[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-121759-en":3,"doc-seo-121759-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},121759,8796095360427,"Lucas Martin","https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d",8,"Research & Report","Counting Triangles in Graph Streams using Machine Learning Predictions","Triangle counting is a key graph problem with applications ranging from social network analysis and anomaly detection to link recommendation, query optimization, and pattern classification. The edge streaming model is popular but becomes difficult at massive scale, where memory limits and long runtimes require sampling to estimate triangle counts. This thesis develops and analyzes two sampling algorithms using a Waiting Room for recent edges and an oracle that predicts heavy edges. By incorporating temporal localities, the oracle reduces estimation error, space, and overhead while preserving rigorous approximation guarantees and improving performance over prior methods.","Master Thesis in Computer Engineering  \nCounting Triangles in Graph Streams using Machine Learning Predictions  \nMaster Candidate Supervisor  \nCristian Boldrin Prof. Fabio Vandin  \nStudent ID 2039313 University of Padova  \nAcademic Year  \nTo my parents and friends  \nAbstract  \nTriangle counting is one of the major graph problems with several applications in social network analysis, anomaly detection, link recommendation, query optimization, pattern classification, and other areas of data mining and big data. One of the most popular computational model considered in the literature is edge streaming. However, as the size of the graph grows and involves millions of edges and nodes, several challenges arise, such as memory space limitations and long running time, and therefore sampling techniques are needed to estimate the number of triangles appearing in our graph. In this work we develop, analyze, and implement two sampling algorithms using Waiting Room to store the most recent edges, and making use of a Machine Learning oracle model that provides predictions about the heaviness of edges to output more accurate estimates, while still maintaining rigorous guarantees on the approximation of our results. We show the importance of capturing features and patterns (temporal localities) in our datasets, and how the oracle model can be used to minimize error, space, and overhead in our experiments, leading to enhancements of performances of our proposed methods compared to the state of the art.  \nContents  \n1 Introduction 1  \n1.1 Practical Applications ......................... 3  \n1.2 Our Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4  \n1.3 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 4  \n2 Related Works 7  \n2.1 Graph Stream Triangles Counting ................... 7  \n2.2 Algorithms with Predictions ...................... 9  \n3 Preliminaries 11  \n3.1 Notation and Symbols ......................... 11  \n3.2 Temporal Localities . . . . . . . . . . . . . . . . . . . . . . . . . . . 13  \n3.3 Fixed Probability vs Fixed Memory .................. 17  \n4 Sampling Approach with Predictions: Fixed Probability 19  \n4.1 Fixed Probability Algorithm ...................... 20  \n4.2 Probabilities Computation ....................... 24  \n4.3 Accuracy Analysis ........................... 26  \n4.3.1 Bias Analysis .......................... 26  \n4.3.2 Variance Analysis ........................ 27  \n4.3.3 (ε,δ)-approximation Analysis ................. 31  \n4.3.4 Complexity Analysis ...................... 36  \n5 Sampling Approach with Predictions: Fixed Memory 39  \n5.1 Fixed Memory Algorithm ....................... 40  \n5.2 Theoretical Analysis .......................... 45  \n5.3 Probabilities Computation ....................... 46  \n5.4 Accuracy Analysis ........................... 49  \nCONTENTS  \n5.4.1 Bias Analysis .......................... 49  \n5.4.2 Variance analysis ........................ 50  \n5.4.3 (ε,δ)-approximation Analysis ................. 54  \n5.4.4 Complexity Analysis ...................... 57  \n6 Experiments 59  \n6. 1 Global Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62  \n6.2 Distribution of Estimates . . . . . . . . . . . . . . . . . . . . . . . 66  \n6.3 Local Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68  \n6.4 Memory Budget ............................. 71  \n6.5 Oracle Accuracy ............................ 73  \n6.6 Overhead of the Oracle . . . . . . . . . . . . . . . . . . . . . . . . . 76  \n7 Conclusions 83  \n7. 1 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84  \nReferences 85  \nAcknowledgments 89  \n1  \nIntroduction  \nAn algorithm is the building block to solve problems or accomplish tasks ina systematic and efficient manner. They play a vital role in various domains, from computer science and data analysis, to artificial intelligence and automation. The continued development of algorithms will shape the technol","cbCaieJveCP84XBZ","https://ap.wps.com/l/cbCaieJveCP84XBZ","pdf",5326072,1,97,"English","en",105,"# Introduction\n## Practical Applications\n## Our Contributions\n## Outline of the Thesis\n# Related Works\n## Graph Stream Triangles Counting\n## Algorithms with Predictions\n# Preliminaries\n## Notation and Symbols\n## Temporal Localities\n## Fixed Probability vs Fixed Memory\n# Sampling Approach with Predictions: Fixed Probability\n## Fixed Probability Algorithm\n## Probabilities Computation\n## Accuracy Analysis\n# Sampling Approach with Predictions: Fixed Memory\n## Fixed Memory Algorithm\n## Theoretical Analysis\n## Accuracy Analysis\n# Experiments\n## Global Error\n## Local Error\n## Memory Budget\n## Oracle Accuracy\n## Overhead of the Oracle\n# Conclusions\n## Future Works","[{\"question\":\"Why is triangle counting challenging in graph streams?\",\"answer\":\"Graph streaming must process very large graphs, and edge streaming faces memory limitations and long runtimes. Sampling is therefore needed to estimate triangle counts effectively.\"},{\"question\":\"How do the proposed methods use machine learning predictions?\",\"answer\":\"The work introduces an oracle model that predicts which edges are likely to be heavy. These predictions guide the sampling so the triangle estimates become more accurate while keeping rigorous approximation guarantees.\"},{\"question\":\"What role do temporal localities play in the results?\",\"answer\":\"The thesis emphasizes capturing dataset features and temporal localities. This improves how the oracle-driven sampling minimizes error and overhead, leading to performance gains versus prior approaches.\"}]","Counting Triangles in Graph Streams using Machine Learning Predictions | PDF",1785806691,244,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"counting-triangles-in-graph-streams-using-machine-learning-predictions","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/counting-triangles-in-graph-streams-using-machine-learning-predictions/121759/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-04",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"Why is triangle counting challenging in graph streams?","Question",{"text":75,"@type":76},"Graph streaming must process very large graphs, and edge streaming faces memory limitations and long runtimes. Sampling is therefore needed to estimate triangle counts effectively.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do the proposed methods use machine learning predictions?",{"text":80,"@type":76},"The work introduces an oracle model that predicts which edges are likely to be heavy. These predictions guide the sampling so the triangle estimates become more accurate while keeping rigorous approximation guarantees.",{"name":82,"@type":73,"acceptedAnswer":83},"What role do temporal localities play in the results?",{"text":84,"@type":76},"The thesis emphasizes capturing dataset features and temporal localities. 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