sign-complex-systems

name

sign-complex-systems

description

Sparse Identification Graph Neural Network (SIGN) for inferring governing equations of complex networked systems. Use when working with: (1) complex systems dynamics prediction, (2) equation discovery from data, (3) graph neural networks for networked systems, (4) interpretable AI for dynamical systems, (5) large-scale network modeling (climate, biological, technological networks), (6) symbolic regression on graphs. Keywords: SIGN, sparse identification, equation discovery, complex systems, graph neural networks, network dynamics, interpretable prediction.

SIGN: Sparse Identification Graph Neural Network

A framework for inferring governing equations of ultra-large complex networked systems from data, combining the interpretability of symbolic discovery with the scalability of neural networks.

Core Innovation

SIGN overcomes the fundamental trade-off in complex systems modeling:

• Equation discovery methods: Interpretable but fail to scale
• Neural networks: Scale but operate as black boxes, lose reliability over long times

Solution: Define symbolic discovery as edge-level information, decoupling sparse identification scalability from network size.

Key Capabilities

• Scalability: Handles networks with >100,000 nodes
• Robustness: Resistant to noise, sparse sampling, missing data
• Interpretability: Recovers governing equations with high precision
• Long-term prediction: Sustains accurate predictions over extended periods

Architecture

Network Data → Graph Neural Network → Edge-level Symbolic Discovery → Governing Equations

Components

1. Graph Neural Network Backbone

   • Processes network structure and dynamics
   • Node embeddings capture local dynamics
   • Edge embeddings capture interaction patterns

2. Sparse Identification Module

   • Edge-level symbolic regression
   • Library of candidate functions (polynomials, trigonometric, etc.)
   • LASSO/STRidge for sparse coefficient selection

3. Equation Assembly

   • Combines edge-level discoveries
   • Generates compact network equations
   • Validates against observed dynamics

Methodology

Step 1: Data Preparation

# Network time series data
# X: node states over time [T, N, D]
# A: adjacency matrix [N, N]

import numpy as np

def prepare_network_data(time_series, adjacency):
    """Prepare network dynamics data for SIGN."""
    # Extract node features
    node_features = extract_features(time_series)
    # Compute edge features
    edge_features = compute_edge_features(time_series, adjacency)
    return node_features, edge_features

Step 2: GNN Encoding

# Graph neural network for encoding
def sign_gnn_encode(node_features, edge_features, adjacency):
    """Encode network dynamics using GNN."""
    # Message passing
    messages = compute_messages(node_features, edge_features)
    # Node updates
    updated_nodes = update_nodes(node_features, messages)
    # Edge representations
    edge_repr = compute_edge_repr(updated_nodes, edge_features)
    return edge_repr

Step 3: Sparse Identification

# Symbolic discovery at edge level
def sparse_identify(edge_repr, library):
    """Discover governing equations via sparse regression."""
    # Build candidate library
    Theta = build_library(edge_repr, library)
    # Sparse regression (STRidge)
    coefficients = stridge(Theta, derivatives)
    # Extract active terms
    equation = extract_equation(coefficients, library)
    return equation

Applications

Climate Networks

• Sea surface temperature prediction
• Climate pattern identification
• Long-term forecasting (2+ years)

Biological Networks

• Neural dynamics modeling
• Gene regulatory networks
• Epidemic spreading

Technological Networks

• Power grid dynamics
• Communication networks
• Traffic flow

Implementation Guide

Dependencies

pip install torch numpy scipy sympy networkx

Basic Usage

from sign import SIGN

# Initialize SIGN model
model = SIGN(
    num_nodes=100000,
    node_dim=3,
    library='polynomial_trigonometric',
    sparsity_threshold=0.01
)

# Fit to network data
model.fit(time_series_data, adjacency_matrix)

# Get discovered equations
equations = model.get_equations()

# Predict future dynamics
predictions = model.predict(horizon=100)

References

• Paper: Predicting Dynamics of Ultra-Large Complex Systems by Inferring Governing Equations
• Detailed methodology: See references/methodology.md
• Benchmark results: See references/benchmarks.md
• Sea surface temperature case study: See references/sst_case_study.md

Scripts

• scripts/sign_model.py - Core SIGN implementation
• scripts/sparse_regression.py - STRidge and sparse identification
• scripts/equation_library.py - Function library construction

Comparison with Alternatives

Method	Scalability	Interpretability	Long-term Accuracy
SINDy	Low	High	Medium
Neural Networks	High	Low	Low
SIGN	High	High	High

Key Insights

1. Edge-level discovery: Key innovation enabling scalability
2. Sparse library: Compact equations from large candidate sets
3. Noise robustness: Statistical methods handle measurement errors
4. Missing data tolerance: GNN structure inference fills gaps