networks
The random graph's threshold
It is 1959 in Budapest, and the object of study looks almost frivolous: take n isolated vertices and add edges uniformly at random, one by one, and simply watch. What emerges is one of mathematics' great phase transitions — below a critical edge density the graph is dust and small trees; passing through it, a giant connected component crystallizes with startling suddenness, and the largest component's size performs its famous double jump. Characterize the threshold precisely and prove the double-jump behavior, with the probabilistic method as the instrument. No application is in view in 1959 — which is the point and the stakes: percolation theory, epidemic thresholds, network robustness, and the analysis of random structures throughout computer science will all be built on exactly this foundation.
Who this problem belongs to
The two figures whose methods fit it best, out of 38 in contention.
Erdos, together with Renyi, is the literal author of this problem: their 1959-1960 papers on random graphs establish precisely the object of study — add edges uniformly at random and watch connectivity emerge — and prove the threshold and double-jump behavior in the size of the largest component using the probabilistic method Erdos himself pioneered and made central to combinatorics. No application was in view, exactly as this problem states, and Erdos's entire mathematical style — existence proofs via random construction, collaborative and rapid-fire across many co-authors — is the origin of the toolkit this problem asks to be deployed. There is no better-matched carrier in this or nearly any pool: this is not analogous to his work, it is his work.
Moore's career centers on phase transitions in random combinatorial structures — exactly the double-jump giant-component behavior this problem asks to be characterized and proved — using statistical-physics and probabilistic-method techniques that are the direct intellectual descendants of the Erdos-Renyi program. His work on inference and computation near critical thresholds gives him deep, hands-on fluency with the mathematics of sudden structural emergence in random graphs, even though he works decades after this problem's 1959 setting. He is essentially working the modern extension of exactly this question throughout his career, making him one of the strongest possible carriers in this pool.
In the mind map
The same ideas, as concepts rather than history — in John's ML knowledge map.
38 figures are scored on this problem. Draw it in a battle to see where you land.