Skill Co-activation Graphs: A Functional Topology of Generalization in Large Language Models

Large language models (LLMs) exhibit strong generalization, yet it remains unclear how reusable predictive mechanisms are organized internally to support this behavior. We introduce the Skill Co-activation Graphs (SCGs), a functional-topology view in which nodes are skill-indexed components induced from token-level loss-gradient alignment and edges encode their empirical co-activation across inputs. Starting from the autoregressive objective, we derive a loss-induced skill-alignment formulation and establish a task-conditioned generalization bound whose complexity depends on $\mathrm{Tr}(L^{+}{\Sigma }_T)$, where $L^{+}$ captures SCG resistance and ${\Sigma }_T$ captures task-induced skill usage. We further extend this analysis to empirically estimated SCGs through a stability-aware graph-estimation bound. Experiments on Pythia with ARC-Challenge and SciQ reveal systematic relationships between SCG structure and downstream generalization. Larger Pythia models tend to exhibit richer and more stable skill organization, while lower held-out task-conditioned resistance is associated with higher task accuracy. Intervention-based analyses support the functional relevance of the induced skill subspaces. Across both Pythia and Qwen2.5, SCG-guided fine-tuning further shows that high-resistance examples provide a useful structural signal for data selection under limited adaptation budgets.