Topological percolation on hyperbolic simplicial complexes

Bianconi, Ginestra; Ziff, Robert M.

doi:10.1103/physreve.98.052308

Cited by 58 publications

(62 citation statements)

References 66 publications

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“…Finally, it has been shown in Ref. [10] that on simplicial complexes of dimension d one can define up to 2d topological percolation problems that can display a critical behavior that cannot be predicted by exclusively studying node and link percolation problems on the same network geometries.…”

Section: Introductionmentioning

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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Section: Introductionmentioning

confidence: 99%

Renormalization group for link percolation on planar hyperbolic manifolds

2019

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“…Therefore, here we characterize the spectral properties of Complex Network Manifolds and study the effect of these properties on the entrained phase synchronization, which is known to display strong spatio-temporal fluctuations of the order parameter [25]. This phase, also called frustruated synchornization [43,44], has a very rich structure and can be interpreted as an extended critical region to be related to the smeared phase observed in critical phenomena on hyperbolic networks, such as percolation [51,52].…”

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Synchronization in network geometries with finite spectral dimension

Millán

Torres²,

Bianconi

2019

Phys. Rev. E

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Recently there is a surge of interest in network geometry and topology. Here we show that the spectral dimension plays a fundamental role in establishing a clear relation between the topological and geometrical properties of a network and its dynamics. Specifically we explore the role of the spectral dimension in determining the synchronization properties of the Kuramoto model. We show that the synchronized phase can only be thermodynamically stable for spectral dimensions above four and that phase entrainment of the oscillators can only be found for spectral dimensions greater than two. We numerically test our analytical predictions on the recently introduced model of network geometry called Complex Network Manifolds which displays a tunable spectral dimension.

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“…We consider link percolation on branching cell complexes where we remove each link independently with probability q = 1 − p. Since the random branching cell complexes that we consider are non-amenable structures [45], link percolation displays at least two percolation thresholds. In particular, as in hyperbolic manifolds [10,11,30], we distinguish between the lower p and the upper p c percolation thresholds leading to the identification of three distinct phases.…”

Section: Percolation In Non-amenable Network Structuresmentioning

confidence: 99%

Percolation on branching simplicial and cell complexes and its relation to interdependent percolation

2019

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Network geometry has strong effects on network dynamics. In particular, the underlying hyperbolic geometry of discrete manifolds has recently been shown to affect their critical percolation properties. Here we investigate the properties of link percolation in non-amenable two-dimensional branching simplicial and cell complexes. We establish the relation between the equations determining the percolation probability in random branching cell complexes and the equation for interdependent percolation in multiplex networks with inter-layer degree correlation equal to one. By using this relation we show that branching cell complexes can display more than two percolation phase transitions: the upper percolation transition, the lower percolation transition, and one or more intermediate phase transitions. At these additional transitions the percolation probability and the fractal exponent both feature a discontinuity. Furthermore, by using the renormalization group theory we show that the upper percolation transition can belong to various universality classes including the Berezinskii-Kosterlitz-Thouless (BKT) transition, the discontinuous percolation transition, and continuous transitions with anomalous singular behavior that generalize the BKT transition. arXiv:1908.09392v1 [cond-mat.dis-nn]

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Topological percolation on hyperbolic simplicial complexes

Cited by 58 publications

References 66 publications

Renormalization group for link percolation on planar hyperbolic manifolds

Renormalization group for link percolation on planar hyperbolic manifolds

Synchronization in network geometries with finite spectral dimension

Percolation on branching simplicial and cell complexes and its relation to interdependent percolation

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