Replicated transfer matrix analysis of Ising spin models on  small world  lattices

Nikoletopoulos, T; Coolen, Anthony C. C.; Castillo, Isaac Pérez; Skantzos, N. S.; Hatchett, J. P. L.; Wemmenhove, B.

doi:10.1088/0305-4470/37/25/003

Cited by 59 publications

(125 citation statements)

References 31 publications

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“…The inclusion of such local interactions can completely change the functioning and the dynamics of * Electronic address: desire.bolle,rob.heylen,nikos.skantzos@fys.kuleuven.be such systems. It was shown in [7], e.g., that this construction significantly enlarges the region in parameter space where ferromagnetism occurs. In particular, for any choice of the value of the average connectivity, however small, the ferromagnetic-paramagnetic transition occurs at a finite temperature.…”

Section: Introductionmentioning

confidence: 99%

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Thermodynamics of spin systems on small-world hypergraphs

2006

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We study the thermodynamic properties of spin systems on small-world hypergraphs, obtained by superimposing sparse Poisson random graphs with p-spin interactions onto a one-dimensional Ising chain with nearest-neighbor interactions. We use replica-symmetric transfer-matrix techniques to derive a set of fixed-point equations describing the relevant order parameters and free energy, and solve them employing population dynamics. In the special case where the number of connections per site is of the order of the system size we are able to solve the model analytically. In the more general case where the number of connections is finite we determine the static and dynamic ferromagneticparamagnetic transitions using population dynamics. The results are tested against Monte-Carlo simulations.

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Section: Introductionmentioning

confidence: 99%

“…on biophysical networks [2]- [4] or social networks [5] and, to a lesser extent, analytically [6,7]. For recent reviews see e.g.…”

Section: Introductionmentioning

confidence: 99%

Thermodynamics of spin systems on small-world hypergraphs

2006

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“…In particular due to the unexpectedly rich and varied range of multi-disciplinary applications of finite connectivity replica techniques which emerged subsequently in, for example, spin-glass modelling [6][7][8][9], error correcting codes [10][11][12][13], theoretical computer science [14][15][16][17], recurrent neural networks [18][19][20] and 'small-world' networks [21], this field is presently enjoying a renewed interest and popularity. Until very recently, analysis was limited to the equilibrium properties of such models, but now attention has also turned to the dynamics of finitely connected spin systems [22][23][24][25], using combinatorial and generating functional methods.…”

Section: Introductionmentioning

confidence: 99%

Finitely connected vector spin systems with random matrix interactions

Coolen¹,

Skantzos²,

Castillo³

et al. 2005

J. Phys. A: Math. Gen.

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We use finite connectivity equilibrium replica theory to solve models of finitely connected unit-length vectorial spins, with random pair-interactions which are of the orthogonal matrix type. Finitely connected spin models, although still of a mean-field nature, can be regarded as a convenient level of description in between fully connected and finite-dimensional ones. Since the spins are continuous and the connectivity c remains finite in the thermodynamic limit, the replica-symmetric order parameter is a functional. The general theory is developed for arbitrary values of the dimension d of the spins, and arbitrary choices of the ensemble of random orthogonal matrices. We calculate phase diagrams and the values of moments of the order parameter explicitly for d = 2 (finitely connected XY spins with random chiral interactions) and for d = 3 (finitely connected classical Heisenberg spins with random chiral interactions). Numerical simulations are shown to support our predictions quite satisfactorily.

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“…We point out however that our results are completely novel. Notice in fact that, without any intention to be exhaustive in citing the large literature on the subject, the state of the art of analytical methods for disordered Ising models defined over Poissonian small-world graphs results nowadays as follows: i) in the case of no short-range couplings, J 0 = 0, and for one community, n = 1, modulo a large use of some population dynamics algorithm for low temperatures, the replica method and the cavity method [25,29,30,31] have established the base to solve exactly the model in any region of the phase diagram, even rigorously in the SK case [32,33] and in unfrustrated cases [34]; ii) for J 0 = 0 and n = 1 these methods have been successfully applied to the one-dimensional case [35,36] but a generalization to higher dimensions (except infinite dimensions [37]) seems impossible due to the presence of loops of any length [53]; on the other hand, even if it is exact only in the P region, the method we have presented in the Ref. [20], modulo solving analytically or numerically a non random Ising model, can be exactly applied in any dimension, and more in general to any underlying pure graph (L 0 , Γ 0 ); iii) for J 0 = 0 and n ≥ 2, the problem was solved only in the limit of infinite connectivity: exactly in the n = 2 CW case in its general form, which includes arbitrary sizes of the two communities, but with no coupling disorder [23]; and, within the replica-symmetric solution, in the generic n SK case, but only in the presence of a same mutual interaction among the n communities of same size [38,39].…”

Section: Introductionmentioning

confidence: 99%

Communication and correlation among communities

Ostilli

Mendes

2009

Phys. Rev. E

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Given a network and a partition in communities, we consider the issues "how communities influence each other" and "when two given communities do communicate". Specifically, we address these questions in the context of small-world networks, where an arbitrary quenched graph is given and long range connections are randomly added. We prove that, among the communities, a superposition principle applies and gives rise to a natural generalization of the effective field theory already presented in [Phys. Rev. E 78, 031102] (n = 1), which here (n > 1) consists in a sort of effective TAP (Thouless, Anderson and Palmer) equations in which each community plays the role of a microscopic spin. The relative susceptibilities derived from these equations calculated at finite or zero temperature, where the method provides an effective percolation theory, give us the answers to the above issues. Unlike the case n = 1, asymmetries among the communities may lead, via the TAP-like structure of the equations, to many metastable states whose number, in the case of negative short-cuts among the communities, may grow exponentially fast with n. As examples we consider the n Viana-Bray communities model and the n one-dimensional small-world communities model. Despite being the simplest ones, the relevance of these models in network theory, as e.g. in social networks, is crucial and no analytic solution were known until now. Connections between percolation and the fractal dimension of a network are also discussed. Finally, as an inverse problem, we show how, from the relative susceptibilities, a natural and efficient method to detect the community structure of a generic network arises.For a short presentation of the main result see arXiv:0812.0608.

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Replicated transfer matrix analysis of Ising spin models on small world lattices

Cited by 59 publications

References 31 publications

Thermodynamics of spin systems on small-world hypergraphs

Thermodynamics of spin systems on small-world hypergraphs

Finitely connected vector spin systems with random matrix interactions

Communication and correlation among communities

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