On semidefinite relaxations for the block model (1406.5647v3)

Abstract: The stochastic block model (SBM) is a popular tool for community detection in networks, but fitting it by maximum likelihood (MLE) involves a computationally infeasible optimization problem. We propose a new semidefinite programming (SDP) solution to the problem of fitting the SBM, derived as a relaxation of the MLE. We put ours and previously proposed SDPs in a unified framework, as relaxations of the MLE over various sub-classes of the SBM, revealing a connection to sparse PCA. Our main relaxation, which we call SDP-1, is tighter than other recently proposed SDP relaxations, and thus previously established theoretical guarantees carry over. However, we show that SDP-1 exactly recovers true communities over a wider class of SBMs than those covered by current results. In particular, the assumption of strong assortativity of the SBM, implicit in consistency conditions for previously proposed SDPs, can be relaxed to weak assortativity for our approach, thus significantly broadening the class of SBMs covered by the consistency results. We also show that strong assortativity is indeed a necessary condition for exact recovery for previously proposed SDP approaches and not an artifact of the proofs. Our analysis of SDPs is based on primal-dual witness constructions, which provides some insight into the nature of the solutions of various SDPs. We show how to combine features from SDP-1 and already available SDPs to achieve the most flexibility in terms of both assortativity and block-size constraints, as our relaxation has the tendency to produce communities of similar sizes. This tendency makes it the ideal tool for fitting network histograms, a method gaining popularity in the graphon estimation literature, as we illustrate on an example of a social networks of dolphins. We also provide empirical evidence that SDPs outperform spectral methods for fitting SBMs with a large number of blocks.

• The paper introduces SDP-1, a tighter semidefinite relaxation that reliably recovers community assignments even under weak assortativity conditions.
• The paper unifies various SDP methods as MLE variants, clarifying how constraint choices impact optimization in stochastic block models.
• The paper demonstrates through empirical analysis that SDP-1 outperforms traditional SDP and spectral methods in detecting block structures in complex networks.

Overview of Semidefinite Relaxations for the Stochastic Block Model

The paper authored by Arash A. Amini and Elizaveta Levina presents a nuanced exploration of semidefinite programming (SDP) relaxations as a computational approach to community detection in networks via the stochastic block model (SBM). The manuscript addresses the longstanding challenges of fitting SBM through maximum likelihood estimation (MLE), which is known to be computationally prohibitive due to the combinatorial nature of the problem.

The authors propose a refined SDP relaxation, termed SDP-1, which tightens constraints on the feasible set compared to previous SDP relaxations. The paper explores the mathematical equivalence between the relaxed problem and the original MLE over specific sub-classes of SBM, revealing an intriguing connection to sparse principal component analysis (PCA). This connection offers insights into potential regularization effects inherent to SDP methods.

Key Contributions

1. Tighter Relaxation with SDP-1: SDP-1 stands as a tighter relaxation than existing SDP methods (SDP-2 and SDP-3). It is shown to exactly recover true community assignments over a larger class of SBM configurations, particularly relaxing the need for strong assortativity to a less restrictive weak assortativity condition, which broadens its applicability.
2. Unified Framework: The authors position various SDP relaxations within a unified framework by conceptualizing them as variations of the MLE across different SBM parameter spaces. This provides a clearer understanding of how constraints influence the optimization process and reveals strong consistency results under specific theoretical conditions.
3. Theoretical Guarantees: Amini and Levina demonstrate that both the proposed SDP-1 and SDP-2 relaxations are strongly consistent for recovering the block structures in SBM, under certain conditions related to assortativity and block model parameters. They also establish the conditions under which SDP-2 fails in the absence of strong assortativity.
4. Numerical Empiricism and Social Network Application: Empirical evidence underscores the superior performance of SDP-1 compared to other SDP approaches and spectral methods, especially in scenarios with a larger number of communities. Additionally, the paper illustrates the utility of SDP relaxations in network histogram construction for graphon estimation—a method gaining traction for its non-parametric approximation capabilities.

Implications

The research extends the methodological toolkit available for community detection in network analysis, particularly improving upon the performance boundaries set by earlier SDP approaches. The assumption relaxation from strong to weak assortativity is significant, allowing broader usage in practical applications where such stringent conditions are not met.

Furthermore, the unified framework paves the way for future explorations to exploit connections with sparse PCA, potentially leading to novel regularization techniques and optimization strategies in high-dimensional settings where community detection is relevant.

The versatility of SDP-1 in handling both assortative and disassortative communities hints at possibilities for tackling mixed networks, an area ripe for exploration given its implications in real-world social, biological, and technological networks.

In summary, this paper's contributions lie at the intersection of theoretical advancement and empirical performance, offering concrete directions for future research and application within the AI landscape.