The Network Structure of Mathlib

2026-04-26

AI Summary

This study investigates the macro-structural complexity arising from the entanglement of logical, mathematical, and infrastructural dependencies in Mathlib, the Lean 4 mathematical library. By constructing a multi-layer dependency graph comprising over 300,000 declarations, 8.4 million edges, and 7,563 modules, and employing graph decomposition algorithms to distinguish explicit from implicit dependencies, the work combines network centrality analysis with dependency scope statistics to quantitatively characterize formalized mathematics for the first time. The findings reveal significant discrepancies between human-assigned classifications and actual logical structure, demonstrate that formalization compresses semantic hierarchies, show a 50.9% inter-namespace coupling rate, indicate that developers utilize on average only 1.6% of their imported scope, and establish that network centrality predominantly reflects language infrastructure rather than mathematical significance.

Abstract

The ongoing development of Lean 4's Mathlib has produced a macroscopic structural complexity that interweaves logical, mathematical, and infrastructural dependencies. We present a network analysis of this library, extracting its dependency structure into a multilayer graph of 308,129 declarations, 8.4 million edges, and 7,563 modules. By introducing graph decompositions that isolate explicit edges from those synthesized by the compiler or driven by proofs, we quantify the structural properties of formalized mathematics. Our analysis reveals three findings. First, taxonomies designed by humans diverge from logical structures, exhibiting a 50.9% coupling across namespaces. Second, developers utilize a median of 1.6% of the imported scope. Third, formalization compresses semantic hierarchies, with network centrality capturing language infrastructure rather than mathematical relevance.

Problem

• network structure
• formalized mathematics
• dependency analysis
• Mathlib
• structural complexity

Innovation

• network analysis
• formalized mathematics
• multilayer graph
• dependency structure
• Lean 4