The Pandemic That Ignored the Map
The big question of this module. Why do we need a science of networks instead of ordinary statistics?
H1N1 Started in Mexico. Spain Was Infected Before Guatemala.
In April 2009 a new strain of influenza, H1N1, began spreading out of Mexico. Nobody had immunity, and every health ministry in the world was asking the same question: when does it get here?
The natural way to answer that is with a map. Disease moves from person to person, people are spread across the ground, so an outbreak ought to grow like an ink stain — neighbours first, then outward in rings. Guatemala shares a land border with Mexico. Spain is nine thousand kilometres away, across an ocean.
The first country outside Mexico to confirm a case was the United States. Spain confirmed its first case before Guatemala did.
One country pair proves nothing, so here is every country at once. Step through the stage below. It opens on the map’s own answer — each country’s distance from Mexico City in kilometres, against the day the virus arrived — and then hands you the ruler.
There is a trend: far-away places do tend to be later. But look along the dashed line at nine thousand kilometres in the first step. Five countries sit within four hundred kilometres of it, and the virus reached the first of them on day 41 and the last on day 121 — six weeks against four months. Kilometres put them in the same place; the epidemic did not. Whatever sets the arrival day, it is not the distance you can measure on a map.
A Route That Carries More Traffic Is a Shorter Route
Dirk Brockmann and Dirk Helbing (Brockmann and Helbing 2013) asked what happens if you stop measuring distance across the ground and start measuring it along the routes people actually travel.
Air travel is the obvious candidate, and it is easy to write down. Take every commercial airport and call it a node. Wherever a scheduled route connects two airports, draw an edge between them. Dots and lines — that is the whole vocabulary of this course, and you have now met all of it.
Here is the move that makes it work. The edges are not all the same length. Of everyone who boards a plane in Mexico City, a large share land in Los Angeles and a very small share land in Ljubljana. A route that carries a big share of the traffic is a short route, because a traveller — and therefore the virus — is likely to take it. A route that carries a trickle is a long one. The effective distance between two airports is then the smallest total you can accumulate along any chain of flights joining them, exactly the way a road atlas gives you the shortest drive, except that “long” and “short” now mean traffic share instead of kilometres.
Put numbers on it. Suppose one in ten passengers leaving Mexico City lands in Madrid, and one in a thousand lands in Guatemala City. On Brockmann and Helbing’s scale the Madrid leg has length 3.3 and the Guatemala City leg has length 7.9. Madrid is the near neighbour, Guatemala City the far one, and the map says the opposite.
If a fraction p of the passengers leaving an airport take a particular route, that leg is given length 1 - \log p (natural log). A leg carrying all the traffic, p = 1, has length 1; halving the share adds about 0.7. The effective distance between two airports is the smallest sum of leg lengths over all routes between them — an ordinary shortest path, on these lengths. That is the whole definition; p = 0.1 gives 1 - \log 0.1 = 3.3 and p = 0.001 gives 7.9.
That is the ruler the stage above switches to at step three, and it is the whole reason the cloud collapsed. Brockmann and Helbing report a fit of R^2 = 0.973. (An R^2 runs from 0, meaning the line tells you nothing about arrival day, to 1, meaning the line is exact.) The same picture holds for the 2003 SARS outbreak. No new data was collected between the two rulers and nothing about the virus changed. Only the ruler changed.
That is the claim the rest of this course rests on: network structure, not geography, determines how things spread.
The Same Skeleton Under a Food Web, a Bank and Your Brain
Nothing in that argument was about influenza. It needed only two things: a set of items, and a record of which pairs are joined. Any system that offers those two things gets the same treatment.
| System | A node is | An edge is |
|---|---|---|
| Air travel | an airport | a scheduled route between two airports |
| The brain | a neuron | a synapse from one neuron to another |
| A cell | a protein | two proteins that bind to each other |
| An ecosystem | a species | “this one eats that one” |
| Banking | a bank | an overnight loan one bank owes another |
| The power grid | a substation | a transmission line |
| Wikipedia | an article | a hyperlink from one article to another |
Read the middle column again. Nothing in the raw data forced any of those choices. From the same airline timetable, you could have made cities or countries the nodes instead of airports. You could have defined an edge to exist if any route connects two places, or only if it carries more than a thousand passengers a week. Deciding what counts as a node and what counts as an edge is a modelling decision, not something the data hands you. It is also the decision that fixes what you will be able to see: effective distance is measurable at all only because the airports, not the countries, are the things that carry a measurable share of the traffic.
