Metastable configurations of small-world networks

Heylen, Rob; Skantzos, N. S.; Blanco, J. Busquets; Bollé, Désiré

doi:10.1103/physreve.73.016138

Cited by 6 publications

(12 citation statements)

References 35 publications

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“…In recent years, ever-increasing attention has been paid to the study of networks. 6,22 There are several questions which can be addressed when dealing with a random graph. 1 Beyond topology one can put variables (e.g., spins) on the nodes and impose rules for their interaction.…”

Section: Resultsmentioning

confidence: 99%

“…In the last few years, while an ever-increasing understanding of disordered statistical mechanics has been achievable -mainly due to the recent breakthrough in mathematical methods (see e.g., Refs. 8,9,[17][18][19][20][21] and mainly focused on the paradigmatic (fully connected) Sherrington-Kirkpatrick (SK) model, 14 -the theory of networks, in particular random, small world and scalefree networks, 6,[22][23][24] revealed surprisingly common features between apparently much different structures in Nature, turning to itself the attention of several scientists, such that, by merging these two fields of research (disordered statistical mechanics and network theory), our study on diluted spin glasses developed.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Driven Transitions at the Onset of Ergodicity Breaking in Gauge-Invariant Complex Networks

Barra

2010

Int. J. Mod. Phys. B

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In the last few years, the statistical mechanics of spin glasses has become me of the major frameworks for analyzing the macroscopical equilibrium properties of complex systems starting from the microscopical dynamics of their components. Recently, many advances in its rigorous formulation without the replica trick have been achieved, highlighting the importance of this field of research in our understanding of complex systems. In this framework we analyze the critical behavior of a Poissonian diluted network with random competitive interactions among gauge-invariant dichotomic variables pasted on the nodes (i.e., a suitable version of the Viana-Bray diluted spin glass). The model is described by an infinite series of order parameters (the multioverlaps) and has two degrees of freedom: the temperature (which can be thought of as the noise level) and the connectivity (the averaged number of links per node in the underlying network). In this paper, we show that there are not several transition lines, one for every order parameter, as a naive approach would suggest but just one corresponding to ergodicity breaking. We explain this scenario within a novel and simple mathematical technique: we show the existence of a driving mechanism such that, as the first order parameter (the two-replica overlap) becomes different from zero due to a real second order phase transition, it enforces all the other multioverlaps toward positive values thanks to the strong correlations which develop among themselves and the two-replica overlap at the critical line. These correlations are ultimately related within our framework to the breaking of the gauge invariance of the Boltzmann state at the boundary of the ergodic region. A discussion on the structure of the free energy, fundamental macroscopical observable by which the whole thermodynamic can be achieved, is also presented

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Driven Transitions at the Onset of Ergodicity Breaking in Gauge-Invariant Complex Networks

Barra

2010

Int. J. Mod. Phys. B

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“…In particular, for any choice of the value of the average connectivity, however small, the ferromagnetic-paramagnetic transition occurs at a finite temperature. Furthermore, a jump in the entropy of metastable configurations has been found [18] exactly at the crossover between the small-world and the Poisson random graph structure due to the formation of disconnected clusters within the graph.…”

Section: Introductionmentioning

confidence: 99%

Thermodynamics of spin systems on small-world hypergraphs

2006

Self Cite

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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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“…This low clustering can indicate a metastable structure. 48 Indeed, the structure of this sample changed slightly during the measurements, either due to the small movements of the objective or very slow sedimentation.…”

Section: Sample Characterisationmentioning

confidence: 97%