The Aberdare projectA neural memory filled with neighbourhoods
Census patterns for 123 local areas, a network that stores them, and the point at which it starts to make things up.
Each memory is a neighbourhood. The learner takes the 123 census output areas that the ONS places in the Aberdare built-up area and describes each one with 36 yes-or-no facts. Is the share of people aged 10 to 14 above the county borough median? Is the share of terraced homes? Of households with two people? Of workers who walk to work? Eighteen age bands, four kinds of home, four bedroom counts, five household sizes and five ways of getting to work give 36 bits, each set to +1 or -1 against the median of all 789 output areas in Rhondda Cynon Taf. No two of the 123 Aberdare patterns are identical, though the closest pair differ in only 2 of the 36 bits.
A Hopfield network stores patterns in a grid of connection weights, one for every pair of bits, using the oldest learning rule in the book: strengthen the link between two bits that agree, weaken it when they disagree. To recall, you hand the network a pattern with some bits flipped and let each bit repeatedly take the sign its neighbours vote for, until nothing changes. The learner stores a few neighbourhoods, flips 4 of the 36 bits in one of them, and checks whether the network settles back on exactly the right pattern. Then the same test is run with random patterns of the same length for comparison, 200 times for each number of stored memories.
Share of damaged patterns recalled exactly, 4 of 36 bits flipped, our Python run| Memories stored | Aberdare census patterns | Random patterns |
|---|
| 1 | 100% | 100% |
| 3 | 96.3% | 99.7% |
| 4 | 80.6% | 97.9% |
| 5 | 63.3% | 87.8% |
| 6 | 44.5% | 79.8% |
| 8 | 18.6% | 51.5% |
| 10 | 7.0% | 27.5% |
Theory predicts trouble. For large random networks, Amit, Gutfreund and Sompolinsky showed in 1985 that reliable recall breaks down at about 0.14 memories per bit, which for 36 bits is about five. The random patterns behave roughly that way. The Aberdare patterns fail much sooner: with five neighbourhoods stored, only 63.3% of damaged ones come back exactly. And when recall fails, the network rarely lands on the wrong neighbourhood. With eight stored, 79.0% of attempts ended in a stable pattern that matches none of them: a spurious memory, invented by the weights.
Why do real patterns fare worse when, on average, two Aberdare neighbourhoods differ in 17.94 of 36 bits, almost exactly the 18 you would expect by chance? Our reading is that the bits are not independent. Across all 789 output areas in the county borough, the average strength of correlation between two bits is 0.12, roughly four times what independent bits would show; the share of under-fives, for example, rises and falls with the share of people aged 30 to 34 (a correlation of 0.42). Bits that move together give the stored patterns shared, hidden structure. The average distance hides that; the network does not.
Ages 8 to 11
A memory game with cards: how many pictures can you hold before you start mixing them up?
Ages 11 to 15
Turn census shares into yes-or-no bits in Python and compare two neighbourhoods bit by bit.
Ages 15 and up
Write the network, measure recall as memories are added, and explain why real data overflows first.
Sources and limits
Census 2021 tables TS007A, TS044, TS050, TS017 and TS061 come from Nomis and are published by the ONS under the Open Government Licence; the output areas counted as Aberdare come from the ONS 2021 output area to built-up area lookup. The 36-bit coding, the network, the random seed (20261001) and every percentage above come from our own run.
