Dense Stochastic Block Models

• Dense Stochastic Block Models are random graph models with constant edge probabilities that rigorously define community structures.
• They employ spectral, convex relaxation, and likelihood approaches to establish clear detectability and recovery thresholds based on signal-to-noise metrics.
• Applications span balanced, heterogeneous, and multi-layer networks, offering efficient algorithmic solutions for community detection.

A dense Stochastic Block Model (SBM) refers to a random graph framework where the connection probabilities between and within node classes (blocks) are bounded away from zero as the system size grows, so that the average degree is proportional to the number of vertices. Dense SBMs serve as key models for understanding community structure in large-scale networks, anchoring spectral, probabilistic, and information-theoretic analyses of detectability, estimation, and recovery thresholds.

1. Model Formulation and Regimes

A dense SBM consists of an undirected graph $G=(V,E)$ on $n$ vertices, partitioned into $K$ communities. Each node $i$ is assigned to a block via a (latent) label $g_i\in\{1,\dots,K\}$. For each pair $(i,j)$, the edge indicator variable $A_{ij}$ is drawn independently as

$A_{ij} \sim \text{Bernoulli}\big(p_{g_i,g_j}\big)$

where the $K\times K$ symmetric matrix $P = [p_{ab}]$ encodes the within- and between-community edge probabilities. Dense regime is characterized by connection probabilities $p_{ab}=\Theta(1)$ for all $a,b$, hence $\mathbb{E}[\deg_i]=\Theta(n)$ for all $i$ (Lei et al., 2013, Han et al., 2023, Bolla et al., 2023).

Important subclasses include:

• Planted partition model: two-block case with $p_{aa}=p_{\mathrm{in}}$, $p_{ab}=p_{\mathrm{out}}$ for $a\neq b$
• Heterogeneous SBM: varying block sizes and connection probabilities (Jalali et al., 2015)
• Balanced SBM: all blocks have $n/K$ nodes

In the limit $n\to\infty$, dense SBMs converge (in homomorphism densities) to step-function graphons (Tran et al., 2020).

2. Detectability, Contiguity, and Non-Reconstruction

The detectability threshold is a central object in the theory of dense SBMs. For the two-block model ($n$ vertices, blocks of size $n/2$),

$p_n = \frac{a_n}{n}, \quad q_n = \frac{b_n}{n}$

the critical metric is $c_n=\frac{(a_n-b_n)^2}{a_n+b_n}$. When $a_n/n, b_n/n \rightarrow p\in (0,1)$, the distinction between the planted model and the Erdős–Rényi random graph with matching mean degree depends on (Banerjee, 2016): $$\frac{c_n}{2(1-\hat p_n)} \underset{n\to\infty}{\lessgtr} 1 \,, \quad \hat p_n = \frac12(p_n+q_n)$$

• Contiguity: If $\frac{c_n}{2(1-\hat p_n)}<1$, no test can distinguish the SBM from Erdős–Rényi, and it is impossible to estimate the node labels with positive correlation to ground truth.
• Singularity/Recoverability: If $\frac{c_n}{2(1-\hat p_n)}>1$, the SBM and Erdős–Rényi are asymptotically singular; consistent estimation of labels and parameters is achievable.

Proofs utilize signed-cycle statistics $C_{n,k}$ and a Gaussian coupling for likelihood-ratio contiguity.

3. Spectral and Statistical Phase Transitions

Dense SBMs exhibit a sharp spectral-phase transition at the Kesten–Stigum bound (KS threshold), formulated in the balanced $K$-block SBM: $$\gamma_N^2 = \frac{N(p_{\mathrm{in}} - p_{\mathrm{out}})^2}{K p_a (1-p_a)}, \quad p_a = \frac{p_{\mathrm{in}} + (K-1)p_{\mathrm{out}}}{K}$$ When $\gamma_N>1$, the top $K-1$ eigenvalues of the normalized adjacency matrix escape the Wigner bulk and cluster around $\gamma_N+\gamma_N^{-1}$, allowing detection via spectral methods. For $\gamma_N\leq 1$, all eigenvalues remain inside the bulk, rendering detection impossible (Han et al., 2023).

Linear spectral statistics such as $L_M(f)$ admit central limit theorems, and can be used for optimal tests of community existence and number.

