A Spectral Theory of Distortion in LLM Graph Reconstruction: Sharp Bounds and Empirical Characterization

Large Language Models (LLMs) have shown promise for reasoning over Text-Attributed Graphs (TAGs). However, applying LLMs to graphs requires linearizing their structure into sequences, introducing distortion rooted in the graph bandwidth problem. While this distortion has been shown to degrade performance, it is often attributed to prompt design or model scale, leaving the underlying mechanism unclear. In this work, we show \textit{how} rotary positional embeddings turn graph linearization into bandwidth-dependent attention decay, suppressing attention between graph-adjacent nodes that are forced far apart in the serialized sequence. This shifts the focus of LLM-based graph reasoning from prompt engineering and scaling toward correcting attention misalignment. Motivated by this analysis, we propose \textbf{G}raph-\textbf{a}ligned \textbf{L}anguage \textbf{A}ttention (\textbf{GaLA}), a lightweight, inference-time modification for LLMs. GaLA biases attention toward graph-adjacent nodes while preserving the LLM's sequential inductive biases. Across TAG benchmarks, GaLA improves performance with negligible overhead, demonstrating that distortion is a correctable bottleneck in LLM-based graph reasoning.

Existing LLM graph benchmarks typically ask models to answer graph-theoretic questions or compute symbolic solutions rather than construct spatial layouts. Within-task difficulty is also primarily stratified by vertex count. However, existing research also suggests that task difficulty is more closely related to the number of constraints imposed by the edges than to the number of vertices being arranged. We introduce PlanarBench, a benchmark that asks models to produce crossing-free ASCII drawings of planar graphs given only an edge list. Across 91 model configurations and 199 non-isomorphic connected planar graphs with 2-7 vertices, edge count is more strongly associated with mean task score than vertex count ($r=-0.85$) versus ($r=-0.47$) and remains strongly associated after controlling for vertex count ($r_p=-0.80$). PlanarBench provides a controlled setting for separating these two difficulty axes. In addition, neither drawing area nor total response length demonstrated a meaningful correlation with score, which is evidence against a simple output-size explanation. Performance varies widely: the best model scores 159.5 out of 199, most models below 30B parameters score under 25, and substantial failures remain among frontier systems.

It remains unclear whether graph language models learn structural regularities or merely memorize training graphs; this cannot be resolved by current aggregate fidelity metrics alone. We develop a calibrated diagnostic protocol that combines frequent subgraph mining, a graph-level bootstrap baseline, and three-level frequency stratification to disentangle memorization from structural alignment. Using this framework, we show that graph language models can acquire structural regularities beyond memorization at scale, primarily in the high-frequency regime. This is supported by the following empirical evidence: On five TU benchmarks, LLaMA-style graph language models reach high subgraph-rank correlation, yet their alignment is matched or exceeded by the memorization bootstrap in most cases. At small scale, under our bootstrap diagnostic, fidelity is largely indistinguishable from verbatim recall. In contrast, at large scale with 3.75M graphs, verbatim memorization drops sharply while rank correlation remains near ceiling. Crucially, in a separate fixed-subsample analysis, frequent subgraph mining restricted to the novel-only subset closely tracks the corresponding all-generation Spearman correlation, providing evidence that the alignment is not driven solely by verbatim recall. Across all scales, high-frequency patterns are well reproduced, while rare patterns remain poorly covered, and this deficit narrows only marginally as capacity increases. We observe the same scale-dependent crossover under two distinct graph serializations (canonical DFS code and action sequences), providing evidence of robustness in our analysis.