Serialization Is Not Enough: From Presentation to Utilization in LLM-Based Graph Prediction

Graph serialization makes structural evidence explicit to large language models (LLMs), whereas this explicit presentation does not ensure that the evidence is reliably used for prediction. We study this distinction through degree profiles, which record category-specific neighbor counts. Constructing and using a controlled diagnostic GraphMACC, which separates recognition of category presence from calculation and comparison of category counts, we find a consistent recognition-utilization gap across LLMs and representative presentation-alignment methods. We further examine a possible contributor to this gap. Under a local exchangeability assumption, normalized attention exhibits marginal response compression, where successive occurrences of the same structural category induce progressively smaller changes in the aggregated representation, reducing the separation between adjacent counts. This motivates utilization alignment, which directly targets how presented structural evidence is accumulated and propagated for prediction. We instantiate this idea with SAIL (Structural Accumulation and Integration for LLMs), which preserves occurrence-level structural information through unnormalized accumulation and uses the resulting structural state to correct the original LLM candidate scores. Under mild conditions, we further show that the accumulated differences can be mapped to improved prediction margins for adjacent counts. Experiments substantiate that SAIL improves category-count utilization across LLM families and betters graph prediction across node-, edge-, and graph-level tasks. Its gains are also complementary to existing presentation-alignment methods and persist under fine-tuning and in-context learning.