Fractal Analysis on Human Behaviors Dynamics

📝 Abstract

The study of human dynamics has attracted much interest from many fields recently. In this paper, the fractal characteristic of human behaviors is investigated from the perspective of time series constructed with the amount of library loans. The Hurst exponents and length of non-periodic cycles calculated through Rescaled Range Analysis indicate that the time series of human behaviors is fractal with long-range correlation. Then the time series are converted to complex networks by visibility graph algorithm. The topological properties of the networks, such as scale-free property, small-world effect and hierarchical structure imply that close relationships exist between the amounts of repetitious actions performed by people during certain periods of time, especially for some important days. Finally, the networks obtained are verified to be not fractal and self-similar using box-counting method. Our work implies the intrinsic regularity shown in human collective repetitious behaviors.

💡 Analysis

The study of human dynamics has attracted much interest from many fields recently. In this paper, the fractal characteristic of human behaviors is investigated from the perspective of time series constructed with the amount of library loans. The Hurst exponents and length of non-periodic cycles calculated through Rescaled Range Analysis indicate that the time series of human behaviors is fractal with long-range correlation. Then the time series are converted to complex networks by visibility graph algorithm. The topological properties of the networks, such as scale-free property, small-world effect and hierarchical structure imply that close relationships exist between the amounts of repetitious actions performed by people during certain periods of time, especially for some important days. Finally, the networks obtained are verified to be not fractal and self-similar using box-counting method. Our work implies the intrinsic regularity shown in human collective repetitious behaviors.

📄 Content

Fractal Analysis on Human Behaviors Dynamics

Chao Fan 1,*, Jin-Li Guo 1,†, Yi-Long Zha2

1. Business School, University of Shanghai for Science and Technology, Shanghai, 200093, PR China
2. Web Sciences Center, University of Electronic Science and Technology of China, Chengdu, 610054, PR China

Abstract: The study of human dynamics has attracted much interest from many fields recently. In this paper, the fractal characteristic of human behaviors is investigated from the perspective of time series constructed with the amount of library loans. The Hurst exponents and length of non-periodic cycles calculated through Rescaled Range Analysis indicate that the time series of human behaviors is fractal with long-range correlation. Then the time series are converted to complex networks by visibility graph algorithm. The topological properties of the networks, such as scale-free property, small-world effect and hierarchical structure imply that close relationships exist between the amounts of repetitious actions performed by people during certain periods of time, especially for some important days. Finally, the networks obtained are verified to be not fractal and self-similar using box-counting method. Our work implies the intrinsic regularity shown in human collective repetitious behaviors.

Key Words: human dynamics, fractal, long-range correlation, time series, visibility graph.                                                               *  Working organization: College of Arts and Science, Shanxi Agricultural University.    †  Corresponding author E‐mail addresses: Phd5816@163.com.    2   

1. Introduction In recent years, the studies of statistical characteristics in human behaviors attract much attention of researchers. In the past, it is generally assumed that human behaviors happen randomly, thus one of the conclusions of this assumption is that human behaviors can be described with Poisson process. Since 2005, as the inter-event time distributions of many human behaviors in daily life and work being investigated, such as E-mail and surface mail communications [1, 2], people found that these behaviors are totally different from the previous assumption: human behaviors exhibit the inhomogeneous feature with bursts and heavy tails, namely, the inter-event time distributions behave the right-skewed power-law shape. From then on, close attention have been paid to the research of human dynamics both in temporal scaling law such as web-browsing [3], short message communication [4], logistics operation [5] and spatial scaling characteristic in human mobility [6]. Moreover, many dynamic mechanisms have been proposed to explain the origin of the power-law distribution [1, 7, 8]. Given that people always repeat some actions in a certain period, the amount of events performed can be seen as a time series. Time series is defined as a set of quantitative observations recorded at a specific time and arranged in chronological order [9, 10]. It can be divided to two kinds, i.e. continuous ones and discrete ones, and generally time is considered as discrete variables in time series analysis which attracts special attention due to its practical and theoretical importance in physics, biology economics and society. Theoretical physics is one of the basic origins of the ideas and the methods. Applications of physical theories have led to fruitful achievements in this field. To cite an example, the complexity theory has played an important role in finding the non-trivial features in time series such as the long-range correlations, the scale invariance and so on. The studies of time series have great help to find the underlying rules of variables and forecast the future trend.
   Complex network theory [11, 12] is a new branch in statistical physics, in which complex systems are described with networks. The nodes and edges represent the   3    elements and the relationships between them respectively. The statistical characteristics of complex networks are mainly reflected by the measures like degree distribution, clustering coefficient and average shortest path length. Many networks in real world such as scientific collaboration network, Internet, air line network, protein interaction network, etc. show partially some of the following attributes like scale-free property, small-world effect, hierarchical structure as well as fractal and self-similar feature. Recently, several efforts have been made to bridge time series and complex networks. Zhang et al. [13, 14] introduced the cycle networks for oscillation series, in which the segments between successive extremes are regarded as nodes and the distances in phase space between the nodes are used to construct the edges. Yang et al. [15] proposed a reliable procedure for constructing complex networks from the correlation matrix of a time series. Then Lacasa et al. [16] proposed the so-called visibility graph algorithm,

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