Annals of Operations Research

, Volume 249, Issue 1–2, pp 55–73

Detecting large risk-averse 2-clubs in graphs with random edge failures

  • Foad Mahdavi Pajouh
  • Esmaeel Moradi
  • Balabhaskar Balasundaram
Pardalos60
  • 169 Downloads

Abstract

Detecting large 2-clubs in biological, social and financial networks can help reveal important information about the structure of the underlying systems. In large-scale networks that are error-prone, the uncertainty associated with the existence of an edge between two vertices can be modeled by assigning a failure probability to that edge. Here, we study the problem of detecting large “risk-averse” 2-clubs in graphs subject to probabilistic edge failures. To achieve risk aversion, we first model the loss in 2-club property due to probabilistic edge failures as a function of the decision (chosen 2-club cluster) and randomness (graph structure). Then, we utilize the conditional value-at-risk (CVaR) of the loss for a given decision as a quantitative measure of risk for that decision, which is bounded in the model. More precisely, the problem is modeled as a CVaR-constrained single-stage stochastic program. The main contribution of this article is a new Benders decomposition algorithm that outperforms an existing decomposition approach on a test-bed of randomly generated instances, and real-life biological and social networks.

Keywords

2-club Graph-based data mining Conditional value-at-risk Benders decomposition 

References

  1. Ahmed, S. (2006). Convexity and decomposition of mean-risk stochastic programs. Mathematical Programming, 106(3), 433–446.CrossRefGoogle Scholar
  2. Andersson, F., Mausser, H., Rosen, D., & Uryasev, S. (2001). Credit risk optimization with conditional value-at-risk criterion. Mathematical Programming, 89(2), 273–291.CrossRefGoogle Scholar
  3. Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9(3), 203–228.CrossRefGoogle Scholar
  4. Balasundaram, B., & Pajouh, F. M. (2013). Graph theoretic clique relaxations and applications. In P. M. Pardalos, D. Z. Du, & R. Graham (Eds.), Handbook of combinatorial optimization (2nd ed., pp. 1559–1598). New York: Springer.CrossRefGoogle Scholar
  5. Balasundaram, B., Butenko, S., & Trukhanov, S. (2005). Novel approaches for analyzing biological networks. Journal of Combinatorial Optimization, 10(1), 23–39.CrossRefGoogle Scholar
  6. Benders, J. F. (1962). Partitioning procedures for solving mixed-variables programming problems. Numerische Mathematik, 4, 238–252.CrossRefGoogle Scholar
  7. Boginski, V., Butenko, S., & Pardalos, P. (2006). Mining market data: A network approach. Computers & Operations Research, 33(11), 3171–3184.CrossRefGoogle Scholar
  8. Bourjolly, J. M., Laporte, G., & Pesant, G. (2002). An exact algorithm for the maximum \(k\)-club problem in an undirected graph. European Journal Of Operational Research, 138, 21–28.CrossRefGoogle Scholar
  9. Center for Complex Networks Research (2007). Network databases. http://www3.nd.edu/~networks/resources.htm. Accessed Dec 2014.
  10. Chung, F., & Lu, L. (2006). Complex graphs and networks. CBMS lecture series. Providence: American Mathematical Society.CrossRefGoogle Scholar
  11. Cook, D. J., & Holder, L. B. (2000). Graph-based data mining. IEEE Intelligent Systems, 15(2), 32–41.CrossRefGoogle Scholar
  12. Fábián, C. I. (2008). Handling CVaR objectives and constraints in two-stage stochastic models. European Journal of Operational Research, 191(3), 888–911.CrossRefGoogle Scholar
  13. Faghih-Roohi, S., Ong, Y. S., Asian, S., & Zhang, A. N. (2015). Dynamic conditional value-at-risk model for routing and scheduling of hazardous material transportation networks. Annals of Operations Research,. doi:10.1007/s10479-015-1909-2.Google Scholar
  14. Grossman, J., Ion, P., & Castro, R.D. (1995). The Erdös number project. Online: http://www.oakland.edu/enp/. Accessed Dec 2014.
  15. Haneveld, W., & van der Vlerk, M. (2006). Integrated chance constraints: Reduced forms and an algorithm. Computational Management Science, 3(4), 245–269.CrossRefGoogle Scholar
  16. Huang, P., & Subramanian, D. (2012). Iterative estimation maximization for stochastic linear programs with conditional value-at-risk constraints. Computational Management Science, 9(4), 441–458.CrossRefGoogle Scholar
  17. Jeong, H., Mason, S. P., Barabási, A. L., & Oltvai, Z. N. (2001). Centrality and lethality of protein networks. Nature, 411, 41–42.CrossRefGoogle Scholar
  18. Kammerdiner, A., Sprintson, A., Pasiliao, E., & Boginski, V. L. (2012). Optimization of discrete broadcast under uncertainty using conditional value-at-risk. Optimization Letters, 8(1), 45–59. doi:10.1007/s11590-012-0542-0.CrossRefGoogle Scholar
  19. KEGG BRITE Database (2014). Biomolecular relations in information transmission and expression. http://www.genome.jp/kegg/brite.html. Accessed Dec 2014.
