The Experts below are selected from a list of 723663 Experts worldwide ranked by ideXlab platform
Nihat A Berker - One of the best experts on this subject based on the ideXlab platform.
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inverted berezinskii kosterlitz thouless singularity and high temperature algebraic order in an ising model on a scale free hierarchical lattice small world Network
Physical Review E, 2006Co-Authors: Michael Hinczewski, Nihat A BerkerAbstract:We have obtained exact results for the Ising model on a hierarchical lattice incorporating three key features characterizing many real-world Networks---a scale-free degree distribution, a high clustering coefficient, and the Small-World effect. By varying the probability $p$ of long-range bonds, the entire spectrum from an unclustered, non-Small-World Network to a highly clustered, Small-World system is studied. Using the self-similar structure of the Network, we obtain analytic expressions for the degree distribution $P(k)$ and clustering coefficient $C$ for all $p$, as well as the average path length $\ensuremath{\ell}$ for $p=0$ and $1$. The ferromagnetic Ising model on this Network is studied through an exact renormalization-group transformation of the quenched bond probability distribution, using up to $562\phantom{\rule{0.2em}{0ex}}500$ renormalized probability bins to represent the distribution. For $pl0.494$, we find power-law critical behavior of the magnetization and susceptibility, with critical exponents continuously varying with $p$, and exponential decay of correlations away from ${T}_{c}$. For $p\ensuremath{\ge}0.494$, in fact where the Network exhibits Small-World character, the critical behavior radically changes: We find a highly unusual phase transition, namely an inverted Berezinskii-Kosterlitz-Thouless singularity, between a low-temperature phase with nonzero magnetization and finite correlation length and a high-temperature phase with zero magnetization and infinite correlation length, with power-law decay of correlations throughout the phase. Approaching ${T}_{c}$ from below, the magnetization and the susceptibility, respectively, exhibit the singularities of $\mathrm{exp}(\ensuremath{-}C∕\sqrt{{T}_{c}\ensuremath{-}T})$ and $\mathrm{exp}(D∕\sqrt{{T}_{c}\ensuremath{-}T})$, with $C$ and $D$ positive constants. With long-range bond strengths decaying with distance, we see a phase transition with power-law critical singularities for all $p$, and evaluate an unusually narrow critical region and important corrections to power-law behavior that depend on the exponent characterizing the decay of long-range interactions.
Klaus Lehnertz - One of the best experts on this subject based on the ideXlab platform.
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multistability local pattern formation and global collective firing in a small world Network of non leaky integrate and fire neurons
arXiv: Computational Physics, 2012Co-Authors: Alexander Rothkegel, Klaus LehnertzAbstract:We investigate numerically the collective dynamical behavior of pulse-coupled non-leaky integrate-and-fire-neurons that are arranged on a two-dimensional Small-World Network. To ensure ongoing activity, we impose a probability for spontaneous firing for each neuron. We study Network dynamics evolving from different sets of initial conditions in dependence on coupling strength and rewiring probability. Beside a homogeneous equilibrium state for low coupling strength, we observe different local patterns including cyclic waves, spiral waves, and turbulent-like patterns, which -- depending on Network parameters -- interfere with the global collective firing of the neurons. We attribute the various Network dynamics to distinct regimes in the parameter space. For the same Network parameters different Network dynamics can be observed depending on the set of initial conditions only. Such a multistable behavior and the interplay between local pattern formation and global collective firing may be attributable to the spatiotemporal dynamics of biological Networks.
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multistability local pattern formation and global collective firing in a small world Network of nonleaky integrate and fire neurons
Chaos, 2009Co-Authors: Alexander Rothkegel, Klaus LehnertzAbstract:We investigate numerically the collective dynamical behavior of pulse-coupled nonleaky integrate-and-fire neurons that are arranged on a two-dimensional Small-World Network. To ensure ongoing activity, we impose a probability for spontaneous firing for each neuron. We study Network dynamics evolving from different sets of initial conditions in dependence on coupling strength and rewiring probability. Besides a homogeneous equilibrium state for low coupling strength, we observe different local patterns including cyclic waves, spiral waves, and turbulentlike patterns, which—depending on Network parameters—interfere with the global collective firing of the neurons. We attribute the various Network dynamics to distinct regimes in the parameter space. For the same Network parameters different Network dynamics can be observed depending on the set of initial conditions only. Such a multistable behavior and the interplay between local pattern formation and global collective firing may be attributable to the spati...
