Title: Hierarchical navigable small world URL Source: https://en.wikipedia.org/wiki/Hierarchical_Navigable_Small_World_graphs Published Time: 2024-02-22T19:54:41Z Markdown Content: Hierarchical navigable small world - Wikipedia =============== [Jump to content](https://en.wikipedia.org/wiki/Hierarchical_Navigable_Small_World_graphs#bodyContent) Main menu Main menu move to sidebar hide Navigation * [Main page](https://en.wikipedia.org/wiki/Main_Page "Visit the main page [z]") * [Contents](https://en.wikipedia.org/wiki/Wikipedia:Contents "Guides to browsing Wikipedia") * [Current events](https://en.wikipedia.org/wiki/Portal:Current_events "Articles related to current events") * [Random article](https://en.wikipedia.org/wiki/Special:Random "Visit a randomly selected article [x]") * [About Wikipedia](https://en.wikipedia.org/wiki/Wikipedia:About "Learn about Wikipedia and how it works") * [Contact us](https://en.wikipedia.org/wiki/Wikipedia:Contact_us "How to contact Wikipedia") Contribute * [Help](https://en.wikipedia.org/wiki/Help:Contents "Guidance on how to use and edit Wikipedia") * [Learn to edit](https://en.wikipedia.org/wiki/Help:Introduction "Learn how to edit Wikipedia") * [Community portal](https://en.wikipedia.org/wiki/Wikipedia:Community_portal "The hub for editors") * [Recent changes](https://en.wikipedia.org/wiki/Special:RecentChanges "A list of recent changes to Wikipedia [r]") * [Upload file](https://en.wikipedia.org/wiki/Wikipedia:File_upload_wizard "Add images or other media for use on Wikipedia") [![](/static/images/icons/wikipedia.png) ![Wikipedia](/static/images/mobile/copyright/wikipedia-wordmark-en.svg) ![The Free Encyclopedia](/static/images/mobile/copyright/wikipedia-tagline-en.svg)](https://en.wikipedia.org/wiki/Main_Page) [Search](https://en.wikipedia.org/wiki/Special:Search "Search Wikipedia [f]") Search * [Donate](https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en "Support us by donating to the Wikimedia Foundation") Appearance * [Create account](https://en.wikipedia.org/w/index.php?title=Special:CreateAccount&returnto=Hierarchical+navigable+small+world "You are encouraged to create an account and log in; however, it is not mandatory") * [Log in](https://en.wikipedia.org/w/index.php?title=Special:UserLogin&returnto=Hierarchical+navigable+small+world "You're encouraged to log in; however, it's not mandatory. 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Please [help improve it](https://en.wikipedia.org/w/index.php?title=Hierarchical_navigable_small_world&action=edit) to [make it understandable to non-experts](https://en.wikipedia.org/wiki/Wikipedia:Make_technical_articles_understandable "Wikipedia:Make technical articles understandable"), without removing the technical details. _(February 2024)_ _([Learn how and when to remove this message](https://en.wikipedia.org/wiki/Help:Maintenance_template_removal "Help:Maintenance template removal"))_ Clustering and community detection algorithm Part of [a series](https://en.wikipedia.org/wiki/Category:Network_science "Category:Network science") on [Network science](https://en.wikipedia.org/wiki/Network_science "Network science") [![Internet_map_1024.jpg](//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Internet_map_1024.jpg/250px-Internet_map_1024.jpg)](https://en.wikipedia.org/wiki/File:Internet_map_1024.jpg) * [Theory](https://en.wikipedia.org/wiki/Network_theory "Network theory") * [Graph](https://en.wikipedia.org/wiki/Graph_/(discrete_mathematics/) "Graph (discrete mathematics)") * [Complex