Dense subgraphs induced by edge labels
Finding densely connected groups of nodes in networks is a widely-used tool for analysis in graph mining. A popular choice for finding such groups is to find subgraphs with a high average degree. While useful, interpreting such subgraphs may be difficult. On the other hand, many real-world networks...
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| Published in | Machine learning Vol. 113; no. 4; pp. 1967 - 1987 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.04.2024
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0885-6125 1573-0565 1573-0565 |
| DOI | 10.1007/s10994-023-06377-y |
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| Abstract | Finding densely connected groups of nodes in networks is a widely-used tool for analysis in graph mining. A popular choice for finding such groups is to find subgraphs with a high average degree. While useful, interpreting such subgraphs may be difficult. On the other hand, many real-world networks have additional information, and we are specifically interested in networks with labels on edges. In this paper, we study finding sets of labels that induce dense subgraphs. We consider two notions of density: average degree and the number of edges minus the number of nodes weighted by a parameter
α
. There are many ways to induce a subgraph from a set of labels, and we study two cases: First, we study conjunctive-induced dense subgraphs, where the subgraph edges need to have all labels. Secondly, we study disjunctive-induced dense subgraphs, where the subgraph edges need to have at least one label. We show that both problems are
NP
-hard. Because of the hardness, we resort to greedy heuristics. We show that we can implement the greedy search efficiently: the respective running times for finding conjunctive-induced and disjunctive-induced dense subgraphs are in
O
p
log
k
and
O
p
log
2
k
, where
p
is the number of edge-label pairs and
k
is the number of labels. Our experimental evaluation demonstrates that we can find the ground truth in synthetic graphs and that we can find interpretable subgraphs from real-world networks. |
|---|---|
| AbstractList | Finding densely connected groups of nodes in networks is a widely-used tool for analysis in graph mining. A popular choice for finding such groups is to find subgraphs with a high average degree. While useful, interpreting such subgraphs may be difficult. On the other hand, many real-world networks have additional information, and we are specifically interested in networks with labels on edges. In this paper, we study finding sets of labels that induce dense subgraphs. We consider two notions of density: average degree and the number of edges minus the number of nodes weighted by a parameter
$$\alpha$$
α
. There are many ways to induce a subgraph from a set of labels, and we study two cases: First, we study conjunctive-induced dense subgraphs, where the subgraph edges need to have all labels. Secondly, we study disjunctive-induced dense subgraphs, where the subgraph edges need to have at least one label. We show that both problems are
NP
-hard. Because of the hardness, we resort to greedy heuristics. We show that we can implement the greedy search efficiently: the respective running times for finding conjunctive-induced and disjunctive-induced dense subgraphs are in
$$\mathcal {O} \mathopen {}\left( p \log k\right)$$
O
p
log
k
and
$$\mathcal {O} \mathopen {}\left( p \log ^2 k\right)$$
O
p
log
2
k
, where
p
is the number of edge-label pairs and
k
