Finding key players in complex networks through deep reinforcement learning


Finding an optimal set of nodes, called key players, whose activation (or removal) would maximally enhance (or degrade) a certain network functionality, is a fundamental class of problems in network science. Potential applications include network immunization, epidemic control, drug design and viral marketing. Due to their general NP-hard nature, these problems typically cannot be solved by exact algorithms with polynomial time complexity. Many approximate and heuristic strategies have been proposed to deal with specific application scenarios. Yet, we still lack a unified framework to efficiently solve this class of problems. Here, we introduce a deep reinforcement learning framework FINDER, which can be trained purely on small synthetic networks generated by toy models and then applied to a wide spectrum of application scenarios. Extensive experiments under various problem settings demonstrate that FINDER significantly outperforms existing methods in terms of solution quality. Moreover, it is several orders of magnitude faster than existing methods for large networks. The presented framework opens up a new direction of using deep learning techniques to understand the organizing principle of complex networks, which enables us to design more robust networks against both attacks and failures.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Fig. 1: Finding key players in a network.
Fig. 2: The process of finding key players in a network using FINDER.
Fig. 3: Overview of the FINDER framework.
Fig. 4: Performance of FINDER on synthetic graphs.
Fig. 5: Performance of FINDER on real-world networks.
Fig. 6: Cost distributions of key players identified by FINDER.

Data availability

All the data analysed in this paper, including synthetic graphs and real-world networks, can be accessed through our Code Ocean compute capsule (

Code availability

All source codes and models (including those that can reproduce all figures and tables analysed in this work) are publicly available through our Code Ocean compute capsule ( or on GitHub (


  1. 1.

    Albert, R. & Barabási, A.-L. Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47 (2002).

    MathSciNet  Article  Google Scholar 

  2. 2.

    Newman, M. E. The structure and function of complex networks. SIAM Rev. 45, 167–256 (2003).

    MathSciNet  Article  Google Scholar 

  3. 3.

    Morone, F. & Makse, H. A. Influence maximization in complex networks through optimal percolation. Nature 524, 65–68 (2015).

    Article  Google Scholar 

  4. 4.

    Kempe, D., Kleinberg, J. & Tardos, É. Influential nodes in a diffusion model for social networks. In International Colloquium on Automata, Languages and Programming 1127–1138 (Springer, 2005).

  5. 5.

    Corley, H. & David, Y. S. Most vital links and nodes in weighted networks. Oper. Res. Lett. 1, 157–160 (1982).

    MathSciNet  Article  Google Scholar 

  6. 6.

    Borgatti, S. P. Identifying sets of key players in a social network. Comput. Math. Org. Theory 12, 21–34 (2006).

    Article  Google Scholar 

  7. 7.

    Lalou, M., Tahraoui, M. A. & Kheddouci, H. The critical node detection problem in networks: a survey. Comput. Sci. Rev. 28, 92–117 (2018).

    MathSciNet  Article  Google Scholar 

  8. 8.

    Arulselvan, A., Commander, C. W., Elefteriadou, L. & Pardalos, P. M. Detecting critical nodes in sparse graphs. Comput. Oper. Res. 36, 2193–2200 (2009).

    MathSciNet  Article  Google Scholar 

  9. 9.

    Kuntz, I. D. Structure-based strategies for drug design and discovery. Science 257, 1078–1082 (1992).

    Article  Google Scholar 

  10. 10.

    Vitoriano, B., Ortuño, M. T., Tirado, G. & Montero, J. A multi-criteria optimization model for humanitarian aid distribution. J. Global Optim. 51, 189–208 (2011).

    MathSciNet  Article  Google Scholar 

  11. 11.

    Kempe, D., Kleinberg, J. & Tardos, É. Maximizing the spread of influence through a social network. In Proceedings of the 9th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining 137–146 (ACM, 2003).

  12. 12.

    Pastor-Satorras, R. & Vespignani, A. Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86, 3200 (2001).

    Article  Google Scholar 

  13. 13.

    Cohen, R., Erez, K., Ben-Avraham, D. & Havlin, S. Breakdown of the internet under intentional attack. Phys. Rev. Lett. 86, 3682 (2001).

    Article  Google Scholar 

  14. 14.

    Braunstein, A., Dall’Asta, L., Semerjian, G. & Zdeborová, L. Network dismantling. Proc. Natl Acad. Sci. USA 113, 12368–12373 (2016).

    Article  Google Scholar 

  15. 15.

    Shen, Y., Nguyen, N. P., Xuan, Y. & Thai, M. T. On the discovery of critical links and nodes for assessing network vulnerability. IEEE/ACM Trans. Netw. 21, 963–973 (2013).

    Article  Google Scholar 

  16. 16.

    Mugisha, S. & Zhou, H.-J. Identifying optimal targets of network attack by belief propagation. Phys. Rev. E 94, 012305 (2016).

    Article  Google Scholar 

  17. 17.

    Zdeborová, L., Zhang, P. & Zhou, H.-J. Fast and simple decycling and dismantling of networks. Sci. Rep. 6, 37954 (2016).

    Article  Google Scholar 

  18. 18.

    Ren, X.-L., Gleinig, N., Helbing, D. & Antulov-Fantulin, N. Generalized network dismantling. Proc. Natl Acad. Sci. USA 116, 6554–6559 (2019).

    MathSciNet  Article  Google Scholar 

  19. 19.

