Spectral Graph Theory for Diagnostics

The graph Laplacian's eigenvectors are the natural basis for sensor fields; spectral methods underlie smoothing, imputation, and anomaly detection on the constellation.

The graph Laplacian

The mathematics behind graph learning starts with the Laplacian L = D - A, where A is the adjacency matrix of the diagnostic graph and D the degree matrix. Its eigenvectors form an orthogonal basis - a Fourier basis on the graph - ordered by smoothness. Signals from the sensor constellation decompose in this basis, separating coherent physical structure from noise.

Graph Laplacian and spectrum:

  L = D - A          (combinatorial)
  L_sym = I - D^-1/2 A D^-1/2   (normalized)

  L u_k = lambda_k u_k
  0 = lambda_0 <= lambda_1 <= ... <= lambda_{n-1}

  small lambda_k -> smooth eigenvectors (low graph frequency)
  large lambda_k -> oscillatory (high graph frequency)

Graph Fourier and filtering

Projecting a sensor signal onto the eigenvectors is the graph Fourier transform. Smoothing keeps low-frequency components (physically coherent fields), while high-frequency energy flags noise or a localized anomaly. This is the formal basis of the Laplacian smoothness term used in imputation, and of spectral anomaly scores.

# graph Fourier smoothing / anomaly energy (schematic)
evals, U = eigh(L_sym)
x_hat = U.T @ x                 # graph Fourier coefficients
x_smooth = U @ (lowpass(evals) * x_hat)
anomaly_energy = sum((evals > cut) * x_hat**2)   # high-freq power

Why it matters for the machines

The diagnostic graph's smoothness prior encodes that neighboring sensors on the same flux surface should read consistently. When one drops, its value is inferred from the smooth part of the field; when one reads an implausible high-frequency spike relative to neighbors, spectral energy localizes the fault. Message-passing GNNs approximate localized spectral filters, which is why they inherit these guarantees while scaling better than full eigendecomposition.

• Laplacian eigenvectors: ordered smooth-to-oscillatory basis.
• Smoothness prior: neighboring sensors agree on coherent fields.
• High graph-frequency energy: localized fault / anomaly indicator.
• GNN layers approximate polynomial spectral filters efficiently.

Spectral theory gives the imputation and anomaly modules their mathematical justification; the GNN is the scalable, learnable realization of these filters on the live constellation.