4. Algorithmic Approaches and Estimation

Spectral Methods

Vanilla spectral clustering: compute $K$ leading eigenvectors of $A$ or Laplacian, and assign communities via $k$-means. In the dense regime this yields misclassification rate $O(K^3/(n\lambda^2))$ for $\lambda=p_{\mathrm{in}}-p_{\mathrm{out}}$ (Lei et al., 2013). Modularity and Laplacian-based embeddings are also effective (Bolla et al., 2023).

Convex Relaxation

Nuclear norm–constrained semidefinite programming (SDP) relaxations enable exact recovery when the minimal relative density $\rho_k=n_k(p_k-q)$ satisfies $\rho_k\gg\log n$, with $\rho_k^2\gtrsim n_k\log n$ in the dense regime (Jalali et al., 2015).

Likelihood and EM Algorithms

Maximum likelihood estimation (MLE) maximizes $\langle A,Y\rangle$ over feasible cluster matrices $Y$ (Jalali et al., 2015). For dense graphons, the likelihood is nontrivial in respondent-driven exploration (random walk on a graphon), leading to bias-corrected estimators via SAEM and debiasing algebraic equations (Tran et al., 2020).

Graph Pencil (Editor’s term)

Given sufficiently many estimates of star and bistar subgraph densities, all parameters of a degree-separated SBM can be recovered by solving matrix pencil and generalized eigenvalue problems, without iterative optimization (Gunderson et al., 31 Jan 2024).

Method	Core principle	Recovery condition
Spectral	Leading eigenvectors + k-means	$p_{\mathrm{in}}-p_{\mathrm{out}}~\gtrsim~1/n$
Convex SDP	Trace-norm relaxation	$\rho_k~\gg~\log n$
Graph Pencil	Star/bistar densities	Degree separation, dense regime
EM/de-biased	Likelihood + correction	Consistent under graphon identifiability

5. Information-Theoretic and Computational Thresholds

Recovery in dense SBMs is fundamentally characterized by information-theoretic and algorithmic limits. For the $K$-block case with per-block probabilities $p_k=O(1)$, the block is detectable if $\rho_k\gg\log n$:

• If not, no estimator can succeed, even with unlimited computation (Jalali et al., 2015).
• Convex relaxations and spectral methods reach this threshold for all clusters except the smallest (size $o(\log n)$) clusters, where only MLE (non-convex) achieves optimal rates. For two blocks, the contiguity criterion encapsulates the detectability threshold (Banerjee, 2016).

In multi-layer dense SBMs, a computational-to-statistical gap emerges: unconstrained recovery is feasible when $nT\rho\to\infty$, but efficient algorithms require $n\sqrt{T}\rho\gg 1$ (Lei et al., 2023).

6. Dense Regime Anomalies: Small, Dense, and Heterogeneous Clusters

In dense heterogeneous SBMs, recovery of clusters with size $o(\log n)$—down to $O(\sqrt{\log n})$—is possible provided the relative density $\rho_k$ is sufficiently large and the number of such small clusters is small. When $p_k\approx 1$, even naive algorithms succeed (Jalali et al., 2015).

Phase diagrams detail the interplay of edge probability scaling, cluster-size scaling, and recovery. The critical role of the quantity $\rho_k=n_k(p_k-q)$ governs all known detection and recovery thresholds in the dense regime (Jalali et al., 2015).

7. Connections to Graphon Theory and Extensions

Dense SBMs are step-graphons, with parameters estimable via subgraph densities. Respondent-driven (random-walk) exploration leads to bias toward high-degree blocks, correctable algebraically (Tran et al., 2020). The finite-forcibility result in the Graph Pencil method demonstrates that for degree-separated dense SBMs, $O(K^2)$ subgraph densities suffice for unique parameter identification (Gunderson et al., 31 Jan 2024).

In summary, dense SBMs represent a regime where statistical recovery of block structure is algorithmically tractable for most practical purposes. Detectability and recoverability are controlled by explicit, interpretable signal-to-noise metrics. The dense setting also allows for efficient algorithmic design (spectral, SDP, star-count pencils), robust theoretical guarantees, and connections to network limit theories.

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References:

(Banerjee, 2016, Jalali et al., 2015, Tran et al., 2020, Gunderson et al., 31 Jan 2024, Lei et al., 2023, Han et al., 2023, Lei et al., 2013, Bolla et al., 2023)