Large language models (LLMs) have demonstrated emergent abilities in text generation, question answering, and reasoning, facilitating various tasks and domains. Despite their proficiency in various tasks, LLMs like PaLM 540B and Llama-3.1 405B face limitations due to large parameter sizes and computational demands, often requiring cloud API use, which raises privacy concerns, limits real-time applications on edge devices, and increases fine-tuning costs. Additionally, LLMs often underperform in specialized domains such as healthcare and law due to insufficient domain-specific knowledge, necessitating specialized models. Therefore, Small Language Models (SLMs) are increasingly favored for their low inference latency, cost-effectiveness, efficient development, and easy customization and adaptability. These models are particularly well-suited for resource-limited environments and domain knowledge acquisition, addressing LLMs’ challenges and proving ideal for applications that require localized data handling for privacy, minimal inference latency for efficiency, and domain knowledge acquisition through lightweight fine-tuning. The rising demand for SLMs has spurred extensive research and development. However, a comprehensive survey investigating issues related to the definition, acquisition, application, enhancement, and reliability of SLM remains lacking, prompting us to conduct a detailed survey on these topics. The definition of SLMs varies widely; thus, to standardize, we propose defining SLMs by their capability to perform specialized tasks and suitability for resource-constrained settings, setting boundaries based on the minimal size for emergent abilities and the maximum size sustainable under resource constraints. For other aspects, we provide a taxonomy of relevant models/methods and develop general frameworks for each category to enhance and utilize SLMs effectively. We have compiled the collected SLM models and related methods on GitHub: https://github.com/FairyFali/SLMs-Survey .

Spectral Graph Neural Networks (GNNs) are gaining attention for their ability to surpass the limitations of message-passing GNNs. They rely on supervision from downstream tasks to learn spectral filters that capture useful graph frequency information. However, some works empirically show that the preferred graph frequency is related to the graph homophily level. The relationship between graph frequency and graph homophily level has not been systematically analyzed and explored in existing spectral GNNs. To mitigate this gap, we conduct theoretical and empirical analyses revealing a positive correlation between low-frequency importance and the homophily ratio, and a negative correlation between high-frequency importance and the homophily ratio. Motivated by this, we propose shape-aware regularization on a Newton Interpolation-based spectral filter that can (i) learn an arbitrary polynomial spectral filter; and (ii) incorporate prior knowledge about the desired shape of the corresponding homophily level. Comprehensive experiments demonstrate that NewtonNet can achieve graph spectral filters with desired shapes and superior performance on both homophilous and heterophilous datasets. Our code is available at https://github.com/junjie-xu/NewtonNet.

Graph self-training (GST), which selects and assigns pseudo-labels to unlabeled nodes, is popular for tackling label sparsity in graphs. However, recent study on homophily graphs show that GST methods could introduce and amplify distribution shift between training and test nodes as they tend to assign pseudo-labels to nodes they are good at. As GNNs typically perform better on homophilic nodes, there could be potential shifts towards homophilic pseudo-nodes, which is underexplored. Our preliminary experiments on heterophilic graphs verify that these methods can cause shifts in homophily ratio distributions, leading to training bias that improves performance on homophilic nodes while degrading it on heterophilic ones. Therefore, we study a novel problem of reducing homophily ratio distribution shifts during self-training on heterophilic graphs. A key challenge is the accurate calculation of homophily ratios and their distributions without extensive labeled data. To tackle them, we propose a novel Heterophily-aware Distribution Consistency-based Graph Self-Training (HC-GST) framework, which estimates homophily ratios using soft labels and optimizes a selection vector to align pseudo-nodes with the global homophily ratio distribution. Extensive experiments on both homophilic and heterophilic graphs show that HC-GST effectively reduces training bias and enhances self-training performance.