HVG–Autoencoder–SINDy: Chaotic Dynamics of Cultural Attention

Official implementation of "Chaotic Dynamics of Tai Chi Public Attention Revealed by an Integrated Framework of Horizontal Visibility Graphs, Autoencoders, and Sparse Identification".

Using daily Baidu Index data (2014–2024) from four Chinese provinces (Beijing, Shanghai, Guangdong, Henan), this pipeline characterizes low-dimensional chaotic attractors, discovers sparse governing equations, and benchmarks against ARIMA, VAR, and LSTM baselines.

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Overview

The framework operates as a sequential analytical pipeline:

Time series → HVG module → Autoencoder → SINDy (STLS) → Lyapunov / Phase space
              (12-dim features) (3-dim latent)  (sparse equations)  (validation)

Key components:

• HVG module — constructs Horizontal Visibility Graphs and extracts 12-dimensional topological feature vectors via a sliding window; estimates chaos criterion λ vs. ln(3/2) ≈ 0.405
• Autoencoder — fully connected encoder [12→64→32→16→3] with symmetric decoder; VAE variant included for comparison
• SINDy (STLS) — discovers sparse discrete-time governing equations $z(t+1) = f(z(t))$ from a 48-term library (constant, linear, quadratic, trigonometric)
• Baselines — ARIMA, VAR, LSTM applied in the same latent space for fair comparison
• Chaos diagnostics — Rosenstein and Wolf Lyapunov exponents, phase portraits, Poincaré sections, correlation dimension

The form of the governing equation is as follows:

$$ \begin{bmatrix} z_1(t+1) \ z_2(t+1) \ z_3(t+1) \end{bmatrix} ^\top= C + A \underbrace{\mathbf{z}}_{\text{linear}} + \underbrace{\begin{bmatrix} \mathbf{z}^\top Q_1 \mathbf{z} \\ \mathbf{z}^\top Q_2 \mathbf{z} \\ \mathbf{z}^\top Q_3 \mathbf{z} \end{bmatrix}}_{\text{quadratic}} + D \underbrace{\boldsymbol{\phi}}_{\text{trigonometric}} $$

where

$$ \mathbf{z} = \begin{bmatrix} z_1 & z_2 & z_3 \end{bmatrix}^\top , \boldsymbol{\phi} = \begin{bmatrix} \sin z_1 & \cos z_1 & \sin z_2 & \cos z_2 & \sin z_3 & \cos z_3 \end{bmatrix}^\top $$

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Key Results

Metric	Value
HVG λ range	0.257 – 0.437 (reference: ln(3/2) ≈ 0.405)
AE reconstruction correlation	0.896 – 0.918 (VAE: 0.821–0.823)
SINDy NRMSE (in-sample)	11.7 – 12.8%
Active SINDy terms	30–38 / 48 candidates
Rosenstein λ_max	0.025 – 0.036 bits/day
Predictability horizon	30 – 70 days

Regional chaos classification

Province	Channel	HVG λ	λ_max (Rosenstein)	λ_max (Wolf)	Regime
Beijing	PC	0.291	0.027	0.171	Strong chaos
Beijing	Mobile	0.316	0.033	0.179	Strong chaos
Guangdong	PC	0.376	0.026	0.020	Strong chaos
Guangdong	Mobile	0.376	0.034	0.174	Strong chaos
Shanghai	Mobile	0.257	0.032	0.178	Strong chaos
Henan	PC	0.415	0.026	0.016	Quasi-periodic
Henan	Mobile	0.434	0.036	0.217	Quasi-periodic
Shanghai	PC	0.437	0.025	0.195	Quasi-periodic

Comparison with baselines (50-day test, Mobile channel, NRMSE %)

Method	Beijing	Guangdong	Henan	Shanghai
ARIMA	21.1	24.2	21.0	18.2
VAR	21.2	24.7	21.5	18.7
LSTM	31.1	28.9	24.2	25.9
SINDy (ours)	20.8	22.4	19.9	17.6

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Project Structure

HVG-SINDy-Autoencoder/
├── data/                  # Baidu Index Excel files
│   ├── Beijing.xlsx
│   ├── Guangdong.xlsx
│   ├── Henan.xlsx
│   └── Shanghai.xlsx
├── results/               # Output figures, tables, JSON reports
├── hvg_sindy.py           # Core: HVG construction, AE/VAE, SINDy, Lyapunov
├── comparison.py          # Baseline methods: ARIMA, VAR, LSTM
├── data_sequence.py       # temporal pattern visualization
├── main.py                # Entry point
└── requirements.txt

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Installation

# Clone the repository
git clone https://github.com/wind-whispered/HVG-SINDy-Autoencoder.git
cd HVG-SINDy-Autoencoder

# Install dependencies
pip install -r requirements.txt
pip install openpyxl

Requirements: Python >= 3.8, PyTorch >= 2.0, CUDA optional.

Core packages:

torch>=2.0
numpy
pandas
scipy
networkx          # HVG construction
statsmodels       # ARIMA / VAR baselines
scikit-learn      # preprocessing and metrics
matplotlib
seaborn
openpyxl          # Excel file reading

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Data Preparation

Place province Excel files in ./data/. Each file should contain three columns:

Column	Format	Description
date	YYYYMMDD	Daily date
PC	integer	PC-based search index
Mobile	integer	Mobile-based search index

Files must be named Beijing.xlsx, Guangdong.xlsx, Henan.xlsx, Shanghai.xlsx. The dataset covers 4,018 daily observations per province per channel (January 1, 2014 – December 31, 2024), yielding 32,144 total observations.

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Usage

Run the full pipeline (HVG analysis → autoencoder training → SINDy identification → comparative evaluation → all figures and tables):

python main.py

All outputs are saved to ./results/:

• figure_*.png / .pdf — all paper figures
• Table*.csv — descriptive statistics, chaos metrics, AE performance, SINDy coefficients
• analysis_results.json — full numerical results

Individual scripts can also be run independently:

# Temporal pattern visualization
python data_sequence.py

# Comparative analysis only (requires saved SINDy models from main.py)
python comparison.py

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SINDy Coefficient Structure

Representative governing equations for Beijing (3-dimensional latent space):

Province	Linear (mean)	Quadratic (mean)	Trigonometric (mean)	Active terms	NRMSE (%)
Beijing	0.337	0.150	0.500	37/48	12.7
Guangdong	0.398	0.111	0.354	38/48	11.9
Henan	0.222	0.115	0.287	38/48	12.8
Shanghai	0.240	0.092	0.349	30/48	11.7

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Citation

@article{kang2026chaotic,
  title={Chaotic dynamics of Tai Chi public attention revealed by an integrated framework of horizontal visibility graphs, autoencoders, and sparse identification},
  author={Kang, Yafeng and Li, Pengchao and Tang, Lu and Zhang, Chao},
  journal={Scientific Reports},
  year={2026},
  publisher={Nature Publishing Group}
}

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Reference

• Brunton et al. (2016) — original SINDy framework (PySINDy)

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License

This project is licensed under the MIT License — see LICENSE for details. Data sourced from Baidu Index; users are responsible for compliance with Baidu's terms of service.

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