is the number of labels. Our experimental evaluation demonstrates that we can find the ground truth in synthetic graphs and that we can find interpretable subgraphs from real-world networks. Finding densely connected groups of nodes in networks is a widely-used tool for analysis in graph mining. A popular choice for finding such groups is to find subgraphs with a high average degree. While useful, interpreting such subgraphs may be difficult. On the other hand, many real-world networks have additional information, and we are specifically interested in networks with labels on edges. In this paper, we study finding sets of labels that induce dense subgraphs. We consider two notions of density: average degree and the number of edges minus the number of nodes weighted by a parameter α. There are many ways to induce a subgraph from a set of labels, and we study two cases: First, we study conjunctive-induced dense subgraphs, where the subgraph edges need to have all labels. Secondly, we study disjunctive-induced dense subgraphs, where the subgraph edges need to have at least one label. We show that both problems are NP-hard. Because of the hardness, we resort to greedy heuristics. We show that we can implement the greedy search efficiently: the respective running times for finding conjunctive-induced and disjunctive-induced dense subgraphs are in Oplogk and Oplog2k, where p is the number of edge-label pairs and k is the number of labels. Our experimental evaluation demonstrates that we can find the ground truth in synthetic graphs and that we can find interpretable subgraphs from real-world networks. Finding densely connected groups of nodes in networks is a widely-used tool for analysis in graph mining. A popular choice for finding such groups is to find subgraphs with a high average degree. While useful, interpreting such subgraphs may be difficult. On the other hand, many real-world networks have additional information, and we are specifically interested in networks with labels on edges. In this paper, we study finding sets of labels that induce dense subgraphs. We consider two notions of density: average degree and the number of edges minus the number of nodes weighted by a parameter α . There are many ways to induce a subgraph from a set of labels, and we study two cases: First, we study conjunctive-induced dense subgraphs, where the subgraph edges need to have all labels. Secondly, we study disjunctive-induced dense subgraphs, where the subgraph edges need to have at least one label. We show that both problems are NP -hard. Because of the hardness, we resort to greedy heuristics. We show that we can implement the greedy search efficiently: the respective running times for finding conjunctive-induced and disjunctive-induced dense subgraphs are in O p log k and O p log 2 k , where p is the number of edge-label pairs and k is the number of labels. Our experimental evaluation demonstrates that we can find the ground truth in synthetic graphs and that we can find interpretable subgraphs from real-world networks. |
| Author | Kumpulainen, Iiro Tatti, Nikolaj |
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| Cites_doi | 10.1007/978-1-4471-2256-2_4 10.1007/s00778-013-0340-z 10.1016/0022-0000(81)90012-X 10.1007/3-540-45995-2_51 10.1145/1401890.1402008 10.1007/BF00139635 10.1093/bioinformatics/btl243 10.1016/0378-8733(83)90028-X 10.1145/3344210 10.1287/opre.1100.0851 10.1007/978-3-031-18840-4_33 10.1007/s00453-008-9238-3 10.1145/2736277.2741098 10.1145/362342.362367 10.1145/3038912.3052619 10.1287/mnsc.13.7.492 10.1145/2517088 10.1007/3-540-44436-X_10 10.1145/3299869.3324962 10.1145/1557019.1557142 10.1007/0-387-23077-7_17 10.1109/SFCS.2002.1181985 |
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| Keywords | Dense subgraphs Label-induced subgraphs Convex hull |
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| References | Håstad, J. (1996). Clique is hard to approximate within n1-ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n^{1 - \epsilon }$$\end{document}. In FOCS, pp 627–636. Charikar, M. (2000). Greedy approximation algorithms for finding dense components in a graph. APPROX. GalbrunEstherGionisAristidesTattiNikolajOverlapping community detection in labeled graphsDMKD2014285158616103250847 Danisch, M., Chan, T-HH., & Sozio, M. 2017). Large scale density-friendly graph decomposition via convex programming. In Proceedings of the 26th International Conference on World Wide Web, pp 