LONDON — I came up out of the Underground this morning at the wrong station, a humbling way to begin a post about the meaning of position. The conference I am here for, the Warwick/Yale/Princeton meeting on political economy, takes its name, at one remove, from an English county whose old town lent its name to an earldom. I will get to the earldom. I want to start with what I was holding when I climbed the stairs at the wrong stop: the map.
The London Underground diagram is, I have come to think, the most honest picture in routine public use anywhere in the world, and almost nobody who relies on it knows why.
The most honest map in the world
In 1931 a draughtsman named Harry Beck, who worked in the railway’s signals office and drew the thing on his own time, did something the Underground at first thought a mistake. He threw the city away.1 Earlier Tube maps placed each station where it sat in geographic London, crushing the centre into an illegible knot while the outskirts trailed off the sheet. Beck spent his days drawing electrical circuit diagrams, and he had noticed something about the passenger underground: she does not care where she is. She cares which line she is on, which stations come in what order, and where she can change lines. Distance, direction, the meander of the river — for her purposes, all of it is noise.
So Beck drew the network the way one draws a circuit. Horizontals, verticals, and diagonals pinned to exactly forty-five degrees; station spacing equalized without regard to the real miles between stops; the Thames flattened into a tidy blue ribbon. The diagram keeps one kind of information — what connects to what, and in what order — and discards everything else without apology. It works not in spite of being a distortion but because it is one.
Here is why that should interest the readers of this series. A force-directed network layout — the standard way political scientists draw the graphs in their papers — also hands you a picture in which every node sits at a coordinate. Those coordinates, I have argued, are very nearly meaningless: they are the resting positions of a simulated physical system, dependent on the random seed, the software defaults, and every other node in the graph. Such a layout fails what I called Network Embedding Independence of Irrelevant Alternatives.2 Move a far-off node and your two nodes of interest slide to new coordinates, though nothing about their relationship has changed. The coordinates carry no warranted local content, and the picture never admits it — it presents them as confidently as its edges, and the reader, reasonably, trusts both.
Beck’s diagram has the same meaningless coordinates — the gap between two stations on the page tells you little about the walk between them — but it announces the fact. It is so flagrantly schematic that no sensible passenger reads a real distance off it. Beck drew the one genuinely invariant quantity, the topology, and made clear the rest was draughtsmanship. That signal is the honesty. A force-directed layout commits the same distortion and hides it.
The distortion is not harmless even on the Underground. Leicester Square and Covent Garden sit one stop apart on the Piccadilly line; on the diagram that looks like a journey, when on the ground they are a four-minute walk. A map that changes how its readers move is not merely describing the city but, in a small way, helping to constitute it.
The man who was an interchange
Now the earldom. The badge on my jacket says Warwick because of a university, which takes its name from the county, whose old seat is the town and castle of Warwick — held for centuries by one of England’s most powerful noble families. The most famous holder of the title was Richard Neville, the sixteenth Earl of Warwick, and history has not remembered him by his name. It remembers him by his function: Warwick the Kingmaker, the man from whom the English word descends.3
Before the storytelling, the mathematics, stated plainly. Consider a voting situation: a set of players, and a rule that declares some coalitions winning and the rest losing. Fix an ordering of the players and admit them one at a time. At some point the admitted group crosses from losing to winning, and the player who causes that crossing is the pivot for that ordering. The Shapley–Shubik index \(\phi_i\) of a player \(i\) is the fraction of all \(n!\) orderings in which \(i\) is the pivot.4 It is a number in \([0,1]\), and it measures one thing: how often you are the swing.
The first thing to notice about \(\phi_i\) is that it is not a property of the player; it is a property of the entire game. Player \(i\) can do nothing at all — not cast a different vote, not lift a finger — and watch her index rise or fall because two other players merged their blocs, or a newcomer entered, or the winning threshold shifted. Pivotality is a global quantity. It only wears the costume of a personal attribute.
Richard Neville’s power was a Shapley–Shubik index, and his mistake was to treat it as a possession. His own hereditary claim to the throne was negligible; what he had was position. In 1461 he was pivotal for the House of York, and Edward of York became Edward IV. By 1470 the configuration had shifted — Edward had married against Warwick’s counsel, built alliances of his own, and stopped needing him — and Warwick was pivotal no longer for York but against it. He changed sides, made common cause with the Lancastrian queen he had fought for a decade, and put the deposed Henry VI back on the throne. For about six months he was again the pivot, the swing, the maker of the king.5
Then, in the spring of 1471, the configuration moved once more, and this time it moved past him. At the Battle of Barnet, fought in heavy fog, Warwick’s army fell into confusion and turned its weapons on its own line; he was unhorsed and killed in the rout. The point is not the moral of the turncoat but a structural one. Warwick’s power had never belonged to Warwick. It was a fact about the relative positions of York and Lancaster, about who needed whom, and the instant that arrangement no longer ran through him he was not a diminished kingmaker. He was a man on foot in a field. Pivotality, like a coordinate in a force-directed layout, is contingent on the rest of the graph — and the rest of the graph does not hold still.
