Evolution of reference networks with aging

We study the growth of a reference network with aging of sites defined in the following way. Each new site of the network is connected to some old site with probability proportional (i) to the connectivity of the old site as in the Barab\'{a}si-Albert's model and (ii) to $\tau^{-\alpha}$, where $\tau$ is the age of the old site. We consider $\alpha$ of any sign although reasonable values are $0 \leq \alpha \leq \infty$. We find both from simulation and analytically that the network shows scaling behavior only in the region $\alpha < 1$. When $\alpha$ increases from $-\infty$ to 0, the exponent $\gamma$ of the distribution of connectivities ($P(k) \propto k^{-\gamma}$ for large $k$) grows from 2 to the value for the network without aging, i.e. to 3 for the Barab\'{a}si-Albert's model. The following increase of $\alpha$ to 1 makes $\gamma$ to grow to $\infty$. For $\alpha>1$ the distribution $P(k)$ is exponentional, and the network has a chain structure.Comment: 4 pages revtex (twocolumn, psfig), 5 figure

Ising model in small-world networks

The Ising model in small-world networks generated from two- and three-dimensional regular lattices has been studied. Monte Carlo simulations were carried out to characterize the ferromagnetic transition appearing in these systems. In the thermodynamic limit, the phase transition has a mean-field character for any finite value of the rewiring probability p, which measures the disorder strength of a given network. For small values of p, both the transition temperature and critical energy change with p as a power law. In the limit p -> 0, the heat capacity at the transition temperature diverges logarithmically in two-dimensional (2D) networks and as a power law in 3D.Comment: 6 pages, 7 figure

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

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 $P(q)$ and the averaged connectivity $\bar{q}(s,t)$ of a site $s$ in the instant $t$ (one site is added per unit of time). At long times $P(q) \sim q^{-\gamma}$ at $q \to \infty$ and $\bar{q}(s,t) \sim (s/t)^{-\beta}$ at $s/t \to 0$, where the exponent $\gamma$ varies from 2 to $\infty$ depending on the initial attractiveness of sites. We show that the relation $\beta(\gamma-1)=1$ between the exponents is universal.Comment: 4 pages revtex (twocolumn, psfig), 1 figur

Full oxide heterostructure combining a high-Tc diluted ferromagnet with a high-mobility conductor

We report on the growth of heterostructures composed of layers of the high-Curie temperature ferromagnet Co-doped (La,Sr)TiO3 (Co-LSTO) with high-mobility SrTiO3 (STO) substrates processed at low oxygen pressure. While perpendicular spin-dependent transport measurements in STO//Co-LSTO/LAO/Co tunnel junctions demonstrate the existence of a large spin polarization in Co-LSTO, planar magnetotransport experiments on STO//Co-LSTO samples evidence electronic mobilities as high as 10000 cm2/Vs at T = 10 K. At high enough applied fields and low enough temperatures (H < 60 kOe, T < 4 K) Shubnikov-de Haas oscillations are also observed. We present an extensive analysis of these quantum oscillations and relate them with the electronic properties of STO, for which we find large scattering rates up to ~ 10 ps. Thus, this work opens up the possibility to inject a spin-polarized current from a high-Curie temperature diluted oxide into an isostructural system with high-mobility and a large spin diffusion length.Comment: to appear in Phys. Rev.

Small-World Networks: Links with long-tailed distributions

Small-world networks (SWN), obtained by randomly adding to a regular structure additional links (AL), are of current interest. In this article we explore (based on physical models) a new variant of SWN, in which the probability of realizing an AL depends on the chemical distance between the connected sites. We assume a power-law probability distribution and study random walkers on the network, focussing especially on their probability of being at the origin. We connect the results to L\'evy Flights, which follow from a mean field variant of our model.Comment: 11 pages, 4 figures, to appear in Phys.Rev.

