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t06 — Network transport efficiency and redundancy

notes.md

# t06 — Network transport efficiency and redundancy

**Design.** Three public networks — OpenFlights airline routes (3,214 airports),
email-Eu-core (986 nodes) and ca-GrQc (4,158) — were cut to their largest
connected component and turned into a 20-point family from a BFS spanning tree to
the full graph, adding shortcut edges in order of tree span. We measured
efficiency (mean reciprocal shortest-path length, BFS from up to 400 sources),
redundancy (E−N+1)/N, algebraic connectivity (100-step Lanczos) and robustness:
the giant-component fraction surviving removal of the top 5% of nodes by degree.

**Result.** Efficiency and robustness rise together in every family: Spearman rho
= +0.98 in all three (mean +0.979, bootstrap 95% CI +0.929 to +0.994, 2,000
resamples of the 20 variants). The tree is both least efficient and least robust
(0.003–0.009); the full graph is most efficient and most robust (0.76–0.92). The
predicted anti-correlation does not appear, so the stated refutation condition is
met. Partial rho controlling redundancy is about zero (+0.04, −0.05, +0.11): the
association is a density effect.

**Choice test.** Thinning the full graph — removing edges in ascending
shortest-path participation while efficiency stayed within 5% — stripped 44–52% of
edges and collapsed robustness in the airline network (0.76 to 0.11) and ca-GrQc
(0.88 to 0.59) but barely in email (0.92 to 0.91). Redundancy is load-bearing
where the graph is hub-like, so a frontier exists — but not between efficiency and
robustness as claimed.

**Limitations.** Family construction raises both mechanically; efficiency is
sampled; one spanning tree fixes the low end.

Included code and results

• tests/t06/code.py
• tests/t06/results.json
• tests/t06/series.csv