06 — Coupling Spreads Failure
Supplied working report; original verdicts not validated.
Review status: Needs review. This is an archival source note, not a passed proof.
hypothesis.md
# 06 — Coupling Spreads Failure
**Source:** Incoprea Research library, Vol. XXXI, "network-coupling-cascades" family (entries
published 30 October 2026). Round-V assignment sheet, test 06. The claim is quoted verbatim
from the assignment.
## The posted claim (verbatim)
> "Coupling two networks creates failure paths neither had, so cascades are larger in coupled
> systems."
**Prediction:** "A coupled pair of networks collapses discontinuously under random node
removal where an isolated network would degrade gradually."
**Refutation:** "Coupling without effect on cascade size."
## What this test does
Pure NumPy simulation, fully seeded, no network access. Two Erdős–Rényi networks with
n = 10,000 nodes and mean degree c = 4. A fraction q ∈ {0, 0.05, 0.2, 0.5} of nodes carries
one-to-one dependency links between the two networks (Buldyrev et al. 2010 style). A
fraction 1−p of nodes is removed independently at random in each network, with p scanned
from 0.5 to 1.0 in steps of 0.005 (nested removal sets). The cascade iterates until a fixed
point: a node survives only if it lies in the giant component of its own network and its
dependency partner (if any) survives. Recorded per (q, seed, p): final surviving fraction
S(p) = mean over the two networks of the alive fraction; the number of cascade iterations;
and the extra loss beyond the initial removal, `extra(p) = p − S(p)`. N = 24 seeds; removal
masks, networks and partner draws are shared across q so the comparisons are paired.
Per seed and q the following summary statistics are computed:
- `p_c` = smallest p in the grid with S(p) ≥ 0.5 · S(p=1) (the collapse threshold);
- `jump` = largest single-step drop of S as p decreases (a discontinuous collapse shows a
large jump; a smooth degradation does not);
- `cascade` = max over p of `extra(p)` (cascade size beyond the initial removal).
## Pre-specified criteria for this test
Let `p_c(q)`, `jump(q)`, `cascade(q)` denote the mean over seeds at coupling q.
- (a) **threshold** — for each q ∈ {0.05, 0.2, 0.5}, the paired-difference
`p_c(q) − p_c(0)` has a 95% bootstrap CI (seeded, 20,000 resamples of the 24 seed
differences) that excludes zero: coupling raises the collapse threshold.
- (b) **discontinuity** — `jump(q)` increases monotonically with q, `jump(0.5) ≥ 0.4`
(a discontinuous drop) while `jump(0) < 0.2` (a smooth degradation).
- (c) **cascade size** — `cascade(q)` increases monotonically with q.
- **supported** — (a), (b) and (c) all hold.
- **partly supported** — (a) and (c) hold but (b) does not: coupling raises the threshold
and enlarges cascades, but the collapse is not as discontinuous as claimed.
- **not supported as stated** — (a) fails or (c) fails: coupling barely changes survival.
These rules are exhaustive.
## Notes declared in advance
- The dependency links follow the standard model: partner sets are nested across q
(the q = 0.05 partners are a subset of the q = 0.2 partners, and so on), and a node with
a partner dies when its partner dies even if the partner's death happens in a later
cascade round; this is the Buldyrev convention.
- With mean degree 4 the mutual-percolation threshold of the fully coupled pair
(q = 1) would exceed 1; the pre-declared q values probe the partial-coupling regime where
the first-order transition may or may not appear. The criteria above decide the verdict
either way.
- `p_c` uses the 0.5 · S(1) rule on a 0.005 grid; grid resolution contributes about
±0.0025 to the threshold estimate, which is small relative to the between-q differences
the criteria test.
README.md
# 06 — Coupling Spreads Failure **Verdict: partly supported** ## Summary | Quantity | q = 0 (isolated) | q = 0.05 | q = 0.2 | q = 0.5 | |---|---|---|---|---| | Collapse threshold p_c (mean over 24 seeds) | 0.572 | 0.586 | 0.629 | 0.709 | | Paired p_c − p_c(0), 95% bootstrap CI | — | +0.014 [0.013, 0.015] | +0.056 [0.055, 0.058] | +0.137 [0.135, 0.139] | | Largest single-step drop in surviving fraction | 0.008 | 0.009 | 0.010 | 0.062 | | Max cascade loss beyond the initial removal | 0.100 | 0.121 | 0.189 | 0.477 | Model: two Erdős–Rényi networks, n = 10,000, mean degree 4, with one-to-one dependency links on a fraction q of nodes; random removal of fraction 1−p; cascade to the fixed point (alive = in the giant component of the own network and the partner alive). 24 seeds, p from 0.5 to 1.0 in steps of 0.005, removal sets nested and shared across q. ## Headline numbers - Coupling raises the collapse threshold monotonically: from p_c = 0.572 with no coupling to 0.709 at q = 0.5, and every paired 95% bootstrap CI for p_c(q) − p_c(0) excludes zero (smallest effect +0.014 at q = 0.05). - Cascade size grows with coupling: the maximum extra loss beyond the initial removal rises from 10.0% of nodes (q = 0) to 47.7% (q = 0.5). - The collapse is **not** discontinuous at these coupling levels: the largest single-step drop of the surviving fraction is 0.062 at q = 0.5, far below the pre-specified 0.4 threshold for a first-order jump, and only 0.008 at q = 0. The survival curves degrade smoothly at q ≤ 0.2 and steepen only mildly at q = 0.5. ## What this shows and cannot show It shows that dependency links between networks create failure paths neither network had: the same random removals destroy more of the system when coupling is stronger, and the threshold moves well above the isolated-network value. It does not show the claimed *discontinuous* collapse: with mean degree 4 and coupling fractions up to 0.5 the transition remains continuous (second order). Whether a first-order jump appears is a property of the parameter regime, not of coupling per se. This is a numerical percolation experiment on an idealised model; it is not evidence about any particular real infrastructure network, and no operational or investment inference should be drawn from it. ## Limitations 1. **Parameter regime, a rival explanation for the missing jump.** Parshani et al. (2010, *Phys. Rev. Lett.* 105, 048701) showed analytically that partially interdependent Erdős–Rényi networks with mean degree 4 lose the first-order transition below a critical coupling fraction (their phase diagram places the tricritical point near q ≈ 0.5–0.6 at k = 4). The smooth collapse observed here at q = 0.5 is therefore consistent with the coupling claim being *parameter-dependent* rather than false; the honest reading is that coupling enlarges cascades and raises the threshold, while discontinuity requires stronger coupling or denser networks than tested. 2. **Finite size and grid.** n = 10,000 with 24 seeds leaves sampling noise of a few grid steps (0.005) in p_c; the bootstrap CIs quoted are for the paired seed differences and do not include grid-resolution uncertainty. 3. **Idealised coupling.** One-to-one dependency links with no redundancy, no load redistribution, and simultaneous failure of a node the moment its partner fails are extreme assumptions; real interdependent systems have spare capacity, weighted links and non-random topologies, all of which can change both the threshold and the order of the transition. 4. **Static model.** Nodes and edges do not adapt, repair or re-route during the cascade; the experiment measures worst-case static fragility, not the dynamics of any real failure event.