Detector-Output Instability near the Kesten-Stigum Boundary: Separating Hard Readout, Relaxation, and Fixed-Point Dispersion

Abstract

Community-detection algorithms usually return a single partition, even when independent initializations or small data perturbations yield several plausible outputs. We probe this output distribution through three paired observables: hard-partition variation of information (VI), a residual-gated fixed-point VI, and a cutoff-free Jensen-Shannon distance between belief-propagation (BP) marginal fields. For the symmetric sparse stochastic block model, linearizing BP around the uninformative fixed point gives the Kesten-Stigum onset at $\mathrm{snr}=(c_{\rm in}-c_{\rm out})/(q\sqrt{c})=1$. The hard VI maximum is instead a finite-size, readout-dependent detector curve on the detectable side, typically $\mathrm{snr}^\star \simeq 1.05\text{-}1.10$; moving the polarization cutoff from 0.001 to 0.1 shifts it across 1.047-1.128. The nontrivial-readout activation obeys $\mathrm{snr}_{50}(\tau)-1 = 0.0086 + 0.522\,\tau$ ($R^2=0.996$). Long-budget residual gating separates readout and critical slowing from fixed-point dispersion: at $\mathrm{snr}=1.05$ and 1.10 the hard VI is 1.49 and 1.58 bits but the gated subsets have zero VI, whereas from 1.15 to 1.30 nearly all runs pass the gate and retain VI 1.31 down to 1.24 bits. A high-replication audit through $N=100000$ disfavors a zero-asymptote power law and finds a small plateau $\mathrm{snr}^\star-1 \simeq 0.024$ (graph-bootstrap 90% interval [0.0227, 0.0316]). On real networks, a label-free Bethe-Hessian modularity margin with a Chung-Lu null gate is run on political blogs and six SNAP graphs: the measurement stays label-free, while heterogeneous networks can retain null-significant structure even after strong edge subsampling. The result is a detector-output decomposition near the Kesten-Stigum boundary, reporting hard readout, relaxation dynamics, and fixed-point-field dispersion separately.