One catastrophic dependency counts less than five mild ones¶

A plain-language companion to the draft working paper "Counting Dependencies: Aggregation Rules and What Official Dependency Indices Identify" (v0.1, July 2026). The paper carries the three propositions with their proofs, the cell-by-cell reconstruction of the published index, the Monte-Carlo report on rank stability, and the caveats; this text carries the ideas.

Read the full draft working paper (PDF)

The short version. European economic-security policy now runs on official dependency indices, and one of them is in the evidence base for the 2028-2034 budget period: thirteen technology areas, eight dependency dimensions, two levels each, aggregated by counting the dimensions rated high and subtracting a matching count of reverse dependencies. The paper asks what that construction identifies, before asking anything about the scores themselves. Counting how many dimensions are bad is a different operation from finding the worst one, and the study's own prose describes a system that fails when a single essential component fails. On a two-level scale the mismatch is worse than approximate. Reconstructing the index from its own published cells then produces the finding: applying the study's own stated rule to its own tables contradicts its own published prose, and the area it ranks most dependent flips on half of all admissible reweightings.

Two different questions¶

A unit with one catastrophic dependency and seven benign ones counts as less dependent than a unit with five moderate ones. That is the whole of the first result, and it is elementary. Counting dimensions above a threshold and taking the worst dimension order the same underlying exposure profiles differently, and neither ordering refines the other.

Nothing prevents a policy instrument from choosing the counting rule, provided the choice is stated and defended. The difficulty arises when the analysis says one thing and the index does another. This study describes what it measures as "a tightly coupled and fragile network, where the absence or failure of one essential component within this pipeline can disrupt the whole system." That sentence describes a weakest link. The footnote defining the aggregate describes a sum. The two disagree about the object they are both claimed to measure.

The scale removes the escape route¶

A reader might hope to recover the weakest-link reading from the published cells: look for the worst dimension rather than the count. On a two-level scale, that collapses to a single question, namely whether any dimension is rated high.

All thirteen areas score high on digital infrastructures and software. So the worst-dimension statistic is the same for every area, it ranks nothing, and the entire published ordering is produced by an additive rule the source neither states as a modelling choice nor defends. Coarseness of the scale is not only a loss of precision here. It removes the structural reading altogether.

Subtracting counts is not comparing bargaining positions¶

The published aggregate nets dependencies against reverse dependencies. Subtracting one count from the other treats a high dependency on raw materials as offsetting a high reverse dependency in skilled labour, at par, and treats every high score as equally intense.

The literature on asymmetric interdependence, from Hirschman onward, locates leverage precisely in the intensities that a two-level scale discards. The paper's third result makes this concrete: there are cost profiles where the net count reports the home party as the more dependent while total adjustment cost falls more heavily on the counterpart. The number can carry the wrong sign relative to the asymmetry it is read as summarising.

What the reconstruction found¶

The paper transcribes both published tables cell by cell and recomputes the rows. Twelve of the thirteen printed totals reproduce exactly. The artificial-intelligence row carries seven high cells and prints six; the discrepancy is flagged and nothing rests on it.

Then it applies the study's own footnote rule to the study's own tables. The accompanying prose states that data analytics, interoperability technologies, photonics, quantum and robotics "show a net dependency of more than three types", and that all other areas exceed four. Neither claim follows from the cells. Every member of the named group scores at or below three, and data analytics scores minus one, a net reverse dependency, so the sign is inverted relative to the group it is placed in. In the second group, microelectronics scores three on any reading; under the strict reading of "more than four", blockchain and cloud fail too.

The sign inversion is the finding to hold on to. It is arithmetic on the source's own cells, it survives both readings of the ambiguous phrase, and no judgment call intervenes anywhere in it.

The ranking is the weighting¶

The index weights the eight dimensions equally, which is a convention rather than a result. Drawing random positive weights and recomputing 20,000 times, the most-dependent area changes on 49.8% of draws, alternating between artificial intelligence and blockchain.

The identity of Europe's most dependent technology area, in the instrument informing a multiannual programme, is set by the decision to weight eight dimensions equally. It does not follow from the recorded scores.

What an instrument would need instead¶

Four requirements come out of the results. Intensity has to survive to the aggregation step, which means more than two levels or a continuous cost measure, otherwise no statement about which dependency binds hardest is available and the netting operation means nothing. The aggregation rule has to be stated and matched to the claimed structure: where the argument is that one failing component disrupts the system, the rule is a minimum and the ranking it produces is the one to publish. Exposure has to be scoped to nodes and composed along paths, because two areas with identical dimension scores can face different exposure when their supply paths run through different controllers. And coverage has to include services: the one indicator the source computes rather than elicits is a goods trade balance, explicitly excluding digital services, which leaves out most of what the exercise is about.

The companion papers in this thread build the instrument that meets them, on a measured 52-category graph with exposure composing along paths.

The small print¶

This paper analyses a rule applied to scores; it does not evaluate the scores, which may be entirely sound. If they are, a better aggregation of them would be informative, and reconstructing the index is a step toward building one. The reconstruction depends on a text extraction of the published tables, hand-checked for the rows carrying the results and shipped in full for inspection. The single printed-total discrepancy may be a source error, an extraction error, or an intentional adjustment, and the paper does not resolve it. The critique of netting applies to the published rule and does not imply that the underlying reverse-dependency scores are uninformative.