    Khalil, E., Dai, H., Zhang, Y., Dilkina, B. & Song, L. Learning combinatorial optimization algorithms over graphs. In Advances in Neural Information Processing Systems 6348–6358 (NIPS, 2017).

  20. 20.

    Nazari, M., Oroojlooy, A., Snyder, L. & Takác, M. Reinforcement learning for solving the vehicle routing problem. In Advances in Neural Information Processing Systems 9839–9849 (NIPS, 2018).

  21. 21.

    Bello, I., Pham, H., Le, Q. V., Norouzi, M. & Bengio, S. Neural combinatorial optimization with reinforcement learning. Preprint at (2016).

  22. 22.

    Bengio, Y., Lodi, A. & Prouvost, A. Machine learning for combinatorial optimization: a methodological tour d’horizon. Preprint at (2018).

  23. 23.

    James, J., Yu, W. & Gu, J. Online vehicle routing with neural combinatorial optimization and deep reinforcement learning. In IEEE Transactions on Intelligent Transportation Systems 1–12 (IEEE, 2019).

  24. 24.

    Li, Z., Chen, Q. & Koltun, V. Combinatorial optimization with graph convolutional networks and guided tree search. In Advances in Neural Information Processing Systems 539–548 (NIPS, 2018).

  25. 25.

    Hamilton, W., Ying, Z. & Leskovec, J. Inductive representation learning on large graphs. In Advances in Neural Information Processing Systems 1024–1034 (NIPS, 2017).

  26. 26.

    Brown, N. & Sandholm, T. Superhuman AI for heads-up no-limit poker: Libratus beats top professionals. Science 359, 418–424 (2018).

    MathSciNet  Article  Google Scholar 

  27. 27.

    Silver, D. et al. Mastering the game of go without human knowledge. Nature 550, 354–359 (2017).

    Article  Google Scholar 

  28. 28.

    Moravčík, M. et al. Deepstack: expert-level artificial intelligence in heads-up no-limit poker. Science 356, 508–513 (2017).

    MathSciNet  Article  Google Scholar 

  29. 29.

    Schneider, C. M., Moreira, A. A., Andrade, J. S., Havlin, S. & Herrmann, H. J. Mitigation of malicious attacks on networks. Proc. Natl Acad. Sci. USA 108, 3838–3841 (2011).

    Article  Google Scholar 

  30. 30.

    Henderson, K. et al. Rolx: structural role extraction & mining in large graphs. In Proceedings of the 18th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining 1231–1239 (ACM, 2012).

  31. 31.

    Kipf, T. N. & Welling, M. Semi-supervised classification with graph convolutional networks. In Proceedings of the International Conference on Learning Representations (ICLR, 2017).

  32. 32.

    Lü, L., Zhang, Y.-C., Yeung, C. H. & Zhou, T. Leaders in social networks, the delicious case. PLoS ONE 6, e21202 (2011).

    Article  Google Scholar 

  33. 33.

    Wang, D., Cui, P. & Zhu, W. Structural deep network embedding. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining 1225–1234 (ACM, 2016).

  34. 34.

    Erdös, P. & Rényi, A. On random graphs. Publ. Math. Debrecen 6, 290–297 (1959).

    MathSciNet  MATH  Google Scholar 

  35. 35.

    Watts, D. J. & Strogatz, S. H. Collective dynamics of ‘small-world’ networks. Nature 393, 440–442 (1998).

    Article  Google Scholar 

  36. 36.

    Barabási, A.-L. & Albert, R. Emergence of scaling in random networks. Science 286, 509–512 (1999).

    MathSciNet  Article  Google Scholar 

  37. 37.

    Barabási, A.-L. Network Science (Cambridge Univ. Press, 2016).

  38. 38.

    Clusella, P., Grassberger, P., Pérez-Reche, F. J. & Politi, A. Immunization and targeted destruction of networks using explosive percolation. Phys. Rev. Lett. 117, 208301 (2016).

    Article  Google Scholar 

  39. 39.

    Rossi, R. A. & Ahmed, N. K. The network data repository with interactive graph analytics and visualization. In Proceedings of 29th AAAI Conference on Artificial Intelligence 4292–4293 (ACM, 2015).

Download references


We are grateful to M. Chen and Z. Liu for the feedback and assistance they provided during the development and preparation of this research. This work is partially supported by NSF III-1705169, NSF CAREER Award 1741634, NSF 1937599, an Okawa Foundation Grant and an Amazon Research Award. C.F. is supported by the CSC Scholarship offered by the China Scholarship Council. Y.-Y.L. is supported by grants from the John Templeton Foundation (award no. 51977) and National Institutes of Health (R01AI141529, R01HD093761, UH3OD023268, U19AI095219 and U01HL089856).

Author information




Y.S. and Y.-Y.L. designed and managed the project. Y.S. and C.F. developed the FINDER algorithm. C.F. and L.Z. performed all the calculations. All authors analysed the results. C.F., Y.-Y.L. and Y.S. wrote the manuscript. All authors edited the manuscript.

Corresponding authors

Correspondence to Yizhou Sun or Yang-Yu Liu.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Finding key players in complex networks through deep reinforcement learnin.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Fan, C., Zeng, L., Sun, Y. et al. Finding key players in complex networks through deep reinforcement learning. Nat Mach Intell 2, 317–324 (2020).

Download citation

Further reading


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing