Urban Heat Democratization · Research wiki

Mathematical Foundations

A careful, equation-level explanation of the spatial and network reasoning used to turn urban-heat evidence into inspectable questions.

1. Start with a spatial field, not a story

Let T(x) denote a temperature-related surface at location x. A raster observes cell summaries, not a continuous city: yᵢ = Aᵢ[T] + εᵢ, where Aᵢ represents the sensor, pixel footprint, resampling, masking, and retrieval choices, and εᵢ collects measurement and processing error. This distinction matters: a map value is a documented estimate, not a direct reading of every person’s exposure.

2. Construct a graph transparently

Partition the study area into cells or features V. Create an edge (i,j) only when the chosen adjacency rule says the units are connected—rook, queen, distance radius, street network, or a documented hybrid. A nonnegative weight wᵢⱼ may express contiguity, similarity, or modeled cooling permeability. The degree is dᵢ = Σⱼ wᵢⱼ.

The graph Laplacian L formalizes smoothness. For any vector f, fᵀLf = ½Σᵢⱼwᵢⱼ(fᵢ−fⱼ)². Large differences across strongly weighted edges cost more; that is a modeling statement about the edge definition, not an independent fact about the city.

3. Locate structural bottlenecks

For a subset S, conductance compares the edge boundary to the modeled volume on its smaller side. A small value identifies a weakly connected region in this particular graph.

The second eigenvalue of the normalized Laplacian, λ₂, is related to the best conductance by Cheeger-type bounds: λ₂/2 ≤ φ* ≤ √(2λ₂). These bounds give a principled reason to inspect a spectral partition; they do not prove that the area has the greatest lived heat burden or that one intervention is optimal.

4. Model pathways with resistance

With conductances collected in L, the effective resistance between nodes a and b is Rₑff(a,b)=(eₐ−e_b)ᵀL⁺(eₐ−e_b), where L⁺ is the Moore–Penrose pseudoinverse. It measures separation in the modeled network. If edge weights represent cooling connectivity, resistance is a useful friction proxy; if they do not, the same number has a different meaning.

5. Make uncertainty travel with the result

For a derived metric g(y,θ) with data y and modeling choices θ, local uncertainty can be approximated by the delta method: Var[g] ≈ ∇gᵀΣ∇g. In practice, rerunning across plausible resolutions, dates, thresholds, adjacency rules, and weights is often more honest than reporting false precision.