Simplicial complexes and complex systems

Salnikov, Vsevolod; Cassese, Daniele; Lambiotte, Renaud

doi:10.1088/1361-6404/aae790

Cited by 158 publications

(120 citation statements)

References 81 publications

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“…(iii) Case γ ∈ (3,4) In this case we consider the asymptotic expansion of Q (T n ) and H(T n ) for ∆p < 0 and ∆T n < 0 with |∆T n | 1 and |∆p| 1 given by…”

Section: Case With Power-law Distribution Qm With Power-law Exponent γmentioning

confidence: 99%

Renormalization group for link percolation on planar hyperbolic manifolds

2019

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Network geometry is currently a topic of growing scientific interest, as it opens the possibility to explore and interpret the interplay between structure and dynamics of complex networks using geometrical arguments. However, the field is still in its infancy. In this work we investigate the role of network geometry in determining the nature of the percolation transition in planar hyperbolic manifolds. In Ref.[1], S. Boettcher, V. Singh, R. M. Ziff have shown that a special type of twodimensional hyperbolic manifolds, the Farey graphs, display a discontinuous transition for ordinary link percolation. Here we investigate using the renormalization group the critical properties of link percolation on a wider class of two-dimensional hyperbolic deterministic and random manifolds constituting the skeletons of two-dimensional cell complexes. These hyperbolic manifolds are built iteratively by subsequently gluing m-polygons to single edges. We show that when the size m of the polygons is drawn from a distribution qm with asymptotic power-law scaling qm Cm −γ for m 1, different universality classes can be observed depending on the value of the power-law exponent γ. Interestingly, the percolation transition is hybrid for γ ∈ (3, 4) and becomes continuous for γ ∈ (2, 3].

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“…(iii) Case γ ∈ (3,4) In this case we consider the asymptotic expansion of Q (T n ) and H(T n ) for ∆p < 0 and ∆T n < 0 with |∆T n | 1 and |∆p| 1 given by…”

Section: Case With Power-law Distribution Qm With Power-law Exponent γmentioning

confidence: 99%

Renormalization group for link percolation on planar hyperbolic manifolds

2019

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“…However, with few exceptions (including [6][7][8]), little attention has been paid to the synchronization dynamics of coupled oscillator systems where interactions are not pair-wise, but rather n-way, with n ≥ 3. Such interactions are called "simplicial", where an n-simplex represents an interaction between n + 1 units, so 2-simplices describe three-way interactions, etc [9]. Recent advances suggest that simplicial interactions may be vital in general oscillator systems [10][11][12] and may play an important role in brain dynamics [13][14][15] and other complex systems phenomena such as, the dynamics of collaborations [16] or social contagion [17].…”

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confidence: 99%

Abrupt Desynchronization and Extensive Multistability in Globally Coupled Oscillator Simplexes

2019

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Collective behavior in large ensembles of dynamical units with non-pairwise interactions may play an important role in several systems ranging from brain function to social networks. Despite recent work pointing to simplicial structure, i.e., higher-order interactions between three or more units at a time, their dynamical characteristics remain poorly understood. Here we present an analysis of the collective dynamics of such a simplicial system, namely coupled phase oscillators with three-way interactions. The simplicial structure gives rise to a number of novel phenomena, most notably a continuum of abrupt desynchronization transitions with no abrupt synchronization transition counterpart, as well as extensive multistability whereby infinitely many stable partially synchronized states exist. Our analysis sheds light on the complexity that can arise in physical systems with simplicial interactions like the human brain and the role that simplicial interactions play in storing information.PACS numbers: 05.45.Xt, 89.75.Hc Research into the macroscopic dynamics of large ensembles of coupled oscillators have extended our understanding of natural and engineered systems ranging from cell cycles to power grids [1][2][3][4][5]. However, with few exceptions (including [6-8]), little attention has been paid to the synchronization dynamics of coupled oscillator systems where interactions are not pair-wise, but rather n-way, with n ≥ 3. Such interactions are called "simplicial", where an n-simplex represents an interaction between n + 1 units, so 2-simplices describe three-way interactions, etc [9]. Recent advances suggest that simplicial interactions may be vital in general oscillator systems [10-12] and may play an important role in brain dynamics [13][14][15] and other complex systems phenomena such as, the dynamics of collaborations [16] or social contagion [17]. In particular, interactions in 2-simplices (named holes or cavities) are important because they can describe correlations in neuronal spiking activity (that can be mapped to phase oscillators [18]) in the brain [19] providing a missing link between structure and function. In fact, coupled oscillator systems that display clustering and multi-branch entrainment have been shown to be useful models for memory and information storage [20][21][22][23][24][25][26]. Despite these findings, the general collective dynamics of coupled oscillator simplices and their utility in storing information are poorly understood.In this work we study large coupled oscillator simplicial complexes, considering the impact of 2-simplices, i.e., threeway interactions, on collective behavior. Specifically, we consider the 2-and 1-simplex multilayer system given bẏThe dynamics in the θ-layer are the natural generalization of the classical Kuramoto model [27] with 2-simplex interactions (namely, coupling is sinusoidal and diffusive), where θ i represents the phase of oscillator i with i = 1, . . . , N , ω i is its natural frequency which is assumed to be drawn from the distribution g(ω), an...

