Islands and Bridges: Why You Can Always Click Your Way to Any Image

Abstract

On a site like Pinterest, an image recommender naturally groups images into clusters — an ant cluster, a space-photography cluster — and links clusters together with a handful of cross-links. We show that if every cluster is internally well connected and these cross-links form a connected chain between clusters, then every image can be reached from every other by clicking suggestions, with certainty, not just typically. The proof is short and direct. This is not just a model: it is the exact design behind Hierarchical Navigable Small World graphs (HNSW), the indexing structure that powers most large-scale recommendation and search systems in production today.