State transfer networks for fGms series.

<p>Segment length is selected to be <i>s</i> = 5. Subplots (a1)-(f1) are the original state transfer networks for the fGm series with <i>H</i> = 0.5, 0.6, 0.65, 0.7, 0.75, and 0.8, respectively. The nodes that have self-links are marked with red color. The label <i>x</i>(<i>y</i>) means the state occurs for the first time at the position <i>x</i> along the time series (the <i>x</i>’th segment), and its identifier number is <i>y</i>; (a2)-(f2) The strong state transfer networks constructed by filtering out weak links (less than 25) in the original state transfer networks; (a3)-(f3) Shuffled networks. One can shuffle each original fGm series, and construct from the resulting series a shuffled network. Each displayed shuffled network is an average over 1000 realizations. Weak links also are filtered out. Except in the networks shown in (a1)-(f1), the size of a node indicates the occurring degree of the state. The width of an edge is the link’s weight.</p

Bridge Successive States

A concept called temporal network of evolutionary matrices is proposed to identify the evolutionary laws for complex systems. The key idea is to separate the trajectory into overlapping segments. The time step between each pair of successive states is assumed to be so short that they can be bridged by a matrix. And the time duration covered by the segment is assumed to be so short that the matrices bridging the pairs of successive states are identical, called evolutionary matrix. The trajectory is them mapped to a temporal network of evolutionary matrices (”bridges”), describing the evolutionary law. Investigations on the series generated with the fractional Brownian motion, and the records for stock markets distributed over the world show that, there exist in all the evolutionary angle series long?range correlations. For the fBm increment series, the series generated with the Heston model, and stock index series, the influences of variables on themselves and the influences between adjacent variables form the backbone of the temporal networks. Non-adjacent impacts can fluctuate simultaneously. The markets in Japan as the center affects Mainland China and is unilaterally affected by the America and Mainland China. After financial crisis, there appear some abrupt and large fluctuations of the evolutionary matrices for the components of the Dow Jones stock market.</p

Long-Range Correlations in Sentence Series from <i>A Story of the Stone</i>

<div><p>A sentence is the natural unit of language. Patterns embedded in series of sentences can be used to model the formation and evolution of languages, and to solve practical problems such as evaluating linguistic ability. In this paper, we apply de-trended fluctuation analysis to detect long-range correlations embedded in sentence series from <i>A Story of the Stone</i>, one of the greatest masterpieces of Chinese literature. We identified a weak long-range correlation, with a Hurst exponent of 0.575±0.002 up to a scale of 10⁴. We used the structural stability to confirm the behavior of the long-range correlation, and found that different parts of the series had almost identical Hurst exponents. We found that noisy records can lead to false results and conclusions, even if the noise covers a limited proportion of the total records (e.g., less than 1%). Thus, the structural stability test is an essential procedure for confirming the existence of long-range correlations, which has been widely neglected in previous studies. Furthermore, a combination of de-trended fluctuation analysis and diffusion entropy analysis demonstrated that the sentence series was generated by a fractional Brownian motion.</p></div

All the states that turn out to be motifs in the stock market index series.

<p>Segment length is selected to be <i>s</i> = 6.</p

Long-range correlations in the noisy and cleaned series.

<p>Noisy records can result in incorrect estimates of the Hurst exponents (solid circles) and consequently false conclusions, even if they only cover a limited proportion of the total series. The effect of noise can be removed using a cleansing procedure (open circles). The X-and E-part of the text are the first to 80th chapters, and the 81th to the 120th chapters, which are currently attributed to Xueqin Cao and E Gao, respectively.</p

All the states occurring in the state transfer networks for the fGm and stock market index series.

<p>Segment length is selected to be <i>s</i> = 5. Each state is assigned an identifier number as presented below it.</p

Degree, degree ratio, and persistent behaviors of motifs for the stock markets.

<p>Segment length is selected to be <i>s</i> = 6. (a1)and (b1) show the occurrence degrees of the states in the original and shuffled S&P500 and Nasdaq index series, respectively. (a2)-(e2) present the degree ratios for all the states in the S&P500 and Nasdaq index series, respectively; (a3)-(e3) Relations of <i>R</i>/<i>S</i> versus <i>n</i> obtained from occurring position series of the motifs, from which one can find persistent behaviors of the motifs’ occurring along the series.</p

Degree, degree ratio, and persistent behaviors of motifs for fGm series.

<p>Segment length is selected to be <i>s</i> = 6. (a1)-(e1) show the occurrence degrees of the states in the original and shuffled fGm series with <i>H</i> = 0.6, 0.65, 0.7, 0.75 and 0.8, respectively; (a2)-(e2) present the degree ratios for all the states (visibility graphs) in the series with <i>H</i> = 0.6, 0.65, 0.7, 0.75, and 0.8, respectively; (a3)-(e3) Relations of <i>R</i>/<i>S</i> versus <i>n</i> obtained from occurring position series of the motifs, from which one can find persistent behaviors of the motifs’ occurring along the series.</p

Degree, degree ratio, and persistent behaviors of motifs for the stock markets.

<p>Segment length is selected to be <i>s</i> = 5. (a1)and (b1) show the occurrence degrees of the states in the original and shuffled S&P500 and Nasdaq index series, respectively. From the occurrence degrees one can easily identify hubs in the original and shuffled series; (a2)-(e2) present the degree ratios for all the states in the S&P500 and Nasdaq index series, respectively. From the ratios one can easily find the the motifs. A state being called a motif means that its degree in the original series is significantly larger than that in the shuffled series; (a3)-(e3) Relations of <i>R</i>/<i>S</i> versus <i>n</i> obtained from occurring position series of the motifs, from which one can find persistent behaviors of the motifs’ occurring along the series.</p

All the states that turn out to be motifs in the fGm series with <i>H</i> = 0.6, 0.65, 0.7, 0.75, 0.8 (the total number of occurrence states is 132).

<p>Segment length is selected to be <i>s</i> = 6.</p