CSDC Seminar - When Is a Spectral Change Observable? Robust Graph Comparison under Geometric Noise

Spatial networks reconstructed from biological images are observed through uncertain geometry but interpreted through topology and graph spectra. This creates a fundamental problem: when does an observed spectral difference represent a meaningful structural change rather than measurement noise? The question has the same character as detecting early-warning signals of critical transitions, posed for network structure rather than for time series.

I will present two complementary approaches to this question. First, geometric vertex-noise models show why independent edge noise is generally inappropriate: perturbing a vertex induces correlated changes in its incident edge weights. By combining geometric constraints, local extremal motifs and spectral perturbation bounds, we obtain a computable description of the resulting spectral noise floor. Second, ROSA applies an order-aware edge-removal filtration to a base spectral distance, allowing weak, localised graph changes to accumulate across multiple comparison scales rather than being diluted in a whole-graph summary.

I will discuss theoretical guarantees, experiments on synthetic and imaging-derived biological networks, and important failure regimes. I will also explore a broader question arising from the two frameworks: amplification may improve the repeatability of a graph-comparison signal without improving its separation from a noise-induced null distribution, or vice versa. This distinction suggests a practical route from spectral amplification to calibrated decisions on previously unseen data.