Community detection with the Bethe-Hessian (2411.02835v2)

Abstract: The Bethe-Hessian matrix, introduced by Saade, Krzakala, and Zdeborov\'a (2014), is a Hermitian matrix designed for applying spectral clustering algorithms to sparse networks. Rather than employing a non-symmetric and high-dimensional non-backtracking operator, a spectral method based on the Bethe-Hessian matrix is conjectured to also reach the Kesten-Stigum detection threshold in the sparse stochastic block model (SBM). We provide the first rigorous analysis of the Bethe-Hessian spectral method in the SBM under both the bounded expected degree and the growing degree regimes. Specifically, we demonstrate that: (i) When the expected degree $d\geq 2$, the number of negative outliers of the Bethe-Hessian matrix can consistently estimate the number of blocks above the Kesten-Stigum threshold, thus confirming a conjecture from Saade, Krzakala, and Zdeborov\'a (2014) for $d\geq 2$. (ii) For sufficiently large $d$, its eigenvectors can be used to achieve weak recovery. (iii) As $d\to\infty$, we establish the concentration of the locations of its negative outlier eigenvalues, and weak consistency can be achieved via a spectral method based on the Bethe-Hessian matrix.

• The paper rigorously proves that the Bethe-Hessian matrix accurately estimates the number of communities in sparse graphs with expected degree d ≥ 2.
• The paper establishes concentration results for negative outlier eigenvalues, achieving both weak recovery and consistency without the need for degree regularization.
• The paper bridges the gap with non-backtracking methods by showing that the eigenvalues of the Bethe-Hessian closely approximate the function of non-backtracking operators for efficient detection.

Community Detection with the Bethe-Hessian

The paper provides a rigorous analysis of the Bethe-Hessian spectral method within the framework of the stochastic block model (SBM). Originally conjectured by Saade, Krzakala, and Zdeborová, the Bethe-Hessian matrix is a Hermitian operator proposed to achieve the Kesten-Stigum detection threshold for community detection in sparse graphs. This operator is of particular interest due to its computational efficiency and stability when compared to non-Hermitian alternatives like the non-backtracking matrix.

Contributions and Results

The authors' contribution is framed through several theoretical results under different degree regimes:

1. Bounded Expected Degree Regime: The paper provides a rigorous proof that the number of negative outlier eigenvalues of the Bethe-Hessian matrix can be used to consistently estimate the number of communities in SBM when the expected degree $d$ is greater than or equal to 2. This result corroborates the conjecture from [SKZ14] and extends previous works that were applicable to the $d \to \infty$ regime.
2. Eigenvalue and Eigenvector Analysis: For growing degree regimes, particularly when $d$ is large, the paper establishes concentration results for the location of negative outlier eigenvalues and shows how spectral algorithms based on the Bethe-Hessian matrix achieve weak recovery and even weak consistency. This holds true without the need for degree regularization, a critical finding given the computational burden associated with regularization in large networks.
3. Connection to Non-backtracking Matrix: A novel angle explored is the relationship between the Bethe-Hessian and the non-backtracking matrix. The authors demonstrate that negative eigenvalues and their associated eigenvectors of the Bethe-Hessian matrix approximate the function of a non-backtracking operator, facilitating community detection effectively under certain conditions.
4. Verification of Conjectures: The paper validates multiple conjectures, specifically confirming that for expected degrees $d \geq 2$, negative outliers of $H(\pm \sqrt{d})$ consistently estimate the number of communities within SBM, presenting a theoretical basis for previously observed empirical phenomena.
5. Real Eigenvalues of the Non-backtracking Matrix: The authors close an existing gap by showing that all outlier eigenvalues of the non-backtracking matrix are real in SBMs, an empirical observation that lacked theoretical backing in earlier studies.

Implications and Future Directions

The implications of this work are significant both theoretically and practically. The consistency results offer a foundation for community detection methods that do not rely heavily on regularization techniques, thereby streamlining computations in large and sparse networks. Furthermore, the connection established between Hermitian operators and their non-backtracking counterparts may pave the way for novel spectral algorithms that leverage the computational efficiencies of symmetric matrices.

Going forward, several avenues present themselves for exploration. Notably, extending these results to weighted graphs and more generalized community structures could provide further insights and broader applicability. Additionally, investigating the robustness and stability of the Bethe-Hessian approach under varying noise levels or model perturbations could further solidify its utility in practical applications. Lastly, this research invites a deeper examination of localization phenomena in sparse matrices, particularly in how informative and uninformative outliers can be systematically distinguished.

Overall, this paper enriches the theoretical landscape of spectral methods in network analysis, offering robust tools and insights for community detection within stochastic frameworks.