metastablex 0.1.0

Metastable dynamics detection for complex systems

🧠 MetastableX

Structural Dynamics Framework for Complex Systems

A computational framework for detecting structural instability, metastable regimes and critical transitions in complex systems using time-series analysis, statistical physics and information theory.

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🧭 Overview

MetastableX is a computational framework designed to analyze the structural dynamics of complex systems.
Rather than focusing only on averages or linear correlations, the framework analyzes structural properties of temporal dynamics.

The goal is to identify when a system transitions between regimes such as:

• Stable equilibrium
• Metastable configurations
• Critical transitions
• Chaotic dynamics

The framework integrates concepts from:

• Statistical Physics
• Nonlinear Dynamical Systems
• Information Theory
• Complex Systems Science
• Epidemiological Data Science

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⚙️ Mathematical Framework

Many real-world systems can be modeled as stochastic dynamical systems:

$$ \frac{dx}{dt} = f(x) + \sigma \eta(t) $$

Where:

• $x(t)$ represents the state of the system
• $f(x)$ represents deterministic dynamics
• $\sigma \eta(t)$ represents stochastic fluctuations

Such systems may evolve through multiple dynamical regimes depending on external perturbations and internal feedback loops.

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🌄 Potential Landscape

Complex systems can be represented through an effective potential landscape.

If the probability density of states is $P(x)$, the effective potential is:

$$ U(x) = -\frac{\sigma^2}{2}\log P(x) $$

Interpretation:

• Deep wells → stable regimes
• Shallow wells → metastable regimes
• Flattening landscape → critical transition

When the landscape flattens, the system becomes highly sensitive to perturbations.

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⚠️ Early Warning Signals

Before abrupt transitions, complex systems often exhibit early warning signals.

Two of the most robust signals are:

Variance increase

$$ \sigma^2 = \frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2 $$

Variance measures the magnitude of fluctuations in the system.

An increase in variance indicates that the system is becoming unstable.

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Autocorrelation increase

$$ AC_1 = \frac{ \sum_{t=2}^{N}(x_t-\mu)(x_{t-1}-\mu) }{ \sum_{t=1}^{N}(x_t-\mu)^2 } $$

Autocorrelation measures temporal memory in the system.

High values indicate critical slowing down, meaning the system takes longer to recover from perturbations.

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🧬 Information-Theoretic Metrics

Shannon Entropy

Entropy measures the informational complexity of the system.

$$ H = -\sum_{i} p_i \log(p_i) $$

Interpretation:

Entropy	Interpretation
Low	rigid deterministic system
Moderate	adaptive system
High	chaotic regime

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📈 Long-Range Correlations (DFA)

Detrended Fluctuation Analysis detects fractal memory in time series.

First, the integrated signal is constructed:

$$ Y(k) = \sum_{i=1}^{k}(x_i - \mu) $$

Then fluctuations across scales are computed:

$$ F(n) = \sqrt{ \frac{1}{N} \sum (Y(k)-Y_n(k))^2 } $$

The scaling exponent is obtained from:

$$ F(n) \sim n^{\alpha} $$

Interpretation:

α	Interpretation
0.5	random noise
0.6–0.8	long-range memory

0.9 | strong structural correlation |

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📊 Distributional Metrics

Skewness

$$ Skew = \frac{1}{N} \sum \left( \frac{x_i - \mu}{\sigma} \right)^3 $$

Measures asymmetry in the distribution.

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Kurtosis

$$ Kurt = \frac{1}{N} \sum \left( \frac{x_i - \mu}{\sigma} \right)^4 $$

Measures the presence of extreme events.

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🔄 Dynamical Regime Classification

The framework classifies regimes using structural indicators.

Regime	Condition
Stable	$AC_1 < 0.5$
Metastable	$0.5 \le AC_1 < 0.7$
Critical	$AC_1 \ge 0.7$

This classification is inspired by phase transitions in statistical physics.

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🏥 Application: Structural Monitoring of Health Systems

Health systems behave as large-scale complex adaptive systems composed of interacting agents:

• hospitals
• patients
• healthcare workers
• policy interventions
• epidemiological dynamics

These interactions generate complex temporal patterns observable in hospitalization data.

Hospitalization time series therefore act as a proxy for the systemic state of healthcare infrastructure.

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📊 Data Source: SIH/SUS

The application of MetastableX described here uses the Hospital Information System of the Brazilian Unified Health System (SIH/SUS).

SIH is one of the largest health administrative datasets in the world.

Each record corresponds to an Authorization for Hospital Admission (AIH).

Key variables include:

Variable	Meaning
DT_INTER	admission date
DIAG_PRINC	primary diagnosis
MUNIC_RES	municipality of residence
DIAS_PERM	length of stay
MORTE	mortality indicator

These records allow the construction of municipal hospitalization time series.

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👥 Population Normalization

Hospitalization counts must be normalized by population.

The hospitalization rate is defined as:

$$ Rate = \frac{\text{Hospitalizations}}{\text{Population}} \times 10,000 $$

This transformation allows meaningful comparison across municipalities.

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📈 Structural Monitoring of Municipal Health Systems

For each municipality the framework computes:

Epidemiological indicators

• hospitalization counts
• hospitalization rate per 10k inhabitants

Statistical indicators

• mean
• variance
• coefficient of variation

Complexity indicators

• entropy
• DFA exponent

Dynamical indicators

• autocorrelation
• rolling variance
• rolling autocorrelation

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⚠️ Detecting Health System Stress

When a health system approaches overload or crisis, hospitalization dynamics may exhibit:

• increase in variance
• increase in autocorrelation
• entropy changes
• volatility spikes

These signals correspond to loss of systemic resilience.

Examples include:

• epidemic waves
• hospital overcrowding
• sudden changes in healthcare demand

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📊 Interactive Dashboard

The MetastableX dashboard allows users to:

• select a municipality
• visualize hospitalization time series
• compute structural metrics
• generate automatic analytical reports

This creates a structural monitoring interface for public health systems.

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🔬 Scientific Implications

The framework bridges:

• health data science
• epidemiology
• complex systems theory
• statistical physics

It enables monitoring healthcare systems not only through averages but through dynamic regime analysis.

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🚀 Future Directions

Potential extensions include:

• nationwide monitoring of all municipalities
• metastable regime maps
• dynamic clustering of healthcare systems
• digital twin simulations of healthcare infrastructure

Such tools may enable structural monitoring of national health systems in real time.

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👨‍🔬 Author

Lucas Amaral Dourado

• Medical Student — Federal University of Tocantins
• Biomedical Engineering Student — UNINTER

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📜 License

MIT License

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MetastableX — Structural monitoring of complex systems applied to public health data.