network](https://en.wikipedia.org/wiki/Complex_network "Complex network") * [Contagion](https://en.wikipedia.org/wiki/Complex_contagion "Complex contagion") * [Small-world](https://en.wikipedia.org/wiki/Small-world_network "Small-world network") * [Scale-free](https://en.wikipedia.org/wiki/Scale-free_network "Scale-free network") * [Community structure](https://en.wikipedia.org/wiki/Community_structure "Community structure") * [Percolation](https://en.wikipedia.org/wiki/Percolation_theory "Percolation theory") * [Evolution](https://en.wikipedia.org/wiki/Evolving_networks "Evolving networks") * [Controllability](https://en.wikipedia.org/wiki/Network_controllability "Network controllability") * [Graph drawing](https://en.wikipedia.org/wiki/Graph_drawing "Graph drawing") * [Social capital](https://en.wikipedia.org/wiki/Social_capital "Social capital") * [Link analysis](https://en.wikipedia.org/wiki/Link_analysis "Link analysis") * [Optimization](https://en.wikipedia.org/wiki/Combinatorial_optimization "Combinatorial optimization") * [Reciprocity](https://en.wikipedia.org/wiki/Reciprocity_/(network_science/) "Reciprocity (network science)") * [Closure](https://en.wikipedia.org/wiki/Triadic_closure "Triadic closure") * [Homophily](https://en.wikipedia.org/wiki/Homophily "Homophily") * [Transitivity](https://en.wikipedia.org/wiki/Transitive_relation "Transitive relation") * [Preferential attachment](https://en.wikipedia.org/wiki/Preferential_attachment "Preferential attachment") * [Balance theory](https://en.wikipedia.org/wiki/Balance_theory "Balance theory") * [Network effect](https://en.wikipedia.org/wiki/Network_effect "Network effect") * [Social influence](https://en.wikipedia.org/wiki/Social_influence "Social influence") Network types * [Informational (computing)](https://en.wikipedia.org/wiki/Computer_network "Computer network") * [Telecommunication](https://en.wikipedia.org/wiki/Telecommunications_network "Telecommunications network") * [Transport](https://en.wikipedia.org/wiki/Transport_network "Transport network") * [Social](https://en.wikipedia.org/wiki/Social_network "Social network") * [Scientific collaboration](https://en.wikipedia.org/wiki/Scientific_collaboration_network "Scientific collaboration network") * [Biological](https://en.wikipedia.org/wiki/Biological_network "Biological network") * [Artificial neural](https://en.wikipedia.org/wiki/Artificial_neural_network "Artificial neural network") * [Interdependent](https://en.wikipedia.org/wiki/Interdependent_networks "Interdependent networks") * [Semantic](https://en.wikipedia.org/wiki/Semantic_network "Semantic network") * [Spatial](https://en.wikipedia.org/wiki/Spatial_network "Spatial network") * [Dependency](https://en.wikipedia.org/wiki/Dependency_network "Dependency network") * [Flow](https://en.wikipedia.org/wiki/Flow_network "Flow network") * [on-Chip](https://en.wikipedia.org/wiki/Network_on_a_chip "Network on a chip") [Graphs](https://en.wikipedia.org/wiki/Graph_/(discrete_mathematics/) "Graph (discrete mathematics)") Features * [Clique](https://en.wikipedia.org/wiki/Clique_/(graph_theory/) "Clique (graph theory)") * [Component](https://en.wikipedia.org/wiki/Connected_component_/(graph_theory/) "Connected component (graph theory)") * [Cut](https://en.wikipedia.org/wiki/Cut_/(graph_theory/) "Cut (graph theory)") * [Cycle](https://en.wikipedia.org/wiki/Cycle_/(graph_theory/) "Cycle (graph theory)") * [Data structure](https://en.wikipedia.org/wiki/Graph_/(abstract_data_type/) "Graph (abstract data type)") * [Edge](https://en.wikipedia.org/wiki/Edge_/(graph_theory/) "Edge (graph theory)") * [Loop](https://en.wikipedia.org/wiki/Loop_/(graph_theory/) "Loop (graph theory)") * [Neighborhood](https://en.wikipedia.org/wiki/Neighbourhood_/(graph_theory/) "Neighbourhood (graph theory)") * [Path](https://en.wikipedia.org/wiki/Path_/(graph_theory/) "Path (graph theory)") * [Vertex](https://en.wikipedia.org/wiki/Vertex_/(graph_theory/) "Vertex (graph theory)") * [Adjacency list](https://en.wikipedia.org/wiki/Adjacency_list "Adjacency list") / [matrix](https://en.wikipedia.org/wiki/Adjacency_matrix "Adjacency matrix") * [Incidence list](https://en.wikipedia.org/wiki/Incidence_list "Incidence list") / [matrix](https://en.wikipedia.org/wiki/Incidence_matrix "Incidence matrix") Types * [Bipartite](https://en.wikipedia.org/wiki/Bipartite_graph "Bipartite graph") * [Complete](https://en.wikipedia.org/wiki/Complete_graph "Complete graph") * [Directed](https://en.wikipedia.org/wiki/Directed_graph "Directed graph") * [Hyper](https://en.wikipedia.org/wiki/Hypergraph "Hypergraph") * [Labeled](https://en.wikipedia.org/wiki/Labeled_graph "Labeled graph") * [Multi](https://en.wikipedia.org/wiki/Multigraph "Multigraph") * [Random](https://en.wikipedia.org/wiki/Random_graph "Random graph") * [Weighted](https://en.wikipedia.org/wiki/Weighted_graph "Weighted graph") * [Metrics](https://en.wikipedia.org/wiki/Metrics_/(networking/) "Metrics (networking)") * [Algorithms](https://en.wikipedia.org/wiki/List_of_algorithms#Networking "List of algorithms") * [Centrality](https://en.wikipedia.org/wiki/Centrality "Centrality") * [Degree](https://en.wikipedia.org/wiki/Degree_/(graph_theory/) "Degree (graph theory)") * [Motif](https://en.wikipedia.org/wiki/Network_motif "Network motif") * [Clustering](https://en.wikipedia.org/wiki/Clustering_coefficient "Clustering coefficient") * [Degree distribution](https://en.wikipedia.org/wiki/Degree_distribution "Degree distribution") * [Assortativity](https://en.wikipedia.org/wiki/Assortativity "Assortativity") * [Distance](https://en.wikipedia.org/wiki/Distance_/(graph_theory/) "Distance (graph theory)") * [Modularity](https://en.wikipedia.org/wiki/Modularity_/(networks/) "Modularity (networks)") * [Efficiency](https://en.wikipedia.org/wiki/Efficiency_/(network_science/) "Efficiency (network science)") Models Topology * [Random graph](https://en.wikipedia.org/wiki/Random_graph "Random graph") * [Erdős–Rényi](https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93R%C3%A9nyi_model "Erdős–Rényi model") * [Barabási–Albert](https://en.wikipedia.org/wiki/Barab%C3%A1si%E2%80%93Albert_model "Barabási–Albert model") * [Bianconi–Barabási](https://en.wikipedia.org/wiki/Bianconi%E2%80%93Barab%C3%A1si_model "Bianconi–Barabási model") * [Fitness model](https://en.wikipedia.org/wiki/Fitness_model_/(network_theory/) "Fitness model (network theory)") * [Watts–Strogatz](https://en.wikipedia.org/wiki/Watts%E2%80%93Strogatz_model "Watts–Strogatz model") * [Exponential random (ERGM)](https://en.wikipedia.org/wiki/Exponential_random_graph_models "Exponential random graph models") * [Random geometric (RGG)](https://en.wikipedia.org/wiki/Random_geometric_graph "Random geometric graph") * [Hyperbolic (HGN)](https://en.wikipedia.org/wiki/Hyperbolic_geometric_graph "Hyperbolic geometric graph") * [Hierarchical](https://en.wikipedia.org/wiki/Hierarchical_network_model "Hierarchical network model") * [Stochastic block](https://en.wikipedia.org/wiki/Stochastic_block_model "Stochastic block model") * [Blockmodeling](https://en.wikipedia.org/wiki/Blockmodeling "Blockmodeling") * [Maximum