Bad local minima exist in the stochastic block model

TL;DR

This work shows that in the disassortative stochastic block model with three communities, maximum-a-posteriori (MAP) inference can yield a bad local minimum that has vanishing agreement with the true partition even when nontrivial detection is statistically possible. The authors adapt the Achlioptas–Moore 3‑coloring algorithm to the SBM and analyze its evolution through Wormald’s differential equations, introducing a flat-white solution under a key ODE Assumption. They prove that, for $d>d_{ ext{sbm}}(\beta)$, there exists a MAP configuration with $\mathfrak{a}(\sigma)=o(1)$ and $L_{\boldsymbol{G}^*}(\sigma)=o(1)$ while the ground-truth loss remains $\Omega(1)$, and that a posterior sample can still achieve substantial overlap with the ground truth. Numerics and evidence indicate the ODE Assumption holds up to $d\approx 4.03$, implying the bad-MAP phenomenon occurs for large $\beta$ just above the SBM threshold, highlighting a fundamental distinction between posterior sampling and MAP in community detection.

Abstract

We study the disassortative stochastic block model with three communities, a well-studied model of graph partitioning and Bayesian inference for which detailed predictions based on the cavity method exist [Decelle et al. (2011)]. We provide strong evidence that for a part of the phase where efficient algorithms exist that approximately reconstruct the communities, inference based on maximum a posteriori (MAP) fails. In other words, we show that there exist modes of the posterior distribution that have a vanishing agreement with the ground truth. The proof is based on the analysis of a graph colouring algorithm from [Achlioptas and Moore (2003)].

Figures (4)

• Figure 1: The black curve shows the maximum $d_{\max}=d_{\max}(\alpha)$ such that the eigenvalue $\lambda(t)$ remains bounded below according to the numerical solution of the ODEs. The orange dots display the corresponding values reported in AchMoore3col. The horizontal line is placed at $d=4.03$.
• Figure 2: Left: The eigenvalue $\lambda(t)$ at each time $t$ for $\alpha = 15$, $d=4.03$ and $\beta = 6$, i.e., slightly above the Kesten-Stigum threshold. The black curve displays the numerical solutions to the ODEs. The orange dotted curve is the empirical value of $\lambda(t)$ averaged over $10$ runs with $n=10^{6}$. The magenta line displays the numerical ODE value of $\gamma(t)$, while the blue dotted shows the experimental value. The dashed vertical line marks the last point $t^*$ where \ref{['eqODEend']} holds. Right: the black/orange lines from the left figure zoomed in on the peak.
• Figure 3: The blue line shows the average empirical agreement (as defined in \ref{['eqagree']}) over 20 runs of $\mathtt{AM}$ with $\beta=6$, $d=4.03$ and $\alpha=15$ on a $(\log,\log)$-scale. The orange dotted line is just a straight line with slope $-1/2$. This is the expected gradient of the blue line, since the agreement of two independent random colourings is of order $n^{-1/2}$.
• Figure 4: Bad vertices in $10^5$ simulated runs with $n=10^6$, $\alpha=15$ and $\beta=6$.

Theorems & Definitions (26)

• Theorem 1.2
• Corollary 1.3
• Proposition 2.2
• proof
• Proposition 2.4
• proof
• Proposition 2.6
• Proposition 2.7
• Corollary 2.8
• proof