Parallel Algorithms for Hierarchical Nucleus Decomposition
Article No.: 32, Pages 1 - 27
Abstract
Nucleus decompositions have been shown to be a useful tool for finding dense subgraphs. The coreness value of a clique represents its density based on the number of other cliques it is adjacent to. One useful output of nucleus decomposition is to generate a hierarchy among dense subgraphs at different resolutions. However, existing parallel algorithms for nucleus decomposition do not generate this hierarchy, and only compute the coreness values. This paper presents a scalable parallel algorithm for hierarchy construction, with practical optimizations, such as interleaving the coreness computation with hierarchy construction and using a concurrent union-find data structure in an innovative way to generate the hierarchy. We also introduce a parallel approximation algorithm for nucleus decomposition, which achieves much lower span in theory and better performance in practice. We prove strong theoretical bounds on the work and span (parallel time) of our algorithms.
On a 30-core machine with two-way hyper-threading, our parallel hierarchy construction algorithm achieves up to a 58.84x speedup over the state-of-the-art sequential hierarchy construction algorithm by Sariyuce et al. and up to a 30.96x self-relative parallel speedup. On the same machine, our approximation algorithm achieves a 3.3x speedup over our exact algorithm, while generating coreness estimates with a multiplicative error of 1.33x on average.
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References
[1]
Esra Akbas and Peixiang Zhao. 2017. Truss-Based Community Search: A Truss-Equivalence Based Indexing Approach. Proc. VLDB Endow., Vol. 10, 11 (Aug. 2017), 1298--1309.
[2]
Sabeur Aridhi, Martin Brugnara, Alberto Montresor, and Yannis Velegrakis. 2016. Distributed k-Core Decomposition and Maintenance in Large Dynamic Graphs. In ACM International Conference on Distributed and Event-Based Systems. 161--168.
[3]
Gary D Bader and Christopher WV Hogue. 2003. An Automated Method for Finding Molecular Complexes in Large Protein Interaction Networks. BMC Bioinformatics, Vol. 4, 1 (2003), 1--27.
[4]
Maciej Besta, Armon Carigiet, Kacper Janda, Zur Vonarburg-Shmaria, Lukas Gianinazzi, and Torsten Hoefler. 2020. High-Performance Parallel Graph Coloring with Strong Guarantees on Work, Depth, and Quality. In ACM/IEEE International Conference for High Performance Computing, Networking, Storage and Analysis (SC).
[5]
Mark Blanco, Tze Meng Low, and Kyungjoo Kim. 2019. Exploration of Fine-Grained Parallelism for Load Balancing Eager k-Truss on GPU and CPU. In IEEE High Performance Extreme Computing Conference (HPEC). 1--7.
[6]
Guy E. Blelloch, Daniel Anderson, and Laxman Dhulipala. 2020. Brief Announcement: ParlayLib -- A Toolkit for Parallel Algorithms on Shared-Memory Multicore Machines. In ACM Symposium on Parallelism in Algorithms and Architectures (SPAA).
[7]
Robert D. Blumofe and Charles E. Leiserson. 1999. Scheduling Multithreaded Computations by Work Stealing. J. ACM, Vol. 46, 5 (Sept. 1999), 720--748.
[8]
Yulin Che, Zhuohang Lai, Shixuan Sun, Yue Wang, and Qiong Luo. 2020. Accelerating Truss Decomposition on Heterogeneous Processors. Proc. VLDB Endow., Vol. 13, 10 (June 2020), 1751--1764.
[9]
Pei-Ling Chen, Chung-Kuang Chou, and Ming-Syan Chen. 2014. Distributed Algorithms for k-Truss Decomposition. In IEEE International Conference on Big Data (BigData). 471--480.
[10]
Norishige Chiba and Takao Nishizeki. 1985. Arboricity and Subgraph Listing Algorithms. SIAM J. Comput., Vol. 14, 1 (Feb. 1985), 210--223.
[11]
Deming Chu, Fan Zhang, Wenjie Zhang, Xuemin Lin, and Ying Zhang. 2022. Hierarchical Core Decomposition in Parallel: From Construction to Subgraph Search. In IEEE International Conference on Data Engineering (ICDE). 1138--1151.
[12]
Jonathan Cohen. 2008. Trusses: Cohesive Subgraphs for Social Network Analysis. National Security Agency Technical Report, Vol. 16, 3.1 (2008).
[13]
Ye Conghuan. 2011. Dense Subgroup Identifying in Social Network. In International Conference on Advances in Social Networks Analysis and Mining. 555--556.