entropy](https://en.wikipedia.org/wiki/Maximum-entropy_random_graph_model "Maximum-entropy random graph model") * [Soft configuration](https://en.wikipedia.org/wiki/Soft_configuration_model "Soft configuration model") * [LFR Benchmark](https://en.wikipedia.org/wiki/Lancichinetti%E2%80%93Fortunato%E2%80%93Radicchi_benchmark "Lancichinetti–Fortunato–Radicchi benchmark") Dynamics * [Boolean network](https://en.wikipedia.org/wiki/Boolean_network "Boolean network") * [agent based](https://en.wikipedia.org/wiki/Agent-based_model "Agent-based model") * [Epidemic](https://en.wikipedia.org/wiki/Epidemic_model "Epidemic model")/[SIR](https://en.wikipedia.org/wiki/SIR_model "SIR model") * Lists * Categories * [Topics](https://en.wikipedia.org/wiki/List_of_network_theory_topics "List of network theory topics") * [Software](https://en.wikipedia.org/wiki/Social_network_analysis_software "Social network analysis software") * [Network scientists](https://en.wikipedia.org/wiki/List_of_network_scientists "List of network scientists") * [Category:Network theory](https://en.wikipedia.org/wiki/Category:Network_theory "Category:Network theory") * [Category:Graph theory](https://en.wikipedia.org/wiki/Category:Graph_theory "Category:Graph theory") * [v](https://en.wikipedia.org/wiki/Template:Network_science "Template:Network science") * [t](https://en.wikipedia.org/wiki/Template_talk:Network_science "Template talk:Network science") * [e](https://en.wikipedia.org/wiki/Special:EditPage/Template:Network_science "Special:EditPage/Template:Network science") The **Hierarchical navigable small world** (**HNSW**) algorithm is a [graph](https://en.wikipedia.org/wiki/Graph_/(discrete_mathematics/) "Graph (discrete mathematics)")\-based [approximate nearest neighbor](https://en.wikipedia.org/wiki/Nearest_neighbor_search#Approximation_methods "Nearest neighbor search") search technique used in many [vector databases](https://en.wikipedia.org/wiki/Vector_database "Vector database").[\[1\]](https://en.wikipedia.org/wiki/Hierarchical_Navigable_Small_World_graphs#cite_note-1)[\[2\]](https://en.wikipedia.org/wiki/Hierarchical_Navigable_Small_World_graphs#cite_note-hnsw-2) Nearest neighbor search without an index involves computing the distance from the query to each point in the database, which for large datasets is computationally prohibitive. For high-dimensional data, tree-based exact vector search techniques such as the [k-d tree](https://en.wikipedia.org/wiki/K-d_tree "K-d tree") and [R-tree](https://en.wikipedia.org/wiki/R-tree "R-tree") do not perform well enough because of the [curse of dimensionality](https://en.wikipedia.org/wiki/Curse_of_dimensionality "Curse of dimensionality"). To remedy this, _approximate_ k-nearest neighbor searches have been proposed, such as [locality-sensitive hashing](https://en.wikipedia.org/wiki/Locality-sensitive_hashing "Locality-sensitive hashing") (LSH) and [product quantization](https://en.wikipedia.org/w/index.php?title=Product_quantization&action=edit&redlink=1 "Product quantization (page does not exist)") (PQ) that trade performance for accuracy.[\[2\]](https://en.wikipedia.org/wiki/Hierarchical_Navigable_Small_World_graphs#cite_note-hnsw-2) The HNSW graph offers an approximate k-nearest neighbor search which scales logarithmically even in high-dimensional data. It is an extension of the earlier work on navigable [small world graphs](https://en.wikipedia.org/wiki/Small_world_graph "Small world graph") presented at the Similarity Search and Applications (SISAP) conference in 2012 with an additional hierarchical navigation to find entry points to the main graph faster.