Abstract
Many models and real complex systems possess critical thresholds at which the systems shift dramatically from one sate to another. The discovery of early-warnings in the vicinity of critical points are of great importance to estimate how far the systems are away from the critical states. Multifractal Detrended Fluctuation analysis (MF-DFA) and visibility graph method have been employed to investigate the multifractal and geometrical properties of the magnetization time series of the two-dimensional Ising model. Multifractality of the time series near the critical point has been uncovered from the generalized Hurst exponents and singularity spectrum. Both long-term correlation and broad probability density function are identified to be the sources of multifractality. Heterogeneous nature of the networks constructed from magnetization time series have validated the fractal properties. Evolution of the topological quantities of the visibility graph, along with the variation of multifractality, serve as new early-warnings of phase transition. Those methods and results may provide new insights about the analysis of phase transition problems and can be used as early-warnings for a variety of complex systems.
Highlights
Complex systems are formed by subunits that interact non-linearly with each other
We have used the multifractal detrended analysis (MF-detrended fluctuation analysis (DFA)) and the visibility graph method to analyze the outputs of the two-dimensional Ising model—magnetization time series
The generalized Hurst exponents uncover the transformation of time series form weak multifractal to strong multifractal when temperature approaches critical region
Summary
Complex systems are formed by subunits that interact non-linearly with each other. There are many examples of critical transitions that pose potential threats to our daily life Such potentially dangerous examples include spontaneous systemic failures disease for human beings, systemic market crashes for global finance, abrupt shifts in ocean circulation or climate and so on. It is not possible, for those complex systems, to fully anticipate their behaviors in terms of behaviors of their components. Characterizing the dynamical process of complex systems from macroscopic quantity, for example time series, is a fundamental problem of significant importance in many research fields [2, 3]
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