233–242. International World Wide Web Conferences Steering Committee. Abello, J., Resende, GC., & Sudarsky S. (2002). Massive quasi-clique detection. In LATIN 2002: Theoretical Informatics, pp 598–612. Pool, S., Bonchi, F., & van Leeuwen, M. (2014). Description-driven community detection. TIST,5(2), 1–28. Goldberg, AV. (1984). Finding a maximum density subgraph. University of California Berkeley Technical report. Li, F., & Klette, R. (2011). Euclidean Shortest Paths: Exact or Approximate Algorithms, chapter Convex Hulls in the Plane, pp 93–125. Springer . Tsourakakis, CE. (2015). The k-clique densest subgraph problem. In WWW, pp 1122–1132. BronCoenKerboschJoepAlgorithm 457: Finding all cliques of an undirected graphCommunications of the ACM197316957557710.1145/362342.362367 MokkenRobert JCliques clubs and clansQuality & Quantity197913216117310.1007/BF00139635 FratkinEugeneNaughtonBrianTBrutlagDouglas LBatzoglouSerafimMotifcut: regulatory motifs finding with maximum density subgraphsBioinformatics20062214e150e15710.1093/bioinformatics/btl243 Kumpulainen, I., & Tatti, N. (2022). Community detection in edge-labeled graphs. In Discovery Science: 25th International Conference, DS 2022, Montpellier, France, October 10–12, 2022, Proceedings, pp 460–475. BalasundaramBalabhaskarButenkoSergiyHicksIllya VClique relaxations in social network analysis: The maximum k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-plex problemOperations Research2011591133142281422410.1287/opre.1100.0851 OvermarsMark HVan LeeuwenJMaintenance of configurations in the planeJournal of computer and System Sciences198123216620464472510.1016/0022-0000(81)90012-X Brodal, GS., Jacob, R. (2002). Dynamic planar convex hull. In FOCS, pp 617–626. Bonchi, F., Khan, A., & Severini, L. (2019). Distance-generalized core decomposition. In SIGMOD, pp 1006–1023. DinkelbachWernerOn nonlinear fractional programmingManagement Science196713749249824248810.1287/mnsc.13.7.492 Du, X., Jin, R., Ding, L., Lee, VE., & Thornton Jr, John H. (2009). Migration motif: a spatial-temporal pattern mining approach for financial markets. In KDD, pp 1135–1144. UnoTakeakiAn efficient algorithm for solving pseudo clique enumeration problemAlgorithmica2010561316257653110.1007/s00453-008-9238-3 AngelAlbertKoudasNickSarkasNikosSrivastavaDiveshSvendsenMichaelTirthapuraSrikantaDense subgraph maintenance under streaming edge weight updates for real-time story identificationThe VLDB Journal201423217519910.1007/s00778-013-0340-z Tang, J., Zhang, J., Yao, L., Li, J., Zhang, L., & Su, Z. (2008). Arnetminer: Extraction and mining of academic social networks. In KDD, pp 990–998. Langston, MA., Lin, L., Peng, X., Baldwin, NE., Symons, CT., Zhang, B., & Snoddy, JR. (2005). A combinatorial approach to the analysis of differential gene expression data. In Methods of Microarray Data Analysis, pp 223–238. Springer. Tatti, N. (2019). Density-friendly graph decomposition. TKDD,13(5), 1–29. SeidmanStephen BNetwork structure and minimum degreeSocial Networks19835326928772129510.1016/0378-8733(83)90028-X 6377_CR17 6377_CR16 6377_CR15 6377_CR14 6377_CR13 Robert J Mokken (6377_CR18) 1979; 13 6377_CR10 Werner Dinkelbach (6377_CR9) 1967; 13 Esther Galbrun (6377_CR12) 2014; 28 Stephen B Seidman (6377_CR21) 1983; 5 Takeaki Uno (6377_CR25) 2010; 56 6377_CR24 6377_CR23 6377_CR22 6377_CR4 Balabhaskar Balasundaram (6377_CR3) 2011; 59 Coen Bron (6377_CR6) 1973; 16 6377_CR1 6377_CR8 6377_CR7 6377_CR5 Eugene Fratkin (6377_CR11) 2006; 22 6377_CR20 Albert Angel (6377_CR2) 2014; 23 Mark H Overmars (6377_CR19) 1981; 23 |
