One number changed and the whole map moved

Six rungs have priced what a cartogram costs to build: that every density can be met, that meeting it can fold the map, that everything except the areas pays, that some requests are cheaper than others, and that a density asking for no room breaks the construction.

All six take one density specification and produce one map. This one takes two specifications that differ in a single number.

The experiment #

The density is twoCities: a base of 0.25 with two bumps of weight 6, at London and at Tokyo. The perturbed one is identical except that London’s bump is weight 9.

Both are solved by the same diffusion construction, both meet their own area targets to the solver’s own tolerance, and neither is wrong in any way. They are answers to two questions that differ in one place.

London is last of seven. The place whose number changed moves less than every other place measured, and Delhi — whose density is exactly what it was — moves four times as far.

Why it is not local #

The mechanism is conservation, and it is the same one that makes a cartogram possible at all.

A cartogram redistributes a fixed total area. Making London bigger means making somewhere else smaller, and the construction — a diffusion of density followed by an advection of the coordinates — has no reason to take that area from London’s own neighbourhood. It takes it from wherever the flow field says, and the flow field is the solution of a global problem.

So a cartogram is not a function of the data near a place. It is a function of all of the data, and every value enters every coordinate.

Two features of that curve are worth stating.

The peak is a ring, not a point. The mean displacement rises from 0.099 in the first ten degrees to 0.201 at twenty to thirty, and only then falls. The place that changed sits at the bottom of a bowl: an enlarged region is pushed outwards symmetrically about itself, so its own centre barely moves while its surroundings are shoved aside.

The tail does not go to zero. The 170°–180° band — the far side of the world from London — moves by 0.031 on average, fifteen per cent of the peak. Nothing in the construction attenuates with distance, because area is conserved globally and a global conservation law has no length scale.

What this means for reading one #

The consequence is about what a reader can conclude from a cartogram, and it is sharper than the usual complaint about shapes.

A map drawn to a density it was handed established the construction’s contract: the areas come out right. This rung is about everything the contract does not cover, and the first item is position.

A place’s position on a cartogram is not about that place. A reader who notices that a region has moved cannot conclude that anything about it has changed, because it may have been pushed. That is a much stronger statement than “the shapes are distorted”, which everybody knows and forgives.

And a sequence of cartograms is not an animation of the data. Cartograms of successive years are solved independently, so a region whose value is flat for a decade moves every year, by amounts set by everybody else’s values. An animated cartogram shows motion that is partly the data and partly the redistribution, and no part of the picture separates the two. That is the same objection every density can be met and none is free raises about a single map, carried into a sequence.

Nor is a comparison of two cartograms a comparison of two datasets. Subtracting one from the other gives the displacement field above, which is dominated by the geometry rather than by the difference in the data — Delhi’s 0.2256 is entirely other people’s numbers.

The same experiment on a busier density #

One perturbation on one density is an anecdote. The check that it is the construction rather than the case is to run it again with more bumps and a different one moved.

More constraints localise the response without making it local: the ring moves inward and the far tail thins, because with four fixed bumps there is less freedom in where the displaced area can be taken from. That is a useful thing to know for anybody reading a cartogram of many regions rather than a few — the effect is smaller on a busy map and it does not go away, and a real cartogram has hundreds of regions rather than four.

What was computed, and how #

The two cartograms come from diffusionCartogram on two specifications differing in one field. The displacement is measured at a 40 × 21 grid of samples in the rectangle both maps are built in, and the landmark figures at seven stated places from the site’s own list.

The distance bands are great-circle distances from London on the sphere, so the decay is measured against ground distance rather than against rectangle distance, which would mix the projection into the answer.

Three assertions carry the rung and each could fail alone.

A place whose own density did not change must move by more than a sixth of the largest movement anywhere. That is the finding stated as a check, and a construction whose effect really was local would fail it.

The largest movement must not be at the place whose number changed, by a factor of two. A solver that pushed the change outwards from its source — which is what an intuition about diffusion suggests — would fail this while passing the first.

And every stated place must move at all, by more than a thousandth. A displacement field with an exact zero somewhere would mean the construction has a region it does not touch, and it does not.

What a producer could do #

Three responses, and the first is the only complete one.

Draw the difference rather than the maps. If the question is what changed between two years, the honest picture is a map of the change in the data on a fixed base — a choropleth, a proportional symbol, anything whose geometry does not move. A choropleth is read by area has its own failures and they are local ones: a region’s colour depends on its own value and its own areal factor, and nothing else in the dataset enters. That is a much weaker requirement than a cartogram makes and it is the reason a cartogram is a poor instrument for comparison.

Or fix the construction’s freedom. A cartogram solved with the previous year’s map as its starting point, and a penalty on displacement, moves less — which is what the animated-cartogram literature does and which converts the problem into a different one: the map now depends on the order the years were processed in, so two analysts starting from different years get different pictures of the same decade.

Or state that positions are not comparable. One line in a caption, and it is the only repair available to somebody who has already drawn the thing. It is also the least likely to be believed, because a map that looks like a map is read like a map.

None of the three is a fix in the sense the earlier rungs’ repairs were. The cheapest map that meets its areas can be found; the non-locality cannot be removed, because it is the conservation law the construction exists to satisfy.

Why the earlier rungs could not see it #

Every rung below this one measures a property of one map: its areal error, its angular deformation, whether it folds, what it costs against the cheapest. All four are functions of a single cartogram, and a single cartogram is a perfectly well defined object whose every property is measurable.

The non-locality is not a property of a map. It is a property of the mapping from data to map, and seeing it requires two maps and a controlled difference between them — which is a different kind of experiment from anything this anchor had run.

