Hysteresis in the Random-Field Ising Model and Bootstrap Percolation

Sabhapandit, Sanjib; Dhar, Deepak; Shukla, Prabodh

doi:10.1103/physrevlett.88.197202

Cited by 56 publications

(67 citation statements)

References 18 publications

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“…Since it has been known that the k-core percolation is related to a kinetically constrained model and a random-field Ising model [32,33,34,35,36], our work might be useful for theoretical analysis of such systems.…”

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confidence: 99%

Critical phenomena in globally coupled excitable elements

Ohta

Sasa

2008

Phys. Rev. E

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Critical phenomena in globally coupled excitable elements are studied by focusing on a saddlenode bifurcation at the collective level. Critical exponents that characterize divergent fluctuations of interspike intervals near the bifurcation are calculated theoretically. The calculated values appear to be in good agreement with those determined by numerical experiments. The relevance of our results to jamming transitions is also mentioned.PACS numbers: 82.40. Bj, 05.10.Gg, 89.75.Da,The understanding of cooperative phenomena in nonequilibrium systems is one of the most important problems in physics. In contrast to those in equilibrium systems, their nature essentially depends on the dynamical properties of the systems. This makes it difficult to perform a systematic study of the problem. Thus, it is necessary to investigate the typical cooperative phenomena in nonequilibrium systems.Recently, critical behaviors have been observed experimentally [1,2,3,4,5,6,7] and numerically [8,9,10,11,12,13,14,15] in typical examples of coupled excitable elements such as neural networks and cardiac tissues. In general, such critical behaviors are classified into several groups on the basis of the exponents of divergences. The classification enables the realization of a universality class for critical phenomena in coupled excitable elements. However, the broad distribution of the phenomena makes it difficult to elucidate the mechanism of the criticality. When we consider the role of the mean-field Ising model in theories of equilibrium statistical mechanics, it becomes apparent that it is necessary to develop an elegant method for theoretical analysis of a minimal model describing critical phenomena in coupled excitable elements.Thus, with this aim in mind, we analyze a previously proposed simple model for globally coupled excitable elements in this Letter [16]. In particular, we focus on a divergent behavior with respect to parameter change around a saddle-node bifurcation because the excitability of this model is related to the bifurcation. It should be noted that such transition properties have been studied for different excitable systems [2,8,10,11,14]. The main achievement of this Letter is a theoretical derivation of the critical exponents that characterize the singular behavior near a saddle-node bifurcation.Model: The excitable nature of a system is characterized by the existence of spikes in a time series. Mathematically speaking, spikes are described by trajectories near a homoclinic orbit in a differential equation. As a simple example, let us consider an ordinary differential equation ∂ t φ = ω−h sin φ for a phase variable φ ∈ [0, 2π], in which there exists the homoclinic orbit when ω = h. Then, when h is slightly larger than ω, a small perturbation for the fixed point φ * = sin −1 (ω/h) yields one spike. On the other hand, when h is slightly less than ω, the system shows an array of spikes with a long interspike interval. The qualitative change in the trajectories is an example of saddle-node bifurcation. By using thi...

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confidence: 99%

Critical phenomena in globally coupled excitable elements

Ohta

Sasa

2008

Phys. Rev. E

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“…Results on non tree-like graphs have been largely numerical [24,25], although some analytic results incorporating clustering have recently been obtained [26,27]. At the same time, bootstrap percolation has emerged as a useful model for a variety of applications such as neuronal activity [28][29][30], jamming and rigidity transitions and glassy dynamics [31,32], and magnetic systems [33]. In bootstrap percolation, a set of seed vertices is initially activated, and other vertices become active if they have k active neighbors.…”

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confidence: 99%

Heterogeneousk-core versus bootstrap percolation on complex networks

et al. 2011

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We introduce the heterogeneous-k-core, which generalizes the k-core, and contrast it with bootstrap percolation. Vertices have a threshold ki which may be different at each vertex. If a vertex has less than ki neighbors it is pruned from the network. The heterogeneous-k-core is the sub-graph remaining after no further vertices can be pruned. If the thresholds ki are 1 with probability f or k ≥ 3 with probability (1 − f ), the process forms one branch of an activation-pruning process which demonstrates hysteresis. The other branch is formed by ordinary bootstrap percolation. We show that there are two types of transitions in this heterogeneous-k-core process: the giant heterogeneousk-core may appear with a continuous transition and there may be a second, discontinuous, hybrid transition. We compare critical phenomena, critical clusters and avalanches at the heterogeneous-kcore and bootstrap percolation transitions. We also show that network structure has a crucial effect on these processes, with the giant heterogeneous-k-core appearing immediately at a finite value for any f > 0 when the degree distribution tends to a power law P (q) ∼ q −γ with γ < 3.PACS numbers: 64.60.aq, 64.60.ah, 05.70.Fh Bootstrap percolation and the k-core are closely related concepts, and in fact it is easy to confuse the two. Both belong to a new class of systems with hybrid phase transitions, yet it can be clearly shown that the two processes do not map onto each other. Here we elucidate the relationship and differences between these two concepts by introducing a generalization of the k-core, the heterogeneous-k-core.The k-core is the maximal sub-graph whose vertices all have internal degree at least k [1]. It has proved a useful tool giving insight into the deep structure of complex networks [2][3][4][5][6] , and has found applications in diverse areas, from rigidity [7] and jamming [8] transitions to real neural networks [9,10] and evolution [11] . The kcore has been extensively studied on tree-like networks, starting with Bethe lattices [12,13] and Random graphs [14][15][16], before finally being extended to arbitrary degree distributions [5,[17][18][19]. Hyperbolic lattices have also been considered [20]. Other studies, mostly numerical, have considered the sizes of culling avalanches [21][22][23]. Results on non tree-like graphs have been largely numerical [24,25], although some analytic results incorporating clustering have recently been obtained [26,27]. At the same time, bootstrap percolation has emerged as a useful model for a variety of applications such as neuronal activity [28][29][30], jamming and rigidity transitions and glassy dynamics [31,32], and magnetic systems [33]. In bootstrap percolation, a set of seed vertices is initially activated, and other vertices become active if they have k active neighbors. This process has been investigated on two and three dimensional lattices (see [34][35][36][37] and references therein). Bootstrap percolation has been studied * gjbaxter@ua.pt on the random regular graph [38,39], on...

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“…We find that the behavior of hysteresis loops depends nontrivially on the coordination number Þ (Sabhapandit et al 2002). For Þ ¿, for continuous unbounded distributions of random fields, the hysteresis loops show no jump discontinuity of magnetization even in the limit of small disorder, but for higher Þ they do.…”

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confidence: 89%