{
 "artifacts": null,
 "category": "architecture",
 "description": "Add a fixed or lightly parameterized graph-level feature extractor based on short non-backtracking cycle counts, and concatenate its Poisson log-likelihood-ratio features with the output of a graph neural network. The feature scaling uses the paper's explicit means, so the network receives statistics that are approximately independent and correctly normalized rather than raw, highly correlated cycle counts.",
 "formulas_latex": [
  "$$\\mu_t=(1-\\delta)^t\\left(\\nu_t-\\frac{(d-1)^t}{2t}\\right)+\\frac{(d-1)^t}{2t},\\qquad \\delta=\\frac{2r}{nd},\\qquad \\nu_t=\\frac{\\operatorname{tr}(B^t)}{2t}.$$",
  "$$C_1,\\ldots,C_D\\ \\approx\\ \\bigotimes_{t=1}^{D}\\operatorname{Pois}(\\mu_t),\\qquad D=\\Omega(\\log n).$$",
  "$$\\mu_t^{(0)}=\\frac{(d-1)^t}{2t},\\qquad \\mu_t^{(1)}=(1-\\delta)^t\\left(\\nu_t-\\mu_t^{(0)}\\right)+\\mu_t^{(0)}.$$",
  "$$\\phi(G)=\\sum_{t=1}^{D}\\left[C_t(G)\\log\\frac{\\mu_t^{(1)}+\\varepsilon}{\\mu_t^{(0)}+\\varepsilon}-\\left(\\mu_t^{(1)}-\\mu_t^{(0)}\\right)\\right],\\qquad z_t=\\frac{C_t-\\mu_t^{(0)}}{\\sqrt{\\mu_t^{(0)}+\\varepsilon}}.$$"
 ],
 "id": 2945,
 "implementation": "Integrate this at the graph-level readout of a message-passing GNN, before the final MLP classifier. The input is a sparse d-regular or approximately regular graph G with N vertices; choose D=4, 6, or min(12, floor(log(N)/2)) for the first experiment. Construct the directed-edge non-backtracking matrix B_G: its rows and columns are directed edges e=(u-\u003ev), and B_G[e,f]=1 when f=(v-\u003ew) with w != u. Estimate C_t(G)=tr(B_G^t)/(2t). For small graphs compute sparse matrix-vector products and the trace exactly; for larger graphs use q Rademacher Hutchinson probes g_j, with C_t approximately equal to (1/(2tq)) times the sum of g_j^T B_G^t g_j, followed by clamping to nonnegative values. Obtain d from the graph and set mu0[t]=(d-1)^t/(2t). If the base H and rematching rate r are known, calculate B_H, nu[t]=tr(B_H^t)/(2t), delta=2r/(Nd), and mu1[t]=(1-delta)^t*(nu[t]-mu0[t])+mu0[t]. If they are unknown, estimate mu1[t] from positive training graphs using a shrinkage mean, while retaining mu0 from matched random-regular null graphs. Form z_t and phi, concatenate [z_1,...,z_D,phi] to the GNN pooled embedding, and train the final classifier normally. Pseudocode: `counts=zeros(D); for probe in probes: v=g; for t in 1..D: v=B_G@v; counts[t]+=g.T@v/(2*t*q); z=(counts-mu0)/sqrt(mu0+eps); phi=sum(counts[t]*log((mu1[t]+eps)/(mu0[t]+eps))-(mu1[t]-mu0[t])); h=concat(GNN_pool(G),z,phi); logits=MLP(h)`. The paper supplies the Poisson means and likelihood; Hutchinson variance, the base spectrum, and unknown rematching rates are estimated empirically. First test on synthetic 2- or 3-regular graphs with N=128, 256, and 512, comparing noisy lifts against uniformly random regular graphs. Use a 3-layer GIN baseline with identical hidden width and matched training FLOPs. Pre-register three mechanism tests: first, at fixed N, fingerprint advantage should decrease as delta increases and become negligible when (1-delta)^t*(nu[t]-mu0[t]) is below sqrt(mu0[t]); second, adding lengths up to D should improve log-likelihood in the ordering predicted by the sum of Poisson KL terms sum_t[mu1[t]log(mu1[t]/mu0[t])-mu1[t]+mu0[t]], with observed AUC ordering matching predictions within 20%; third, replacing non-backtracking counts with ordinary closed-walk traces should lose advantage at larger t because of backtracking noise. Remove only the fingerprint features for the ablation. Falsify the transfer if the fingerprint does not improve over the matched GNN at delta=0.1 or if its advantage fails to decline with the predicted signal gap.",
 "math_summary": "Theorem 1.3 states that for a base d-regular graph H on k vertices, the cycle counts C_t of a noisy random lift are close in total variation to independent Poisson variables with means mu_t. Here t is the cycle length, r is the number of rematched edges, n is the lift size, delta=2r/(nd) is the rematching fraction, B is the weighted non-backtracking matrix of the base graph, and nu_t=tr(B^t)/(2t) is its base-cycle contribution. The theorem gives mu_t=(1-delta)^t(nu_t-(d-1)^t/(2t))+(d-1)^t/(2t), with null random-regular mean mu_t^(0)=(d-1)^t/(2t). For independent Poisson counts, the log likelihood ratio is sum_t [C_t log(mu_t^(1)/mu_t^(0))-(mu_t^(1)-mu_t^(0))], where mu_t^(1)=mu_t. The total-variation approximation is asserted for lengths up to D=Omega(log n), motivating a short-cycle feature extractor. The implementation uses exact or stochastic estimates of C_t from powers of the observed graph's non-backtracking matrix.",
 "math_tags": [
  "graph-theory",
  "spectral-theory",
  "probability",
  "statistics"
 ],
 "ml_areas": [
  "graph-nn",
  "embedding",
  "loss"
 ],
 "paper": {
  "arxiv_id": "2608.28539",
  "arxiv_url": "https://arxiv.org/abs/2608.28539",
  "summary_what_math_gives_to_ml": "The paper provides a constructive low-degree statistical representation of noisy random lifts: short cycle counts are approximately independent Poisson variables whose means are explicit functions of the base graph's weighted non-backtracking spectrum. This suggests a graph-neural-network front end that computes non-backtracking cycle statistics and feeds a Poisson likelihood-ratio embedding into a graph classifier, anomaly detector, or routing module. The transferable asset is not the random-lift application itself, but the principled conversion of many correlated local subgraph observations into approximately independent, variance-calibrated features with an explicit signal-to-noise scale. A clean test is whether these features improve detection of planted graph structure at matched parameters and whether performance follows the predicted dependence on rematching rate and cycle length.",
  "title": "Analysis of Polynomial Threshold Functions on Random Regular Graphs: Computational Complexity of Detecting Noisy Random Lift",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 6,
  "usefulness": 6
 },
 "solves": [
  "accuracy",
  "sample-efficiency",
  "scalability"
 ],
 "title": "Poisson Non-Backtracking Fingerprint",
 "url": "https://synthcore.org/idea/2945/poisson-non-backtracking-fingerprint",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)",
  "verification_axes": {
   "benchmark_mechanism": {
    "confirmed": null,
    "tested": false
   },
   "practical_benchmark": {
    "beats_baseline": null,
    "tested": false
   },
   "toy_mechanism_gate": {
    "confirmed": null,
    "tested": false
   }
  }
 }
}