Thermodynamics of spin systems on small-world hypergraphs

Bollé, Désiré; Heylen, Rob; Skantzos, N. S.

doi:10.1103/physreve.74.056111

Cited by 13 publications

(23 citation statements)

References 19 publications

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“…SADDLE-POINT EQUATIONS As usual we replicate the partition function in order to calculate the free energy. Analytically, the calculation is analogous to the one found in [11]:…”

Section: The Modelmentioning

confidence: 85%

Small-world hypergraphs on a bond-disordered Bethe lattice

Bollé

Heylen

2008

Phys. Rev. E

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We study the thermodynamic properties of spin systems with bond-disorder on small-world hypergraphs, obtained by superimposing a one-dimensional Ising chain onto a random Bethe graph with p-spin interactions. Using transfer-matrix techniques, we derive fixed-point equations describing the relevant order parameters and the free energy, both in the replica symmetric and one step replica symmetry breaking approximation. We determine the static and dynamic ferromagnetic transition and the spinglass transition within replica symmetry for all temperatures, and demonstrate corrections to these results when one step replica symmetry breaking is taken into account. The results obtained are in agreement with Monte-Carlo simulations.

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“…SADDLE-POINT EQUATIONS As usual we replicate the partition function in order to calculate the free energy. Analytically, the calculation is analogous to the one found in [11]:…”

Section: The Modelmentioning

confidence: 85%

Small-world hypergraphs on a bond-disordered Bethe lattice

Bollé

Heylen

2008

Phys. Rev. E

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“…where Θ 0 (η c ) and ρ(η c ) are calculated using Eqs. (15) and (19), respectively. Therefore, the transition is hybrid with the exponent β = 1/2.…”

Section: Order Parametermentioning

confidence: 99%

“…where G(Θ) is the self-consistency function of the infinite system provided in Eq. (15). The solution of G N (Θ) = 0 yields the density of infected nodes in finite systems.…”

Section: Correlation Sizementioning

confidence: 99%

Simplicial SIS model in scale-free uniform hypergraph

Jhun¹,

Jo²,

Kahng³

2019

J. Stat. Mech.

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The hypergraph offers a platform to study structural properties emerging from more complicated and higher-order than pairwise interactions among constituents and dynamical behavior such as the spread of information or disease. Recently, a simplicial contagion problem was introduced and considered using a simplicial susceptible-infected-susceptible (SIS) model. Although recent studies have investigated random hypergraphs with a Poisson-type facet degree distribution, hypergraphs in the real world can have a power-law type of facet degree distribution. Here, we consider the SIS contagion problem on scale-free uniform hypergraphs and find that a continuous or hybrid epidemic transition occurs when the hub effect is dominant or weak, respectively. We determine the critical exponents analytically and numerically. We discuss the underlying mechanism of the hybrid epidemic transition.

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“…Explicitly, by applying Eqs. (123), (9) and (121), the transformations (127) and (128) become, respectively…”

Section: B Random Models Defined On Unconstrained Random Graphsmentioning

confidence: 99%

Effective field theory for models defined over small-world networks: First- and second-order phase transitions

Ostilli

Mendes

2008

Phys. Rev. E

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We present an effective field theory to analyze, in a very general way, models defined over small-world networks. Even if the exactness of the method is limited to the paramagnetic regions and to some special limits, it provides, yielding a clear and immediate (also in terms of calculation) physical insight, the exact critical behavior and the exact critical surfaces and percolation thresholds. The underlying structure of the nonrandom part of the model-i.e., the set of spins filling up a given lattice L0 of dimension d_{0} and interacting through a fixed coupling J0 -is exactly taken into account. When J_{0}> or = 0 , the small-world effect gives rise, as is known, to a second-order phase transition that takes place independently of the dimension d_{0} and of the added random connectivity c . When J0<0 , a different and novel scenario emerges in which, besides a spin-glass transition, multiple first- and second-order phase transitions may take place. As immediate analytical applications we analyze the Viana-Bray model (d_{0}=0) , the one-dimensional chain (d_{0}=1) , and the spherical model for arbitrary d_{0} .

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Thermodynamics of spin systems on small-world hypergraphs

Cited by 13 publications

References 19 publications

Small-world hypergraphs on a bond-disordered Bethe lattice

Small-world hypergraphs on a bond-disordered Bethe lattice

Simplicial SIS model in scale-free uniform hypergraph

Effective field theory for models defined over small-world networks: First- and second-order phase transitions

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