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Register a new userDiscrete surface growth process as a synchronization mechanism for scale free complex networks
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Added by
Lidia A. Braunstein
Publication date
2007
fields
Physics
and research's language is
English
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We consider the discrete surface growth process with relaxation to the minimum [F. Family, J. Phys. A {bf 19} L441, (1986).] as a possible synchronization mechanism on scale-free networks, characterized by a degree distribution $P(k) sim k^{-lambda}$, where $k$ is the degree of a node and $lambda$ his broadness, and compare it with the usually applied Edward-Wilkinson process [S. F. Edwards and D. R. Wilkinson, Proc. R. Soc. London Ser. A {bf 381},17 (1982) ]. In spite of both processes belong to the same universality class for Euclidean lattices, in this work we demonstrate that for scale-free networks with exponents $lambda<3$ this is not true. Moreover, we show that for these ubiquitous cases the Edward-Wilkinson process enhances spontaneously the synchronization when the system size is increased, which is a non-physical result. Contrarily, the discrete surface growth process do not present this flaw and is applicable for every $lambda$.
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In this letter, we proposed an ungrowing scale-free network model, wherein the total number of nodes is fixed and the evolution of network structure is driven by a rewiring process only. In spite of the idiographic form of $G$, by using a two-order master equation, we obtain the analytic solution of degree distribution in stable state of the network evolution under the condition that the selection probability $G$ in rewiring process only depends on nodes degrees. A particular kind of the present networks with $G$ linearly correlated with degree is studied in detail. The analysis and simulations show that the degree distributions of these networks can varying from the Possion form to the power-law form with the decrease of a free parameter $alpha$, indicating the growth may not be a necessary condition of the self-organizaton of a network in a scale-free structure.
We discuss the concept of discrete scale invariance and how it leads to complex critical exponents (or dimensions), i.e. to the log-periodic corrections to scaling. After their initial suggestion as formal solutions of renormalization group equations in the seventies, complex exponents have been studied in the eighties in relation to various problems of physics embedded in hierarchical systems. Only recently has it been realized that discrete scale invariance and its associated complex exponents may appear ``spontaneously in euclidean systems, i.e. without the need for a pre-existing hierarchy. Examples are diffusion-limited-aggregation clusters, rupture in heterogeneous systems, earthquakes, animals (a generalization of percolation) among many other systems. We review the known mechanisms for the spontaneous generation of discrete scale invariance and provide an extensive list of situations where complex exponents have been found. This is done in order to provide a basis for a better fundamental understanding of discrete scale invariance. The main motivation to study discrete scale invariance and its signatures is that it provides new insights in the underlying mechanisms of scale invariance. It may also be very interesting for prediction purposes.
It was recently conjectured that in generic quantum many-body systems, the spectral density of local operators has the slowest high-frequency decay as permitted by locality. We show that the infinite-temperature version of this universal operator growth hypothesis holds for the quantum Ising spin model in $d ge 2$ dimensions, and for the chaotic Ising chain (with longitudinal and transverse fields) in one dimension. Moreover, the disordered chaotic Ising chain that exhibits many-body localization can have the same high-frequency spectral density decay as thermalizing models. Our argument is statistical in nature, and is based on the observation that the moments of the spectral density can be written as a sign-problem-free sum over paths of Pauli string operators.
We describe in detail and extend a recently introduced nonperturbative renormalization group (RG) method for surface growth. The scale invariant dynamics which is the key ingredient of the calculation is obtained as the fixed point of a RG transformation relating the representation of the microscopic process at two different coarse-grained scales. We review the RG calculation for systems in the Kardar-Parisi-Zhang universality class and compute the roughness exponent for the strong coupling phase in dimensions from 1 to 9. Discussions of the approximations involved and possible improvements are also presented. Moreover, very strong evidence of the absence of a finite upper critical dimension for KPZ growth is presented. Finally, we apply the method to the linear Edwards-Wilkinson dynamics where we reproduce the known exact results, proving the ability of the method to capture qualitatively different behaviors.
Contrary to many recent models of growing networks, we present a model with fixed number of nodes and links, where it is introduced a dynamics favoring the formation of links between nodes with degree of connectivity as different as possible. By applying a local rewiring move, the network reaches equilibrium states assuming broad degree distributions, which have a power law form in an intermediate range of the parameters used. Interestingly, in the same range we find non-trivial hierarchical clustering.
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