ACDC Seminar by Riccardo Muolo on the 6th of January 2027.

On the 6th of January at 16.00, Riccardo Muolo will give an Amsterdam Dynamics Seminar. We will go for drinks at Bar Boele after the talk.

The talk will take place in the Maryam seminar room (9A-46).

Title: Dynamics beyond nodes: from topology to synchronization and Turing patterns in higher-order networks

Abstract:
In recent years, increasing attention has been given to dynamical processes taking place on higher-order networks, where interactions are not limited to links, but may involve also higher-dimensional simplices [1]. While classical network models assume that state variables live on nodes and interact through links, many real systems — including brain, climate, and transportation systems — cannot be fully described within this node-centric perspective [2]. In this talk, I will introduce the framework of higher-order networks and the concept of topological signals, namely, dynamical variables defined on simplices of higher dimensions. I will briefly present the basic tools required for this setting, including elementary notions of discrete calculus, discrete topology and geometric algebra, which serve as the mathematical foundation for modeling dynamical processes beyond the node-based paradigm.Next, I will discuss models of oscillatory dynamics and reaction-diffusion extended to this framework. First, I will present the topological Kuramoto model [2,3], in which phases are not restricted to nodes but may also be associated with links, and where the coupling arises from the combinatorial structure of the simplicial complex. Then, I will introduce the discrete Hodge Laplacian and the Dirac operator [4], the former generalizing diffusive interactions to the higher-order setting, while the latter provides cross-talk between signals defined on simplices of different dimensions. Finally, I will introduce reaction-diffusion equations and Turing pattern formation [5], discussing the extension to this framework [6].

References

[1] Bianconi G., Higher‑Order Networks: An Introduction to Simplicial Complexes. Elements in the Structure and Dynamics of Complex Networks, Cambridge University Press, 2021.

[2] Millán A.P., Sun H., Giambagli L., Muolo R., Carletti T., Torres J.J., Radicchi F., Kurths J., Bianconi G., Topology shapes dynamics of higher-order networks. Nat. Phys., 21: 353–361, 2025.

[3] Millán A.P., Torres J.J., Bianconi G., Explosive Higher-Order Kuramoto Dynamics on Simplicial Complexes. Phys. Rev. Lett., 124(21): 218301, 2020.

[4] Bianconi G., The topological Dirac equation of networks and simplicial complexes. J. Phys. Complex., 2(3): 035022, 2021.

[5] Muolo R., Giambagli L., Nakao H., Fanelli D., Carletti T., Turing patterns on discrete topologies: from networks to higher-order structures, Proc. Royal Soc. A, 480, 20240235, 2024.

[6] Giambagli L., Calmon L., Muolo R., Carletti T., Bianconi G., Diffusion-driven instability of topological signals coupled by the Dirac operator, Phys. Rev. E, 106(6), 2022.

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