Complexity-Entropy Plane Analysis

Updated 27 July 2025

The Complexity-Entropy Plane is a computational framework that quantifies and discriminates systems based on normalized Shannon entropy and statistical complexity.
It employs symbolic dynamics and parameterized entropic measures like Tsallis and Rényi entropy to distinguish between ordered, chaotic, and stochastic regimes.
The methodology transforms raw data into symbolic sequences, estimates probability distributions, and maps them onto bounded planes for effective system classification.

The complexity-entropy plane is a mathematical and computational framework for the quantitative description and discrimination of complex systems, processes, and structures based on joint analysis of entropy (disorder, unpredictability) and statistical complexity (organization, correlational structure). It encompasses a family of two-dimensional diagrams, parameterized by suitable definitions of entropy and complexity, that are capable of distinguishing between regimes such as order, chaos, stochasticity, and functional/non-functional organizational patterns. The framework has been adapted to time series, spatial data, networks, quantum systems, and more.

1. Mathematical Formulation and Variants

The most widely used instantiations of the complexity-entropy plane rely on symbolic dynamics and information-theoretic quantification. Common choices for the axes are normalized Shannon entropy ( $H_S$ ) along the abscissa (quantifying randomness, uncertainty) and a statistical complexity measure ( $C$ ) on the ordinate (capturing disequilibrium or correlational structure).

Permutation entropy and statistical complexity (e.g., Jensen–Shannon complexity, $C_{JS}$ ) are central to many applications (1105.4550, Weck et al., 2014, 1112.2316, Wiedermann et al., 2017):

For a symbolic sequence (e.g., series of ordinal patterns via the Bandt–Pompe method), let $P = \{ p(\pi) \}$ be the distribution over $n$ possible patterns.
The normalized Shannon entropy:

$H_S[P] = -\frac{1}{\ln n} \sum_{i=1}^n p_i \ln p_i$

The statistical complexity measure (Jensen–Shannon form):

$C_{JS}[P] = Q_J(P,P_e) \cdot H_S[P]$

where $P_e$ is the uniform distribution, and $Q_J(P,P_e)$ is a normalized Jensen–Shannon divergence.

Generalizations exploit parameterized entropic forms:

Tsallis $q$ -entropy, yielding the $q$ -complexity-entropy curve with

$S_q(P) = \sum_{j=1}^{n} p_j \log_q (1/p_j)$

and corresponding generalized complexity (Ribeiro et al., 2017).

Rényi $\alpha$ -entropy, producing Rényi complexity-entropy curves with

$S_\alpha(P) = \frac{1}{1-\alpha} \ln \left( \sum_{i=1}^n p_i^\alpha \right)$

and a properly normalized complexity measure (Jauregui et al., 2018).

In quantum systems, the complexity-entropy plane is specialized by defining entropy and complexity through phase-space distributions (e.g., Wigner function harmonics entropy (1009.0560)) or via basis-independent quantum statistical measures (e.g., difference between Shannon entropy and second-order Rényi entropy (Varga, 10 Aug 2024)).

2. Methodology and Algorithmic Steps

The construction of the complexity–entropy plane generally requires:

Symbolization: Transformation of raw data (e.g., numerical time series, images, network adjacency matrix) into a discrete set of symbols or patterns, typically via embedding and ordinal analysis (Bandt–Pompe method for time series (1105.4550, 1112.2316, Weck et al., 2014); block analysis for binary series (Pinto et al., 24 Mar 2025); subarray analysis for images (Zunino et al., 2016, Kim et al., 19 Aug 2024)).
Estimation of Probability Distribution: Empirical probabilities $p(\pi)$ reflect the frequencies of the extracted symbols/patterns.
Entropy Calculation: Compute entropy using the chosen functional (Shannon, Tsallis, Rényi, etc.).
Complexity Calculation: Quantify disequilibrium via divergence to the uniform distribution—typically Jensen–Shannon divergence or alternatives—combined multiplicatively or otherwise with the entropy.
Normalization and Mapping: Normalize both entropy and complexity so the resulting plane is bounded, typically $0 \le H, C \le 1$ .
Analysis and Interpretation: Each data set, dynamical state, or network is plotted as a point in the plane. Trajectories (time or parameter dependence), statistical distributions, or entire curves (for parameterized entropy/compexity families) can also be considered.

3. Discriminative Power and Classification

A central strength of the complexity-entropy plane is its ability to distinguish qualitatively different regimes:

Random processes: Occupy the upper right (maximal entropy, minimal complexity).
Periodic or highly ordered dynamics: Found at lower left (minimal entropy, minimal complexity).
Deterministic chaos/complex organization: Populate interior regions at intermediate entropy and maximal complexity (1105.4550, 1112.2316, Weck et al., 2014).
Distinct structural classes in images (textures), networks, musical signals, or quantum many-body dynamics are separated as clusters or distinct trajectories in the plane (Zunino et al., 2016, 1112.2316, Wiedermann et al., 2017, Kim et al., 19 Aug 2024).

