Common mistake 1 with Flow map

Reading a thick arc as a strong relationship. Big economies trade heavily with each other because they are big; that is what the gravity model exists to control for.

Common mistake 2 with Flow map

Drawing every flow. A thousand arcs is a solid block of ink and shows nothing - which is why the default draws the largest and tells you what you are missing.

Common mistake 3 with Flow map

Forgetting the direction. The arcs bend consistently so a flow and its reverse are distinguishable, but the dot is the only marker of which end is which.

Common mistake 4 with Flow map

Mapping a bilateral panel without filtering to a period, then wondering why the numbers are several times the annual figures.

Common mistake 5 with Flow map

Treating the arcs as routes. They are not shipping lanes; the curve is a drawing convention that keeps opposite flows apart.

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Flow map

Draws a curved line for every flow between two places, with the width carrying the size. Made for bilateral data - exports from one country to another, migrants between states, remittance corridors - which a shaded map cannot show at all, because the quantity belongs to a PAIR of places rather than to one.

What it needs

Origin - Where the flow starts.
Destination
Size of the flow - Rows with the same origin and destination are added together first.

The intuition

Thematic maps shade places. That works when the quantity belongs to a place - a population, an unemployment rate - and fails completely when it belongs to a PAIR of places, which is what nearly all interesting economic geography is about. Trade, migration, remittances, commuting, investment: none of them can be shaded onto a country, because the number is not a property of the country, it is a property of the link. Drawing the link is the only honest option, and once you do, two facts about economic flows become visible immediately that no table makes obvious: how extraordinarily concentrated they are, and how much of the structure is just distance.

When to use it

Bilateral data of any kind: exports from one country to another, migrants between states, remittance corridors, commuting, freight, capital flows, even citations between institutions. A choropleth simply cannot show these, because the number belongs to a PAIR.

Where the estimator comes from

Each drawn arc is a quadratic Bezier from origin centroid p_0 to destination centroid p_1: B(t)=(1-t)^2p_0+2(1-t)t\,c+t^2p_1,\qquad c=\tfrac{p_0+p_1}{2}+\beta\,R\!\left(p_1-p_0\right), where R rotates by ninety degrees and \beta is the bend. Rotating the offset rather than displacing it vertically is what makes every arc bow the SAME way relative to its own direction of travel, so X_{ij} and X_{ji} separate instead of overplotting. Width is linear in the flow, w_{ij}=w_0+w_{\max}X_{ij}/\max_{kl}X_{kl} - linear because a line has only one dimension that reads as size, unlike a circle, where area must be used instead.

The model

\text{width}_{ij}\propto X_{ij},\qquad X_{ij}=\sum_{t}x_{ijt}

What it assumes

Each row is one directed flow. A row from A to B and a row from B to A are different flows, and the map draws them as two arcs bending the same way so they sit side by side.
Rows sharing an origin and a destination are added together before drawing, so a panel over several years becomes one arc per pair unless you filter to a period first.
A flow of zero has nothing to draw. A NEGATIVE flow is a flow in the other direction, and is dropped rather than guessed at - swap the columns for those rows if that is what they mean.
Only the largest flows are drawn. The share of the total they represent is printed above the map, and it is usually far more than the share of the flows.
Both places need a boundary on the map. Aggregates - "World", "EU", "OECD" - have none, and the flows involving them are named and excluded.

Statistical properties

Flows are concentrated to a degree that surprises people every time. In bilateral trade the top ten per cent of country pairs routinely carry two thirds to three quarters of the total value, which is why drawing the largest few hundred arcs shows most of the money.
The map draws DIRECTED flows. A and B trading in both directions is two arcs, and because both bend the same way relative to their own direction, they sit side by side rather than on top of each other.
Overplotting is used rather than fought: arcs are semi-transparent, so a busy corridor darkens in proportion to how many flows share it. Density becomes a second, unlabelled encoding.
Within-place flows - a country with itself, a state with itself - have no line to draw and are excluded. They are frequently the largest rows in a real file, so the total shown is not the total in the data and the count is reported.
A flow map shows magnitude and geography but no counterfactual. Two countries trading heavily because they are large and adjacent looks identical to two trading heavily for any other reason.
Centroids are a simplification. A flow to Russia is drawn to the middle of Russia, which is nowhere near where the trade arrives; for large or oddly shaped units the endpoint is a label, not a location.

How to read the output

Line width is the size of the flow. The dot marks the destination.
The subtitle is the honest part: "40 of 1,071 flows drawn · 51% of the total" means you are looking at half the trade in four per cent of the lines.
The concentration line in the summary - what share the top tenth of flows carry - is usually the most striking number on the page. In bilateral trade it is routinely over two thirds.
Dense bundles are a pattern, not a failure: where many arcs overlap, opacity builds up and the corridor darkens.
Within-place flows are counted and excluded. In trade data they are often the largest rows in the file, so check the total is the one you meant.

A worked reading

On 34 countries of bilateral trade, 1,071 non-zero directed flows survive aggregation. Drawing the largest 150 of them - fourteen per cent of the lines - shows fifty-one per cent of the total value, and the top tenth of all flows carries about seventy per cent. The largest single pair is Germany-China at seven per cent of world trade in the sample, with the reverse flow at 6.7% drawn beside it rather than underneath it. Nothing in that picture separates gravity from policy, which is exactly why the gravity model sits next to it in the library.

Symptom → diagnosis → what to do

If you see this	It means	Do this
A dense unreadable bundle	Too many arcs drawn	Lower the count - the subtitle says what share you keep
One arc dominates everything	A within-place row, or a units mismatch	Check the largest-flows table against the file
Half the flows missing	Aggregates like "World" or "EU" have no polygon	The warning names them; filter them out
Arcs overlapping in pairs	Bend set to zero	A straight line and its return journey are the same line
Totals several times what you expect	A panel summed over every year	Filter to one period first
Thick arcs all between big economies	That is what gravity predicts	Run the gravity model; the map cannot answer this

Common mistakes

Reading a thick arc as a strong relationship. Big economies trade heavily with each other because they are big; that is what the gravity model exists to control for.
Drawing every flow. A thousand arcs is a solid block of ink and shows nothing - which is why the default draws the largest and tells you what you are missing.
Forgetting the direction. The arcs bend consistently so a flow and its reverse are distinguishable, but the dot is the only marker of which end is which.
Mapping a bilateral panel without filtering to a period, then wondering why the numbers are several times the annual figures.
Treating the arcs as routes. They are not shipping lanes; the curve is a drawing convention that keeps opposite flows apart.

Where to read this properly

Tobler (1987), "Experiments in Migration Mapping by Computer", The American Cartographer 14(2).
Holten and van Wijk (2009), "Force-Directed Edge Bundling for Graph Visualization", Computer Graphics Forum 28(3).
Head and Mayer (2014), "Gravity Equations: Workhorse, Toolkit, and Cookbook", Handbook of International Economics 4.
Brewer (2016), Designing Better Maps, 2nd edition, Esri Press.
Slocum, McMaster, Kessler and Howard (2022), Thematic Cartography and Geovisualization, 4th edition.

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Flow map

What it needs

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

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