Hierarchical Bayesian models are increasingly used in large, inhomogeneous complex network dynamical systems by modeling parameters as draws from a hyperparameter-governed distribution. However, theoretical guarantees for these estimates as the system size grows have been lacking. A critical concern is that hyperparameter estimation may diverge for larger networks, undermining the model's reliability. Formulating the system's evolution in a measure transport perspective, we propose a theoretical framework for estimating hyperparameters with mean-type observations, which are prevalent in many scientific applications. Our primary contribution is a nonasymptotic bound for the deviation of estimate of hyperparameters in inhomogeneous complex network dynamical systems with respect to network population size, which is established for a general family of optimization algorithms within a fixed observation duration. While we firstly establish a consistency result for systems with independent nodes, our main result extends this guarantee to the more challenging and realistic setting of weakly-dependent nodes. We validate our theoretical findings with numerical experiments on two representative models: a Susceptible-Infected-Susceptible model and a Spiking Neuronal Network model. In both cases, the results confirm that the estimation error decreases as the network population size increases, aligning with our theoretical guarantees. This research proposes the foundational theory to ensure that hierarchical Bayesian methods are statistically consistent for large-scale inhomogeneous systems, filling a gap in this area of theoretical research and justifying their application in practice.

In this paper, the solution to the empirical risk minimization problem with $f$-divergence regularization (ERM-$f$DR) is presented and conditions under which the solution also serves as the solution to the minimization of the expected empirical risk subject to an $f$-divergence constraint are established. The proposed approach extends applicability to a broader class of $f$-divergences than previously reported and yields theoretical results that recover previously known results. Additionally, the difference between the expected empirical risk of the ERM-$f$DR solution and that of its reference measure is characterized, providing insights into previously studied cases of $f$-divergences. A central contribution is the introduction of the normalization function, a mathematical object that is critical in both the dual formulation and practical computation of the ERM-$f$DR solution. This work presents an implicit characterization of the normalization function as a nonlinear ordinary differential equation (ODE), establishes its key properties, and subsequently leverages them to construct a numerical algorithm for approximating the normalization factor under mild assumptions. Further analysis demonstrates structural equivalences between ERM-$f$DR problems with different $f$-divergences via transformations of the empirical risk. Finally, the proposed algorithm is used to compute the training and test risks of ERM-$f$DR solutions under different $f$-divergence regularizers. This numerical example highlights the practical implications of choosing different functions $f$ in ERM-$f$DR problems.

A key challenge in network science is the detection of communities, which are sets of nodes in a network that are densely connected internally but sparsely connected to the rest of the network. A fundamental result in community detection is the existence of a nontrivial threshold for community detectability on sparse graphs that are generated by the planted partition model (PPM). Below this so-called ``detectability limit'', no community-detection method can perform better than random chance. Spectral methods for community detection fail before this detectability limit because the eigenvalues corresponding to the eigenvectors that are relevant for community detection can be absorbed by the bulk of the spectrum. One can bypass the detectability problem by using special matrices, like the non-backtracking matrix, but this requires one to consider higher-dimensional matrices. In this paper, we show that the difference in graph energy between a PPM and an Erdős--Rényi (ER) network has a distinct transition at the detectability threshold even for the adjacency matrices of the underlying networks. The graph energy is based on the full spectrum of an adjacency matrix, so our result suggests that standard graph matrices still allow one to separate the parameter regions with detectable and undetectable communities.

Existing Grammatical Error Correction (GEC) systems suffer from limited reference diversity, leading to underestimated evaluation and restricted model generalization. To address this issue, we introduce the Judge of Edit-Level Validity (JELV), an automated framework to validate correction edits from grammaticality, faithfulness, and fluency. Using our proposed human-annotated Pair-wise Edit-level Validity Dataset (PEVData) as benchmark, JELV offers two implementations: a multi-turn LLM-as-Judges pipeline achieving 90% agreement with human annotators, and a distilled DeBERTa classifier with 85% precision on valid edits. We then apply JELV to reclassify misjudged false positives in evaluation and derive a comprehensive evaluation metric by integrating false positive decoupling and fluency scoring, resulting in state-of-the-art correlation with human judgments. We also apply JELV to filter LLM-generated correction candidates, expanding the BEA19's single-reference dataset containing 38,692 source sentences. Retraining top GEC systems on this expanded dataset yields measurable performance gains. JELV provides a scalable solution for enhancing reference diversity and strengthening both evaluation and model generalization.

Bridging is an anaphoric phenomenon where the referent of an entity in a discourse is dependent on a previous, non-identical entity for interpretation, such as in "There is 'a house'. 'The door' is red," where the door is specifically understood to be the door of the aforementioned house. While there are several existing resources in English for bridging anaphora, most are small, provide limited coverage of the phenomenon, and/or provide limited genre coverage. In this paper, we introduce GUMBridge, a new resource for bridging, which includes 16 diverse genres of English, providing both broad coverage for the phenomenon and granular annotations for the subtype categorization of bridging varieties. We also present an evaluation of annotation quality and report on baseline performance using open and closed source contemporary LLMs on three tasks underlying our data, showing that bridging resolution and subtype classification remain difficult NLP tasks in the age of LLMs.