Community structure of the pseudofractal web

The Ramsey community number rκ is the smallest network size at which a graph is better described by a partition into communities than by no partition, under a prescribed detection rule and at certainty q. On a scale-free graph this question is confounded: a block model can split a network on its degree distribution alone, separating hubs from leaves that are no community at all. I compute rκ analytically for the deterministic pseudofractal scale-free web of Dorogovtsev, Goltsev, and Mendes, separating genuine community structure from degree heterogeneity with two closed-form detection rules. Under a plain Bernoulli stochastic block model, the web’s natural recursive bipartition is unpreferred while small and breaks at rκ = 1095 nodes, with a log-evidence growing as (ln 3-2/3ln 2)n. Under a degree-corrected model tested against the configuration-model null, the same partition survives, breaking far earlier at rκ = 123, with a log-evidence growing as (2ln 3-4/3ln 2)n—exactly twice the plain slope, and independent of the prior. Both figures are quoted at the default certainty q = 0.99; the transitions are sharp and the web exists only at discrete sizes, so of the values tabulated only q=1/2 changes an answer, lowering the degree-corrected one to 42. Degree correction reverses the ordering of the candidate cuts, demoting the hub–leaf split and elevating the recursive one. Because the web is self-similar, the best description is not two communities but a nested hierarchy: the degree-corrected evidence keeps rising as the partition is refined, and is maximized at of order √n communities of ~√n nodes. A purely local recursive rule thus builds true hierarchical community structure, over and above the scale-free degree sequence it also produces, in an exactly solvable setting. The appendices give every derivation.