Essential Simplices Dominate in Harmonic Representatives of One-Dimensional Persistent Classes

What the math gives to ML

This paper gives a sharp localization principle for one-dimensional persistent homology: under genericity, the minimum-norm harmonic representative of a persistent class assigns equal absolute coefficients to all essential edges and strictly smaller coefficients to every non-essential edge. The transferable asset is a topology-derived edge-ranking mechanism with a provable coefficient gap, stronger than merely maximizing aggregate essential energy. A graph neural network can use these coefficients as topology-aware attention biases or regularizers, encouraging message passing to preserve persistent cycle structure. The most credible evaluation target is graph data containing predictive loops, where the method can be compared against ordinary Laplacian features and unregularized graph attention.