Urban Heat Democratization · Research wiki

Spectral Theory Contract

What the graph mathematics establishes exactly, what it cannot establish, and the evidence required to cross that boundary.

The object is a graph—not a city in full

The model analyses an undirected weighted graph of valid raster cells. Its edge weights encode documented assumptions about local gradient barriers and optional greenness. The resulting mathematics is rigorous about that graph; it is not a direct measurement of heat transport, lived access, health risk, or intervention performance.

L = I − D⁻¹ᐟ²WD⁻¹ᐟ² ; φ(S) = cut(S,V∖S) / min(vol(S),vol(V∖S))

What is guaranteed

For a connected, nonnegative weighted graph, the implementation computes the second normalized-Laplacian eigenpair and evaluates threshold sets along the Fiedler-vector ordering. The selected set is the lowest-conductance sweep cut—a reproducible, inspectable candidate, not an exhaustive search over all subsets.

λ₂/2 ≤ φ* ≤ √(2λ₂)

Cheeger-type bounds explain why spectral connectivity and conductance are related. They do not validate the raster, weights, sink definition, or a causal theory of intervention.

What requires evidence beyond the graph

Before any public decision: test reasonable parameter choices, document sources and spatial support, seek field and domain review, incorporate community knowledge, and use a pre-specified evaluation design for outcome claims.

Read the complete Spectral Theory Contract, then inspect the canonical platform alongside its methods and caveats.

Explore on the Canonical Platform

Connect this research guide to live interactive modules on urban-heat.ai-aarti.com:

Boston Heat & Cooling Atlas

Inspect satellite thermal layers, bottleneck cuts, and localized cooling access.

Mitigation Scenarios

Explore transparent what-if cooling options, budgets, and equity trade-offs.

Solution Suite & GIS Tools

Interactive OSM queries, microclimate sensor bench, and federal grant generator.

Interactive Mitigation Lab

Model reflective roofs, urban canopy, and cool pavements with live feedback.

Robustness Lab

Explore spectral graph conductance, Fiedler vectors, and percolation dynamics.

Collaborate & Partner

Partner with research mentors, community advocates, and public agencies.