Talk 1: Synchronization of topological signals defined on simplicial complexes
Talk 2: Turing patterns on adaptive networks
Talk 3: A robust method for classification of chimera states
Synchronization of topological signals defined on simplicial complexes
Timoteo Carletti, University of Namur, Belgium
15 April 2026, 3:30 PM
Synchronization is a widespread phenomenon at the root of several biological rhythms or humanmade technological systems [1,2]. Synchronization refers to the spontaneous ability of coupled oscillators to operate at unison and thus exhibit a coherent collective behavior. Global synchronization is the resulting phenomenon where all oscillators behave in the same way.
Traditionally synchronization has been studied when identical or nonidentical oscillators are defined on the nodes of a network and are coupled by the network links. However, to capture the function of many complex systems, e.g., brain networks, social networks and protein interaction networks, it is important to go beyond pairwise interactions and consider higher-order interactions [3].
Simplicial complexes can be used to encode higher-order interactions by modeling many-body coupling. They can also sustain topological signals [4,5-7], i.e., dynamical variables that can be defined not only on nodes but also on links, triangles, and higher-dimensional simplices. Examples of real topological signals are edge signals such as biological transportation fluxes or traffic signals [7], synaptic and brain edge signals [8]. Finally, topological signals are attracting increasing attention in signal processing and machine learning [7].
The goal of this lecture is to provide an introduction to simplicial complexes by focusing on their representation via incidence boundary matrices, allowing to define Hodge Laplace matrices and Dirac operator. We then introduce topological signals and their temporal evolution; in particular we will interested in synchronization and the role of the geometrical properties of the simplicial complex in the dynamics outcome.
References
[1] A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization (Cambridge University Press, Cambridge, England, 2001).
[2] A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou, Synchronization in complex networks, Phys. Rep. 469, 93 (2008).
[3] F. Battiston, E. Amico, A. Barrat, G. Bianconi, G. Ferraz de Arruda, B. Franceschiello, I. Iacopini, S. Kéfi, V. Latora, Y. Moreno et al., The physics of higher-order interactions in complex systems, Nat. Phys. 17, 1093 (2021).
[4] G. Bianconi, Higher-Order Networks: An Introduction to Simplicial Compelxes (Cambridge University Press, Cambridge, England, 2021).
[5] A. P. Millán, J. J. Torres, and G. Bianconi, Explosive Higher-Order Kuramoto Dynamics on Simplicial Complexes, Phys. Rev. Lett. 124, 218301 (2020).
[6] R. Ghorbanchian, J. G. Restrepo, J. J. Torres, and G. Bianconi, Higher-order simplicial synchronization of coupled topological signals, Commun. Phys. 4, 120 (2021).
[7] S. Sardellitti, S. Barbarossa, and L. Testa, Topological signal processing over cell complexes, in Proceedings of the 2021 55th Asilomar Conference on Signals, Systems, and Computers (IEEE, New York, 2021), pp. 1558.
[8] J. Faskowitz, R. F. Betzel, and O. Sporns, Edges in brain networks: Contributions to models of structure and function, Network Neurosci. 6, 1 (2022).
[9] T. Carletti, L. Giambagli, and G. Bianconi, Global Topological Synchronization on Simplicial and Cell Complexes, Phys. Rev. Lett., 130, 187401 (2023).
Turing patterns on adaptive networks
Timoteo Carletti, University of Namur, Belgium
16 April 2026, 3:30 PM
Self-organized phenomena are widespread in Nature and have been studied for long time in various domains, be it physics, chemistry, biology, ecology, neurophysiology, to name a few [1]. Despite the rich literature on the subject, there is still need for understanding, analyzing and predicting their emergence and behavior. Patterns are commonly based on local interaction rules that determine the creation and destruction of the basic entities – species – at spatial locations, upon which the action of a diffusion process determines the migration of the species. For this reason reaction-diffusion systems are a common framework for modeling such systems [2].
In a pioneer article, Turing considered a two-species model of morphogenesis [3]. For the first time, he established the conditions for a stable spatially homogeneous state, to migrate towards a new heterogeneous, spatially patched, equilibrium under the driving effect of diffusion, at odd with the idea that diffusion is a source of homogeneity. Nowadays, Turing instability goes beyond this initial framework and it can be used to explain emergence of self-organized collective patterns.
In several applications the underlying domain can be supposed to be divided into local patches where reactions occurs and diffusion across patches is realized via the links existing among the latter; this framework leads naturally to the introduction of reaction-diffusion systems defined on complex networks [4]. More recently Turing theory has been generalized on higher-order structures, by also considering time evolving networks, nevertheless in those cases the network dynamics is exogenous, namely links weights evolution is prescribed a priori, thus independently from the nodes dynamics. There are on the other hand several relevant phenomena where links weights follow an endogenous process, namely they adapt to the nodes dynamics. The prototype example comes from brain dynamics where synapses between neurons adjust their weight according to some function of neuronal activity.
In this talk, we investigate how the co-evolution of network topology and node dynamics lead to the spontaneous emergence of Turing patterns in adaptive systems, where, i.e., links weights evolve as a function of the nodes states according to some adaptive response function [5].
Our results are rooted on the study of the spectrum of the Laplace matrix associated to the network and allows to determine conditions for the emergence of Turing patterns depending on models parameters and on the network topology as well. The resulting dynamics is very rich, we have proven the existence of stationary, wave-like Turing patterns but also of “bursty” Turing patterns, where the system switches between a well defined pattern and an homogenous state. We have also shown the existence of parameters and adaptive response functions for which Turing patterns emerge, even once they could not in the counterpart of static network.
References
[1] G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems: From Dissipative Structures to Order through Fluctuations, Wiley (1977).
[2] P. Grindrod, Patterns and waves: The theory and applications of reaction-diffusion equations, Clarendon Press Oxford, (1991).
[3] A. Turing, The Chemical Basis of Morphogenesis, Phils Trans R Soc London Ser B, 237, (1952), 37.
[4] H. Nakao and A.S. Mikhailov, Turing patterns in network-organized activator-inhibitor systems, Nature Physics, 6, (2010), 544.
[5] M. Dorchain, S. N. Jenifer and Timoteo Carletti, Turing patterns on adaptive networks, arXiv:2509.10124, (2025).
A robust method for classification of chimera states
S. Nirmala Jenifer, University of Namur, Belgium
16 April 2026, 5:00 PM
Chimera states are one of the most intriguing phenomena in nonlinear dynamics, characterized by the coexistence of coherent and incoherent behavior in systems of coupled identical oscillators. Despite extensive studies and numerous observations in different settings, the development of reliable and systematic methods to classify chimera states and distinguish them from other dynamical patterns remains a challenging task. Existing approaches are often limited in scope and lack robustness. In this work, we propose a method based on Fourier analysis combined with statistical classification to characterize chimera behavior. The method is applied to a system of topological signals coupled via the Dirac operator, where it successfully captures the rich dynamical regimes exhibited by the model. We demonstrate that the proposed approach is robust with respect to variations in network topology and system parameters. Beyond the specific model considered, the framework provides a general and automated tool for distinguishing different dynamical regimes in complex systems.
Ref: arxiv.org/pdf/2603.22026