[\[3\]](https://en.wikipedia.org/wiki/Hierarchical_Navigable_Small_World_graphs#cite_note-3) HNSW-based libraries are among the best performers in the approximate nearest neighbors benchmark.[\[4\]](https://en.wikipedia.org/wiki/Hierarchical_Navigable_Small_World_graphs#cite_note-4)[\[5\]](https://en.wikipedia.org/wiki/Hierarchical_Navigable_Small_World_graphs#cite_note-5)[\[6\]](https://en.wikipedia.org/wiki/Hierarchical_Navigable_Small_World_graphs#cite_note-6) Use in vector databases ----------------------- \[[edit](https://en.wikipedia.org/w/index.php?title=Hierarchical_navigable_small_world&action=edit§ion=1 "Edit section: Use in vector databases")\] HNSW is a key method for approximate nearest neighbor search in high-dimensional vector databases, for example in the context of embeddings from neural networks in large language models. Databases that use HNSW as search index include: * [Apache Lucene](https://en.wikipedia.org/wiki/Apache_Lucene "Apache Lucene") Vector Search * Chroma * [FAISS](https://en.wikipedia.org/w/index.php?title=FAISS&action=edit&redlink=1 "FAISS (page does not exist)") * Qdrant * Vespa * Vearch Gamma * Weaviate * pgvector * MariaDB[\[7\]](https://en.wikipedia.org/wiki/Hierarchical_Navigable_Small_World_graphs#cite_note-7) * MongoDB Atlas[\[8\]](https://en.wikipedia.org/wiki/Hierarchical_Navigable_Small_World_graphs#cite_note-8) * Milvus[\[9\]](https://en.wikipedia.org/wiki/Hierarchical_Navigable_Small_World_graphs#cite_note-9) Several of these use the hnswlib library[\[10\]](https://en.wikipedia.org/wiki/Hierarchical_Navigable_Small_World_graphs#cite_note-10) provided by the original authors. References ---------- \[[edit](https://en.wikipedia.org/w/index.php?title=Hierarchical_navigable_small_world&action=edit§ion=2 "Edit section: References")\] 1. **[^](https://en.wikipedia.org/wiki/Hierarchical_Navigable_Small_World_graphs#cite_ref-1)** Malkov, Yu A.; Yashunin, D. A. (2016). "Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs". [arXiv](https://en.wikipedia.org/wiki/ArXiv_/(identifier/) "ArXiv (identifier)"):[1603.09320](https://arxiv.org/abs/1603.09320) \[[cs.DS](https://arxiv.org/archive/cs.DS)\]. 2. ^ [_**a**_](https://en.wikipedia.org/wiki/Hierarchical_Navigable_Small_World_graphs#cite_ref-hnsw_2-0) [_**b**_](https://en.wikipedia.org/wiki/Hierarchical_Navigable_Small_World_graphs#cite_ref-hnsw_2-1) Malkov, Yury A; Yashunin, Dmitry A (1 April 2020). "Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs". _IEEE Transactions on Pattern Analysis and Machine Intelligence_. **42** (4): 824–836. [arXiv](https://en.wikipedia.org/wiki/ArXiv_/(identifier/) "ArXiv (identifier)"):[1603.09320](https://arxiv.org/abs/1603.09320). [doi](https://en.wikipedia.org/wiki/Doi_/(identifier/) "Doi (identifier)"):[10.1109/TPAMI.2018.2889473](https://doi.org/10.1109%2FTPAMI.2018.2889473). [PMID](https://en.wikipedia.org/wiki/PMID_/(identifier/) "PMID (identifier)") [30602420](https://pubmed.ncbi.nlm.nih.gov/30602420). 3. **[^](https://en.wikipedia.org/wiki/Hierarchical_Navigable_Small_World_graphs#cite_ref-3)** Malkov, Yury; Ponomarenko, Alexander; Logvinov, Andrey; Krylov, Vladimir (2012). ["Scalable Distributed Algorithm for Approximate Nearest Neighbor Search Problem in High Dimensional General Metric Spaces"](https://link.springer.com/chapter/10.1007/978-3-642-32153-5_10). In Navarro, Gonzalo; Pestov, Vladimir (eds.). _Similarity Search and Applications_. Lecture Notes in Computer Science. Vol. 7404\. Berlin, Heidelberg: Springer. pp. 132–147. [doi](https://en.wikipedia.org/wiki/Doi_/(identifier/) "Doi (identifier)"):[10.1007/978-3-642-32153-5\_10](https://doi.org/10.1007%2F978-3-642-32153-5_10). [ISBN](https://en.wikipedia.org/wiki/ISBN_/(identifier/) "ISBN (identifier)") [978-3-642-32153-5](https://en.wikipedia.org/wiki/Special:BookSources/978-3-642-32153-5 "Special:BookSources/978-3-642-32153-5"). 