AI summaryⓘ
The authors studied complex systems by looking at their network structures to see if important parts (hubs) in one measure stay important in another, revealing real organizational patterns. They tested this idea across seven different systems, like mathematical proofs, software, brains, and circuits, finding that some network measures (betweenness) better capture these consistent hubs than others (degree). Their results support the idea that certain patterns of key elements persist across layers in these systems, which experts recognize as important functional structures. They confirmed their main hypotheses with solid statistical support and preregistered all their tests beforehand to increase confidence.
multilayer network analysishub persistencedegree centralitybetweenness centralityfunctional proximity lawISCAS85 benchmarkformal systemsprebiotic autocatalytic networkSpearman correlationproof assistants
Abstract
We ask whether dependency topology correlates with functional load-bearing organization as recoverable geometry -- not as a metaphor, but as a measurable structural property detectable by multilayer network analysis. Across seven independent substrates, we show that hub persistence and rank divergence under the Functional Proximity Law recover operational organization that domain experts describe as logic: axiomatic load-bearing structure in formal mathematics, control and contract structure in legacy software, conserved hub grammar across approx. 600 million years of neural evolution, catalytic role organization in a published prebiotic autocatalytic network, carry-path dominance in a 4-bit digital circuit, betweenness persistence in the ISCAS85 c432 standard benchmark (n=196), and a directional formal-systems replication in the Coq Corelib (n=17). A key methodological finding: degree-based hub persistence is weak between physical wiring and simulation state-correlation layers (r=0.21 in c432), while betweenness-based persistence is stronger (r=0.77 in the 4-bit ALU post-hoc; r=0.34 in c432). The ISCAS85 pre-registered primary hypothesis was CONFIRMED (degree r=0.426, p=0.002, Spearman r=0.551). The formal-systems claim is supported by two proof-assistant corpora: Lean 4 mathlib4 (CONFIRMED, r=0.777, p=0.004) and Coq Corelib (PARTIAL, direction confirmed, r=0.288, p=0.287, n=17, underpowered). All seven experiments were pre-registered before analysis.