A Power Law in Logarithm's Clothing: On the Scalability of Graph-Based Vector Search

Most vector databases rely on graph-based indexes, notably HNSW and Vamana, for approximate nearest neighbor search. With embedding models widely adopted, the datasets these databases store grow rapidly. At a fixed accuracy, how does search cost scale with dataset size? The prevailing answer is poly-logarithmic growth. Yet the claim is proven only under special conditions and asserted without proof for the indexes used in practice. It is also largely untested: standard benchmarks measure cost at one dataset size, not across sizes. We put the claim to the test. The answer depends on the scale itself. While the dataset size $N$ is small relative to the data's intrinsic dimensionality, search cost grows as $N^c$ for a constant $0<c<1$. We call this scaling the Sublinear Power Law. Once $N$ is large enough, growth slows to subpolynomial, consistent with the poly-logarithmic claim. The Sublinear Power Law appears on every dataset, mostly up to its full size, at every recall target, query hardness level, and index configuration we test. The transition to subpolynomial growth appears on the two datasets that grow large enough relative to their intrinsic dimensionality. One mechanism underlies both behaviors: a dataset's intrinsic dimensionality grows with its size until the data resolves its underlying distribution. Higher intrinsic dimensionality packs more vectors into the query neighborhood the search must examine. We present a unifying theory of beam-search cost that explains our observations. For exact and bounded-degree constructions, we prove the Sublinear Power Law and the eventual transition to poly-logarithmic scaling, and derive the scale at which it occurs. We also develop models that predict the power-law exponents for any recall target and index configuration. These models give a principled way to navigate trade-offs among search cost, insertion cost, and recall as data grows.

From Feature Interaction to Feature Transport - A Unified Block for Scalable Recommendation Models

Unified recommendation models aim to jointly model non-sequential multi-field features and sequential user behaviors, but existing interaction-centric designs mainly focus on mixing heterogeneous tokens within each layer. We argue that scalable unified recommendation also requires controlling how intent information is carried, filtered, and preserved across stacked blocks. Inspired by flow-based representation dynamics, we introduce feature transport, a view that treats deep unified recommendation as a discrete context-conditioned representation evolution process. We propose CRAFT, a Contextual Residual Adaptive Feature Transport block, which summarizes non-sequential features into a reliability-aware contextual field and uses it to generate residual displacement and memory-preserving signals for intent and sequence representations. In this way, non-sequential context acts as an active controller of representation evolution rather than a passive object of interaction. In the TAAC2026 advertising recommendation competition, CRAFT achieves a test AUC of 0.838090, surpassing the previous leaderboard-best score of 0.83798. Scaling experiments further show that CRAFT benefits from both depth and width expansion: stacking CRAFT to six blocks improves test AUC to 0.838148, while increasing the hidden dimension reaches 0.838106. These results demonstrate the effectiveness, scalability, and generalization potential of the feature transport paradigm. Source code: https://github.com/AshleyLuo001/CRAFT

Scaling Graph Neural Networks for Friend Recommendation: Multi-Hash User Embeddings and Temporal Neighbor Sampling

Friend recommendation is inherently graph-structured: the relevance of a potential connection depends on multi-hop social context rather than user attributes alone. However, deploying message-passing GNNs on a production-scale social graph with hundreds of millions of users and tens of billions of edges requires addressing numerous modeling and systems challenges. We present a scalable end-to-end GNN ranking system for production social graphs, focusing on two design choices that are critical in this setting: multi-hash ID embeddings and temporal neighbor sampling. Multi-hash embeddings are common for high-cardinality features, but industrial GNN systems typically either ignore trainable IDs or accept full embedding tables, exceeding 200 GB for our graph. We integrate multi-hash as the primary node representation, reducing the ID-embedding table size by more than 98 percent while preserving ranking quality. Temporal neighbor sampling is well understood in principle, but existing implementations scan full adjacency lists, which is a non-starter for users with tens of thousands of friends. We implement timestamp-sorted CSR storage with binary search, reducing the per-node temporal sampling cost from $O(deg(v) + k)$ to $O(\log(deg(v)) + k)$. Beyond these components, we show that this combination scales and yields measurable production impact. On a graph with 194M users and 28B edges, offline ablations isolate each design choice's contribution. In an online A/B test, our system increases friend additions from recommendations by 16 percent and unique friend adders by 11.5 percent over a strong production baseline. We release our framework for distributed training and inference on large temporal graphs.

Efficient Recommendations via Graph Coarsening and Label Propagation

Graph-based recommendations are widely adopted in real-world industrial applications. However, graphs in these systems often reach a massive scale, posing notable scalability and efficiency challenges. This requires techniques that can effectively balance predictive quality with computational cost. One promising approach is graph coarsening, an adaptive graph reduction technique that offers a way to systematically construct smaller, yet structurally representative, versions of the original large-scale graphs. In this work, we propose a flexible two-stage diffusion framework that combines graph coarsening with multi-step label propagation in the telecommunications domain. Domain-specific heuristics are applied to first aggregate nodes into meaningful communities, reducing graph size while preserving essential business-relevant relationships. An initial diffusion process done by a Label Propagation Algorithm (LPA) or a Graph Neural Network (GNN) propagates labels across the coarsened graph to produce coarse-grained predictions. Finally, a second LPA within subgraphs generates the final recommendations for individual users. On a real-world telecommunications dataset, when using LPA in both stages, our method achieves up to +24% NDCG@5 over the full-graph LPA baseline. Incorporating a lightweight GNN in the first stage further boosts NDCG@5 by more than 50%, but requires substantial training and inference time. Through extensive experiments and a detailed ablation, we quantify these trade-offs and demonstrate that our coarsening-driven approach delivers an optimal balance between scalability, latency, and recommendation quality.