[14]
Alessio Conte, Daniele De Sensi, Roberto Grossi, Andrea Marino, and Luca Versari. 2018. Discovering k-Trusses in Large-Scale Networks. In IEEE High Performance Extreme Computing Conference (HPEC). 1--6.
[15]
Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. 2009. Introduction to Algorithms (3. ed.). MIT Press.
[16]
Laxman Dhulipala, Guy Blelloch, and Julian Shun. 2017. Julienne: A Framework for Parallel Graph Algorithms Using Work-efficient Bucketing. In ACM Symposium on Parallelism in Algorithms and Architectures (SPAA). 293--304.
[17]
Laxman Dhulipala, Quanquan C. Liu, Sofya Raskhodnikova, Jessica Shi, Julian Shun, and Shangdi Yu. 2022. Differential Privacy from Locally Adjustable Graph Algorithms: k-Core Decomposition, Low Out-Degree Ordering, and Densest Subgraphs. In IEEE Annual Symposium on Foundations of Computer Science. 754--765.
[18]
Fatemeh Esfahani, Venkatesh Srinivasan, Alex Thomo, and Kui Wu. 2022. Nucleus Decomposition in Probabilistic Graphs: Hardness and Algorithms. In IEEE International Conference on Data Engineering (ICDE). 218--231.
[19]
Yixiang Fang, Kaiqiang Yu, Reynold Cheng, Laks V. S. Lakshmanan, and Xuemin Lin. 2019. Efficient Algorithms for Densest Subgraph Discovery. Proc. VLDB Endow., Vol. 12, 11 (July 2019), 1719--1732.
[20]
Martin Farach-Colton and Meng-Tsung Tsai. 2014. Computing the Degeneracy of Large Graphs. In Latin American Symposium on Theoretical Informatics. 250--260.
[21]
Eugene Fratkin, Brian T Naughton, Douglas L Brutlag, and Serafim Batzoglou. 2006. MotifCut: Regulatory Motifs Finding with Maximum Density Subgraphs. Bioinformatics, Vol. 22, 14 (2006), e150--e157.
[22]
Hillel Gazit. 1991. An Optimal Randomized Parallel Algorithm for Finding Connected Components in a Graph. SIAM J. Comput., Vol. 20, 6 (1991), 1046--1067.
[23]
Mohsen Ghaffari, Silvio Lattanzi, and Slobodan Mitrović. 2019. Improved Parallel Algorithms for Density-Based Network Clustering. In Proceedings of the 36th International Conference on Machine Learning. 2201--2210.
[24]
David Gibson, Ravi Kumar, and Andrew Tomkins. 2005. Discovering Large Dense Subgraphs in Massive Graphs. In Proc. VLDB Endow. 721--732.
[25]
J. Gil, Y. Matias, and U. Vishkin. 1991. Towards a Theory of Nearly Constant Time Parallel Algorithms. In IEEE Symposium on Foundations of Computer Science (FOCS). 698--710.
[26]
Q. Hua, Y. Shi, D. Yu, H. Jin, J. Yu, Z. Cai, X. Cheng, and H. Chen. 2020. Faster Parallel Core Maintenance Algorithms in Dynamic Graphs. IEEE Transactions on Parallel and Distributed Systems (TPDS), Vol. 31, 6 (2020), 1287--1300.
[27]
Xin Huang, Hong Cheng, Lu Qin, Wentao Tian, and Jeffrey Xu Yu. 2014. Querying k-Truss Community in Large and Dynamic Graphs. In ACM SIGMOD International Conference on Management of Data. 1311--1322.
[28]
Xin Huang, Laks V. S. Lakshmanan, Jeffrey Xu Yu, and Hong Cheng. 2015. Approximate Closest Community Search in Networks. Proc. VLDB Endow., Vol. 9, 4 (Dec. 2015), 276--287.
[29]
Yihao Huang, Claire Wang, Jessica Shi, and Julian Shun. 2023. Efficient Algorithms for Parallel Bi-core Decomposition. In Symposium on Algorithmic Principles of Computer Systems (APOCS). 17--32.
[30]
J. Jaja. 1992. Introduction to Parallel Algorithms. Addison-Wesley Professional.
[31]
Siddhartha V. Jayanti and Robert E. Tarjan. 2016. A Randomized Concurrent Algorithm for Disjoint Set Union. In ACM Symposium on Principles of Distributed Computing (PODC). 75--82.
[32]
H. Jin, N. Wang, D. Yu, Q. Hua, X. Shi, and X. Xie. 2018. Core Maintenance in Dynamic Graphs: A Parallel Approach Based on Matching. IEEE Transactions on Parallel and Distributed Systems (TPDS), Vol. 29, 11 (2018), 2416--2428.
