Sampling frequency dependent visibility graphlet approach to time series

Recent years have witnessed special attention on complex network based time series analysis. To extract evolutionary behaviors of a complex system, an interesting strategy is to separate the time series into successive segments, map them further to graphlets as representatives of states, and extract from the state (graphlet) chain transition properties, called graphlet based time series analysis. Generally speaking, properties of time series depend on the time scale. In reality, a time series consists of records that are sampled usually with a specific frequency. A natural question is how the evolutionary behaviors obtained with the graphlet approach depend on the sampling frequency? In the present paper, a new concept called the sampling frequency dependent visibility graphlet is proposed to answer this problem. The key idea is to extract a new set of series in which the successive elements have a specified delay and obtain the state transition network with the graphlet based approach. The dependence of the state transition network on the sampling period (delay) can show us the characteristics of the time series at different time scales. Detailed calculations are conducted with time series produced by the fractional Brownian motion, logistic map and Rössler system, and the empirical sentence length series for the famous Chinese novel entitled A Story of the Stone. It is found that the transition networks for fractional Brownian motions with different Hurst exponents all share a backbone pattern. The linkage strengths in the backbones for the motions with different Hurst exponents have small but distinguishable differences in quantity. The pattern also occurs in the sentence length series; however, the linkage strengths in the pattern have significant differences with that for the fractional Brownian motions. For the period-eight trajectory generated with the logistic map, there appear three different patterns corresponding to the conditions of the sampling period being odd/even-fold of eight or not both. For the chaotic trajectory of the logistic map, the backbone pattern of the transition network for sampling 1 saturates rapidly to a new structure when the sampling period is larger than 2. For the chaotic trajectory of the Rössler system, the backbone structure of the transition network is initially formed with two self-loops, the linkage strengths of which decrease monotonically with the increase of the sampling period. When the sampling period reaches 9, a new large loop appears. The pattern saturates to a complex structure when the sampling period is larger than 11. Hence, the new concept can tell us new information on the trajectories. It can be extended to analyze other series produced by brains, stock markets, and so on.