Deterministic Topological Retrieval-Augmented Generation via Discrete Dynamical Basins

Abstract

Retrieval-Augmented Generation (RAG) systems navigate a fundamental trade-off between the computational simplicity of flat vector indices and the structural richness of graph representations. Standard dense retrieval collapses document discourse into isolated embeddings, causing semantic fragmentation and context loss. Conversely, unconstrained graph construction methods that induce bipartite entity-document networks or co-occurrence graphs frequently suffer from the hub collapse athology, wherein high-frequency domain tokens become overconnected hubs that drown out long-tail signals during random walks, all while incurring substantial construction overhead. In this work, we propose BasinRAG, a high performance, deterministic topological framework for document retrieval grounded in the theory of discrete dynamical systems. We model sequential text as a discrete functional graph ϕ : V → V ∪{∅} with out-degree bounded by unity, strictly decoupling the primary structural backbone from auxiliary virtual semantic synapses. We prove that structural sinks naturally induce an exact, disjoint partition of the corpus into basins of attraction B(Ai) and inverse arborescences (ρ-trees) parameterized by a topological depth function h(v). Retrieval is performed via an intelligent query router, a weighted Reciprocal Rank Fusion (RRF) modulated by an exponential topological decay prior exp(−λh(v)), and localized spectral diffusion via mass-conserving Personalized PageRank over induced subgraphs. On the standard BEIR SciFact benchmark (5,183 scientific documents), BasinRAG achieves an nDCG@10 of 0.771 and an MRR@10 of 0.750, significantly outperforming competitive dense bi-encoders, neural sparse methods, and heuristic graph baselines, while achieving O(N) linear index construction in 90.8 seconds on commodity hardware. On Princeton SWE-bench Lite, BasinRAG achieves 84.6% Hit@10 in software fault localization with zero external inference API expenditure.