# Signal Through Structure — Portfolio Narrative

## Question

How does graph topology determine whether a message-passing neural network transmits, crushes, or erases information?

## Problem

Graph neural network depth has two competing effects. More layers expand the receptive field, but distant information can be compressed through narrow graph cuts (oversquashing), while repeated neighbor mixing can make node representations increasingly alike (oversmoothing). Aggregate accuracy curves and equations rarely make both mechanisms visible at once.

## Contribution

Signal Through Structure is an editable, deterministic observatory for this tradeoff. A viewer injects a source signal, steps through normalized message-passing layers, and watches the signal move across path, tree, bottleneck, and rewired graphs. Exact influence, variance, energy, density, and spectral-gap diagnostics update with the graph.

The signature interaction is counterfactual topology editing: select any two nodes to add or remove an edge, then see how that structural intervention changes both reach and representation collapse.

## Locked result

The pre-build spike compared a 12-node barbell graph with the same graph plus two cross-community edges. At layer 8, rewiring increased source-to-target influence by 10.76× while reducing node variance more quickly. This validated the central product story: widening the information channel improves transmission while accelerating mixing.

## Implementation depth

- exact normalized-adjacency propagation in a dependency-free ES module;
- deterministic graph fixtures and replay across 17 layers;
- Jacobi eigensolver for normalized-Laplacian spectral gap;
- accessible SVG graph rendering with keyboard-operable node selection;
- responsive metric timelines and dynamic causal explanation;
- no server, model download, training corpus, or runtime dependency.

## Evidence boundary

- The observatory illustrates mechanisms on small linear fixtures.
- It does not establish trained nonlinear GNN behavior on a real dataset.
- It does not propose a new graph-rewiring algorithm.
- It does not claim that a larger spectral gap is universally better.
- It does not provide production debugging, model safety, or state-of-the-art performance evidence.

## Research grounding

- Topping et al., *Understanding over-squashing and bottlenecks on graphs via curvature* (2021).
- Wu et al., *A Non-Asymptotic Analysis of Oversmoothing in Graph Neural Networks* (2022).
- Karhadkar et al., *FoSR: First-order spectral rewiring for addressing oversquashing in GNNs* (2022).
- Chen et al., *Residual connections provably mitigate oversmoothing in graph neural networks* (2025).