Implicit graph theory: Scaling graph mining with vector arithmetic
Date
2026
Authors
Yu, Tengkai
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
The rapid ascent of representation learning has transformed how we model complex systems, replacing discrete identifiers with continuous, high-dimensional vectors. In modern pipelines, relationships between entities, whether users in a recommender system or text chunks in a retrieval corpus, are defined not by explicit links but by geometric proximity in a dense vector space. Because every pair of embeddings yields a non-zero similarity, the induced graph is fully connected by construction, containing O(N^2) weighted edges that no standard adjacency representation can store at scale. Traditional graph mining algorithms, designed for sparse networks, remain tied to the "Explicit Paradigm" of materializing this adjacency matrix before performing computation. For the dense semantic graphs produced by modern embedding pipelines, this imposes a quadratic O(N^2) memory bottleneck that renders standard methods such as Label Propagation and Louvain intractable beyond a few thousand nodes. Common workarounds, such as k-nearest-neighbour pruning, discard precisely the long-range structure that semantic embeddings are meant to capture.
This dissertation introduces Implicit Graph Theory, a unified framework that redefines the graph not as a static container of edges, but as a linear algebraic operator defined by the inner product of node embeddings (A = VV⊤). By exploiting the low-rank structure of semantic data, we derive exact matrix-vector formulations for fundamental graph primitives that operate in linear memory in N, without approximation error relative to the inner-product adjacency model.
We validate this framework through three contributions. First, we introduce Vector-Based Label Propagation (VLP), which addresses the memory wall for semi-supervised learning on dense graphs. We prove that VLP is mathematically equivalent to explicit label propagation and show that it scales to 1.5 million nodes on a single GPU, in a setting where standard dense-graph libraries exhaust memory beyond roughly 15,000 nodes. Second, we present VLouvain, a structure-discovery algorithm that detects hierarchical communities directly from embedding vectors, without ever generating an edge list. The key result, the Vector Supernode Theorem, shows that modularity-driven graph coarsening reduces to vector addition, reproducing the modularity scores of standard Louvain exactly. VLouvain clusters 1.57 million product embeddings in minutes on a single GPU, a scale at which cuGraph and NetworKit exhaust memory entirely. Finally, we apply these principles to retrieval-augmented generation with GraphRAG-V, which uses VLouvain to organize text-chunk embeddings into semantic communities, replacing the expensive LLM-based entity extraction pipeline used by prior graph-RAG systems. On a multi-hop reasoning benchmark, GraphRAG-V reduces indexing time from 3.0 hours to 5.3 minutes (34x) while outperforming both a vector-store baseline and Microsoft's GraphRAG in answer recall.
Together, these results argue for a different way of scaling graph mining to the era of dense representations: rather than storing the graph, we compute it on demand from the embeddings that already exist.
Description
Keywords
implicit graph theory, graph mining, node embeddings, label propagation, community detection, Louvain algorithm, modularity, retrieval-augmented generation, low-rank approximation, GPU computing