  20. Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of Risk, 4, 43–68.CrossRefGoogle Scholar
  21. Künzi-Bay, A., & Mayer, J. (2006). Computational aspects of minimizing conditional value-at-risk. Computational Management Science, 3(1), 3–27.CrossRefGoogle Scholar
  22. Lim, C., Sherali, H. D., & Uryasev, S. (2010). Portfolio optimization by minimizing conditional value-at-risk via nondifferentiable optimization. Computational Optimization and Applications, 46(3), 391–415.CrossRefGoogle Scholar
  23. Luce, R. D. (1950). Connectivity and generalized cliques in sociometric group structure. Psychometrika, 15(2), 169–190.CrossRefGoogle Scholar
  24. Ma, J., Pajouh, F. M., Balasundaram, B., & Boginski, V. (2016). The minimum spanning \(k\)-core problem with bounded CVaR under probabilistic edge failures. INFORMS Journal on Computing, 28(2), 295–307.CrossRefGoogle Scholar
  25. Mansini, R., Ogryczak, W., & Speranza, M. G. (2006). Conditional value at risk and related linear programming models for portfolio optimization. Annals of Operations Research, 152(1), 227–256. doi:10.1007/s10479-006-0142-4.CrossRefGoogle Scholar
  26. Moazeni, S., Powell, W. B., & Hajimiragha, A. H. (2015). Mean-conditional value-at-risk optimal energy storage operation in the presence of transaction costs. IEEE Transactions on Power Systems, 30(3), 1222–1232. doi:10.1109/TPWRS.2014.2341642.CrossRefGoogle Scholar
  27. Mokken, R. J. (1979). Cliques, clubs and clans. Quality and Quantity, 13(2), 161–173.CrossRefGoogle Scholar
  28. Pajouh, F. M., & Balasundaram, B. (2012). On inclusionwise maximal and maximum cardinality \(k\)-clubs in graphs. Discrete Optimization, 9(2), 84–97.CrossRefGoogle Scholar
  29. Pattillo, J., Youssef, N., & Butenko, S. (2013). On clique relaxation models in network analysis. European Journal of Operational Research, 226(1), 9–18.CrossRefGoogle Scholar
  30. Pavlikov, K., & Uryasev, S. (2014). CVaR norm and applications in optimization. Optimization Letters, 8(7), 1999–2020. doi:10.1007/s11590-013-0713-7.CrossRefGoogle Scholar
  31. Quaranta, A. G., & Zaffaroni, A. (2008). Robust optimization of conditional value at risk and portfolio selection. Journal of Banking & Finance, 32(10), 2046–2056.CrossRefGoogle Scholar
  32. Rain, J.C., Selig, L., Reuse, H.D., Battaglia, V., Reverdy, C., Simon, S., Lenzen, G., Petel, F., Wojcik, J., Schachter, V., Chemama, Y., Labigne, A., & Legrain, P. (2004). The protein-protein interaction map of helicobacter pylori. Nature 409(6817):211–215, erratum in: Nature 409(6820):553 and 409(6821):743, 2001.Google Scholar
  33. Rockafellar, R., & Uryasev, S. (2000). Optimization of conditional value-at-risk. The Journal of Risk, 2(3), 21–41.CrossRefGoogle Scholar
  34. Rockafellar, R., & Uryasev, S. (2002). Conditional value-at-risk for general loss distributions. Journal of Banking & Finance, 26(7), 1443–1471.CrossRefGoogle Scholar
  35. Schultz, R., & Tiedemann, S. (2006). Conditional value-at-risk in stochastic programs with mixed-integer recourse. Mathematical Programming, 105(2–3), 365–386.CrossRefGoogle Scholar
  36. Soleimani, H., & Govindan, K. (2014). Reverse logistics network design and planning utilizing conditional value at risk. European Journal of Operational Research, 237(2), 487–497. doi:10.1016/j.ejor.2014.02.030, http://www.sciencedirect.com/science/article/pii/S0377221714001635.
  37. Spirin, V., & Mirny, L. A. (2003). Protein complexes and functional modules in molecular networks. Proceedings of the National Academy of Sciences, 100(21), 12,123–12,128.CrossRefGoogle Scholar
  38. Subramanian, D., & Huang, P. (2009). An efficient decomposition algorithm for static, stochastic, linear and mixed-integer linear programs with conditional value-at-risk constraints. Tech. Rep. RC24752, IBM Research Report.Google Scholar
  39. Uryasev, S. (2000). Conditional value-at-risk: optimization algorithms and applications. In: Computational Intelligence for Financial Engineering, 2000. (CIFEr) Proceedings of the IEEE/IAFE/INFORMS 2000 Conference on, IEEE, pp. 49–57.Google Scholar
  40. Van Slyke, R., & Wets, R. (1969). L-shaped linear programs with applications to control and stochastic programming. SIAM Journal on Applied Mathematics, 17, 638–663.CrossRefGoogle Scholar
  41. Yezerska, O., Butenko, S., & Boginski, V. L. (2016). Detecting robust cliques in graphs subject to uncertain edge failures. Annals of Operations Research,. doi:10.1007/s10479-016-2161-0.Google Scholar
  42. Zheng, Q. P., Wang, J., & Liu, A. L. (2015). Stochastic optimization for unit commitment-a review. IEEE Transactions on Power Systems, 30(4), 1913–1924.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Management Science and Information Systems DepartmentUniversity of Massachusetts BostonBostonUSA
  2. 2.School of Industrial Engineering and ManagementOklahoma State UniversityStillwaterUSA
  3. 3.Lee Scott Logistics ComplexBentonvilleUSA

Personalised recommendations