Simplicial complexes: higher-order spectral dimension and dynamics

Torres, Joaquı́n J.; Bianconi, Ginestra

doi:10.1088/2632-072x/ab82f5

Cited by 81 publications

(82 citation statements)

References 47 publications

(99 reference statements)

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“…A stream of research has recently focused on correctly characterizing the structure of systems with higher-order interactions [14][15][16][17][18][19][20]. Interestingly, considering this additional level of complexity sometimes leads to changes in the emerging dynamics of complex systems, including social contagions [21,22], activity-driven models [23], diffusion [24,25], random walks [26,27], and evolutionary games [28].…”

Section: Introductionmentioning

confidence: 99%

Multiorder Laplacian for synchronization in higher-order networks

Lucas

Cencetti

Battiston³

2020

Phys. Rev. Research

138

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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

138

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“…Additionally, this framework generates simplicial networks whose underlying network structure displays all the statistical properties of complex networks including scale-free degree distribution, high clustering coefficient, small-world diameter and significant community structure. The resulting simplicial complexes can reveal distinct geometrical features including a finite spectral dimension [32,33]. The spectral dimension [34] characterizes the slow relaxation of diffusion processes to their equilibrium steady-state distribution, similarly to what happens for finite-dimensional Euclidean networks.…”

Section: A Geometrical Approach To Higher-order Networkmentioning

confidence: 99%

Higher-Order Networks

Bianconi

2021

Self Cite

145

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Higher-order networks describe the many-body interactions of a large variety of complex systems, ranging from the the brain to collaboration networks. Simplicial complexes are generalized network structures which allow us to capture the combinatorial properties, the topology and the geometry of higher-order networks. Having been used extensively in quantum gravity to describe discrete or discretized space-time, simplicial complexes have only recently started becoming the representation of choice for capturing the underlying network topology and geometry of complex systems. This Element provides an in-depth introduction to the very hot topic of network theory, covering a wide range of subjects ranging from emergent hyperbolic geometry and topological data analysis to higher-order dynamics. This Elements aims to demonstrate that simplicial complexes provide a very general mathematical framework to reveal how higher-order dynamics depends on simplicial network topology and geometry.

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“…For example, simplicial complexes have been used to represent neural recordings [87,54], classify images [205,56,67], and describe the mesoscale architecture of brain networks [203,204,169,193,161]. Even more recent work has focused on defining generative models to construct simplicial complexes with given topological features [52,51] and investigating dynamics that could take place upon nodes or higher-dimensional simplices [211,132].…”

Section: Simplicial Complexesmentioning

confidence: 99%

The Why, How, and When of Representations for Complex Systems

Torres¹,

Blevins²,

Bassett³

et al. 2021

SIAM Rev.

180

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Complex systems, composed at the most basic level of units and their interactions, describe phenomena in a wide variety of domains, from neuroscience to computer science and economics. The wide variety of applications has resulted in two key challenges: the generation of many domain-specific strategies for complex systems analyses that are seldom revisited, and the compartmentalization of representation and analysis ideas within a domain due to inconsistency in complex systems language. In this work we propose basic, domain-agnostic language in order to advance toward a more cohesive vocabulary. We use this language to evaluate each step of the complex systems analysis pipeline, beginning with the system under study and data collected, then moving through different mathematical frameworks for encoding the observed data (i.e., graphs, simplicial complexes, and hypergraphs), and relevant computational methods for each framework. At each step we consider different types of dependencies; these are properties of the system that describe how the existence of an interaction among a set of units in a system may affect the possibility of the existence of another relation. We discuss how dependencies may arise and how they may alter the interpretation of results or the entirety of the analysis pipeline. We close with two real-world examples using coauthorship data and email communications data that illustrate how the system under study, the dependencies therein, the research question, and the choice of mathematical representation influence the results. We hope this work can serve as an opportunity for reflection for experienced complex systems scientists, as well as an introductory resource for new researchers.

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Simplicial complexes: higher-order spectral dimension and dynamics

Cited by 81 publications

References 47 publications

Multiorder Laplacian for synchronization in higher-order networks

Multiorder Laplacian for synchronization in higher-order networks

Higher-Order Networks

The Why, How, and When of Representations for Complex Systems

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