# Incidence-Mixed Simplicial Diffusion

- ID: 2445
- Canonical URL: https://synthcore.org/idea/2445/incidence-mixed-simplicial-diffusion
- API JSON: https://synthcore.org/api/idea/2445.json
- API Markdown: https://synthcore.org/api/idea/2445.md
- Verification status: unverified
- Source: [arXiv:2608.17229](https://arxiv.org/abs/2608.17229)
- Category: architecture
- Solves: accuracy, stability, scalability
- ML areas: graph-nn, training, optimizer
- Math tags: dynamical-systems, spectral-theory, graph-theory
- Ratings: usefulness 6/10; difficulty 5/10; novelty 7/10

## Idea description

Represent edge or pair-token features and propagate them with a convex mixture of two normalized channels: transitions through shared vertices and transitions through shared triangles. This preserves higher-order connectivity that an ordinary graph convolution loses, while the mixing coefficient q controls whether information follows pairwise support or genuine triangular structure.

## Mathematical statement

Let B1 be the binary vertex-edge incidence matrix and B2 the binary edge-triangle incidence matrix. The vertex degree is dV(v) = sum over e of B1[v,e], and the triangle degree is dT(e) = sum over f of B2[e,f]. Form the vertex channel PV by multiplying B1 transpose, the inverse vertex-degree matrix DV inverse, and B1, followed by row normalization. Form the triangle channel PT from B2 times B2 transpose, followed by row normalization. The mixed walk is P(q) = q PV + (1 minus q) PT, with q between zero and one. For edge features H, use H next = (1 minus alpha) H plus alpha P(q) H Theta, where alpha is a diffusion step size and Theta is a trainable feature map. The second-largest eigenvalue modulus mu2 controls relaxation, tau = minus one divided by log of the absolute value of mu2 is the relaxation time, and IPR = sum over e of pi[e] squared measures stationary localization.

## Key formulas

- dV(v) = sum over e of B1[v,e], and dT(e) = sum over f of B2[e,f]
- PV = row_normalize(B1 transpose times DV inverse times B1), PT = row_normalize(B2 times B2 transpose), and P(q) = q PV + (1 minus q) PT
- H next = (1 minus alpha) H + alpha P(q) H Theta
- tau(q,p) = minus 1 divided by log of absolute value of mu2(q,p), and IPR(pi) = sum over e of pi[e] squared; P(q,p) to the power t times x minus pi decays approximately as absolute value of mu2(q,p) to the power t

## Implementation notes

Integrate the mechanism as an edge-level message-passing block before pooling edge features to nodes or to a graph representation. For every input graph, construct sparse B1 and B2. Compute the vertex channel by multiplying B1 transpose, the inverse diagonal vertex-degree matrix, and B1, then row-normalize the resulting sparse matrix. Compute the triangle channel from B2 times B2 transpose and row-normalize it. Add self-loops before normalization so edges with no retained triangles remain numerically valid. Set P = q PV + (1 minus q) PT and return H next = (1 minus alpha) H + alpha P H Theta. Begin with q = 0.5, alpha = 0.5, and a shared Theta; later test a learned q using q = sigmoid(a), or a graph-dependent q predicted from pooled features. The incidence matrices, degrees, and sparse propagation are computed exactly. Estimate mu2 with 10 to 20 Lanczos iterations and estimate the stationary distribution pi by repeatedly applying P to a probability vector. The first cheap experiment should generate 100-vertex complexes with candidate triangles independently retained at densities p from 0.05 to 0.95. Assign edge labels using a latent triangle-dependent rule and compare an ordinary edge GCN with this layer at matched parameter count. Measure validation accuracy, mu2, tau, IPR, and feature autocorrelation decay. The predicted signature is an interior maximum of mu2, tau, and IPR at intermediate p, while empirical diffusion decay should match absolute value of mu2 to the power t within approximately 20 percent.

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.