[33]
H. Kabir and K. Madduri. 2017a. Parallel k-Core Decomposition on Multicore Platforms. In IEEE International Parallel and Distributed Processing Symposium Workshops (IPDPSW). 1482--1491.
[34]
Humayun Kabir and Kamesh Madduri. 2017b. Parallel k-Truss Decomposition on Multicore Systems. In IEEE High Performance Extreme Computing Conference (HPEC). 1--7.
[35]
Wissam Khaouid, Marina Barsky, Venkatesh Srinivasan, and Alex Thomo. 2015. k-Core Decomposition of Large Networks on a Single PC. Proc. VLDB Endow., Vol. 9, 1 (2015), 13--23.
[36]
Kartik Lakhotia, Rajgopal Kannan, Viktor K. Prasanna, and Cé sar A. F. De Rose. 2020. RECEIPT: REfine CoarsE-grained IndePendent Tasks for Parallel Tip decomposition of Bipartite Graphs. Proc. VLDB Endow., Vol. 14, 3 (2020), 404--417.
[37]
Jure Leskovec and Andrej Krevl. 2019. SNAP Datasets: Stanford Large Network Dataset Collection. http://snap.stanford.edu/data.
[38]
R. Li, J. Yu, and R. Mao. 2014. Efficient Core Maintenance in Large Dynamic Graphs. IEEE Transactions on Knowledge & Data Engineering (TKDE), Vol. 26, 10 (oct 2014), 2453--2465.
[39]
Zhe Lin, Fan Zhang, Xuemin Lin, Wenjie Zhang, and Zhihong Tian. 2021. Hierarchical Core Maintenance on Large Dynamic Graphs. Proc. VLDB Endow., Vol. 14, 5 (2021), 757--770.
[40]
Boge Liu, Long Yuan, Xuemin Lin, Lu Qin, Wenjie Zhang, and Jingren Zhou. 2020. Efficient ((α), (β))-Core Computation in Bipartite Graphs. Proc. VLDB Endow., Vol. 29, 5 (2020), 1075--1099.
[41]
Quanquan C. Liu, Jessica Shi, Shangdi Yu, Laxman Dhulipala, and Julian Shun. 2022. Parallel Batch-Dynamic Algorithms for k-Core Decomposition and Related Graph Problems. In ACM Symposium on Parallelism in Algorithms and Architectures. 191--204.
[42]
Tze Meng Low, Daniele G. Spampinato, Anurag Kutuluru, Upasana Sridhar, Doru Thom Popovici, Franz Franchetti, and Scott McMillan. 2018. Linear Algebraic Formulation of Edge-Centric k-Truss Algorithms with Adjacency Matrices. In IEEE High Performance Extreme Computing Conference (HPEC). 1--7.
[43]
Qi Luo, Dongxiao Yu, Xiuzhen Cheng, Zhipeng Cai, Jiguo Yu, and Weifeng Lv. 2020. Batch Processing for Truss Maintenance in Large Dynamic Graphs. IEEE Transactions on Computational Social Systems, Vol. 7, 6 (2020), 1435--1446.
[44]
Qi Luo, Dongxiao Yu, Hao Sheng, Jiguo Yu, and Xiuzhen Cheng. 2021. Distributed Algorithm for Truss Maintenance in Dynamic Graphs. In Parallel and Distributed Computing, Applications and Technologies (PDCAT). 104--115.
[45]
David W. Matula and Leland L. Beck. 1983. Smallest-Last Ordering and Clustering and Graph Coloring Algorithms. J. ACM, Vol. 30, 3 (July 1983).
[46]
Alberto Montresor, Francesco De Pellegrini, and Daniele Miorandi. 2012. Distributed k-Core Decomposition. IEEE Transactions on Parallel and Distributed Systems (TPDS), Vol. 24, 2 (2012), 288--300.
[47]
Ahmet Erdem Sariyü ce. 2021. Motif-Driven Dense Subgraph Discovery in Directed and Labeled Networks. In The Web Conference (WWW). 379--390.
[48]
Ahmet Erdem Sariyüce, Buug ra Gedik, Gabriela Jacques-Silva, Kun-Lung Wu, and Ümit V cC atalyürek. 2016. Incremental k-Core Decomposition: Algorithms and Evaluation. Proc. VLDB Endow., Vol. 25, 3 (2016), 425--447.
[49]
Ahmet Erdem Sariyüce and Ali Pinar. 2016. Fast Hierarchy Construction for Dense Subgraphs. Proc. VLDB Endow., Vol. 10, 3 (Nov. 2016), 97--108.
[50]
Ahmet Erdem Sariyü ce and Ali Pinar. 2018. Peeling Bipartite Networks for Dense Subgraph Discovery. In ACM International Conference on Web Search and Data Mining (WSDM). 504--512.