That is worth naming because it is a general gap. A collection that measures objects will not find the failures that live in the operator producing them, and the only way in is to perturb the input and look. This anchor had six rungs of measurements on outputs before anybody varied an input, which is not carelessness; it is what measuring an object rather than a process looks like from the inside.

Where the model stops #

One construction. Everything here is the diffusion cartogram. The triangular constructions this anchor also carries are separable — one coordinate is solved from a marginal and the other conditionally — so their displacement fields have a different structure, and a change confined to one longitude band moves only that band’s conditional. That is a genuinely different answer and it is not measured here. It is also not an argument for using them, because everything else on the page pays for the areas prices what the triangular constructions cost instead.

A stated density rather than real counts. The bumps are closed forms, as everything in this anchor is, so the perturbation is exactly a change to one parameter. A real dataset’s revision changes many numbers at once and the displacements superpose, which makes the effect larger rather than smaller.

Nothing here concerns folding. A map that meets its target can fold is a different failure of the same construction, and neither of the two maps compared here folds anywhere — checked, because a fold would make the displacement field meaningless at the point it happened.

And the size of the change is arbitrary. Raising a weight from 6 to 9 is a 50 per cent revision, which is large for a population and small for a stock price. The displacement scales roughly with it, so the numbers here are a statement about the shape of the response rather than about its magnitude.

The displacement’s own conservation #

One more property of the field is worth stating, because it explains the ring.

Area is conserved, so the map’s total area is the same in both cartograms, and a displacement field that everywhere pointed away from London would have to expand the map. It cannot, so the field has to turn over somewhere: places move outward near the change and inward beyond it, and the ring of largest displacement is where the outward and inward parts meet.

That is a constraint on the shape of the response, not merely on its size, and it is the reason the peak is not at the source.

The generalisation #

The rule is about which operations on geography have a local answer, and the answer this anchor gives is unusually stark.

Almost every quantity in this collection is local or nearly so. A distortion is a derivative, so it depends on the map at a point. An area depends on a boundary. A generalisation depends on a feature and its neighbours. Even the failures that spread — the modifiable areal unit problem, a datum shift, a snapping tolerance — spread by a bounded distance or by a stated rule.

A cartogram is the exception, and it is the exception by construction. It solves a global constraint, and a global constraint means every input touches every output. Nothing about the picture says so: it looks like a map, places sit where places sit, and a reader tracks a region by finding it.

The practical form of the rule is a caution about differencing. Two maps that differ in their inputs can be subtracted, and the difference is informative only when the operation producing them is local. For a projection it is: two projections of the same data differ by a function of position. For a cartogram it is not, and the difference is a conserved-quantity redistribution whose largest values are wherever the flow decided.

What a reader can still conclude #

The rung is negative about position and it is not negative about everything, and the difference is worth stating so the anchor’s own results survive it.

An area is exactly right. That is the construction’s contract and it is met at every point of both maps. A reader comparing two regions’ sizes on a cartogram is reading the data and nothing else, which is what the instrument is for.

A shape is wrong in a way that is measured. Everything else on the page pays for the areas prices that, and its numbers are properties of a single map and are unaffected by any of this.

Only position is contaminated, and it is contaminated by everybody else’s data rather than by the map’s own construction. That is a narrow claim and it is the one that matters, because position is what a reader uses to find a region at all.

What it forbids: the cartogram as a difference display #

One use follows directly from the measurement and it is a use cartograms are frequently put to, so it is worth naming as forbidden rather than merely awkward.

Two cartograms of the same regions at two dates cannot be compared by looking at them. Every region has moved, including every region whose own value is identical between the two, because the displacement each one receives is dominated by its neighbours’ values rather than by its own. A reader shown a pair of maps and invited to see what changed will see change everywhere and will attribute it locally, which is the one inference the method cannot support.

The strength of that statement is the point. It is not that the comparison is noisy or that small changes are hard to see; it is that the visual signal a reader is trained to read — this shape grew, so its value grew — is systematically wrong, and wrong hardest for the regions that did nothing.

The display that does work keeps the geometry fixed. Draw the cartogram once, from one date’s density, and show the other date’s values as a colour on that same geometry. The frame is then a constant, every visual difference is a real difference in the quantity being shown, and the cartogram is doing what it is good at — allocating area by population, so that a colour is weighted by the people it describes — without also being asked to encode change.

That costs the reader the thing an animated pair seems to offer, which is the sense of watching regions swell. What it buys is that everything they see is true.

Who found it, and when #

Gastner and Newman’s diffusion method, from 2004, is the construction used here, and its paper is explicit that the density is diffused over the whole domain — the non-locality is in the method’s statement rather than hidden in it. Every cartogram construction has the property, because every one of them solves a global area constraint; the older rubber-sheet methods of Tobler and of Dougenik, Chrisman and Niemeyer are iterative relaxations, which is a global solve by another name.

What is not in the literature is the measurement. Cartograms are compared with each other, scored for shape preservation, and assessed for readability, and the question how much does one region’s value move another region appears not to have been asked — presumably because the answer is obviously “some” and the interesting question seemed to be how much shape survives.

It is worth asking because the answer is not “some” but “more than the region’s own value moves it”, and that reverses what a reader is entitled to conclude from seeing something move.

Where the ladder goes next #

Seven rungs have taken the cartogram from a construction to a cost to a failure mode and now to its response to its own input. Every one of them has treated the density as given. What a cartogram is actually drawn from is a set of counts attached to regions — a discrete thing — and turning counts into the continuous density this anchor’s machinery requires is an interpolation with choices in it, which is a step nobody in this anchor has looked at and which has the same conservation law running through it.