Small-world networks: Evidence for a crossover picture

Watts and Strogatz [Nature 393, 440 (1998)] have recently introduced a model for disordered networks and reported that, even for very small values of the disorder $p$ in the links, the network behaves as a small-world. Here, we test the hypothesis that the appearance of small-world behavior is not a phase-transition but a crossover phenomenon which depends both on the network size $n$ and on the degree of disorder $p$. We propose that the average distance $\ell$ between any two vertices of the network is a scaling function of $n / n^*$. The crossover size $n^*$ above which the network behaves as a small-world is shown to scale as $n^*(p \ll 1) \sim p^{-\tau}$ with $\tau \approx 2/3$.Comment: 5 pages, 5 postscript figures (1 in color), Latex/Revtex/multicols/epsf. Accepted for publication in Physical Review Letter

Impurity Scattering from δ\delta-layers in Giant Magnetoresistance Systems

The properties of the archetypal Co/Cu giant magnetoresistance (GMR) spin-valve structure have been modified by the insertion of very thin (sub-monolayer) $\delta$-layers of various elements at different points within the Co layers, and at the Co/Cu interface. Different effects are observed depending on the nature of the impurity, its position within the periodic table, and its location within the spin-valve. The GMR can be strongly enhanced or suppressed for various specific combinations of these parameters, giving insight into the microscopic mechanisms giving rise to the GMR.Comment: 5 pages, 2 figure

Path Integral Approach to Strongly Nonlinear Composite

We study strongly nonlinear disordered media using a functional method. We solve exactly the problem of a nonlinear impurity in a linear host and we obtain a Bruggeman-like formula for the effective nonlinear susceptibility. This formula reduces to the usual Bruggeman effective medium approximation in the linear case and has the following features: (i) It reproduces the weak contrast expansion to the second order and (ii) the effective medium exponent near the percolation threshold are $s=1$, $t=1+\kappa$, where $\kappa$ is the nonlinearity exponent. Finally, we give analytical expressions for previously numerically calculated quantities.Comment: 4 pages, 1 figure, to appear in Phys. Rev.

Spectra of "Real-World" Graphs: Beyond the Semi-Circle Law

Many natural and social systems develop complex networks, that are usually modelled as random graphs. The eigenvalue spectrum of these graphs provides information about their structural properties. While the semi-circle law is known to describe the spectral density of uncorrelated random graphs, much less is known about the eigenvalues of real-world graphs, describing such complex systems as the Internet, metabolic pathways, networks of power stations, scientific collaborations or movie actors, which are inherently correlated and usually very sparse. An important limitation in addressing the spectra of these systems is that the numerical determination of the spectra for systems with more than a few thousand nodes is prohibitively time and memory consuming. Making use of recent advances in algorithms for spectral characterization, here we develop new methods to determine the eigenvalues of networks comparable in size to real systems, obtaining several surprising results on the spectra of adjacency matrices corresponding to models of real-world graphs. We find that when the number of links grows as the number of nodes, the spectral density of uncorrelated random graphs does not converge to the semi-circle law. Furthermore, the spectral densities of real-world graphs have specific features depending on the details of the corresponding models. In particular, scale-free graphs develop a triangle-like spectral density with a power law tail, while small-world graphs have a complex spectral density function consisting of several sharp peaks. These and further results indicate that the spectra of correlated graphs represent a practical tool for graph classification and can provide useful insight into the relevant structural properties of real networks.Comment: 14 pages, 9 figures (corrected typos, added references) accepted for Phys. Rev.

Self-avoiding walks and connective constants in small-world networks

Long-distance characteristics of small-world networks have been studied by means of self-avoiding walks (SAW's). We consider networks generated by rewiring links in one- and two-dimensional regular lattices. The number of SAW's $u_n$ was obtained from numerical simulations as a function of the number of steps $n$ on the considered networks. The so-called connective constant, $\mu = \lim_{n \to \infty} u_n/u_{n-1}$, which characterizes the long-distance behavior of the walks, increases continuously with disorder strength (or rewiring probability, $p$). For small $p$, one has a linear relation $\mu = \mu_0 + a p$, $\mu_0$ and $a$ being constants dependent on the underlying lattice. Close to $p = 1$ one finds the behavior expected for random graphs. An analytical approach is given to account for the results derived from numerical simulations. Both methods yield results agreeing with each other for small $p$, and differ for $p$ close to 1, because of the different connectivity distributions resulting in both cases.Comment: 7 pages, 5 figure