Description of stochastic and chaotic series using visibility graphs

Area of Science:

• Complex systems
• Nonlinear dynamics
• Network theory

Background:

• Nonlinear time series analysis is crucial for understanding complex signals.
• Mapping time series to network representations offers new analytical approaches.
• The visibility algorithm captures time series correlations within associated graphs.

Purpose of the Study:

• To characterize and distinguish between correlated stochastic, uncorrelated, and chaotic processes using network theory.
• To investigate the properties of time series through graph theoretical tools.
• To explore connections between time series analysis, nonlinear dynamics, and graph theory.

Main Methods:

• Application of the horizontal visibility algorithm to time series data.
• Analysis of the resulting graph's degree distribution.
• Comparison with theoretical predictions and simulation results.

Main Results:

• Time series consistently map to graphs with an exponential degree distribution P(k)∼exp(-λk).
• The parameter λ uniquely characterizes the underlying process (stochastic or chaotic).
• The exact frontier between chaotic and correlated stochastic processes is determined as λ=ln(3/2).

Conclusions:

• The horizontal visibility algorithm provides a robust method for distinguishing between different types of time series processes.
• Network properties, specifically the degree distribution, offer insights into the dynamics of the generating process.
• This approach bridges time series analysis, nonlinear dynamics, and complex network theory.