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Exact results and new insights for models defined over small world networks. First and second order phase transitions. II: Applications  [PDF]
M. Ostilli,J. F. F. Mendes
Physics , 2008,
Abstract: We apply a novel method (presented in part I) to solve several small-world models for which the method can be applied analytically: the Viana-Bray model (which can be seen as a 0 or infinite dimensional small-world model), the one-dimensional chain small-world model, and the small-world spherical model in generic dimension. In particular, we analyze in detail the one-dimensional chain small-world model with negative short-range coupling showing that in this case, besides a second-order spin glass phase transition, there are two critical temperatures corresponding to first- or second-order phase transitions.
Phase transitions in some epidemic models defined on small-world networks  [PDF]
H. N. Agiza,A. S. Elgazzar,S. A. Youssef
Quantitative Biology , 2003, DOI: 10.1142/S0129183103004978
Abstract: Some modified versions of susceptible-infected-recovered-susceptible (SIRS) model are defined on small-world networks. Latency, incubation and variable susceptibility are included, separately. Phase transitions in these models are studied. Then inhomogeneous models are introduced. In some cases, the application of the models to small-world networks is shown to increase the epidemic region.
Effective field theory for models defined over small-world networks. First and second order phase transitions  [PDF]
Massimo Ostilli,José F. F. Mendes
Physics , 2008, DOI: 10.1103/PhysRevE.78.031102
Abstract: We present an effective field theory method 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 non random part of the model, i.e., the set of spins filling up a given lattice L_0 of dimension d_0 and interacting through a fixed coupling J_0, is exactly taken into account. When J_0\geq 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 J_0<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.
First- and second-order phase transitions in Ising models on small world networks, simulations and comparison with an effective field theory  [PDF]
A. L. Ferreira,J. F. F. Mendes,M. Ostilli
Physics , 2010, DOI: 10.1103/PhysRevE.82.011141
Abstract: We perform simulations of random Ising models defined over small-world networks and we check the validity and the level of approximation of a recently proposed effective field theory. Simulations confirm a rich scenario with the presence of multicritical points with first- or second-order phase transitions. In particular, for second-order phase transitions, independent of the dimension d_0 of the underlying lattice, the exact predictions of the theory in the paramagnetic regions, such as the location of critical surfaces and correlation functions, are verified. Quite interestingly, we verify that the Edwards-Anderson model with d_0=2 is not thermodynamically stable under graph noise.
Extended navigability of small world networks: exact results and new insights  [PDF]
Cecile Caretta Cartozo,Paolo De Los Rios
Physics , 2009, DOI: 10.1103/PhysRevLett.102.238703
Abstract: Navigability of networks, that is the ability to find any given destination vertex starting from any other vertex, is crucial to their usefulness. In 2000 Kleinberg showed that optimal navigability could be achieved in small-world networks provided that a special recipe was used to establish long range connections, and that a greedy algorithm, that ensures that the destination will be reached, is used. Here we provide an exact solution for the asymptotic behavior of such a greedy algorithm as a function of the system's parameters. Our solution enables us to show that the original claim that only a very special construction is optimal can be relaxed depending on further criteria, such as, for example, cost minimization, that must be satisfied.
Nonequilibrium Phase Transitions in Directed Small-World Networks  [PDF]
Alejandro D. Sanchez,Juan M. Lopez,Miguel A. Rodriguez
Physics , 2001, DOI: 10.1103/PhysRevLett.88.048701
Abstract: Many social, biological, and economic systems can be approached by complex networks of interacting units. The behaviour of several models on small-world networks has recently been studied. These models are expected to capture the essential features of the complex processes taking place on real networks like disease spreading, formation of public opinion, distribution of wealth, etc. In many of these systems relations are directed, in the sense that links only act in one direction (outwards or inwards). We investigate the effect of directed links on the behaviour of a simple spin-like model evolving on a small-world network. We show that directed networks may lead to a highly nontrivial phase diagram including first and second-order phase transitions out of equilibrium.