What a Brain and a Power Grid Have in Common Is a Table
Once the nodes and edges are chosen, a network can be put down in two forms. You can draw it — dots with lines between them. Or you can write it down as a list of the joined pairs, one pair per row, which is called an edge table. They are the same object, and it is worth checking that claim by hand rather than taking it on trust: every line on the left is one row on the right.
| Source | Target |
|---|---|
| 1 | 2 |
| 1 | 3 |
| 2 | 3 |
| 2 | 4 |
| 3 | 5 |
Five nodes fit in a picture. The air-travel network behind the H1N1 result does not: it has thousands of airports and tens of thousands of routes, and drawn as dots and lines it is a black smudge in which no individual connection can be seen. Zooming does not help, and neither does a better layout algorithm — there is simply more ink than the page has room for. The edge table is unbothered by size. It just gets longer, and a computer reads a long table as happily as a short one.
The table is also what makes this one field instead of ten. Two columns of labels look identical whether the labels are airports, neurons, banks or Wikipedia articles, so a method that works on one works on all of them. That is the bargain of the abstraction, and it is worth stating what each side of it costs: the table keeps exactly one kind of information — which pairs are joined — and discards everything else: how far apart the nodes are, how big they are, what they are made of, what year it is. The H1N1 story is what that bargain buys. Throwing away the kilometres is precisely what let the arrival times line up.
If It Fits in a Spreadsheet, Why Is There a Whole Field?
A two-column table is exactly the sort of thing statistics and machine learning already handle. So why not load the edge table into the usual tools and be done?
Because nothing interesting about a network lives in a row. It lives in how the rows combine.
In 1739 Jacques de Vaucanson exhibited a mechanical duck in Paris. It flapped its wings, took grain from a visitor’s hand, swallowed it, and some minutes later excreted pellets. It was the great advertisement for reductionism: understand each part, build each part, put the parts together, and the behaviour of the whole follows.
It was also a fraud. The duck digested nothing. The swallowed grain went into one compartment, and the pellets it produced had been loaded into a second compartment before the show — the conjurer Jean-Eugène Robert-Houdin, who restored the machine a century later, reported the hidden mechanism. Keep that image, because it is the honest one: a machine assembled from duck-shaped parts produced duck-shaped output, and there was still no duck inside it. (Digesting Duck)
Networks break the reductionist recipe in a specific and repeatable way: the properties that matter belong to the pattern of connections, not to any component. You can know every neuron in a brain and still not know how a memory is stored. You can profile every person in a movement and still not predict which slogan spreads. You can own every computer on the internet and still not explain a traffic jam that appears on it.
Two things go wrong at once. Scale overwhelms intuition: you can hold the relationships among three people in your head, and you cannot do it for three million. And the effects of a change are wildly uneven. Go back to the five-node network above and take away a single edge.
Cut edge 2–3 and nothing is lost: 2 still reaches 3, by way of 1. Cut edge 3–5 instead and node 5 drops off the network entirely — nothing reaches it and it reaches nothing. Same network, same size of edit, opposite consequence. Both edges look identical in the edge table: one row of two numbers each. You cannot tell them apart by looking at a row; you have to look at the pattern the rows make together.
Now scale that question up. Of the hundred thousand transmission lines in a power grid, which ones are the 3–5? That question has no answer in ordinary statistics, and it is the reason this field exists.
What You Can Now Do
- Explain why distance measured through a network beats distance measured across a map when predicting arrival times, and say what effective distance counts.
- Take a system you care about and state what its nodes and edges are — while knowing that you decided that, and the data did not.
- Write any network as an edge table, and say what the table keeps (which pairs are joined) and what it throws away (everything else).
- Say why a table of pairs is where network science starts rather than where it stops.
Next, Module 1: A Stroll, Seven Bridges, and a Mathematical Revolution, where a Sunday walk through an 18th-century city turns into the first theorem of this subject — and where the act of throwing information away becomes a deliberate technique with a name.