| References_xml | – reference: Kumpulainen, I., & Tatti, N. (2022). Community detection in edge-labeled graphs. In Discovery Science: 25th International Conference, DS 2022, Montpellier, France, October 10–12, 2022, Proceedings, pp 460–475. – reference: FratkinEugeneNaughtonBrianTBrutlagDouglas LBatzoglouSerafimMotifcut: regulatory motifs finding with maximum density subgraphsBioinformatics20062214e150e15710.1093/bioinformatics/btl243 – reference: Danisch, M., Chan, T-HH., & Sozio, M. 2017). Large scale density-friendly graph decomposition via convex programming. In Proceedings of the 26th International Conference on World Wide Web, pp 233–242. International World Wide Web Conferences Steering Committee. – reference: Brodal, GS., Jacob, R. (2002). Dynamic planar convex hull. In FOCS, pp 617–626. – reference: MokkenRobert JCliques clubs and clansQuality & Quantity197913216117310.1007/BF00139635 – reference: BalasundaramBalabhaskarButenkoSergiyHicksIllya VClique relaxations in social network analysis: The maximum k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-plex problemOperations Research2011591133142281422410.1287/opre.1100.0851 – reference: Tang, J., Zhang, J., Yao, L., Li, J., Zhang, L., & Su, Z. (2008). Arnetminer: Extraction and mining of academic social networks. In KDD, pp 990–998. – reference: Håstad, J. (1996). Clique is hard to approximate within n1-ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n^{1 - \epsilon }$$\end{document}. In FOCS, pp 627–636. – reference: GalbrunEstherGionisAristidesTattiNikolajOverlapping community detection in labeled graphsDMKD2014285158616103250847 – reference: UnoTakeakiAn efficient algorithm for solving pseudo clique enumeration problemAlgorithmica2010561316257653110.1007/s00453-008-9238-3 – reference: Bonchi, F., Khan, A., & Severini, L. (2019). Distance-generalized core decomposition. In SIGMOD, pp 1006–1023. – reference: Tatti, N. (2019). Density-friendly graph decomposition. TKDD,13(5), 1–29. – reference: Du, X., Jin, R., Ding, L., Lee, VE., & Thornton Jr, John H. (2009). Migration motif: a spatial-temporal pattern mining approach for financial markets. In KDD, pp 1135–1144. – reference: DinkelbachWernerOn nonlinear fractional programmingManagement Science196713749249824248810.1287/mnsc.13.7.492 – reference: BronCoenKerboschJoepAlgorithm 457: Finding all cliques of an undirected graphCommunications of the ACM197316957557710.1145/362342.362367 – reference: Abello, J., Resende, GC., & Sudarsky S. (2002). Massive quasi-clique detection. In LATIN 2002: Theoretical Informatics, pp 598–612. – reference: Charikar, M. (2000). Greedy approximation algorithms for finding dense components in a graph. APPROX. – reference: Tsourakakis, CE. (2015). The k-clique densest subgraph problem. In WWW, pp 1122–1132. – reference: SeidmanStephen BNetwork structure and minimum degreeSocial Networks19835326928772129510.1016/0378-8733(83)90028-X – reference: Li, F., & Klette, R. (2011). Euclidean Shortest Paths: Exact or Approximate Algorithms, chapter Convex Hulls in the Plane, pp 93–125. Springer . – reference: OvermarsMark HVan LeeuwenJMaintenance of configurations in the planeJournal of computer and System Sciences198123216620464472510.1016/0022-0000(81)90012-X – reference: Goldberg, AV. (1984). Finding a maximum density subgraph. University of California Berkeley Technical report. – reference: Langston, MA., Lin, L., Peng, X., Baldwin, NE., Symons, CT., Zhang, B., & Snoddy, JR. (2005). A combinatorial approach to the analysis of differential gene expression data. In Methods of Microarray Data Analysis, pp 223–238. Springer. – reference: Pool, S., Bonchi, F., & van Leeuwen, M. (2014). Description-driven community detection. 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| Snippet | Finding densely connected groups of nodes in networks is a widely-used tool for analysis in graph mining. A popular choice for finding such groups is to find... |
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| SubjectTerms | Algorithms Artificial Intelligence Computer Science Control Graph theory Graphs Labels Machine Learning Mechatronics Natural Language Processing (NLP) Networks Nodes Robotics Simulation and Modeling Social networks Special Issue on Discovery Science 2022 Subject specialists |
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| Title | Dense subgraphs induced by edge labels |
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