Random graph model with power-law distributed triangle subgraphs

Clustering is well-known to play a prominent role in the description and understanding of complex networks, and a large spectrum of tools and ideas have been introduced to this end. In particular, it has been recognized that the abundance of small subgraphs is important. Here, we study the arrangement of triangles in a model for scale-free random graphs and determine the asymptotic behavior of the clustering coefficient, the average number of triangles, as well as the number of triangles attached to the vertex of maximum degree. We prove that triangles are power-law distributed among vertices and characterized by both vertex and edge coagulation when the degree exponent satisfies $2<\beta<2.5$; furthermore, a finite density of triangles appears as $\beta=2+1/3$.Comment: 4 pages, 2 figure; v2: major conceptual change

Scale-Free Networks are Ultrasmall

We study the diameter, or the mean distance between sites, in a scale-free network, having N sites and degree distribution p(k) ~ k^-a, i.e. the probability of having k links outgoing from a site. In contrast to the diameter of regular random networks or small world networks which is known to be d ~ lnN, we show, using analytical arguments, that scale free networks with 2<a<3 have a much smaller diameter, behaving as d ~ lnlnN. For a=3, our analysis yields d ~ lnN/lnlnN, as obtained by Bollobas and Riordan, while for a>3, d ~ lnN. We also show that, for any a>2, one can construct a deterministic scale free network with d ~ lnlnN, and this construction yields the lowest possible diameter.Comment: Latex, 4 pages, 2 eps figures, small corrections, added explanation

Sampling properties of random graphs: the degree distribution

We discuss two sampling schemes for selecting random subnets from a network: Random sampling and connectivity dependent sampling, and investigate how the degree distribution of a node in the network is affected by the two types of sampling. Here we derive a necessary and sufficient condition that guarantees that the degree distribution of the subnet and the true network belong to the same family of probability distributions. For completely random sampling of nodes we find that this condition is fulfilled by classical random graphs; for the vast majority of networks this condition will, however, not be met. We furthermore discuss the case where the probability of sampling a node depends on the degree of a node and we find that even classical random graphs are no longer closed under this sampling regime. We conclude by relating the results to real {\it E.coli} protein interaction network data.Comment: accepted for publication in Phys.Rev.

Prevalence of Escherichia coli carriage in the oropharynx of ambulatory children and adults with and without upper respiratory symptoms

[No abstract available

Optical and Near-IR Imaging of the Dark Globule CB 52

The internal structure of the dark globule CB 52 is investigated by means of broadband imaging in the optical and near-IR spectral range. By exploiting the extinction of the stellar light within this object, we derive observational parameters suitable to infer the internal structure of this cloud. Extinction maps were obtained at different wavelengths by using both stellar counts and two-color diagrams. While in the optical region the extinction is better evaluated at the cloud boundaries, the internal regions are more conveniently probed in the near-IR, so a combined map was derived. The total-to-selective extinction ratio RV was also observed to increase toward the inner regions, and a plot versus the extinction AV suggests that grain growth processes are active in this cloud. The statistical fluctuation of the AV, estimated in the line of sight of the background stars, is investigated by comparing the observed stellar colors with those of the unreddened stars. The dispersion σAV, derived from optical observations, is found to be almost independent of the mean extinction, AV, while by using near-IR data we find a more complex behavior: the σAV versus AV relation increases until AV ~ 6 and then decreases for larger extinctions. This is discussed in the framework of a simple model, suggesting that a clumpy and clustered structure can explain the observations in the inner regions, while outside the cloud the mass distribution remains more homogeneous

Cathodoluminescence-based nanoscopic thermometry in a lanthanide-doped phosphor

Crucial to analyze phenomena as varied as plasmonic hot spots and the spread of cancer in living tissue, nanoscale thermometry is challenging: probes are usually larger than the sample under study, and contact techniques may alter the sample temperature itself. Many photostable nanomaterials whose luminescence is temperature-dependent, such as lanthanide-doped phosphors, have been shown to be good non-contact thermometric sensors when optically excited. Using such nanomaterials, in this work we accomplished the key milestone of enabling far-field thermometry with a spatial resolution that is not diffraction-limited at readout. We explore thermal effects on the cathodoluminescence of lanthanide-doped NaYF$_4$ nanoparticles. Whereas cathodoluminescence from such lanthanide-doped nanomaterials has been previously observed, here we use quantitative features of such emission for the first time towards an application beyond localization. We demonstrate a thermometry scheme that is based on cathodoluminescence lifetime changes as a function of temperature that achieves $\sim$ 30 mK sensitivity in sub-$\mu$m nanoparticle patches. The scheme is robust against spurious effects related to electron beam radiation damage and optical alignment fluctuations. We foresee the potential of single nanoparticles, of sheets of nanoparticles, and also of thin films of lanthanide-doped NaYF$_4$ to yield temperature information via cathodoluminescence changes when in the vicinity of a sample of interest; the phosphor may even protect the sample from direct contact to damaging electron beam radiation. Cathodoluminescence-based thermometry is thus a valuable novel tool towards temperature monitoring at the nanoscale, with broad applications including heat dissipation in miniaturized electronics and biological diagnostics.Comment: Main text: 30 pages + 4 figures; supplementary information: 22 pages + 8 figure

Solution of the 2-star model of a network

The p-star model or exponential random graph is among the oldest and best-known of network models. Here we give an analytic solution for the particular case of the 2-star model, which is one of the most fundamental of exponential random graphs. We derive expressions for a number of quantities of interest in the model and show that the degenerate region of the parameter space observed in computer simulations is a spontaneously symmetry broken phase separated from the normal phase of the model by a conventional continuous phase transition.Comment: 5 pages, 3 figure

Nanostructured Lead Electrodes with Reduced Graphene Oxide for High-Performance Lead–Acid Batteries

Nanostructured Pb electrodes consisting of nanowire arrays were obtained by electrodeposition, to be used as negative electrodes for lead–acid batteries. Reduced graphene oxide was added to improve their performances. This was achieved via the electrochemical reduction of graphene oxide directly on the surface of nanowire arrays. The electrodes with and without reduced graphene oxide were tested in a 5 M sulfuric acid solution using a commercial pasted positive plate and an absorbed glass mat separator in a zero-gap configuration. The electrodes were tested in deep cycling conditions with a very low cut-off potential. Charge–discharge tests were performed at 5C. The electrode with reduced graphene oxide outperformed the electrode without reduced graphene oxide, as it was able to work with a very high utilization of active mass and efficiency. A specific capacity of 258 mAhg−1–very close to the theoretical one–was achieved, and the electrode lasted for more than 1000 cycles. On the other hand, the electrode without reduced graphene oxide achieved a capacity close to 230 mAhg−1, which corresponds to a 90% of utilization of active mass