Title: Transformers Can Learn Connectivity in Some Graphs but Not Others
Primary area: learning on graphs and other geometries & topologies
Keywords: Graph Reasoning with LLMs, Scaling Law for LLMs

Abstract: Highly competent reasoning capability is essential to ensure the factual correctness of the responses of transformer-based Large Language Models (LLMs), and robust reasoning about transitive relations is instrumental in many settings, such as causal inference. Therefore, it is essential to investigate the capability of transformers in the task of inferring transitive relations (e.g., knowing A causes B and B causes C, we can infer that A causes C). The task of inferring transitive relations is *equivalent* to the task of connectivity in directed graphs (e.g., knowing there is a path from A to B, and there is a path from B to C, we can infer that there is a path from A to C). Past research focused on whether transformers can learn to infer transitivity from in-context examples provided in the input prompt. However, transformers' capability to infer transitive relations from training examples and how scaling affects this ability is unexplored. In this study, we endeavor to answer this question by generating directed graphs to train transformer models of varying sizes and evaluate their ability to infer transitive relations for various graph sizes. Our findings suggest that transformers are capable of learning connectivity on "grid-like'' directed graphs where each node can be embedded in a low-dimensional subspace, and connectivity is easily inferable from the embeddings of the nodes. We find that the dimensionality of the underlying grid graph is a strong predictor of transformers' ability to learn the connectivity task, where higher-dimensional grid graphs pose a greater challenge than low-dimensional grid graphs. In addition, we observe that increasing the model scale leads to increasingly better generalization to infer connectivity over grid graphs. However, if the graph is not a grid graph and contains many disconnected components, transformers struggle to learn the connectivity task, especially when the number of components is large. We also find that transformers benefit more from increasing the graph size than increasing the model size. The code of our experiments is publicly available at [github.com/anonymoususer437/transformers_graph_connectivity](https://github.com/anonymoususer437/transformers_graph_connectivity)

TL;DR: This work investigates how transformers learn to infer transitive relations equivalently learning connectivity on grid graphs and disconnected chain graphs and how scaling impact the performance of transformers on learning connectivity on graphs.