Sparse Law Calculator — DTN Delivery Rate Predictor

Predicted Delivery Rate ?

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S_T

reachability

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exp(E[H]·λ)

chain survival

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interpolated

How this works

All three factors are interpolated from a 17,280-configuration simulation sweep across 8 solar system targets (Mercury–Titan), 4 constellation sizes, 12 link-quality levels, 3 attenuation exponents, 5 epochs, and 3 seeds.

Chain attenuation exp(E[H]·λ) is from Dijkstra oracle-path data — NOT p^E[H]. λ is the Lyapunov exponent (mean ln p_hop).

Φ (policy distortion) = η_sim / η_lyap. Φ > 1 means greedy outperforms time-optimal by exploiting survivable paths.

Dashed curve on the chart is the closed-form prediction Φ = exp[−γ·E[H]·λ / (1+α·p)] with γ estimated from this cell's data and α = 0.73 (CRAWDAD inter-contact tail exponent).

Three-Factor Decomposition

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β_myopic = —
formula: 0.311 · E[H] · (1 − e^−p/0.042)
Φ_pred = exp(−β · ln n) = —

Closed-Form Φ — Competing Risk Theory

Φ = exp[ −γ · E[H] · λ / (1 + α · p_eff) ]

Full chain: DR = S_T · exp[ E[H]·λ · (1 − γ/(1+α·p_eff)) ]

γ is the order parameter classifying temporal graphs into two universality classes: trap-dominated (γ < 0, gateway-scarce networks) vs cluster-dominated (γ > 0, connectivity-redundant networks). The gap between classes is +0.75 with zero overlap across 154,000+ configurations. Four independent confirmations: γ sign, χ vulnerability, achievability gap, forwarding ratio.

α ≈ 0.73 is the inter-contact tail exponent (power-law P(gap > t) ~ t^−α). Heavy tails (α < 1) create infinite mean waiting time — this is the microscopic origin of γ. Validated R² = 0.941 on 20 CRAWDAD cells (4 real human-mobility traces).