4. **[^](https://en.wikipedia.org/wiki/Hierarchical_Navigable_Small_World_graphs#cite_ref-4)** Aumüller, Martin; Bernhardsson, Erik; Faithfull, Alexander (2017). ["ANN-Benchmarks: A Benchmarking Tool for Approximate Nearest Neighbor Algorithms"](https://link.springer.com/chapter/10.1007/978-3-319-68474-1_3). In Beecks, Christian; Borutta, Felix; Kröger, Peer; Seidl, Thomas (eds.). _Similarity Search and Applications_. Lecture Notes in Computer Science. Vol. 10609\. Cham: Springer International Publishing. pp. 34–49. [arXiv](https://en.wikipedia.org/wiki/ArXiv_/(identifier/) "ArXiv (identifier)"):[1807.05614](https://arxiv.org/abs/1807.05614). [doi](https://en.wikipedia.org/wiki/Doi_/(identifier/) "Doi (identifier)"):[10.1007/978-3-319-68474-1\_3](https://doi.org/10.1007%2F978-3-319-68474-1_3). [ISBN](https://en.wikipedia.org/wiki/ISBN_/(identifier/) "ISBN (identifier)") [978-3-319-68474-1](https://en.wikipedia.org/wiki/Special:BookSources/978-3-319-68474-1 "Special:BookSources/978-3-319-68474-1"). 5. **[^](https://en.wikipedia.org/wiki/Hierarchical_Navigable_Small_World_graphs#cite_ref-5)** Aumüller, Martin; Bernhardsson, Erik; Faithfull, Alexander (2020). ["ANN-Benchmarks: A benchmarking tool for approximate nearest neighbor algorithms"](https://linkinghub.elsevier.com/retrieve/pii/S0306437918303685). _Information Systems_. **87**: 101374. [arXiv](https://en.wikipedia.org/wiki/ArXiv_/(identifier/) "ArXiv (identifier)"):[1807.05614](https://arxiv.org/abs/1807.05614). [doi](https://en.wikipedia.org/wiki/Doi_/(identifier/) "Doi (identifier)"):[10.1016/j.is.2019.02.006](https://doi.org/10.1016%2Fj.is.2019.02.006). 6. **[^](https://en.wikipedia.org/wiki/Hierarchical_Navigable_Small_World_graphs#cite_ref-6)** ["ANN-Benchmarks"](https://ann-benchmarks.com/). _ann-benchmarks.com_. Retrieved 2024-03-19. 7. **[^](https://en.wikipedia.org/wiki/Hierarchical_Navigable_Small_World_graphs#cite_ref-7)** ["MariaDB Vector"](https://mariadb.org/projects/mariadb-vector/). _MariaDB.org_. Retrieved 2024-07-30. 8. **[^](https://en.wikipedia.org/wiki/Hierarchical_Navigable_Small_World_graphs#cite_ref-8)** ["MongoDB Atlas Vector Search"](https://www.mongodb.com/products/platform/atlas-vector-search). _mongodb.com_. Retrieved 2024-09-17. 9. **[^](https://en.wikipedia.org/wiki/Hierarchical_Navigable_Small_World_graphs#cite_ref-9)** ["How to Pick a Vector Index in Your Milvus Instance: A Visual Guide"](https://zilliz.com/learn/how-to-pick-a-vector-index-in-milvus-visual-guide). _zilliz.com_. Retrieved 2024-10-10. 10. **[^](https://en.wikipedia.org/wiki/Hierarchical_Navigable_Small_World_graphs#cite_ref-10)** [_nmslib/hnswlib_](https://github.com/nmslib/hnswlib), nmslib, 2024-03-18, retrieved 2024-03-19 [![Stub icon](//upload.wikimedia.org/wikipedia/commons/thumb/f/f7/Binary_tree.svg/36px-Binary_tree.svg.png)](https://en.wikipedia.org/wiki/File:Binary_tree.svg) This [algorithms](https://en.wikipedia.org/wiki/Algorithm "Algorithm") or [data structures](https://en.wikipedia.org/wiki/Data_structure "Data structure")\-related article is a [stub](https://en.wikipedia.org/wiki/Wikipedia:Stub "Wikipedia:Stub"). 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