13,432 research outputs found

    Scale-free networks with tunable degree distribution exponents

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    We propose and study a model of scale-free growing networks that gives a degree distribution dominated by a power-law behavior with a model-dependent, hence tunable, exponent. The model represents a hybrid of the growing networks based on popularity-driven and fitness-driven preferential attachments. As the network grows, a newly added node establishes mm new links to existing nodes with a probability pp based on popularity of the existing nodes and a probability 1p1-p based on fitness of the existing nodes. An explicit form of the degree distribution P(p,k)P(p,k) is derived within a mean field approach. For reasonably large kk, P(p,k)kγ(p)F(k,p)P(p,k) \sim k^{-\gamma(p)}{\cal F}(k,p), where the function F{\cal F} is dominated by the behavior of 1/ln(k/m)1/\ln(k/m) for small values of pp and becomes kk-independent as p1p \to 1, and γ(p)\gamma(p) is a model-dependent exponent. The degree distribution and the exponent γ(p)\gamma(p) are found to be in good agreement with results obtained by extensive numerical simulations.Comment: 12 pages, 2 figures, submitted to PR

    Scaling Behaviour of Developing and Decaying Networks

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    We find that a wide class of developing and decaying networks has scaling properties similar to those that were recently observed by Barab\'{a}si and Albert in the particular case of growing networks. The networks considered here evolve according to the following rules: (i) Each instant a new site is added, the probability of its connection to old sites is proportional to their connectivities. (ii) In addition, (a) new links between some old sites appear with probability proportional to the product of their connectivities or (b) some links between old sites are removed with equal probability.Comment: 7 pages (revtex

    Signature Characters for A_2 and B_2

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    The signatures of the inner product matrices on a Lie algebra's highest weight representation are encoded in the representation's signature character. We show that the signature characters of a finite-dimensional Lie algebra's highest weight representations obey simple difference equations that have a unique solution once appropriate boundary conditions are imposed. We use these results to derive the signature characters of all A2A_2 and B2B_2 highest weight representations. Our results extend, and explain, signature patterns analogous to those observed by Friedan, Qiu and Shenker in the Virasoro algebra's representation theory.Comment: 22 p

    Analysis of Nonlinear Synchronization Dynamics of Oscillator Networks by Laplacian Spectral Methods

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    We analyze the synchronization dynamics of phase oscillators far from the synchronization manifold, including the onset of synchronization on scale-free networks with low and high clustering coefficients. We use normal coordinates and corresponding time-averaged velocities derived from the Laplacian matrix, which reflects the network's topology. In terms of these coordinates, synchronization manifests itself as a contraction of the dynamics onto progressively lower-dimensional submanifolds of phase space spanned by Laplacian eigenvectors with lower eigenvalues. Differences between high and low clustering networks can be correlated with features of the Laplacian spectrum. For example, the inhibition of full synchoronization at high clustering is associated with a group of low-lying modes that fail to lock even at strong coupling, while the advanced partial synchronizationat low coupling noted elsewhere is associated with high-eigenvalue modes.Comment: Revised version: References added, introduction rewritten, additional minor changes for clarit

    Structure of Growing Networks: Exact Solution of the Barabasi--Albert's Model

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    We generalize the Barab\'{a}si--Albert's model of growing networks accounting for initial properties of sites and find exactly the distribution of connectivities of the network P(q)P(q) and the averaged connectivity qˉ(s,t)\bar{q}(s,t) of a site ss in the instant tt (one site is added per unit of time). At long times P(q)qγP(q) \sim q^{-\gamma} at qq \to \infty and qˉ(s,t)(s/t)β\bar{q}(s,t) \sim (s/t)^{-\beta} at s/t0s/t \to 0, where the exponent γ\gamma varies from 2 to \infty depending on the initial attractiveness of sites. We show that the relation β(γ1)=1\beta(\gamma-1)=1 between the exponents is universal.Comment: 4 pages revtex (twocolumn, psfig), 1 figur

    Random matrix analysis of complex networks

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    We study complex networks under random matrix theory (RMT) framework. Using nearest-neighbor and next-nearest-neighbor spacing distributions we analyze the eigenvalues of adjacency matrix of various model networks, namely, random, scale-free and small-world networks. These distributions follow Gaussian orthogonal ensemble statistic of RMT. To probe long-range correlations in the eigenvalues we study spectral rigidity via Δ3\Delta_3 statistic of RMT as well. It follows RMT prediction of linear behavior in semi-logarithmic scale with slope being 1/π2\sim 1/\pi^2. Random and scale-free networks follow RMT prediction for very large scale. Small-world network follows it for sufficiently large scale, but much less than the random and scale-free networks.Comment: accepted in Phys. Rev. E (replaced with the final version

    Theory of Networked Minority Games based on Strategy Pattern Dynamics

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    We formulate a theory of agent-based models in which agents compete to be in a winning group. The agents may be part of a network or not, and the winning group may be a minority group or not. The novel feature of the present formalism is its focus on the dynamical pattern of strategy rankings, and its careful treatment of the strategy ties which arise during the system's temporal evolution. We apply it to the Minority Game (MG) with connected populations. Expressions for the mean success rate among the agents and for the mean success rate for agents with kk neighbors are derived. We also use the theory to estimate the value of connectivity pp above which the Binary-Agent-Resource system with high resource level goes into the high-connectivity state.Comment: 24 pages, 3 figures, submitted to PR

    The Number of Incipient Spanning Clusters in Two-Dimensional Percolation

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    Using methods of conformal field theory, we conjecture an exact form for the probability that n distinct clusters span a large rectangle or open cylinder of aspect ratio k, in the limit when k is large.Comment: 9 pages, LaTeX, 1 eps figure. Additional references and comparison with existing numerical results include

    High-Q-factor Al [subscript 2]O[subscript 3] micro-trench cavities integrated with silicon nitride waveguides on silicon

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    We report on the design and performance of high-Q integrated optical micro-trench cavities on silicon. The microcavities are co-integrated with silicon nitride bus waveguides and fabricated using wafer-scale silicon-photonics-compatible processing steps. The amorphous aluminum oxide resonator material is deposited via sputtering in a single straightforward post-processing step. We examine the theoretical and experimental optical properties of the aluminum oxide micro-trench cavities for different bend radii, film thicknesses and near-infrared wavelengths and demonstrate experimental Q factors of > 10[superscript 6]. We propose that this high-Q micro-trench cavity design can be applied to incorporate a wide variety of novel microcavity materials, including rare-earth-doped films for microlasers, into wafer-scale silicon photonics platforms
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