日時: 2026年6月15日 (月) 16:30-17:15, 17:30-18:15
場所: 京都教育大学 1A402教室 (Zoom同時配信)
講演者: Riccardo Muolo氏 (理化学研究所 数理創造研究センター/東京科学大学)
題目: An introduction to reaction-diffusion dynamics and pattern formation in discrete spaces
Pattern formation is a key feature of many natural and engineered systems, ranging from ecosystems to fluid dynamics and neural dynamics, just to mention a few. Turing instability provides one of the most famous theories for pattern formation in a continuous domain [1]. On networks, Turing patterns were extended by Nakao and Mikhailov [2], allowing to model systems where the topology is intrinsically discrete, and, dynamical variables (species) are defined on the nodes, i.e., they interact in the nodes and flow among nodes by using network links. Currently, growing interest is addressing problems related to the formation of Turing patterns of species located on the nodes of simple hyperbolic [3] and higher-order networks [4]. However, in a number of real systems, including the brain and the climate, dynamical variables are not only defined on nodes but also on links, triangles and higher-dimensional simplexes, leading to topological signals [5]. The discrete topological Dirac operator [6] is emerging as the key operator that allows cross-talk between signals defined on simplices of different dimensions, for instance among nodes and links signals of a network, which allows to further extend Turing theory [7,8]. In this seminar, I will introduce reaction-diffusion systems and Turing pattern formation on discrete spaces, from networks to more complex structures such as hypergraphs and simplicial complexes, mainly following a recent review on the topic [9]. In the first part, I will extend the PDEs framework of reaction diffusion to networks and discuss Turing pattern formation on such structures, including multi-layer, directed, and temporal ones. In the second part, I will introduce higher-order structures and discuss some recent extentions of Turing theory on that direction.
References
[1] Turing A., The chemical basis of morphogenesis. Proc. Royal Soc. A, 37(237), 1952.
[2] Nakao H. and Mikhailov A., Turing patterns in network-organized activator–inhibitor systems. Nat. Phys., 544(6), 2010.
[3] van der Kolk, J., García-Pérez, G., Kouvaris, N.E., Serrano, M.Á ., and Boguñá, M., Emergence of geometric Turing patterns in complex networks. Physical Review X, 13(2), p.021038, 2023.
[4] Muolo R., Gallo L., Latora V., Frasca M., and Carletti T., Turing patterns in systems with high-order interactions. Chaos Sol. Frac., 166(112912), 2023.
[5] Bianconi G., Higher-Order Networks: An introduction to simplicial complexes. Cambridge University Press, 2021.
[6] Bianconi, G. The topological Dirac equation of networks and simplicial complexes. Journal of Physics: Complexity, 2(3), p.035022, 2021.
[7] Giambagli, L., Calmon, L., Muolo, R., Carletti, T., and Bianconi, G., Diffusion-driven instability of topological signals coupled by the Dirac operator. Phys. Rev. E, 106:064314, 2022.
[8] Muolo R., Carletti T. ,and Bianconi G., The three way Dirac operator and dynamical Turing and Dirac induced patterns on nodes and links. Chaos Sol. Frac., 178(114312), 2024.
[9] R. Muolo, L. Giambagli, H. Nakao, D. Fanelli, and T. Carletti, Turing patterns on discrete topologies: from networks to higher-order structures, Proceedings of the Royal Society A 480, 20240235, 2024.