Table: Representative Regions in the Complexity-Entropy Plane | Regime | Normalized Entropy | Statistical Complexity | |-----------------------|--------------------|-----------------------| | Random (white noise) | $\approx 1$ | $\approx 0$ | | Periodic/Ordered | $\approx 0$ | $\approx 0$ | | Chaotic/Complex | $0 < H < 1$ | High |

4. Generalizations and Extensions

The complexity–entropy framework has been significantly extended:

Parametric families: Construction of $q$ -complexity–entropy (using Tsallis $q$ -entropy) and Rényi complexity–entropy curves, providing greater discriminative sensitivity and the ability to elucidate scaling regimes and distinct dynamical classes (chaos, stochasticity, periodicity) via geometric properties of these curves (e.g., looped vs. open, positive/negative curvature) (Ribeiro et al., 2017, Jauregui et al., 2018).
Multiscale analysis: Varying the embedding delay or patch size in images yields information about correlations and structure at different spatial or temporal scales, facilitating discrimination of roughness in textures or crossovers in fractals (Zunino et al., 2016).
Binary sequences: Adaptation to binary time series for financial time series and cryptocurrencies by defining block entropy and block statistical complexity (BiCEP), with a tailored inefficiency score for market efficiency ranking (Pinto et al., 24 Mar 2025).
Networks: Definition of network entropy (average per-node entropy) and statistical complexity (product with network-level Jensen–Shannon divergence), yielding discrimination of real-world and model network classes (Wiedermann et al., 2017).
Quantum systems: Definition of complexity via Wigner harmonics entropy or basis-independent differences of quantum entropies (Shannon–Rényi), enabling basis-free diagnosis of quantum phase transitions, entanglement, and dynamical complexity (1009.0560, Varga, 10 Aug 2024, Bergamasco et al., 2019, Shekhar et al., 2023).

5. Applications

The complexity-entropy plane has found diverse application domains:

Time series analysis: Distinguishing chaos from noise in logistical maps (decay of missing ordinal patterns, robustness to noise) (1105.4550), characterization of financial, physiological, seismic, geophysical, and musical data (1112.2316, Ribeiro et al., 2017, Jauregui et al., 2018).
Turbulence: Differentiation of laboratory plasma turbulence from fully-developed solar wind turbulence; association of low complexity/high entropy with high-degree-of-freedom turbulence (Weck et al., 2014).
Image texture discrimination: Multiscale 2D planes reveal differences in natural and artificial textures, identification of crossovers in fractal images, and quantification of surface roughness in physical and biological samples (Zunino et al., 2016).
Networks: Mapping and discriminating social, infrastructural, brain, and model networks by their entropy and complexity; threshold optimization to maximize functional complexity (Wiedermann et al., 2017).
Quantum many-body systems and phase transitions: Detecting quantum phase transitions in Ising chains by observing peaks in entropy growth rate (harmonics entropy), and distinguishing between integrable and chaotic regimes via long-term entropy fluctuations (1009.0560).
Physiology: Differentiating healthy vs. pathological dynamics in cardiac signals and EEGs on the basis of entropy-complexity coordinates (Omidvarnia et al., 2018, Letellier et al., 2019).
Financial and crypto markets: Binary complexity analysis of price fluctuation series reveals market efficiency and the impact of consensus design in cryptocurrencies (Pinto et al., 24 Mar 2025).
Artistic evolution: Visualization of evolution in user-generated visual arts by mapping over 149,000 images into the (C,H) plane, linking the emergence of novel styles to increased diversity in a specific region (Kim et al., 19 Aug 2024).

6. Interpretative and Theoretical Insights

Intermediate complexity: Maximum statistical complexity arises neither at perfect order nor complete randomness, but at intermediate, optimally mixed states—a principle that traces back to the Gell-Mann notion of complexity, formalized via e.g., nonlinear transformations of entropy (Klamut et al., 2020).
Disentangling chaos and randomness: The open/closed geometry of parametric complexity-entropy curves (e.g., q-complexity or Rényi curves) provides an operational criterion to distinguish deterministic chaos from stochasticity (Ribeiro et al., 2017, Jauregui et al., 2018).
Criticality and universality: Scaling analysis in quantum and classical systems (e.g., entanglement entropy scaling, infinite universality classes parameterized by a complexity parameter, or critical points identified as intersection curves in finite-size scaling (Shekhar et al., 2023)) links the complexity-entropy plane to phase transitions and universality theory.
Robustness and basis-independence: Measures constructed from density matrices averaged over disorder (correlational entropy) and differences between entropies of different orders (e.g., Shannon–Rényi) yield basis-free quantification suitable for both thermal and quantum-coherent regimes (Varga, 10 Aug 2024).
Dynamics and time evolution: In dynamical systems, complexity is time-dependent, typically rising rapidly during the increase of inter-component correlations and saturating once the system equilibrates; for quantum chaotic systems, the suppression of long-term fluctuations in complexity is diagnostic of the chaotic regime (1009.0560).

7. Limitations and Future Directions

Parameter selection: The diagnostic power of the complexity-entropy plane depends critically on the embedding dimension, delay, choice of entropy/complexity functionals, and partitioning schemes; unsuitable parameterization can impede discrimination.
Scalability and coarse-graining: High-dimensional systems and large networks entail factorial or exponential growth in pattern space, necessitating efficient coarse-graining or sampling.
Integration with machine learning: The interface with machine-learning based feature extraction, as in deep neural network representations for image analysis, suggests a fruitful path for combining physics-inspired descriptors with data-driven classification (Kim et al., 19 Aug 2024).
Multivariate and multi-mode complexity: Extension beyond single-variable or fixed-dimension symbolic representations, e.g., via vector symbolic dynamics or multiscale windows, is an open research frontier.

The complexity–entropy plane thus emerges as a unifying diagnostic and analytic tool for probing organization, randomness, and structure in a broad spectrum of complex systems, operating at the interface of information theory, statistical physics, dynamical systems, and data science.