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In order to explore the characteristics of the evolution behavior of the time-varying relationships between multivariate time series, this paper proposes an algorithm to transfer this evolution process to a complex network. We take the causality patterns as nodes and the succeeding sequence relations between patterns as edges. We used four time series as sample data. The results of the analysis reveal some statistical evidences that the causalities between time series is in a dynamic process. It implicates that stationary long-term causalities are not suitable for some special situations. Some short-term causalities that our model recognized can be referenced to the dynamic adjustment of the decisions. The results also show that weighted degree of the nodes obeys power law distribution. This implies that a few types of causality patterns play a major role in the process of the transition and that international crude oil market is statistically significantly not random. The clustering effect appears in the transition process and different clusters have different transition characteristics which provide probability information for predicting the evolution of the causality. The approach presents a potential to analyze multivariate time series and provides important information for investors and decision makers.

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Aiming to investigate the evolution of mean and volatility spillovers between oil and stock marketsin the time and frequency dimensions,we employed WTI crude oil prices, the S&P 500 (USA) index and the MICEX index (Russia) for the period Jan. 2003-Dec. 2014 as sample data. We first applied a wavelet-based GARCH-BEKK method to examine the spillover features in frequency dimension.To consider the evolution of spillover effectsin time dimension at multiple-scales, we then divided the full sample period into three sub-periods, pre-crisis period, crisis period, and post-crisis period. The results indicate thatspillover effects varyacross wavelet scales in terms of strength and direction.By analysis the time-varying linkage, we found the different evolution features of spillover effects between the Oil-US stock market and Oil-Russia stock market. The spillover relationship between oil and US stock market is shifting to short-termwhile the spillover relationship between oil and Russia stock market is changing to all time scales. That result implies that the linkage between oil and US stock market is weakening in the long-term, and the linkage between oil and Russia stock market is getting close in all time scales. This may explain the phenomenon that the US stock index and the Russia stock index showed the opposite trend with the falling of oil price in the post-crisis period. .

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