[Paper Review] Complexity, information transfer and collective behavior in chaotic dynamical networks

M. Escalona-Morán, G. Paredes+1 more|arXiv (Cornell University)|Oct 22, 2010

Nonlinear Dynamics and Pattern Formation13 references3 citations

TL;DR

This study investigates how complexity, information transfer, and collective behaviors—such as nontrivial collective behavior and chaos synchronization—emerge in globally coupled chaotic map networks. Using a measure of statistical complexity and information-theoretic quantities, the authors show that nontrivial collective behavior correlates with peak complexity and that information transfer from the global mean field to local units is maximized just before the onset of collective states, suggesting a predictive role for information flow in emergence transitions.

ABSTRACT

We investigate the relationship between complexity, information transfer and the emergence of collective behaviors, such as synchronization and nontrivial collective behavior, in a network of globally coupled chaotic maps as a simple model of a complex system. We calculate various quantities for this system: the mean field, a measure of statistical complexity, the information transfer, as well as the information shared, between the macroscopic and local levels as functions of the strength of a coupling parameter in the system. Our results show that the emergence of nontrivial collective behavior is associated to higher values of complexity. Little transference of information from the global to the local level occurs when the system settles into nontrivial collective behavior while no information at all flows between these two scales in a synchronized collective state. As the parameter values for the onset of nontrivial collective behavior or chaos synchronization are approached, the information transfer from the macroscopic level to the local level is higher, in comparison to the situation where those collective states are already established in the system. Our results add support to the view of complexity as an emergent collective property that is absent at the local level in systems of interacting elements.

Motivation & Objective

• To understand the relationship between complexity, information transfer, and collective behavior in networks of interacting chaotic elements.
• To determine whether statistical complexity and information flow can serve as indicators of emergent collective phenomena in spatiotemporal systems.
• To investigate how information transfers between macroscopic (mean field) and microscopic (local map) levels during transitions to collective states.
• To assess whether the onset of nontrivial collective behavior or chaos synchronization is preceded by distinct patterns in information transfer.
• To support the view that complexity is an emergent, collective property absent at the local level in interacting dynamical systems.

Proposed method

• A network of N globally coupled chaotic maps is modeled using a mean-field coupling scheme: $ x_{t+1}^i = (1-\varepsilon)f_i(x_t^i) + \frac{\varepsilon}{N}\sum_{j=1}^N f(x_t^j) $.
• Local dynamics are defined by singular maps $ f(x_t) = b - |x_t|^z $ with $ |z| < 1 $, which exhibit robust chaos without periodic windows.
• Statistical complexity is calculated using the measure proposed by López-Ruiz et al. (1995), quantifying the balance between disorder and correlation in the system.
• Information transfer $ T_{S \rightarrow x^i} $ from the mean field $ S_t $ to a local map $ x_t^i $ is computed using information-theoretic formalism to assess directional flow between scales.
• Mutual information $ M_{S,x^i} $ between the mean field and local states is calculated to quantify shared information across scales.
• The system is analyzed across varying coupling strength $ \varepsilon $, with results mapped to identify transitions between turbulent, nontrivial collective behavior (NTCB), and synchronized states.

Experimental results

Research questions

• RQ1 How does statistical complexity vary across different collective states—turbulent, nontrivial collective behavior, and synchronized—of a globally coupled chaotic map network?
• RQ2 What is the role of information transfer from the global mean field to local elements in the emergence of nontrivial collective behavior?
• RQ3 How does information flow from the macroscopic to the local level change as the system approaches the onset of synchronization or nontrivial collective behavior?
• RQ4 Is there a measurable difference in information sharing between global and local variables during synchronized versus non-synchronized collective states?
• RQ5 Can information transfer serve as a predictor for the emergence of collective behavior in complex dynamical networks?

Key findings

• Nontrivial collective behavior (NTCB) is associated with the highest values of statistical complexity, indicating that complexity peaks when global order coexists with local chaos.
• Information transfer $ T_{S \rightarrow x^i} $ from the mean field to local maps reaches a maximum just before the onset of nontrivial collective behavior or chaos synchronization, suggesting a predictive role for information flow.
• Once chaos synchronization is achieved, information transfer $ T_{S \rightarrow x^i} $ vanishes because the local maps evolve identically to the mean field, eliminating directional information flow.
• Mutual information $ M_{S,x^i} $ is maximized in the synchronized state, confirming that global and local variables become identical and fully correlated.
• In the NTCB region, information transfer decreases significantly, indicating that self-organization reduces the need for top-down information flow from the global level.
• The results support the view that complexity is an emergent, collective property not present at the local level, and that information transfer dynamics can signal impending transitions to collective states.