  Many models and real complex systems possess critical thresholds at which the
systems shift from one sate to another. The discovery of the early warnings of
the systems in the vicinity of critical point are of great importance to
estimate how far a system is from a critical threshold. Multifractal Detrended
Fluctuation analysis (MF-DFA) and visibility graph method have been employed to
investigate the fluctuation and geometrical structures of magnetization time
series of two-dimensional Ising model around critical point. The Hurst exponent
has been confirmed to be a good indicator of phase transition. Increase of the
multifractality of the time series have been observed from generalized Hurst
exponents and singularity spectrum. Both Long-term correlation and broad
probability density function are identified to be the sources of
multifractality of time series near critical regime. Heterogeneous nature of
the networks constructed from magnetization time series have validated the
fractal properties of magnetization time series from complex network
perspective. Evolution of the topology quantities such as clustering
coefficient, average degree, average shortest path length, density,
assortativity and heterogeneity serve as early warnings of phase transition.
Those methods and results can provide new insights about analysis of phase
transition problems and can be used as early warnings for various complex
systems.
