The spectral dimension of simplicial complexes: a renormalization group theory

Bianconi, Ginestra; Dorogovstev, Sergey N.

doi:10.1088/1742-5468/ab5d0e

Cited by 29 publications

(35 citation statements)

References 58 publications

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“…This result generalizes the RG equations that were found in Ref. [28] and can be derived using a similar procedure. The results derived in Ref.…”

Section: The Integralsupporting

confidence: 87%

“…As was the case in Ref. [28], where the spectrum of the 0-Laplacian was derived using the RG flow, the parameters p and µ are renormalized differently for faces of different type when we study the spectrum of the m-dimensional up-Laplacian. The partition function Z n (ω) corresponding to the Gaussian model of the simplicial complex evolved up to generation n is a function of the parameters ω = ({µ }, {p }), and can be expressed as…”

Section: The General Rg Approachmentioning

confidence: 99%

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The higher-order spectrum of simplicial complexes: a renormalization group approach

Reitz¹,

Bianconi²

2020

J. Phys. A: Math. Theor.

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Network topology is a flourishing interdisciplinary subject that is relevant for different disciplines including quantum gravity and brain research. The discrete topological objects that are investigated in network topology are simplicial complexes. Simplicial complexes generalize networks by not only taking pairwise interactions into account, but also taking into account many-body interactions between more than two nodes. Higher-order Laplacians are topological operators that describe higher-order diffusion on simplicial complexes and constitute the natural mathematical objects that capture the interplay between network topology and dynamics. We show that higher-order up and down Laplacians can have a finite spectral dimension, characterizing the long time behaviour of the diffusion process on simplicial complexes that depends on their order m. We provide a renormalization group theory for the calculation of the higher-order spectral dimension of two deterministic models of simplicial complexes: the Apollonian and the pseudo-fractal simplicial complexes. We show that the RG flow is affected by the fixed point at zero mass, which determines the higher-order spectral dimension d S of the up-Laplacians of order m with m ⩾ 0.

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“…This result generalizes the RG equations that were found in Ref. [28] and can be derived using a similar procedure. The results derived in Ref.…”

Section: The Integralsupporting

confidence: 87%

Section: The General Rg Approachmentioning

confidence: 99%

The higher-order spectrum of simplicial complexes: a renormalization group approach

Reitz¹,

Bianconi²

2020

J. Phys. A: Math. Theor.

Self Cite

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“…For example, to give new insights over the finite size effects [58][59][60], critical phenomena [61,62], and randomlink matching problems on random regular graphs [63]. On the other hand we can use the DZFM on multiplex networks to investigate critical phenomena and collective behavior [64], and finally use our formalism to enlarge the set of statistical field theory toolbox that is been currently used for simplicial complex [65,66]. These issues are under investigation by the authors.…”

Section: Discussionmentioning

confidence: 99%

The Sherrington-Kirkpatrick model for spin glasses: A new approach for the solution

Rodríguez-Camargo,

Mojica-Nava,

Svaiter

2021

Preprint

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We discuss the Sherrington-Kirkpatrick mean-field version of a spin glass within the distributional zeta-function method (DZFM). In the DZFM, since the dominant contribution to the average free energy is written as a series of moments of the partition function of the model, the spin-glass multivalley structure is obtained. Also, an exact expression for the saddle points corresponding to each valley and a global critical temperature showing the existence of many stables or at least metastables equilibrium states is presented. Near the critical point we obtain analytical expressions of the order parameters that are in agreement with phenomenological results. We evaluate the linear and nonlinear susceptibility and we find the expected singular behavior at the spin-glass critical temperature. Furthermore, we obtain a positive definite expression for the entropy and we show that ground-state entropy tends to zero as the temperature goes to zero. We show that our solution is stable for each term in the expansion. Finally, we analyze the behavior of the overlap distribution, where we find a general expression for each moment of the partition function.

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“…Although the original definition of these models is in a given dimension, i. Both simplicial complex models display finite spectral dimensions of the graph Laplacian [155] and of higher-order up-Laplacians [88].…”

Section: A2 Canonical Ensembles Of Simplicial Complexes With a Given Sequence Of Generalized Degree Of The Nodesmentioning

confidence: 99%

Higher-Order Networks

Bianconi

2021

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143

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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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The spectral dimension of simplicial complexes: a renormalization group theory

Cited by 29 publications

References 58 publications

The higher-order spectrum of simplicial complexes: a renormalization group approach

The higher-order spectrum of simplicial complexes: a renormalization group approach

The Sherrington-Kirkpatrick model for spin glasses: A new approach for the solution

Higher-Order Networks

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