ABSTRACT
Network analysis has played a key role in knowledge discovery and data mining. In many real-world applications in recent years, we are interested in mining multilayer networks, where we have a number of edge sets called layers, which encode different types of connections and/or time-dependent connections over the same set of vertices. Among many network analysis techniques, dense subgraph discovery, aiming to find a dense component in a network, is an essential primitive with a variety of applications in diverse domains. In this paper, we introduce a novel optimization model for dense subgraph discovery in multilayer networks. Our model aims to find a stochastic solution, i.e., a probability distribution over the family of vertex subsets, rather than a single vertex subset, whereas it can also be used for obtaining a single vertex subset. For our model, we design an LP-based polynomial-time exact algorithm. Moreover, to handle large-scale networks, we also devise a simple, scalable preprocessing algorithm, which often reduces the size of the input networks significantly and results in a substantial speed-up. Computational experiments demonstrate the validity of our model and the effectiveness of our algorithms.
Supplemental Material
- A. Angel, N. Sarkas, N. Koudas, and D. Srivastava. 2012. Dense subgraph maintenance under streaming edge weight updates for real-time story identification. In Proceedings of VLDB. 574--585.Google Scholar
- T. Bacsar and G. J. Olsder. 1999. Dynamic Noncooperative Game Theory. Classics in Applied Mathematics, Vol. 23. SIAM.Google Scholar
- G. D. Bader and C. W. V. Hogue. 2003. An automated method for finding molecular complexes in large protein interaction networks. BMC Bioinformatics, Vol. 4, 1 (2003), 1--27.Google ScholarCross Ref
- O. D. Balalau, F. Bonchi, T-H. H. Chan, F. Gullo, and M. Sozio. 2015. Finding subgraphs with maximum total density and limited overlap. In Proceedings of WSDM. 379--388.Google Scholar
- P. Basaras, G. Iosifidis, D. Katsaros, and L. Tassiulas. 2019. Identifying influential spreaders in complex multilayer networks: A centrality perspective. IEEE Transactions on Network Science and Engineering, Vol. 6, 1 (2019), 31--45.Google ScholarCross Ref
- M. Bazzi, M. A. Porter, S. Williams, M. McDonald, D. J. Fenn, and S. D. Howison. 2016. Community detection in temporal multilayer networks, with an application to correlation networks. Multiscale Modeling & Simulation, Vol. 14, 1 (2016), 1--41.Google ScholarDigital Library
- S. Boccaletti, G. Bianconi, R. Criado, C.I. del Genio, J. Gómez-Gardeñes, M. Romance, I. Sendiña-Nadal, Z. Wang, and M. Zanin. 2014. The structure and dynamics of multilayer networks. Physics Reports, Vol. 544, 1 (2014), 1--122.Google ScholarCross Ref
- D. Boob, Y. Gao, R. Peng, S. Sawlani, C. E. Tsourakakis, D. Wang, and J. Wang. 2020. Flowless: Extracting densest subgraphs without flow computations. In Proceedings of The Web Conference 2020. 573--583.Google Scholar
- M. Charikar. 2000. Greedy approximation algorithms for finding dense components in a graph. In Proceedings of APPROX. 84--95.Google ScholarCross Ref
- M. Charikar, Y. Naamad, and J. Wu. 2018. On finding dense common subgraphs. arXiv preprint arXiv:1802.06361 (2018).Google Scholar
- C. Chekuri, K. Quanrud, and M. R. Torres. 2022. Densest subgraph: Supermodularity, iterative peeling, and flow. In Proceedings of SODA. 1531--1555.Google Scholar
- F. Chung and L. Lu. 2002. The average distances in random graphs with given expected degrees. Internet Mathematics, Vol. 1 (2002), 15879--15882.Google Scholar
- C. De Bacco, E. A. Power, D. B. Larremore, and C. Moore. 2017. Community detection, link prediction, and layer interdependence in multilayer networks. Physical Review E, Vol. 95 (2017), 042317.Google ScholarCross Ref
- M. De Domenico, C. Granell, M. A. Porter, and A. Arenas. 2016. The physics of spreading processes in multilayer networks. Nature Physics, Vol. 12, 10 (2016), 901--906.Google ScholarCross Ref
- M. De Domenico, A. Solé-Ribalta, E. Cozzo, M. Kivel"a, Y. Moreno, M. A. Porter, S. Gómez, and A. Arenas. 2013. Mathematical formulation of multilayer networks. Physical Review X, Vol. 3 (2013), 041022.Google ScholarCross Ref
- M. De Domenico, A. Solé-Ribalta, E. Omodei, S. Gómez, and A. Arenas. 2015. Ranking in interconnected multilayer networks reveals versatile nodes. Nature Communications, Vol. 6 (2015), 6868.Google ScholarCross Ref
- Y. Dourisboure, F. Geraci, and M. Pellegrini. 2007. Extraction and classification of dense communities in the web. In Proceedings of WWW. 461--470.Google Scholar
- J. A. Firth and B. C. Sheldon. 2015. Experimental manipulation of avian social structure reveals segregation is carried over across contexts. Proceedings of the Royal Society B: Biological Sciences, Vol. 282, 1802 (2015), 20142350.Google ScholarCross Ref
- K. J. Friston. 2011. Functional and effective connectivity: A review. Brain Connectivity, Vol. 1, 1 (2011), 13--36.Google ScholarCross Ref
- E. Galimberti, F. Bonchi, and F. Gullo. 2017. Core decomposition and densest subgraph in multilayer networks. In Proceedings of CIKM. 1807--1816.Google Scholar
