Concept Bottleneck Models (CBMs) are interpretable machine learning models that ground their predictions on human-understandable concepts, allowing for targeted interventions in their decision-making process. However, when intervened on, CBMs assume the availability of humans that can identify the need to intervene and always provide correct interventions. Both assumptions are unrealistic and impractical, considering labor costs and human error-proneness. In contrast, Learning to Defer (L2D) extends supervised learning by allowing machine learning models to identify cases where a human is more likely to be correct than the model, thus leading to deferring systems with improved performance. In this work, we gain inspiration from L2D and propose Deferring CBMs (DCBMs), a novel framework that allows CBMs to learn when an intervention is needed. To this end, we model DCBMs as a composition of deferring systems and derive a consistent L2D loss to train them. Moreover, by relying on a CBM architecture, DCBMs can explain the reasons for deferring on the final task. Our results show that DCBMs can achieve high predictive performance and interpretability by deferring only when needed.

Community detection is a fundamental task in data analysis. Block models form a standard approach to partition nodes according to a graph model, facilitating the analysis and interpretation of the network structure. By grouping nodes with similar connection patterns, they enable the identification of a wide variety of underlying structures. The degree-corrected block model (DCBM) is an established model that accounts for the heterogeneity of node degrees. However, existing inference methods for the DCBM are heuristics that are highly sensitive to initialization, typically done randomly. In this work, we show that DCBM inference can be reformulated as a constrained nonnegative matrix factorization problem. Leveraging this insight, we propose a novel method for community detection and a theoretically well-grounded initialization strategy that provides an initial estimate of communities for inference algorithms. Our approach is agnostic to any specific network structure and applies to graphs with any structure representable by a DCBM, not only assortative ones. Experiments on synthetic and real benchmark networks show that our method detects communities comparable to those found by DCBM inference, while scaling linearly with the number of edges and communities; for instance, it processes a graph with 100,000 nodes and 2,000,000 edges in approximately 4 minutes. Moreover, the proposed initialization strategy significantly improves solution quality and reduces the number of iterations required by all tested inference algorithms. Overall, this work provides a scalable and robust framework for community detection and highlights the benefits of a matrix-factorization perspective for the DCBM.

This paper reviews statistical methods for hypothesis testing and clustering in network models. We analyze the method by Bickel et al. (2016) for deriving the asymptotic null distribution of the largest eigenvalue, noting its slow convergence and the need for bootstrap corrections. The SCORE method by Jin et al. (2015) and the NCV method by Chen et al. (2018) are evaluated for their efficacy in clustering within Degree-Corrected Block Models, with NCV facing challenges due to its time-intensive nature. We suggest exploring eigenvector entry distributions as a potential efficiency improvement.

The degree-corrected block model (DCBM), latent space model (LSM), and $\beta$-model are all popular network models. We combine their modeling ideas and propose the logit-DCBM as a new model. Similar as the $\beta$-model and LSM, the logit-DCBM contains nonlinear factors, where fitting the parameters is a challenging open problem. We resolve this problem by introducing a cancellation trick. We also propose R-SCORE as a recursive community detection algorithm, where in each iteration, we first use the idea above to update our parameter estimation, and then use the results to remove the nonlinear factors in the logit-DCBM so the renormalized model approximately satisfies a low-rank model, just like the DCBM. Our numerical study suggests that R-SCORE significantly improves over existing spectral approaches in many cases. Also, theoretically, we show that the Hamming error rate of R-SCORE is faster than that of SCORE in a specific sparse region, and is at least as fast outside this region.

The truncated singular value decomposition is a widely used methodology in music recommendation for direct similar-item retrieval or embedding musical items for downstream tasks. This paper investigates a curious effect that we show naturally occurring on many recommendation datasets: spiking formations in the embedding space. We first propose a metric to quantify this spiking organization's strength, then mathematically prove its origin tied to underlying communities of items of varying internal popularity. With this new-found theoretical understanding, we finally open the topic with an industrial use case of estimating how music embeddings' top-k similar items will change over time under the addition of data.

How to detect a small community in a large network is an interesting problem, including clique detection as a special case, where a naive degree-based $\chi^2$-test was shown to be powerful in the presence of an Erd\H{o}s-Renyi background. Using Sinkhorn's theorem, we show that the signal captured by the $\chi^2$-test may be a modeling artifact, and it may disappear once we replace the Erd\H{o}s-Renyi model by a broader network model. We show that the recent SgnQ test is more appropriate for such a setting. The test is optimal in detecting communities with sizes comparable to the whole network, but has never been studied for our setting, which is substantially different and more challenging. Using a degree-corrected block model (DCBM), we establish phase transitions of this testing problem concerning the size of the small community and the edge densities in small and large communities. When the size of the small community is larger than $\sqrt{n}$, the SgnQ test is optimal for it attains the computational lower bound (CLB), the information lower bound for methods allowing polynomial computation time. When the size of the small community is smaller than $\sqrt{n}$, we establish the parameter regime where the SgnQ test has full power and make some conjectures of the CLB. We also study the classical information lower bound (LB) and show that there is always a gap between the CLB and LB in our range of interest.

There exist various types of network block models such as the Stochastic Block Model (SBM), the Degree Corrected Block Model (DCBM), and the Popularity Adjusted Block Model (PABM). While this leads to a variety of choices, the block models do not have a nested structure. In addition, there is a substantial jump in the number of parameters from the DCBM to the PABM. The objective of this paper is formulation of a hierarchy of block model which does not rely on arbitrary identifiability conditions, treats the SBM, the DCBM and the PABM as its particular cases with specific parameter values and, in addition, allows a multitude of versions that are more complicated than DCBM but have fewer unknown parameters than the PABM. The latter allows one to carry out clustering and estimation without preliminary testing to see which block model is really true.

In this paper, we present an optimized version of the previously developed Block Modularity algorithm (Anthony,2009). The original algorithm was a fast, greedy method that effectively discovered a structured clustering in linked data and scaled very well with the number of nodes and edges. The optimized version is scalable in terms of the model complexity; the technique can now be used effectively to discover thousands of clusters in data sets with hundreds of thousands (and possibly more) nodes and edges. The optimization leads to an improvement of the runtime per iteration from cubic to quadratic with a small increase in the constant factor. The algorithm compares favorably with Karrer and Newman's Degree-Corrected Block Model (DCBM) in both runtime and quality of results.