Attention as Differentiable Sparsity: A Graph-Theoretic Foundation for Dynamic Token Interaction

This work establishes a rigorous mathematical framework that models transformer attention mechanisms as structured graph sparsification processes, where attention scores define edge weights in token-interaction graphs. By treating sparsity as a differentiable graph property, the analysis provides theoretical foundations for understanding sparse attention approximations through spectral graph theory. The framework yields several conditional theoretical contributions under clearly stated assumptions: (1) approximation bounds for top-$k$ attention mechanisms based on spectral properties, demonstrating that sparse attention preserves essential connectivity under low-rank structural conditions with error bounds proportional to the spectral gap {chung1997spectral}; (2) characterization of sparsity as implicit spectral regularization through effective resistance minimization {ghosh2008minimizing}, potentially influencing model generalization; and (3) derivation of hardware-efficiency bounds via graph partitioning theory {karypis1998multilevel}, providing theoretical justification for structured sparse attention designs. The mathematical analysis assumes attention matrices exhibit spectral decay properties and maintains graph connectivity under sparsification. These theoretical insights offer new perspectives on attention mechanisms through established results in spectral graph theory {spielman2011spectral}, potentially informing principled sparse attention variants. The framework establishes conditional guarantees rather than universal claims, with results explicitly dependent on structural assumptions about attention patterns. All theoretical results are presented with complete proofs and clearly delineated assumptions to ensure mathematical rigor and reproducibility.