The failure of locality
So I have collected two doctrines of position in a single morning’s walk, from a Tube platform to picking up my conference badge at the Shard. The first, Beck’s, holds that position is meaningless and should be thrown away: draw the topology, equalize the spacing, tell the reader plainly that the coordinates are not to be trusted. The second, Warwick’s, holds that position is the only thing that matters: a man with no claim and no crown ran a kingdom for a decade because of where he sat in the structure.
These sound like opposite lessons. They are not. They are the same lesson, stated once by a draughtsman and once by a corpse at Barnet. In each case a quantity that looks like a property of a single object — the coordinate of a station, the power of an earl — is really a property of the whole structure that object sits inside. Beck’s response was honesty: he isolated the part that survived a redrawing, the connectivity, and drew only that. Warwick’s response, if a career can be called one, was to mistake a structural quantity for a personal possession, and to be astonished when the structure took it back.
The force-directed network layout makes Warwick’s mistake on the reader’s behalf. It takes coordinates that belong to the whole graph and prints them as though they belonged to the nodes. Every time a paper shows two groups drifting closer across a sequence of network snapshots and calls it convergence, it is doing what Warwick did: treating a borrowed position as an owned one. The position was only ever lent, and the lender is the rest of the graph.
The honest move is the old one, and it is not hard. Draw like Beck. Find the quantity that genuinely survives a redrawing — and such quantities exist; a node’s degree does not move when you move the page — report that, and label the rest draughtsmanship. The argument that node coordinates carry no warranted inferential content is one Maggie and I first put in print more than a decade ago, in a paper on which centrality measures hold up under a redrawing and which collapse.6 The Underground map got there first, in 1931, with considerably better graphic design.
This dispatch has a sequel. In a few days I file again, from Nancy, where Maggie gives a keynote on three paradoxes of optimal evaluation, and where the mathematics turns from the geometry of networks to something older and stranger — the point at which adding one more body, or one more voter, or one more alternative tips an orderly system into chaos. Position will give way to motion.
With that, I leave you with this.
Notes
1 Henry Beck (1902–1974) produced his diagram of the Underground in 1931 while employed as an engineering draughtsman in the railway’s Signals Office. The company at first rejected it as too radical a departure, then relented far enough to print a small trial run of pocket maps in 1933, which the public took to at once. The fee Beck received for one of the most influential pieces of information design of the twentieth century has become a small parable in its own right: the sum was a handful of guineas, and he spent decades afterward revising the map for not much more.
2 See “The Physics of Political Networks,” April 1, 2026, where NEIIA is defined formally and force-directed layouts are shown to fail it by construction, alongside a three-tier account of which network quantities survive a redrawing and which do not. Node coordinates are the least invariant object of the three; degree centrality is the most.
3 The University of Warwick, despite the name, sits on the edge of Coventry rather than in the town of Warwick itself; it draws the name from the wider county. The chain from a present-day political economy conference back to a fifteenth-century earl is therefore real but pleasantly indirect, which is the only kind of provenance a blog post should ever lean on.
4 The index is due to Lloyd Shapley and Martin Shubik, “A Method for Evaluating the Distribution of Power in a Committee System,” American Political Science Review 48(3): 787–792, 1954. A close relative, the Banzhaf index, counts the coalitions in which a player is critical rather than the orderings in which she is pivotal. The two indices can disagree, sometimes substantially, on how power is distributed across the same set of players — itself a quiet lesson in how much a “measure of power” depends on the choice of what to count.
5 Richard Neville, 16th Earl of Warwick (1428–1471). His restoration of the Lancastrian Henry VI in 1470, displacing the Yorkist Edward IV whom he had himself installed nine years earlier, is known as the Readeption; it lasted roughly six months. Warwick was killed at the Battle of Barnet on 14 April 1471. The epithet attached to him firmly only after his death — a later historian described him as the setter-up and plucker-down of kings, and the shorter word eventually won out.
6 Patty, John W. and Elizabeth Maggie Penn. “Analyzing Big Data: Social Choice and Measurement.” PS: Political Science & Politics 48(1): 95–101, 2015. That paper works out, among other things, the Independence of Irrelevant Edges condition for centrality measures: a combinatorial demand that a measure not depend on edges elsewhere in the network. NEIIA, the geometric condition invoked above, is the embedding-level analogue — the same independence demand, made of coordinates instead of rankings.