- D. Gibson, R. Kumar, and A. Tomkins. 2005. Discovering large dense subgraphs in massive graphs. In Proceedings of VLDB. 721--732.Google Scholar
- A. Gionis and C. E. Tsourakakis. 2015. Dense subgraph discovery: KDD 2015 Tutorial. In Proceedings of KDD. 2313--2314.Google Scholar
- A. V. Goldberg. 1984. Finding a maximum density subgraph. Technical Report. University of California Berkeley.Google Scholar
- F. Hashemi, A. Behrouz, and L. V.S. Lakshmanan. 2022. FirmCore decomposition of multilayer networks. In Proceedings of The Web Conference. 1589--1600.Google ScholarDigital Library
- R. Interdonato, A. Tagarelli, D. Ienco, A. Sallaberry, and P. Poncelet. 2017. Local community detection in multilayer networks. Data Mining and Knowledge Discovery, Vol. 31, 5 (2017), 1444--1479.Google ScholarDigital Library
- M. Jalili, Y. Orouskhani, M. Asgari, N. Alipourfard, and M. Perc. 2017. Link prediction in multiplex online social networks. Royal Society Open Science, Vol. 4, 2 (2017), 160863.Google ScholarCross Ref
- V. Jethava and N. Beerenwinkel. 2015. Finding dense subgraphs in relational graphs. In Proceedings of ECML PKDD. 641--654.Google Scholar
- Y. Kawase, Y. Kuroki, and A. Miyauchi. 2019. Graph mining meets crowdsourcing: Extracting experts for answer aggregation. In Proceedings of IJCAI. 1272--1279.Google Scholar
- Y. Kawase and A. Miyauchi. 2018. The densest subgraph problem with a convex/concave size function. Algorithmica, Vol. 80, 12 (2018), 3461--3480.Google ScholarDigital Library
- M. Kivelä, A. Arenas, M. Barthelemy, J. P. Gleeson, Y. Moreno, and M. A. Porter. 2014. Multilayer networks. Journal of Complex Networks, Vol. 2, 3 (2014), 203--271.Google ScholarCross Ref
- G. Kortsarz and D. Peleg. 1994. Generating sparse 2-spanners. Journal of Algorithms, Vol. 17, 2 (1994), 222--236.Google ScholarDigital Library
- V. E. Lee, N. Ruan, R. Jin, and C. Aggarwal. 2010. A survey of algorithms for dense subgraph discovery. 303--336.Google Scholar
- J. Leskovec, J. Kleinberg, and C. Faloutsos. 2005. Graphs over time: Densification laws, shrinking diameters and possible explanations. In Proceedings of KDD. 177--187.Google Scholar
- A. Miyauchi and N. Kakimura. 2018. Finding a dense subgraph with sparse cut. In Proceedings of CIKM. 547--556.Google Scholar
- A. Miyauchi and A. Takeda. 2018. Robust densest subgraph discovery. In Proceedings of ICDM. 1188--1193.Google Scholar
- E. Omodei, M. De Domenico, and A. Arenas. 2015. Characterizing interactions in online social networks during exceptional events. Frontiers in Physics, Vol. 3 (2015), 59.Google ScholarCross Ref
- M. Salehi, R. Sharma, M. Marzolla, M. Magnani, P. Siyari, and D. Montesi. 2015. Spreading processes in multilayer networks. IEEE Transactions on Network Science and Engineering, Vol. 2, 2 (2015), 65--83.Google ScholarCross Ref
- K. Semertzidis, E. Pitoura, E. Terzi, and P. Tsaparas. 2019. Finding lasting dense subgraphs. Data Mining and Knowledge Discovery, Vol. 33, 5 (2019), 1417--1445.Google ScholarDigital Library
- A. Tagarelli, A. Amelio, and F. Gullo. 2017. Ensemble-based community detection in multilayer networks. Data Mining and Knowledge Discovery, Vol. 31, 5 (2017), 1506--1543.Google ScholarDigital Library
- C. E. Tsourakakis. 2015. The k-clique densest subgraph problem. In Proceedings of WWW. 1122--1132.Google ScholarDigital Library
- C. E. Tsourakakis, T. Chen, N. Kakimura, and J. Pachocki. 2019. Novel dense subgraph discovery primitives: Risk aversion and exclusion queries. In Proceedings of ECML PKDD. 378--394.Google Scholar
- R. J. Vanderbei. 2020. Linear Programming: Foundations and Extensions. International Series in Operations Research & Management Science, Vol. 196. Springer.Google ScholarCross Ref
- N. Veldt, A. R. Benson, and J. Kleinberg. 2021. The generalized mean densest subgraph problem. In Proceedings of KDD. 1604--1614.Google Scholar
- Z. Zou. 2013. Polynomial-time algorithm for finding densest subgraphs in uncertain graphs. In Proceedings of MLG. No page numbers.Google Scholar
Index Terms
- Stochastic Solutions for Dense Subgraph Discovery in Multilayer Networks
Recommendations
Peeling Bipartite Networks for Dense Subgraph Discovery
WSDM '18: Proceedings of the Eleventh ACM International Conference on Web Search and Data MiningFinding dense bipartite subgraphs and detecting the relations among them is an important problem for affiliation networks that arise in a range of domains, such as social network analysis, word-document clustering, the science of science, internet ...
Near-optimal fully dynamic densest subgraph
STOC 2020: Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of ComputingWe give the first fully dynamic algorithm which maintains a (1−є)-approximate densest subgraph in worst-case time poly(logn, є−1) per update. Dense subgraph discovery is an important primitive for many real-world applications such as community detection,...
Finding densest k-connected subgraphs
AbstractDense subgraph discovery is an important graph-mining primitive with a variety of real-world applications. One of the most well-studied optimization problems for dense subgraph discovery is the densest subgraph problem, where given an ...
Comments