Robustness of scale-free spatial networks

A growing family of random graphs is called robust if it retains a giant component after percolation with arbitrary positive retention probability. We study robustness for graphs, in which new vertices are given a spatial position on the $d$-dimensional torus and are connected to existing vertices with a probability favouring short spatial distances and high degrees. In this model of a scale-free network with clustering we can independently tune the power law exponent $\tau$ of the degree distribution and the rate $\delta d$ at which the connection probability decreases with the distance of two vertices. We show that the network is robust if $\tau<2+1/\delta$, but fails to be robust if $\tau>3$. In the case of one-dimensional space we also show that the network is not robust if $\tau<2+1/(\delta-1)$. This implies that robustness of a scale-free network depends not only on its power-law exponent but also on its clustering features. Other than the classical models of scale-free networks our model is not locally tree-like, and hence we need to develop novel methods for its study, including, for example, a surprising application of the BK-inequality.Comment: 34 pages, 4 figure

Spatial preferential attachment networks: Power laws and clustering coefficients

We define a class of growing networks in which new nodes are given a spatial position and are connected to existing nodes with a probability mechanism favoring short distances and high degrees. The competition of preferential attachment and spatial clustering gives this model a range of interesting properties. Empirical degree distributions converge to a limit law, which can be a power law with any exponent $\tau>2$. The average clustering coefficient of the networks converges to a positive limit. Finally, a phase transition occurs in the global clustering coefficients and empirical distribution of edge lengths when the power-law exponent crosses the critical value $\tau=3$. Our main tool in the proof of these results is a general weak law of large numbers in the spirit of Penrose and Yukich.Comment: Published in at http://dx.doi.org/10.1214/14-AAP1006 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Physics of Transport and Traffic Phenomena in Biology: from molecular motors and cells to organisms

Traffic-like collective movements are observed at almost all levels of biological systems. Molecular motor proteins like, for example, kinesin and dynein, which are the vehicles of almost all intra-cellular transport in eukayotic cells, sometimes encounter traffic jam that manifests as a disease of the organism. Similarly, traffic jam of collagenase MMP-1, which moves on the collagen fibrils of the extracellular matrix of vertebrates, has also been observed in recent experiments. Traffic-like movements of social insects like ants and termites on trails are, perhaps, more familiar in our everyday life. Experimental, theoretical and computational investigations in the last few years have led to a deeper understanding of the generic or common physical principles involved in these phenomena. In particular, some of the methods of non-equilibrium statistical mechanics, pioneered almost a hundred years ago by Einstein, Langevin and others, turned out to be powerful theoretical tools for quantitaive analysis of models of these traffic-like collective phenomena as these systems are intrinsically far from equilibrium. In this review we critically examine the current status of our understanding, expose the limitations of the existing methods, mention open challenging questions and speculate on the possible future directions of research in this interdisciplinary area where physics meets not only chemistry and biology but also (nano-)technology.Comment: 33 page Review article, REVTEX text, 29 EPS and PS figure

β\beta--Radioactive Cosmic Rays in a diffusion model: test for a local bubble?

In the present paper, we extend the analysis of Maurin et al. (2001) and Donato et al. (2001) to the $\beta$-radioactive nuclei $^{10}$Be, $^{26}$Al and $^{36}$Cl. These species are be shown to be particularly sensitive to the properties of the local interstellar medium (LISM). As studies of the LISM suggest that we live in an underdense bubble of extent r_{hole} \sim 50 - 200 \unit{pc}, this local feature must be taken into account. We present a modified version of our diffusion model which describes the underdensity as a hole in the galactic disc. It is found that the presence of the bubble leads to a decrease in the radioactive fluxes which can be approximated by a simple factor $\exp(-r_{hole}/l_{rad})$ where $l_{rad}=\sqrt{K \gamma \tau_0}$ is the typical distance travelled by a radioactive nucleus before it decays. We find that each of the radioactive nuclei independently point towards a bubble radius \lesssim 100 \unit{pc}. If these nuclei are considered simultaneously, only models with a bubble radius r_{hole} \sim 60 - 100 \unit{pc} are marginally consistent with data. In particular, the standard case r_{hole}=0 \unit{pc} is disfavoured. Our main concern is about the consistency of the currently available data, especially $^{26}$Al/$^{27}$Al.Comment: 21 pages, 11 figures, Latex, macro aa.cls, to appear in A&

Geometric Graph Properties of the Spatial Preferred Attachment model

The spatial preferred attachment (SPA) model is a model for networked information spaces such as domains of the World Wide Web, citation graphs, and on-line social networks. It uses a metric space to model the hidden attributes of the vertices. Thus, vertices are elements of a metric space, and link formation depends on the metric distance between vertices. We show, through theoretical analysis and simulation, that for graphs formed according to the SPA model it is possible to infer the metric distance between vertices from the link structure of the graph. Precisely, the estimate is based on the number of common neighbours of a pair of vertices, a measure known as {\sl co-citation}. To be able to calculate this estimate, we derive a precise relation between the number of common neighbours and metric distance. We also analyze the distribution of {\sl edge lengths}, where the length of an edge is the metric distance between its end points. We show that this distribution has three different regimes, and that the tail of this distribution follows a power law

Three-dimensional cell to tissue assembly process

The present invention relates a 3-dimensional cell to tissue and maintenance process, more particularly to methods of culturing cells in a culture environment, either in space or in a gravity field, with minimum fluid shear stress, freedom for 3-dimensional spatial orientation of the suspended particles and localization of particles with differing or similar sedimentation properties in a similar spatial region