Influence Maximization in Ising Models

Authors Zongchen Chen, Elchanan Mossel

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Zongchen Chen
  • Department of Computer Science and Engineering, University at Buffalo, NY, USA
Elchanan Mossel
  • Department of Mathematics, MIT, Cambridge, MA, USA

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Zongchen Chen and Elchanan Mossel. Influence Maximization in Ising Models. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 30:1-30:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


Given a complex high-dimensional distribution over {± 1}ⁿ, what is the best way to increase the expected number of +1’s by controlling the values of only a small number of variables? Such a problem is known as influence maximization and has been widely studied in social networks, biology, and computer science. In this paper, we consider influence maximization on the Ising model which is a prototypical example of undirected graphical models and has wide applications in many real-world problems. We establish a sharp computational phase transition for influence maximization on sparse Ising models under a bounded budget: In the high-temperature regime, we give a linear-time algorithm for finding a small subset of variables and their values which achieve nearly optimal influence; In the low-temperature regime, we show that the influence maximization problem cannot be solved in polynomial time under commonly-believed complexity assumption. The critical temperature coincides with the tree uniqueness/non-uniqueness threshold for Ising models which is also a critical point for other computational problems including approximate sampling and counting.

Subject Classification

ACM Subject Classification
  • Theory of computation → Discrete optimization
  • Theory of computation → Randomness, geometry and discrete structures
  • Mathematics of computing → Markov networks
  • Mathematics of computing → Probabilistic inference problems
  • Influence maximization
  • Ising model
  • phase transition
  • correlation decay


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  1. Nima Anari, Kuikui Liu, and Shayan Oveis Gharan. Spectral independence in high-dimensional expanders and applications to the hardcore model. In Proceedings of the 61st Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 1319-1330, 2020. Google Scholar
  2. Kristine Eia S Antonio, Chrysline Margus N Pinol, and Ronald S Banzon. An ising model approach to malware epidemiology. arXiv preprint, 2010. URL:
  3. Alexander Barvinok. Combinatorics and Complexity of Partition Functions, volume 30. Springer Algorithms and Combinatorics, 2016. Google Scholar
  4. Christian Borgs, Jennifer Chayes, Jeff Kahn, and László Lovász. Left and right convergence of graphs with bounded degree. Random Structures & Algorithms, 42(1):1-28, 2013. Google Scholar
  5. Guy Bresler, Frederic Koehler, and Ankur Moitra. Learning restricted Boltzmann machines via influence maximization. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (STOC), pages 828-839, 2019. Google Scholar
  6. Zongchen Chen, Kuikui Liu, and Eric Vigoda. Rapid mixing of Glauber dynamics up to uniqueness via contraction. In Proceedings of the 61st Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 1307-1318, 2020. Google Scholar
  7. Byron Chin, Ankur Moitra, Elchanan Mossel, and Colin Sandon. The power of an adversary in Glauber dynamics. arXiv preprint, 2023. URL:
  8. Andreas Galanis, Daniel Štefankovič, and Eric Vigoda. Inapproximability of the partition function for the antiferromagnetic Ising and hard-core models. Combinatorics, Probability and Computing, 25(4):500-559, 2016. Google Scholar
  9. David Kempe, Jon Kleinberg, and Éva Tardos. Maximizing the spread of influence through a social network. In Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining, pages 137-146, 2003. Google Scholar
  10. David Kempe, Jon Kleinberg, and Éva Tardos. Influential nodes in a diffusion model for social networks. In Automata, Languages and Programming: 32nd International Colloquium, ICALP 2005, Lisbon, Portugal, July 11-15, 2005. Proceedings 32, pages 1127-1138. Springer, 2005. Google Scholar
  11. Cristina Gabriela Aguilar Lara, Eduardo Massad, Luis Fernandez Lopez, and Marcos Amaku. Analogy between the formulation of Ising-Glauber model and Si epidemiological model. Journal of Applied Mathematics and Physics, 7(05):1052, 2019. Google Scholar
  12. Liang Li, Pinyan Lu, and Yitong Yin. Correlation decay up to uniqueness in spin systems. In Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 67-84, 2013. Google Scholar
  13. Adam Lipowski. Ising model: Recent developments and exotic applications. Entropy, 24(12):1834, December 2022. Google Scholar
  14. Shihuan Liu, Lei Ying, and Srinivas Shakkottai. Influence maximization in social networks: An Ising-model-based approach. In Proceedings of the 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pages 570-576. IEEE, 2010. Google Scholar
  15. Jacek Majewski, Hao Li, and Jurg Ott. The Ising model in physics and statistical genetics. The American Journal of Human Genetics, 69(4):853-862, 2001. Google Scholar
  16. Andrea Montanari and Amin Saberi. The spread of innovations in social networks. Proceedings of the National Academy of Sciences, 107(47):20196-20201, 2010. Google Scholar
  17. Elchanan Mossel and Sébastien Roch. Submodularity of influence in social networks: From local to global. SIAM Journal on Computing, 39(6):2176-2188, 2010. Google Scholar
  18. Viresh Patel and Guus Regts. Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials. SIAM Journal on Computing, 46(6):1893-1919, 2017. Google Scholar
  19. Alistair Sinclair, Piyush Srivastava, and Marc Thurley. Approximation algorithms for two-state anti-ferromagnetic spin systems on bounded degree graphs. Journal of Statistical Physics, 155(4):666-686, 2014. Google Scholar
  20. Allan Sly. Computational transition at the uniqueness threshold. In Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 287-296, 2010. Google Scholar
  21. Allan Sly and Nike Sun. The computational hardness of counting in two-spin models on d-regular graphs. The Annals of Probability, 42(6):2383-2416, 2014. Google Scholar
  22. Dror Weitz. Counting independent sets up to the tree threshold. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing (STOC), pages 140-149, 2006. Google Scholar