Abstract
As effective representations of complex systems, complex networks have attracted scholarly attention for their many practical applications. They also represent a new tool for time series analysis. In order to characterize the underlying dynamic features, the structure of transformed networks should be encoded with the systematic evolution information that always hides behind the time series data. Thus, the way of mapping segments of the time series into nodes of the network is particularly crucial, but it is liable to be unstable under noise and missing values. In this paper, we propose a coarse-graining based on statistics of segments (CBS) founded complex network method, which can make it immune to interference to a certain degree. The time series is divided into many segments by a slide window, of which the width is determined by the multi-scale entropy of the data. We use a multi-dimensional symbol to represent the motion state of every segment. Due to the utilization of the distribution information of the fragments’ numerical characteristics, the coarse-graining process is self-adaptive to some extent. The complex network is then established based on the adjacent relations of the symbolic sequence. With our method, the differences in the network measurements between the periodic and chaotic motion is easily observable. Furthermore, we investigated the robustness of CBS by adding noise and missing values. We found that CBS is still valid, even with strong noise and 15% missing values, and simulation shows that it is more robust than the VG and LS approaches. By mapping a time series into a complex network, we provide a new tool for understanding the dynamic evolution mechanism of a complex system. This method has been applied in various fields, such as physics, engineering, medicine and economics. However, the interference of noise may greatly affects the reliability of judgment, which is based on the structures of transformed networks. An insufficient robustness is mostly to blame for the transformation from a time series to a symbolic sequence. In this paper, we suggest a new approach to the coarse-graining process which is self-adaptive for threshold choosing. Simulations show that even with strong disturbances, our network structure is easily distinguishable under different dynamic mechanisms.
Highlights
In recent decades, complex networks have attracted interest from scholars in many different disciplines.1–3 Current research mainly includes the following 3 aspects: 1. In the preliminary stage, researchers focused on complex network evolution mechanisms,4–6 such as BarabasiAlbert model (BA model),7 which uses a dynamic model of connection preferences to explain that the degree distribution of nodes in complex networks follows a power law; 2
We propose a coarse-graining based on statistics of segments (CBS) founded complex network method, which can make it immune to interference to a certain degree
It includes 3 parts: dividing a time series into segments based on multi-scale entropy; coarse-graining fragments into nodes based on its statistics information; constructing the transition networks
Summary
Complex networks have attracted interest from scholars in many different disciplines. Current research mainly includes the following 3 aspects: 1. In the preliminary stage, researchers focused on complex network evolution mechanisms, such as BarabasiAlbert model (BA model), which uses a dynamic model of connection preferences to explain that the degree distribution of nodes in complex networks follows a power law; 2. The method of mapping a time series into complex networks can be roughly divided into 3 steps: the segmentation according to the specified length, coarse granulation of the fragments, define edges and weights specification. Zhang and Small proposed a surrogate generation method for complex networks that is similar to that of a nonlinear time series analysis based on an efficient algorithm to generate representative samples from the space of all of the networks with a scale-free degree distribution.. To convert a time series into a complex network, the following 3 steps are needed: determine the length of each fragment, apply coarse graining according to the statistical data of each segment, and weight the edges based on the probability transition.
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