Accelerating returns in gene panel design: A completion theory for regulatory network identification

Measurement panel design for regulatory network inference requires choosing which genes to observe. We prove that under random gene selection, expected marginal identifiability is non-decreasing and strictly increasing for sufficiently large k (once nontrivial regulatory interactions can be completed)-a property we term acceleration, meaning that E[Δ(k+1)]>E[Δ(k)], rooted in order statistics of random permutations. Formally, the expected number of regulatory edges completed at step k equals ∑_j(k-1s_j-1)/(Ns_j), which is strictly increasing in k for k≥max_js_j whenever any regulatory interaction involves two or more genes. From this theorem we motivate a lookahead panel selection strategy that achieves 10 percentage points more completed regulatory edges than greedy set cover at 50% panel size (Cohen's d=0.77, p=9.8×10^-8, paired Wilcoxon, n=50 networks with N≤40), evaluated across all 285 published Boolean regulatory networks from the Biodivine Boolean Models database. Across 285 BBM networks, the mean minimum panel size for eliminating all observational blind spots is 26.4% of model nodes, while the median 50%-full-observability threshold is 65.2% (under two-phase greedy ordering), and 90% full observability requires a mean of 95.1% of model nodes (median 95.2%). With the lookahead strategy, 50% completion is reached at 61.2% panel size in the strategy-comparison cohort (n=50, N≤40). The theorem further establishes that, under random ordering, intrinsic identifiability potential depends solely on the in-degree distribution, while extractable identifiability under a topology-aware algorithm depends on the interaction between the selection algorithm and network topology. Biological wiring is significantly better exploited by topology-aware algorithms than degree-preserved random wiring (Cohen's d=0.29, p=2×10^-7), a small but robust effect consistent with biological regulatory architecture being exploitable by systematic measurement design.