Explosive Higher-Order Kuramoto Dynamics on Simplicial Complexes

Millán, Ana P.; Torres, Joaquı́n J.; Bianconi, Ginestra

doi:10.1103/physrevlett.124.218301

Cited by 195 publications

(129 citation statements)

References 40 publications

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“…Yet, for all these studies, analytical insights are limited to all-to-all coupling settings, disregarding the rich architecture of interactions of real-world systems. Interestingly, a different type of model was introduced in [43], where a Kuramoto phase oscillator is associated with each simplex.…”

Section: Introductionmentioning

confidence: 99%

Multiorder Laplacian for synchronization in higher-order networks

Lucas

Cencetti

Battiston³

2020

Phys. Rev. Research

123

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The emergence of synchronization in systems of coupled agents is a pivotal phenomenon in physics, biology, computer science, and neuroscience. Traditionally, interaction systems have been described as networks, where links encode information only on the pairwise influences among the nodes. Yet, in many systems, interactions among the units take place in larger groups. Recent work has shown that the presence of higher-order interactions between oscillators can significantly affect the emerging dynamics. However, these early studies have mostly considered interactions up to four oscillators at time, and analytical treatments are limited to the all-to-all setting. Here, we propose a general framework that allows us to effectively study populations of oscillators where higher-order interactions of all possible orders are considered, for any complex topology described by arbitrary hypergraphs, and for general coupling functions. To this end, we introduce a multiorder Laplacian whose spectrum determines the stability of the synchronized solution. Our framework is validated on three structures of interactions of increasing complexity. First, we study a population with all-to-all interactions at all orders, for which we can derive in a full analytical manner the Lyapunov exponents of the system, and for which we investigate the effect of including attractive and repulsive interactions. Second, we apply the multiorder Laplacian framework to synchronization on a synthetic model with heterogeneous higher-order interactions. Finally, we compare the dynamics of coupled oscillators with higher-order and pairwise couplings only, for a real dataset describing the macaque brain connectome, highlighting the importance of faithfully representing the complexity of interactions in real-world systems. Taken together, our multiorder Laplacian allows us to obtain a complete analytical characterization of the stability of synchrony in arbitrary higher-order networks, paving the way toward a general treatment of dynamical processes beyond pairwise interactions.

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Section: Introductionmentioning

confidence: 99%

Multiorder Laplacian for synchronization in higher-order networks

Lucas

Cencetti

Battiston³

2020

Phys. Rev. Research

123

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“…This field is undergoing a rapid expansion thanks to its rooting in the powerful languages of homological algebra and category theory, which provide strong formal foundations, as well as to the wide variety of applications it found, that span material science 25,26 , biology and chemistry [27][28][29][30][31][32][33] , sensor networks 34 , cosmology 35 , medicine and neuroscience [36][37][38][39][40][41][42][43][44] , manufacturing and engineering [45][46][47] , social sciences 13,48 , and network science itself 17,[49][50][51][52][53][54] .…”

Section: Homological Scaffold Via Minimal Homology Bases Marco Guerramentioning

confidence: 99%

Homological scaffold via minimal homology bases

Guerra

Gregorio

Fugacci

et al. 2021

Sci Rep

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The homological scaffold leverages persistent homology to construct a topologically sound summary of a weighted network. However, its crucial dependency on the choice of representative cycles hinders the ability to trace back global features onto individual network components, unless one provides a principled way to make such a choice. In this paper, we apply recent advances in the computation of minimal homology bases to introduce a quasi-canonical version of the scaffold, called minimal, and employ it to analyze data both real and in silico. At the same time, we verify that, statistically, the standard scaffold is a good proxy of the minimal one for sufficiently complex networks.

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“…One can also calculate the numbers of zero eigenvalues of higher-order Hodge-Laplacian matrices, so as to find the Betti numbers. To do so, it needs to follow some algebraic topology rules to form oriented cliques 16 . As an example, consider the network shown in Figure 2, which is a subnetwork of the one shown in Figure 1, with the node-edge boundary matrix B 1 of rank as follows, where the shaded row is linearly dependent on the others:…”

Section: Computing Betti Numbersmentioning

confidence: 99%

“…In addition, the advance from pairwise interactions to higher-order interactions in dynamics of complex systems requires the knowledge of higher-order cliques and cavities of networks 15 . The numbers of zero eigenvalues of higher-order Hodge-Laplacian matrices are equal to the corresponding Betti numbers, while their associate eigenvectors are closely related to higher-order cavities 16 .…”

Section: Introductionmentioning

confidence: 99%

Computing Cliques and Cavities in Networks

Shi

Chen

Sun

et al. 2021

Preprint

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Complex networks have complete subgraphs such as nodes, edges, triangles, etc., referred to as cliques of different orders. Notably, cavities consisting of higher-order cliques have been found playing an important role in brain functions. Since searching for the maximum clique in a large network is an NP-complete problem, we propose using k-core decomposition to determine the computability of a given network subject to limited computing resources. For a computable network, we design a search algorithm for finding cliques of different orders, which also provides the Euler characteristic number. Then, we compute the Betti number by using the ranks of the boundary matrices of adjacent cliques. Furthermore, we design an optimized algorithm for finding cavities of different orders. Finally, we apply the algorithm to the neuronal network of C. elegans in one dataset, and find all of its cliques and some cavities of different orders therein, providing a basis for further mathematical analysis and computation of the structure and function of the C. elegans neuronal network.

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Explosive Higher-Order Kuramoto Dynamics on Simplicial Complexes

Cited by 195 publications

References 40 publications

Multiorder Laplacian for synchronization in higher-order networks

Multiorder Laplacian for synchronization in higher-order networks

Homological scaffold via minimal homology bases

Computing Cliques and Cavities in Networks

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