Scaling of Clusters near Discontinuous Percolation Transitions in Hyperbolic Networks  [PDF]
Vijay Singh,Stefan Boettcher
Physics , 2014, DOI: 10.1103/PhysRevE.90.012117
Abstract: We investigate the onset of the discontinuous percolation transition in small-world hyperbolic networks by studying the systems-size scaling of the typical largest cluster approaching the transition, $p\nearrow p_{c}$. To this end, we determine the average size of the largest cluster $\left\langle s_{{\rm max}}\right\rangle \sim N^{\Psi\left(p\right)}$ in the thermodynamic limit using real-space renormalization of cluster generating functions for bond and site percolation in several models of hyperbolic networks that provide exact results. We determine that all our models conform to the recently predicted behavior regarding the growth of the largest cluster, which found diverging, albeit sub-extensive, clusters spanning the system with finite probability well below $p_{c}$ and at most quadratic corrections to unity in $\Psi\left(p\right)$ for $p\nearrow p_{c}$. Our study suggest a large universality in the cluster formation on small-world hyperbolic networks and the potential for an alternative mechanism in the cluster formation dynamics at the onset of discontinuous percolation transitions.
Brane World as a Result of Spontaneous Symmetry Breaking  [PDF]
Boris E. Meierovich
Physics , 2009,
Abstract: The theories of brane world and multidimensional gravity are widely discussed in the literature in connection with problems of evolution of early Universe, including dark matter and energy. A natural physical concept is that a distinguished surface in the space-time manifold is a topological defect appeared as a result of a phase transition with spontaneous symmetry breaking. The macroscopic theory of phase transitions allows considering the brane world concept self-consistently, even without the knowledge of the nature of physical vacuum. Gravitational properties of topological defects (cosmic strings, monopoles,...) in extra dimensions are studied in General Relativity considering the order parameter as a vector and a multiplet in a plane target space of scalar fields. The common results and differences of these two approaches are analyzed and demonstrated in detail. Among the variety of regular solutions, there are those having brane features, including solutions with multiple branes, as well as the ones of potential interest from the standpoint of the dark matter and hierarchy problems. Regular configurations have a growing gravitational potential and are able to trap the matter on the brane. If the energy of spontaneous symmetry breaking is high, the attracting potential can have several points of minimum. Identical in the uniform bulk spin-less particles, being trapped within the separate points of minimum, acquire different masses and appear to an observer on brane as different particles with integer spins.
Cosmological Phase Transitions in a Brane World  [PDF]
Stephen C. Davis,Warren B. Perkins,Anne-Christine Davis,Ian R. Vernon
Physics , 2000, DOI: 10.1103/PhysRevD.63.083518
Abstract: In brane world scenarios the Friedmann equation is modified, resulting in an increased expansion at early times. This has important effects on cosmological phase transitions which we investigate, elucidating significant differences to the standard case. First order phase transitions require a higher nucleation rate to complete; baryogenesis and particle abundances could be suppressed. Topological defect evolution is also affected, though the current defect densities are largely unchanged. In particular, the increased expansion does not solve the usual monopole and domain wall problems.
Dynamic Transitions in Small World Networks: Approach to Equilibrium  [PDF]
Prashant M. Gade,Sudeshna Sinha
Physics , 2005, DOI: 10.1103/PhysRevE.72.052903
Abstract: We study the transition to phase synchronization in a model for the spread of infection defined on a small world network. It was shown (Phys. Rev. Lett. {\bf 86} (2001) 2909) that the transition occurs at a finite degree of disorder $p$, unlike equilibrium models where systems behave as random networks even at infinitesimal $p$ in the infinite size limit. We examine this system under variation of a parameter determining the driving rate, and show that the transition point decreases as we drive the system more slowly. Thus it appears that the transition moves to $p=0$ in the very slow driving limit, just as in the equilibrium case.
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