong Stock Exchange Market. It is separated into successive segments with a one-year length and a one-month step. The corresponding redundancy networks form a temporal network. The dependence between the stocks and the subsequent properties such as the cliques, the fission/fusion of the communities, the star-structures starting from the stocks each, turn out to be closely related with the onset, the spreading’s acceleration, the spreading’s deceleration, and the disappearance of the COVID-19’s pandemic. The properties of temporal network jointly provide a multi-dimensional portrait for every individual stock, by which one can evaluate its contribution to the whole system.