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“…In particular, our understanding of both natural and man-made systems has significantly improved by studying how network structures and dynamical processes combined shape the overall systems' behavior. Recently, the network science community has turned its attention to network geometry [6][7][8][9] to better represent the kinds of interactions that one can find beyond typical pairwise interactions.…”

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Abrupt phase transition of epidemic spreading in simplicial complexes

Matamalas

Gómez

Arenas

2020

Phys. Rev. Research

138

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Recent studies on network geometry, a way of describing network structures as geometrical objects, are revolutionizing our way to understand dynamical processes on networked systems. Here, we cope with the problem of epidemic spreading, using the Susceptible-Infected-Susceptible (SIS) model, in simplicial complexes. In particular, we analyze the dynamics of SIS in complex networks characterized by pairwise interactions (links), and three-body interactions (filled triangles, also known as 2-simplices). This higher-order description of the epidemic spreading is analytically formulated using a microscopic Markov chain approximation. The analysis of the fixed point solutions of the model, reveal an interesting phase transition that becomes abrupt with the infectivity parameter of the 2-simplices. Our results are of outmost importance for network theorists to advance in our physical understanding of epidemic spreading in real scenarios where diseases are transmitted among groups as well as among pairs, and to better understand the behavior of dynamical processes in simplicial complexes.The collective behavior of dynamical systems on networks, has been a major subject of research in the physics community during the last decades [1][2][3][4][5]. In particular, our understanding of both natural and man-made systems has significantly improved by studying how network structures and dynamical processes combined shape the overall systems' behavior. Recently, the network science community has turned its attention to network geometry [6-9] to better represent the kinds of interactions that one can find beyond typical pairwise interactions.These higher-order interactions are encoded in geometrical structures that describe the different kinds of simplex structure present in the network: a filled clique of m + 1 nodes is known as an m-simplex, and together a set of 1-simplexes (links), 2-simplexes (filled triangles), etc., comprise the simplicial complex. While simplicial complexes have been proven to be very useful for the analysis and computation in high dimensional data sets, e.g., using persistent homologies [10][11][12][13][14], little is understood about their role in shaping dynamical processes, save for a handful of examples [15][16][17][18].A more accurate description of dynamical processes on complex systems necessarily requires a new paradigm where the network structure representation helps to include higher-order interactions [19]. Simplicial geometry of complex networks is a natural way to extend manybody interactions in complex systems. The standard approach so far consists in understanding the coexistence of two-body (link) interactions and three-body interactions (filled triangles). Note that this approach is different from considering pairwise interactions among three elements of a triangle, it refers to the interaction of the three elements, in the filled triangle, at unison.Here we present a probabilistic formalization of the higher-order interactions of an epidemic process, represented by the well-known Suscep...

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Simplicial complexes and complex systems

Cited by 158 publications

References 81 publications

Renormalization group for link percolation on planar hyperbolic manifolds

Renormalization group for link percolation on planar hyperbolic manifolds

Abrupt Desynchronization and Extensive Multistability in Globally Coupled Oscillator Simplexes

Abrupt phase transition of epidemic spreading in simplicial complexes

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