# Prism: Structural Symmetry Scanning via Duality-Constrained Laplacian Projection

arXiv: https://arxiv.org/abs/2605.20245 · HTML: https://mlsanity.com/paper/2605.20245/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 13 references that are in the map.
- First posted on arXiv 18 May 2026.

## Builds on (most cited first, 4 of 4)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (2018)
- [Cooperative Game Theory Approaches for Network Partitioning](https://mlsanity.com/paper/1707.03587/) (2017)

## Abstract (arXiv)

We introduce Prism, a framework for structural symmetry diagnosis in complex networks. Given a graph Laplacian L and a duality operator P (a symmetric involution), Prism computes the duality defect δ(L,P) = |LP - PL|F / |L|F -- a scalar measuring how far the network deviates from structural self-consistency. When P encodes the network's true symmetry, δ starts near zero and rises monotonically as structure degrades; an arbitrary P gives noise. We prove that the optimal L' satisfying [L', P] = 0 is given by a closed-form block-diagonal projection, and provide an unsupervised alternating optimization that learns P from the graph's own Fiedler vector. Experiments on synthetic networks show the true-P defect is 3.38× more sensitive to structural degradation than an index-reversal baseline and more sensitive than modularity. On Zachary's Karate Club with edge noise, Prism achieves 94.5% community detection accuracy at 5% noise versus 76.6% for the raw Laplacian baseline. Applied to live S&P~500 data (2026-05-17), Prism detects rising structural stress (defect 0.43 → 0.73 over 90 days) while surface correlations remain low -- a signal invisible to correlation-based methods. In a historical backtest spanning five major stress events (2011--2020), the duality defect exhibits a consistent pattern: it reaches elevated levels before the correlation spike that accompanies each crisis, and sustains high readings during periods of structural fragility that conventional metrics classify as calm. The duality defect is a first-principles structural admissibility condition, requiring no training data and computable in milliseconds.