Graph Spectral Stability, Diffusion and Community Robustness Study
A completed mathematics study of normalized-Laplacian perturbations, invariant-subspace bounds, graph diffusion and spectral community recovery.
Software compatibility
The retained release results use Python 3.14.6. The Docker workflow uses Python 3.12.14 and reproduces the scientific evidence within the declared tolerance.
Project definition
Problem statement
Weighted network data can contain missing, added or uncertain links, so an engineering conclusion should not depend unpredictably on one exact edge list.
The mathematical problem is to determine when a small graph edit remains small after degree normalization, eigendecomposition, diffusion and spectral clustering.
Project objectives
- Generate connected weighted graphs with three planted communities and controlled separation.
- Implement edge deletion, edge addition, weight noise, bridge strengthening and within-community removal.
- Verify full-spectrum eigenvalue drift against the Weyl spectral-norm bound.
- Measure low-frequency invariant-subspace rotation and its Davis-Kahan bound.
- Compare heat-kernel sensitivity across five diffusion times.
- Measure ARI, NMI and planted conductance across mechanisms and graph regimes.
- Measure dense eigendecomposition runtime from 60 to 240 vertices.
Project structure
Project components
Graph model
Generates connected weighted stochastic block graphs with retained planted labels.
Perturbation library
Applies five reproducible structural and weight changes without self-loops or asymmetry.
Spectral analysis
Computes Laplacian spectra, eigengaps, principal angles, Weyl ratios and Davis-Kahan bounds.
Diffusion analysis
Compares complete heat kernels at local and global graph time scales.
Community evaluation
Runs normalized spectral clustering and retains ARI, NMI and conductance.
Evidence builder
Writes replicate-level CSV files, summary JSON, checksums and seventeen labelled figures.
Methodology
Project workflow
- 01Load the study
The program reads graph size, probabilities, weights, perturbation fractions, repetitions and diffusion times.
- 02Generate baselines
Thirty seeded weighted graphs are created across three community-separation regimes.
- 03Apply perturbations
Five mechanisms and six fractions produce 900 altered-graph comparisons.
- 04Check spectra
Operator norms, eigenvalue drift, eigengaps, principal angles and theorem bounds are retained.
- 05Evaluate outcomes
Heat diffusion, conductance and community recovery are compared against each baseline.
- 06Export evidence
Replicate tables, summary values, runtime measurements, figures and checksums are written.
Demonstration scenario
Thirty 120-node weighted graphs are perturbed across five mechanisms and six fractions. All 900 Weyl and Davis-Kahan checks pass. Mean positive-fraction ARI is 0.98885, while the worst retained run reaches 0.67848 after 20 percent within-community edge removal in the weakest separation regime.
Engineering
Tools and method
- Tools
- The project uses Python, NumPy, SciPy, NetworkX, scikit-learn, Pandas, Matplotlib, Jupyter for subject analysis, simulation, and results.
- Graph family
- Three balanced weighted stochastic blocks with controlled within- and between-community probabilities.
- Graph operator
- Symmetric normalized Laplacian for production experiments with the combinatorial form available for verification.
- Perturbation theory
- Spectral-norm eigenvalue bounds and gap-conditioned invariant-subspace bounds.
- Dynamics
- Relative Frobenius comparison of complete matrix-exponential heat kernels.
- Clustering
- Row-normalized low-frequency embedding, fixed three-cluster K-means, ARI and NMI.
- Verification
- Automated tests, branch coverage, static checks, dependency audit, Docker reproduction and document validation.
Testing
Evaluation
Evaluation measures
- Normalized-Laplacian perturbation spectral norm
- Maximum ordered eigenvalue drift and Weyl ratio
- Baseline spectral gap and largest principal-angle sine
- Davis-Kahan bound and pass condition
- Relative heat-kernel distance across five diffusion times
- Planted-community conductance
- Adjusted Rand index and normalized mutual information
- Dense eigendecomposition runtime scaling
Project boundaries
- All graphs, weights, labels and perturbations are synthetic.
- The retained graph family has three balanced, undirected, nonnegative communities.
- The theorem bounds establish matrix consistency rather than real-network validity.
- Dense runtime results do not represent large sparse production networks.
Included
- 01Complete Python source code
- 02Connected weighted stochastic block graph generator
- 03Combinatorial and normalized-Laplacian implementations
- 04Five random and targeted graph perturbation operators
- 05Nine hundred spectral and clustering perturbation runs
- 06Nine hundred time-resolved heat-kernel comparisons
- 07Weyl and Davis-Kahan theorem diagnostics for every run
- 08Seventeen reproducible project figures and complete retained evidence
- 0994-page project report in PDF and editable Word formats
- 1012-page setup and usage guide in PDF and editable Word formats
- 1145 annotated references and two attributed literature figures
- 1216 automated tests with 98.40 percent branch-aware coverage
Project record
No information is collected on this page.
- Permanent project ID
- GP-MA-1TAK6BD
- Catalogued
- 21 Aug 2026
- Completed
- 28 Aug 2026
- Verified
- 28 Aug 2026
- Demonstration
- Included in repository
Handover
After purchase
- 01Payment is confirmed
The project is marked unavailable and cannot be purchased again.
- 02Repository access is granted
The buyer's submitted GitHub account receives access to the private repository.
- 03The purchase record is delivered
The certification sheet is prepared from the reviewed buyer details and sent privately by email.