Alexander Rothkegel - One of the best experts on this subject based on the ideXlab platform.
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multistability local pattern formation and global collective firing in a small world Network of non leaky integrate and fire neurons
arXiv: Computational Physics, 2012Co-Authors: Alexander Rothkegel, Klaus LehnertzAbstract:We investigate numerically the collective dynamical behavior of pulse-coupled non-leaky integrate-and-fire-neurons that are arranged on a two-dimensional Small-World Network. To ensure ongoing activity, we impose a probability for spontaneous firing for each neuron. We study Network dynamics evolving from different sets of initial conditions in dependence on coupling strength and rewiring probability. Beside a homogeneous equilibrium state for low coupling strength, we observe different local patterns including cyclic waves, spiral waves, and turbulent-like patterns, which -- depending on Network parameters -- interfere with the global collective firing of the neurons. We attribute the various Network dynamics to distinct regimes in the parameter space. For the same Network parameters different Network dynamics can be observed depending on the set of initial conditions only. Such a multistable behavior and the interplay between local pattern formation and global collective firing may be attributable to the spatiotemporal dynamics of biological Networks.
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multistability local pattern formation and global collective firing in a small world Network of nonleaky integrate and fire neurons
Chaos, 2009Co-Authors: Alexander Rothkegel, Klaus LehnertzAbstract:We investigate numerically the collective dynamical behavior of pulse-coupled nonleaky integrate-and-fire neurons that are arranged on a two-dimensional Small-World Network. To ensure ongoing activity, we impose a probability for spontaneous firing for each neuron. We study Network dynamics evolving from different sets of initial conditions in dependence on coupling strength and rewiring probability. Besides a homogeneous equilibrium state for low coupling strength, we observe different local patterns including cyclic waves, spiral waves, and turbulentlike patterns, which—depending on Network parameters—interfere with the global collective firing of the neurons. We attribute the various Network dynamics to distinct regimes in the parameter space. For the same Network parameters different Network dynamics can be observed depending on the set of initial conditions only. Such a multistable behavior and the interplay between local pattern formation and global collective firing may be attributable to the spati...
Duncan J Watts - One of the best experts on this subject based on the ideXlab platform.
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mean field solution of the small world Network model
Physical Review Letters, 2000Co-Authors: M E J Newman, Cristopher Moore, Duncan J WattsAbstract:The Small-World Network model is a simple model of the structure of social Networks, which simultaneously possesses characteristics of both regular lattices and random graphs. The model consists of a one-dimensional lattice with a low density of shortcuts added between randomly selected pairs of points. These shortcuts greatly reduce the typical path length between any two points on the lattice. We present a mean-field solution for the average path length and for the distribution of path lengths in the model. This solution is exact in the limit of large system size and either large or small number of shortcuts.
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mean field solution of the small world Network model
Physical Review Letters, 2000Co-Authors: M E J Newman, Cristopher Moore, Duncan J WattsAbstract:The Small-World Network model is a simple model of the structure of social Networks, which possesses characteristics of both regular lattices and random graphs. The model consists of a one-dimensional lattice with a low density of shortcuts added between randomly selected pairs of points. These shortcuts greatly reduce the typical path length between any two points on the lattice. We present a mean-field solution for the average path length and for the distribution of path lengths in the model. This solution is exact in the limit of large system size and either a large or small number of shortcuts.
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renormalization group analysis of the small world Network model
Physics Letters A, 1999Co-Authors: M E J Newman, Duncan J WattsAbstract:We study the Small-World Network model, which mimics the transition between regular-lattice and random-lattice behavior in social Networks of increasing size. We contend that the model displays a normal continuous phase transition with a divergent correlation length as the degree of randomness tends to zero. We propose a real-space renormalization group transformation for the model and demonstrate that the transformation is exact in the limit of large system size. We use this result to calculate the exact value of the single critical exponent for the system, and to derive the scaling form for the average number of "degrees of separation" between two nodes on the Network as a function of the three independent variables. We confirm our results by extensive numerical simulation. Appears in Phys. Lett. A 263, 341-346 (1999).