[51]
Ahmet Erdem Sariyü ce, C. Seshadhri, and Ali Pinar. 2018. Local Algorithms for Hierarchical Dense Subgraph Discovery. Proc. VLDB Endow., Vol. 12, 1 (2018), 43--56.
[52]
Ahmet Erdem Sariyüce, C. Seshadhri, Ali Pinar, and Ümit V. cCatalyürek. 2017. Nucleus Decompositions for Identifying Hierarchy of Dense Subgraphs. ACM Trans. Web, Vol. 11, 3, Article 16 (July 2017), 16:1--16:27 pages.
[53]
Stephen B. Seidman. 1983. Network Structure and Minimum Degree. Soc. Networks, Vol. 5, 3 (1983), 269 -- 287.
[54]
Jessica Shi, Laxman Dhulipala, and Julian Shun. 2021. Parallel Clique Counting and Peeling Algorithms. In SIAM Conference on Applied and Computational Discrete Algorithms (ACDA). 135--146.
[55]
Jessica Shi, Laxman Dhulipala, and Julian Shun. 2022. Theoretically and Practically Efficient Parallel Nucleus Decomposition. Proc. VLDB Endow., Vol. 15, 3 (feb 2022), 583--596.
[56]
Jessica Shi, Laxman Dhulipala, and Julian Shun. 2023. Parallel Algorithms for Hierarchical Nucleus Decomposition. arxiv: 2306.08623 [cs.DC]
[57]
Jessica Shi and Julian Shun. 2020. Parallel Algorithms for Butterfly Computations. In SIAM Symposium on Algorithmic Principles of Computer Systems (APoCS). 16--30.
[58]
Shaden Smith, Xing Liu, Nesreen K Ahmed, Ancy Sarah Tom, Fabrizio Petrini, and George Karypis. 2017. Truss Decomposition on Shared-Memory Parallel Systems. In IEEE High Performance Extreme Computing Conference (HPEC). 1--6.
[59]
Bintao Sun, T.-H. Hubert Chan, and Mauro Sozio. 2020. Fully Dynamic Approximate k-Core Decomposition in Hypergraphs. ACM Trans. Knowl. Discov. Data (TKDD), Vol. 14, 4, Article 39 (May 2020).
[60]
Charalampos Tsourakakis. 2015. The k-Clique Densest Subgraph Problem. In The Web Conference (WWW). 1122--1132.
[61]
Liptia Venica and Gusti Ayu Putri Saptawati. 2021. Finding Dense Subgraph for Community Detection on Social Network Based on Information Diffusion. In International Conference on Data and Software Engineering (ICoDSE). 1--6.
[62]
Jia Wang and James Cheng. 2012. Truss Decomposition in Massive Networks. Proc. VLDB Endow., Vol. 5, 9 (May 2012), 812--823.
[63]
Kai Wang, Xuemin Lin, Lu Qin, Wenjie Zhang, and Ying Zhang. 2020. Efficient Bitruss Decomposition for Large-Scale Bipartite Graphs. In IEEE International Conference on Data Engineering (ICDE). 661--672.
[64]
D. Wen, L. Qin, Y. Zhang, X. Lin, and J. Yu. 2019. I/O Efficient Core Graph Decomposition: Application to Degeneracy Ordering. IEEE Transactions on Knowledge & Data Engineering (TKDE), Vol. 31, 01 (jan 2019), 75--90.
[65]
Yang Zhang and Srinivasan Parthasarathy. 2012. Extracting Analyzing and Visualizing Triangle k-Core Motifs within Networks. In IEEE International Conference on Data Engineering (ICDE). 1049--1060.
[66]
Yikai Zhang and Jeffrey Xu Yu. 2019. Unboundedness and Efficiency of Truss Maintenance in Evolving Graphs. In ACM SIGMOD International Conference on Management of Data. 1024--1041.
[67]
Y. Zhang, J. X. Yu, Y. Zhang, and L. Qin. 2017. A Fast Order-Based Approach for Core Maintenance. In IEEE International Conference on Data Engineering (ICDE). 337--348.
[68]
Feng Zhao and Anthony KH Tung. 2012. Large Scale Cohesive Subgraphs Discovery for Social Network Visual Analysis. Proc. VLDB Endow., Vol. 6, 2 (2012), 85--96.
[69]
Zhaonian Zou. 2016. Bitruss Decomposition of Bipartite Graphs. In Database Systems for Advanced Applications (DASFAA). 218--233.
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Published: 26 March 2024
Published in PACMMOD Volume 2, Issue 1
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