The Ramsey community number as a renormalization-group crossing
Abstract
The Ramsey community number rk is the smallest size at which a network is better described by communities than by none, under a Bayesian detection rule. On the diamond hierarchical lattice we show that rk is an exact renormalization-group crossing: the block-model sufficient statistics obey a linear map with eigenvalues {bs,b}, the degree-corrected evidence density flows to ln K at a community fixed point, and rk is the generation at which the running evidence clears the detection threshold. Degree correction advances detection by two generations. We derive rk(b,s;q) in closed form for the whole family. Finally, placing on the lattice the Reichardt--Bornholdt community Hamiltonian -- whose ground state is the partition itself -- we find an exact community-ordered phase: below the ferromagnetic critical temperature the two hubs lock into opposite communities for any resolution γ>0, a staggered order that persists as n→∞. Allowing each nested sub-community its own label, the optimal partition is a hierarchy of qopt∼√n communities, so the number of Potts states that best describes the network grows with the network. This hierarchy orders thermally level by level, through a cascade of first-order transitions whose temperatures fall as 1/ln q, so every stable level persists as n→∞: the emergent partition is detectable, optimal, and thermodynamically ordered.
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BibTeX
@misc{vazquez2026ramsey,
title = {{The Ramsey community number as a renormalization-group crossing}},
author = {Alexei Vazquez},
howpublished = {arXiv},
year = {2026},
month = jul,
doi = {10.48550/arxiv.2607.05954},
eprint = {2607.05954},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.05954},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - The Ramsey community number as a renormalization-group crossing
AU - Alexei Vazquez
PY - 2026
DA - 2026/07/07
PB - arXiv
DO - 10.48550/arxiv.2607.05954
UR - https://arxiv.org/abs/2607.05954
AB - The Ramsey community number r_k is the smallest size at which a network is better described by communities than by none, under a Bayesian detection rule. On the diamond hierarchical lattice we show that r_k is an exact renormalization-group crossing: the block-model sufficient statistics obey a linear map with eigenvalues {bs,b}, the degree-corrected evidence density flows to ln K at a community fixed point, and r_k is the generation at which the running evidence clears the detection threshold. Degree correction advances detection by two generations. We derive r_k(b,s;q) in closed form for the whole family. Finally, placing on the lattice the Reichardt--Bornholdt community Hamiltonian -- whose ground state is the partition itself -- we find an exact community-ordered phase: below the ferromagnetic critical temperature the two hubs lock into opposite communities for any resolution γ>0, a staggered order that persists as n→∞. Allowing each nested sub-community its own label, the optimal partition is a hierarchy of q_opt∼√n communities, so the number of Potts states that best describes the network grows with the network. This hierarchy orders thermally level by level, through a cascade of first-order transitions whose temperatures fall as 1/ln q, so every stable level persists as n→∞: the emergent partition is detectable, optimal, and thermodynamically ordered.
ER -