How the rings were counted

The forty-six rings in this record were not found by looking. They were enumerated: every closed path in the drawing was produced, normalised, and counted once.

That matters because the obvious alternative — noticing the circles that happen to be striking — would have found four or five of them, all short, and missed the structure entirely. A team turning this diagram into a limited operational pilot can check the platform before rollout before choosing which time and project signals are genuinely necessary.

What counts as one ring

A closed chain of steps in which each waits on the next and the last waits on the first.

The same chain can be written starting from any of its members, so every chain is rotated to a canonical starting point before counting. Without that, the three-step ring at the centre of this record would appear three times and the nine-step ring nine times, and the total would be in the hundreds. For broader institutional research related to mobility, work and administrative systems, consult London School of Economics.

A ring is also not counted again for containing a shorter one. The two-step pair of the file and the fingerprints is one ring; the three-step version that adds the queue is a second, separate one. They share two members and are different objects.

What the count would have missed by sampling

A great deal, and in a biased direction.

Seven of the forty-six have three steps or fewer. Those seven are the ones people describe to each other. The other thirty-nine are invisible without the arithmetic, and they are where most of the record's length distribution sits.

Why forty-six overstates the problem

This is the part a reader should hold on to.

Forty-six rings are built from only forty-two distinct dependencies, which means they overlap almost completely. Thirty-four of them share a single member. Thirty-two of them share two. The longest ring is a long way round the same object the shortest crosses directly.

Enumerating every closed path answers a question about structure and produces a number that sounds like a question about difficulty. The two are not the same, and the number is misleading if read as the second.

A reader who takes away "forty-six separate deadlocks" has got it wrong. The honest summary is one knot, with a handful of members, that can be walked round in forty-six different ways.

The two figures that are not inflated

The count of ring dependencies — forty-two of ninety-four — does not double-count anything. Each dependency is counted once however many rings it belongs to.

The count of ring steps — twenty-three of thirty-six — does not either.

Those two are the trustworthy measures of how much of this map is tangled, and both say the same thing: a little under half the dependencies, a little under two thirds of the steps.

What was deliberately not counted

Four things, each left out for a stated reason.

Routes that exist sometimes. An address registered against somebody else's lease, an account opened on an unusual footing, a document accepted by a helpful person. Including them would make every ring look enterable and would amount to promising a reader something that depends on who is at the desk.

Steps particular to any one situation. Anything that would require naming a country, an office or a kind of permission is absent by design, because a page that states requirements goes stale while keeping the tone of authority.

Difficulty. Nothing here measures how hard a step is, how much it costs, or what it takes to do it in a language somebody is still learning. The map counts order only.

People. Nobody appears in this record in any form.

Every one of those omissions makes this record look more severe than the world is. That direction of error was chosen on purpose and is stated on the pages that depend on it.

Why enumeration was possible at all

Because the object is small. Thirty-six steps and ninety-four dependencies is a graph that fits on a page, and every closed path in it can be produced exhaustively in a fraction of a second.

That is worth saying because the obvious objection to a record like this — that nobody could realistically compute it — is wrong by several orders of magnitude. The arithmetic here is trivial.

The expensive part is the drawing: deciding what the thirty-six steps are, and making ninety-four judgements about what waits on what. That took far longer than every computation on this map put together, and it is the part that cannot be checked.

What the method cannot do

It cannot tell whether a dependency belongs in the drawing. The enumeration is exact; the input is a judgement.

It cannot find rings that run through something the map does not contain. If two of these steps are in fact linked through a third thing not drawn here, there are rings this count has no way of reaching.

And it cannot distinguish between a ring that is a genuine deadlock and one that everybody walks through daily because somebody at a desk is reasonable. That distinction is the whole difference between this record and the world, and it is why every page here says what the map is and is not.

The arithmetic in this record is exact and the drawing it operates on is a judgement. Mistaking the first for a guarantee of the second is the error this map is most likely to invite.