{
 "artifacts": null,
 "category": "training",
 "description": "Use the paper's analytic merging threshold to choose the Scaled-NAP exponent from an intended community size rather than treating alpha as an arbitrary hyperparameter. A warm-started schedule can begin with persistence-like fine structure and increase alpha only when the model has learned reliable local groups.",
 "formulas_latex": [
  "$$T_{\\alpha}^{\\mathrm{bin}}\\xrightarrow[q\\to\\infty]{}\\bigl(p(p-1)-2\\bigr)\\left(2^{1-\\alpha}-1\\right).$$",
  "$$\\alpha^*(p)=1-\\log_2\\left(\\frac{p(p-1)-1}{p(p-1)-2}\\right),\\qquad\\text{merge adjacent caves iff }\\alpha\u003e\\alpha^*(p).$$",
  "$$\\alpha_t=\\operatorname{clip}\\left(\\alpha_0+(\\alpha_{max}-\\alpha_0)\\left(\\frac{t}{T}\\right)^\\gamma,\\,0,\\,1\\right),\\qquad \\alpha_0=\\max\\{0,\\alpha^*(\\hat p)-\\delta\\}.$$"
 ],
 "id": 2912,
 "implementation": "Use this at the Scaled-NAP loss layer rather than in the encoder's message-passing operation. Choose a target group size p_target from graph metadata; if unavailable, use p_hat=n/R, where R is the desired number of pools, or recompute the median nonempty soft-cluster size every few epochs. Evaluate alpha*(p_hat)=1-log_2((p_hat(p_hat-1)-1)/(p_hat(p_hat-1)-2)), clamp p_hat to at least 3, and initialize alpha_0=max(0, alpha*-0.05). During the first T optimizer steps, update alpha_t with the displayed power schedule toward alpha_max=1; use gamma=1 for a linear schedule or gamma=2 for a slower initial increase. At every step compute Q_alpha_t from the soft assignments using the formulas in the first idea. The Caveman formula is a calibration heuristic derived under clique-like assumptions, so test it rather than assuming that its guarantee transfers to arbitrary graphs. Run a cheap synthetic experiment with 1,000-node planted-partition graphs containing community sizes 8, 16, 32, and a four-size heterogeneous mixture. Compare fixed alpha values with the calibrated schedule in a GCN community-classification model. Measure ARI, NMI, premature cluster merging, final cluster-size error, and training variance across seeds. Success is fewer early merges and higher recovery on heterogeneous graphs at equal final cluster count; failure is no improvement over a fixed alpha=0.5 baseline.",
 "math_summary": "In a connected Caveman graph with q caves, each cave formed from a clique of p\u003e=3 vertices by rewiring one internal edge, the paper states that as q tends to infinity, the adjacent-cave merging threshold converges to T_alpha^bin=[p(p-1)-2](2^(1-alpha)-1). Two adjacent caves merge if and only if alpha\u003ealpha*, where alpha*=1-log_2((p(p-1)-1)/(p(p-1)-2)). Here p is the desired local group size and alpha* is the resolution boundary at which two such groups become preferable to keeping them separate. For a neural model, estimate a target size p_hat from the median predicted cluster size or domain metadata, then initialize alpha just below the boundary so communities of size p_hat are not immediately merged. A differentiable schedule can increase alpha toward one as training progresses.",
 "math_tags": [
  "graph-theory",
  "combinatorics",
  "optimization"
 ],
 "ml_areas": [
  "graph-nn",
  "loss",
  "scheduler"
 ],
 "paper": {
  "arxiv_id": "2608.30934",
  "arxiv_url": "https://arxiv.org/abs/2608.30934",
  "summary_what_math_gives_to_ml": "The paper provides a tunable graph-partition objective that continuously interpolates between a fine-grained persistence criterion and standard modularity. Its transferable asset is the explicit volume exponent: cluster quality is weighted by cluster volume raised to alpha, allowing a model to control whether small cohesive groups or globally balanced groups are preferred. This can become a differentiable auxiliary objective for graph neural networks that produce soft communities, pooling assignments, or routing groups. The Caveman-graph threshold also gives a principled way to initialize or schedule alpha from the target community scale rather than selecting it blindly.",
  "title": "Scaled Null-Adjusted Persistence: A Multiscale Bridge between Modularity and Persistence",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 3,
  "novelty": 8,
  "usefulness": 5
 },
 "solves": [
  "accuracy",
  "generalization",
  "stability"
 ],
 "title": "Caveman-calibrated resolution schedule",
 "url": "https://synthcore.org/idea/2912/caveman-calibrated-resolution-schedule",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}