Abstract
The Relevance Index has been introduced to detect key features of the organisation of complex dynamical systems. It is based upon Shannon entropies and can be used to identify groups of variables that change in a coordinated fashion, while they are less integrated with the rest of the system. In previous work, we have shown that the average Relevance Index attains its maximum at the phase transition in both Ising model and random Boolean networks. In this contribution we present a further study on the Ising model, showing that the relevance index is maximised for large groups of variables at criticality. These results provide further evidence to the hypothesis that this index is a powerful measure for capturing criticality.
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Notes
- 1.
See Roli et al. (to appear) for a detailed review on the subject.
References
Binney, J., Dowrick, N., Fisher, A., & Newman, M. (1992). The theory of critical phenomena. New York: Oxford University.
Brush, S. (1967). History of the Lenz-Ising model. Reviews of Modern Physics, 39(4), 883–893.
Christensen, K., & Moloney, R. (2005). Complexity and criticality. London: Imperial College.
Filisetti, A., Villani, M., Roli, A., Fiorucci, M., & Serra, R. (2015). Exploring the organisation of complex systems through the dynamical interactions among their relevant subsets. In P. Andrews, et al. (Eds.), Proceedings of the European Conference on Artificial Life 2015 (ECAL 2015) (pp. 286–293). Cambridge: The MIT.
Onsager, L. (1944). Crystal statistics. I. A two-dimensional model with an order-disorder transition. Physical Review, 65(3 and 4), 117–149.
Roli, A., Villani, M., Caprari, R., & Serra, R. (2017). Identifying critical states through the relevance index. Entropy, 19(2), 73.
Roli, A., Villani, M., Filisetti, A., & Serra, R. (2016). Beyond networks: Search for relevant subsets in complex systems. In G. Minati, M. Abram, & E. Pessa (Eds.), Towards a post-bertalanffy systemics (pp. 127–134). Cham: Springer.
Roli, A., Villani, M., Filisetti, A., & Serra, R. (to appear). Dynamical criticality: Overview and open questions. Journal of Systems Science and Complexity (A preliminary version of the paper is available as arXiv:1512.05259v2)
Solé, R. (2011). Phase transitions. Princeton: Princeton University.
Stanley, H. E. (1971). Introduction to phase transitions and critical phenomena. New York: Oxford University.
Tononi, G., McIntosh, A., Russel, D., & Edelman, G. (1998). Functional clustering: Identifying strongly interactive brain regions in neuroimaging data. Neuroimage, 7, 133–149.
Villani, M., Filisetti, A., Graudenzi, A., Damiani, C., Carletti, T., & Serra, R. (2014). Growth and division in a dynamic protocell model. Life, 4, 837–864.
Villani, M., Roli, A., Filisetti, A., Fiorucci, M., Poli, I., & Serra, R. (2015). The search for candidate relevant subsets of variables in complex systems. Artificial Life, 21(4), 412–431.
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Roli, A., Villani, M., Serra, R. (2019). A View of Criticality in the Ising Model Through the Relevance Index. In: Minati, G., Abram, M., Pessa, E. (eds) Systemics of Incompleteness and Quasi-Systems. Contemporary Systems Thinking. Springer, Cham. https://doi.org/10.1007/978-3-030-15277-2_12
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