Spectral-gap of audience-signal Laplacian predicts time-to-adoption-saturation: t_sat * gamma_2 in [0.7, 1.3] across panels
A single number from network math could predict how fast any market 'goes viral' — before it happens.
Spectral graph theory (Chung 1997) and PDE-on-graph diffusion (heat semigroup) imported into adoption science, predicting a panel-invariant dimensionless product testable on existing datasets.
4 bridge concepts›
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6-Dimension Weighted Scoring
Each hypothesis is scored across 6 dimensions by the Ranker agent, then verified by a 10-point Quality Gate rubric. A +0.5 bonus applies for hypotheses crossing 2+ disciplinary boundaries.
Is the connection unexplored in existing literature?
How concrete and detailed is the proposed mechanism?
How far apart are the connected disciplines?
Can this be verified with existing methods and data?
If true, how much would this change our understanding?
Are claims supported by retrievable published evidence?
Composite = weighted average of all 6 dimensions. Confidence and Groundedness are assessed independently by the Quality Gate agent (35 reasoning turns of Opus-level analysis).
Empirical Evidence
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The Empirical Evidence Score measures independent real-world signals that converge with a hypothesis — not cited by the pipeline, but discovered through separate search.
Convergence (45% weight): Clinical trials, grants, and patents found by independent search that align with the hypothesis mechanism. Strong = direct mechanism match.
Dataset Evidence (55% weight): Molecular claims verified against public databases (Human Protein Atlas, GWAS Catalog, ChEMBL, UniProt, PDB). Confirmed = data matches the claim.
Two fields are quietly intersecting here. The first is the study of 'weak social signals' — the faint, early behavioral traces people leave when they're starting to pay attention to something new, like subtle shifts in what they click, share, or search for. The second is spectral graph theory, a branch of mathematics originally developed to understand how things spread across networks — think of it like figuring out how heat flows through a weirdly shaped metal object. The hypothesis borrows a specific mathematical tool from that second world and asks whether it can predict something very practical from the first: how long does it take for a new product, idea, or behavior to go from 'gaining steam' to 'everyone's doing it'? The core idea is elegant. You take a map of an audience — clusters of people grouped by how their early-adoption signals overlap — and you build a mathematical network from it. Then you compute a specific property of that network called the 'spectral gap' (essentially, how well-connected the network is at its weakest link). The claim is that if you multiply this spectral gap by the actual time it takes for adoption to plateau, you always get a number between roughly 0.7 and 1.3, regardless of which market or panel you're looking at. That would make it a universal constant of sorts — a dimensionless fingerprint of how fast social spread works. Why does this matter? Because right now, predicting adoption curves is mostly art and hindsight. If this dimensionless product holds across real datasets, it would mean you could measure one mathematical property of early audience signals and immediately estimate how long until saturation — before you've spent the marketing budget, before you've scaled up production, before the wave has crested.
This is an AI-generated summary. Read the full mechanism below for technical detail.
Why This Matters
If confirmed, this could give marketers, product managers, and investors a quantitative early-warning system for adoption timing — derived purely from network math applied to weak pre-adoption signals, without waiting for sales data to accumulate. It could also change how A/B testing panels are designed, since the spectral gap of your audience graph would become a meaningful variable to control for. For social scientists, it would validate importing heat-diffusion physics into human behavior modeling in a concrete, testable way. The hypothesis is speculative enough (confidence 5/10) that it absolutely warrants empirical testing on existing panel datasets before anyone builds a product around it — but the math is grounded and the prediction is falsifiable, which makes it genuinely worth the experiment.
Mechanism
Build per-panel signal-co-occurrence graph (vertices = audience clusters, edges weighted by signal-co-occurrence x Gaussian similarity); compute Laplacian L = D - W spectrum lambda_1=0 < lambda_2 = gamma_2 <= lambda_3 ... Adoption indicator a(t) on graph evolves under reaction-diffusion: a(t) ~ sum_k c_k e^{-gamma_k t} v_k. Under reaction-rate uniformity across panels, t_sat (time from inflection onset to within 10% of plateau) satisfies t_sat gamma_2 = 1/(1 - r/gamma_2). For r/gamma_2 in [-0.43, 0.23], t_sat gamma_2 in [0.7, 1.3]. Cross-panel reaction-rate heterogeneity is the dominant risk; wider [0.5, 2.0] window adopted as primary prediction per QG conditional caveat.
Supporting Evidence
Chung 1997 Spectral Graph Theory: L = D - W; gamma_2 controls slowest diffusion; e^{-tL} as heat semigroup. Kempe-Kleinberg-Tardos 2003 KDD Maximizing the Spread of Influence (10.1145/956750.956769; SIGKDD Test of Time Award 2013) -- distinguished from H11 by graph type (KKT uses social network; H11 uses signal-co-occurrence Laplacian). Reaction-diffusion linearization on graphs: standard PDE-on-graph result.
How to Test
3-5 adoption panels (e.g., SNAP Memetracker for memes, social-bookmarking dataset, financial-product-adoption from broker disclosures). For each: (1) K-means cluster vertices; (2) build edge weights = signal-co-occurrence x Gaussian similarity; (3) sparse Laplacian eigendecomposition (K = 200-500 vertices, tractable); (4) extract gamma_2; (5) operationalize t_sat from observed adoption curve; (6) compute t_sat * gamma_2 per panel; (7) test invariance via mean in [0.5, 2.0] AND CV < 0.5.
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Can you test this?
This hypothesis needs real scientists to validate or invalidate it. Both outcomes advance science.