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scaling and percolation in the small world Network model
Physical Review E, 1999Co-Authors: M E J Newman, Duncan J WattsAbstract:In this paper we study the Small-World Network model of Watts and Strogatz, which mimics some aspects of the structure of Networks of social interactions. We argue that there is one nontrivial length-scale in the model, analogous to the correlation length in other systems, which is well-defined in the limit of infinite system size and which diverges continuously as the randomness in the Network tends to zero, giving a normal critical point in this limit. This length-scale governs the crossover from large- to Small-World behavior in the model, as well as the number of vertices in a neighborhood of given radius on the Network. We derive the value of the single critical exponent controlling behavior in the critical region and the finite size scaling form for the average vertex-vertex distance on the Network, and, using series expansion and Pad\'e approximants, find an approximate analytic form for the scaling function. We calculate the effective dimension of Small-World graphs and show that this dimension varies as a function of the length-scale on which it is measured, in a manner reminiscent of multifractals. We also study the problem of site percolation on Small-World Networks as a simple model of disease propagation, and derive an approximate expression for the percolation probability at which a giant component of connected vertices first forms (in epidemiological terms, the point at which an epidemic occurs). The typical cluster radius satisfies the expected finite size scaling form with a cluster size exponent close to that for a random graph. All our analytic results are confirmed by extensive numerical simulations of the model.
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scaling and percolation in the small world Network model
1999Co-Authors: M E J Newman, Duncan J WattsAbstract:In this paper we study the Small-World Network model of Watts and Strogatz, which mimics some aspects of the structure of Networks of social interactions. We argue that there is one non-trivial length-scale in the model, analogous to the correlation length in other systems, which is well-defined in the limit of infinite system size and which diverges continuously as the randomness in the Network tends to zero, giving a normal critical point in this limit. This length-scale governs the cross-over from large- to Small-World behavior in the model, as well as the number of vertices in a neighborhood of given radius on the Network. We derive the value of the single critical exponent controlling behavior in the critical region and the finite size scaling form for the average vertex-vertex distance on the Network, and, using series expansion and Pade approximants, find an approximate analytic form for the scaling function. We calculate the effective dimension of Small-World graphs and show that this dimension varies as a function of the length-scale on which it is measured, in a manner reminiscent of multifractals. We also study the problem of site percolation on Small-World Networks as a simple model of disease propagation, and derive an approximate expression for the percolation probability at which a giant component of connected vertices first forms (in epidemiological terms, the point at which an epidemic occurs). The typical cluster radius satisfies the expected finite size scaling form with a cluster size exponent close to that for a random graph. All our analytic results are confirmed by extensive numerical simulations of the model. Appears in Phys. Rev. E 60, 7332-7342 (1999).
James H Fowler - One of the best experts on this subject based on the ideXlab platform.
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legislative success in a small world social Network analysis and the dynamics of congressional legislation
The Journal of Politics, 2010Co-Authors: James H FowlerAbstract:We examine the social Network structure of Congress from 1973 to 2004. We treat two Members of Congress as directly linked if they have cosponsored at least one bill together. We then construct explicit Networks for each year using data from all forms of legislation, including resolutions, public and private bills, and amendments. We show that Congress exemplifies the characteristics of a “small world” Network and that the varying Small-World properties during this time period are related to the number of important bills passed.
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legislative success in a small world social Network analysis and the dynamics of congressional legislation
2007Co-Authors: Wendy Tam K Cho, James H FowlerAbstract:We examine the social Network structure of Congress from 1973-2004. We treat two Members of Congress as directly linked if they have cosponsored a bill together. We then construct explicit Networks for each year using data from all forms of legislation, including resolutions, public and private bills, and amendments. We show that Congress exemplifies the characteristics of a "small world" Network and that the varying small world properties during this time period are strongly related to the number of important bills passed.
