Geometric Visualizations of Spectral Graph Conservation
A collection of Python scripts that make the Conservation Spectral framework visible. Each script produces a different geometric visualization — Laplacian as rubber sheet, eigenvalue shells, Dirichlet spring systems, phase portraits, and alignment heatmaps. Together they form a visual atlas of how graph Laplacians interact with attributes.
Conservation is inherently geometric: the Laplacian is a "rubber sheet" that pulls a surface toward its smoothest shape, eigenvalues are shell radii that define conservation bounds, and the Dirichlet energy is literally spring potential energy. These visualizations make that geometry explicit. When you see a graph embedded as a 3D surface where height = attribute and the Laplacian flattens it, conservation isn't an equation — it's a shape. When you see eigenvalue shells where inner spheres are much smaller than outer ones, the conservation ratio isn't a number — it's a volume ratio. Seeing is understanding.
Embeds a 5-node graph in 2D, shows attribute as height (3D surface), then applies the Laplacian to "flatten" it. Compares smooth (conserved) vs rough (not conserved) attributes.
For each eigenvalue λ_k, draws a sphere of radius √λ_k. Conservation = inner shells much smaller than outer shells. Compares structured (music-like) graphs vs random graphs.
Each edge is a spring, attribute values are displacements. Dirichlet energy f^TLf = total spring energy. Shows energy propagation when perturbing a single node.
X-axis: conservation ratio, Y-axis: spectral gap. Plots ALL domains (music, ecology, code, geometry) as colored clusters, revealing the "conservation frontier."
Computes and visualizes α(G,a) for various graph types × attribute types. Shows the boundary α = 0.5 (conjecture C1). High α = conservation works, low α = doesn't.
pip install numpy matplotlib scipy
# Generate all visualizations
python laplacian_rubber_sheet.py
python eigenvalue_shells.py
python dirichlet_springs.py
python phase_portrait.py
python alignment_coefficient.pyEach script writes .png files alongside itself.
These visualizations are the geometric intuition layer for the conservation ratio α(G,a) = (a^T L a) / (λ_max ‖a‖²). Every picture is a geometric interpretation of this equation:
- Rubber sheet: L applied to f smooths the surface
- Eigenvalue shells: λ_max bounds the energy, inner eigenvalues bound conservation
- Springs: Dirichlet energy is physical potential energy
- Phase portrait: Where different domains cluster in (α, λ₂) space
- Alignment: Which graph structures conserve which attribute shapes
- conservation-spectral-v2 — Reference Python implementation
- ecosystem-conservation — Ecological application
- conservation-art — Artistic visualizations
- code-conservation — Software dependency analysis
- flux-flow-state — Interactive flow visualization
MIT