Abstract Graphs are ubiquitous, and graph learning has long been a fundamental topic in machine learning. While Graph Neural Networks (GNNs) have achieved remarkable results, they are typically designed for specific tasks or graphs, requiring retraining to adapt the diversity of real-world graph data. Recently, foundation models, such as Large Language Models (LLMs), have driven revolutionary progress in the language domain through universal pretraining. Their success has sparked growing interest in designing Graph Foundation Models (GFMs), a novel family of graph neural networks pre-trained on large-scale, diverse graph data, which are capable of supporting a wide range of downstream tasks on different graphs. Early efforts in the literature often adapt LLMs by transforming graph data into sequential representations. However, unlike word sequences in natural language, graphs are inherently non-Euclidean structures that encapsulate complex intercorrelations among entities. Existing GFMs often trivialize the structural diversity and complexity inherent in graph data. To address this gap, we propose to study graph foundation model from the perspective of Riemannian geometry, and design a novel Curvature-guided Riemannian Graph Foundation Model (CRGFM). To the best of our knowledge, CRGFM is the first GFM to introduce a curvature-based graph description along with geometric standardization. Specifically, to capture structural diversity, the input graph is represented using a mixture of geometric experts. A novel geometric standardization is then introduced via an augmented Lorentz transformation. To model structural complexity, we design a Riemannian graph transformer within a standardized product bundle that disentangles graph structure from node attributes. Finally, we introduce graph prompt learning on the manifold to bridge contrastive learning with downstream tasks. Extensive experiment on a diverse set of real-world graphs demonstrates the superiority of CRGFM.

Abstract Recently, the reasoning capabilities of Large Reasoning Models (LRMs), such as DeepSeek-R1, have witnessed significant advancements through computationally intensive “slow thinking” processes. These models have demonstrated impressive performance across a variety of complex reasoning tasks. However, despite their remarkable success, LRMs come with substantial computational demands that pose considerable challenges in terms of resource consumption, scalability, and accessibility. In contrast, Small Reasoning Models (SRMs), which are often distilled from larger models, offer a more efficient alternative while still achieving competitive performance. Beyond their efficiency, SRMs frequently exhibit distinct capabilities and cognitive trajectories compared with their larger counterparts, making them particularly interesting from both practical and theoretical perspectives. In this work, we provide a timely and comprehensive survey of recently published research focused on SRMs. We first review the current landscape of SRMs. Then, we analyze diverse training paradigms and inference techniques tailored to enhance the reasoning capabilities of SRMs. Furthermore, we offer an extensive review of domain-specific applications where SRMs have been effectively leveraged. Finally, we discuss promising future research directions that aim to bridge existing gaps. By consolidating recent advances, this survey serves as an essential reference for researchers and practitioners interested in leveraging or developing SRMs to unlock advanced reasoning functionalities with improved efficiency.