The Eccles projectCounting triangles in the Eccles street network
A graph of 1,543 points, three counting methods from two minutes to two milliseconds, and a comparison with chance.
One query to OpenStreetMap returned 1,401 mapped roads, from the motorway to residential streets, around Eccles. The learner turns them into a graph: every junction or road end is a point, and every stretch of road between two of them is a link. Kept inside a rectangle around the town, that gives 1,543 points and 1,781 links. Most points, 773 of them, have three links, the mark of a T-junction; 439 have only one, which are dead ends or roads leaving the rectangle; 68 have four and just 2 have five.
A triangle is three points that are all linked to one another. The first method checks every possible trio, and there are 611,085,091 of them. Timed on five million trios, that approach would take about two minutes in Python for the full count. The second method walks each link and asks which points are neighbours of both ends, using Python sets: 4,119 lookups and 0.002 seconds. The third writes the network as a table of zeros and ones, multiplies it by itself three times with NumPy and reads off the triangles from the diagonal. All three give the same answer: 43 triangles.
Eccles street network, our Python run| Measure | Value |
|---|
| Triangles | 43 |
| Junctions in at least one triangle | 123 of 1,543 |
| Connected trios (two links sharing a point) | 3,008 |
| Clustering coefficient (share of trios that close) | 0.0429 |
| Triangles in the same network rewired at random, 20 runs | 0.70 on average, never more than 4 |
Is 43 a lot? To find out, the learner shuffles the network: repeatedly take two links and swap their ends, so that every junction keeps exactly its number of roads but the connections become random. Across 20 shuffles the random networks had 0.70 triangles on average, never more than 4, which matches the textbook formula for this degree pattern. So Eccles has around sixty times as many triangles as chance would give it. Looking at which roads close each triangle explains why: 14 involve a roundabout, 11 a slip road or other link road, and 18 ordinary streets. Many of the triangles are made by the way junctions are built and mapped, not by the street plan itself.
Ages 8 to 11
Draw six dots and some lines, then count the triangles two ways and see which way misses fewer.
Ages 11 to 15
Store a small network as Python sets and find triangles by checking shared neighbours.
Ages 15 and up
Build the Eccles graph, count triangles three ways, time each, and compare with shuffled networks.
Sources and limits
Roads are from OpenStreetMap under the Open Database Licence, as mapped on 30 September 2026. The rectangle is our approximation of the town, not its boundary, and roads that leave it become dead ends. The clustering coefficient follows Watts and Strogatz (1998). Timings are from one laptop.
