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However, state-of-the-art monotonic graph engines are restricted to in-memory execution and cannot scale to graphs that exceed main memory capacity. In contrast, existing out-of-core graph engines are designed for general-purpose workloads and lack effective pruning mechanisms tailored to monotonic graph algorithms. To bridge this gap, we present Gem, an out-of-core graph engine designed for monotonic graph algorithms. Gem introduces a PageRank-based graph sketch that captures key topological features inmemory with minimal preprocessing overhead. Building on this sketch, we propose a novel graph abstraction that enables the direct derivation of tight bounds for monotonic graph algorithms, supporting effective pruning at both the vertex and partition levels. Comprehensive evaluations on six real-world datasets, including the 42.5-billion-edge ClueWeb graph, show that Gem significantly outperforms existing systems. It achieves up to 135.40\u00d7 speedup over GridGraph and 12.58\u00d7 over Wonderland in out-of-core settings, and also delivers substantial improvements in other modes: up to 10.41\u00d7 over RisGraph in memory and 20.64\u00d7 over CGgraph out-of-GPU memory.<\/jats:p>","DOI":"10.1145\/3769795","type":"journal-article","created":{"date-parts":[[2025,12,6]],"date-time":"2025-12-06T04:32:13Z","timestamp":1764995533000},"page":"1-30","source":"Crossref","is-referenced-by-count":0,"title":["Gem: Scalable Monotonic Graph Processing Beyond Billion-Scale on a Single Machine"],"prefix":"10.1145","volume":"3","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3154-3580","authenticated-orcid":false,"given":"Chengying","family":"Huan","sequence":"first","affiliation":[{"name":"State Key Laboratory for Novel Software Technology, Nanjing University, Nanjing, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1772-6863","authenticated-orcid":false,"given":"Zhengyi","family":"Yang","sequence":"additional","affiliation":[{"name":"University of New South Wales, Sydney, Australia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0001-7896-5019","authenticated-orcid":false,"given":"Haoshen","family":"Yang","sequence":"additional","affiliation":[{"name":"Rutgers, The State University of New Jersey, New Brunswick, NJ, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5718-5657","authenticated-orcid":false,"given":"Shaonan","family":"Ma","sequence":"additional","affiliation":[{"name":"Qiyuan Lab, Beijing, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1565-9997","authenticated-orcid":false,"given":"Rong","family":"Gu","sequence":"additional","affiliation":[{"name":"State Key Laboratory for Novel Software Technology, Nanjing University, Nanjing, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-0277-1136","authenticated-orcid":false,"given":"Fang","family":"Xi","sequence":"additional","affiliation":[{"name":"Qiyuan Lab, Beijing, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3440-9675","authenticated-orcid":false,"given":"Yongchao","family":"Liu","sequence":"additional","affiliation":[{"name":"Ant Group, Hangzhou, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6934-1685","authenticated-orcid":false,"given":"Guihai","family":"Chen","sequence":"additional","affiliation":[{"name":"State Key Laboratory for Novel Software Technology, Nanjing University, Nanjing, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2710-7628","authenticated-orcid":false,"given":"Chen","family":"Tian","sequence":"additional","affiliation":[{"name":"State Key Laboratory for Novel Software Technology, Nanjing University, Nanjing, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"320","published-online":{"date-parts":[[2025,12,5]]},"reference":[{"key":"e_1_2_1_1_1","unstructured":"[n.d.]. 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