Effect of the accelerating growth of communications networks on their structure

Motivated by data on the evolution of the Internet and World Wide Web we consider scenarios of self-organization of the nonlinearly growing networks into free-scale structures. We find that the accelerating growth of the networks establishes their structure. For the growing networks with preferential linking and increasing density of links, two scenarios are possible. In one of them, the value of the exponent $\gamma$ of the connectivity distribution is between 3/2 and 2. In the other, $\gamma>2$ and the distribution is necessarily non-stationary.Comment: 4 pages revtex, 3 figure

Topological phase transition in a network model with preferential attachment and node removal

Preferential attachment is a popular model of growing networks. We consider a generalized model with random node removal, and a combination of preferential and random attachment. Using a high-degree expansion of the master equation, we identify a topological phase transition depending on the rate of node removal and the relative strength of preferential vs. random attachment, where the degree distribution goes from a power law to one with an exponential tail.Comment: The final publication is available at http://www.epj.or

Abundant Trimethylornithine Lipids and Specific Gene Sequences Are Indicative of Planctomycete Importance at the Oxic/Anoxic Interface in <i>Sphagnum</i>-Dominated Northern Wetlands

Northern wetlands make up a substantial terrestrial carbon sink and are often dominated by decay-resistant Sphagnum mosses.Recent studies have shown that planctomycetes appear to be involved in degradation of Sphagnum-derived debris. Novel trimethylornithine(TMO) lipids have recently been characterized as abundant lipids in various Sphagnum wetland planctomyceteisolates, but their occurrence in the environment has not yet been confirmed. We applied a combined intact polar lipid (IPL) andmolecular analysis of peat cores collected from two northern wetlands (Saxnäs Mosse [Sweden] and Obukhovskoye [Russia]) inorder to investigate the preferred niche and abundance of TMO-producing planctomycetes. TMOs were present throughout theprofiles of Sphagnum bogs, but their concentration peaked at the oxic/anoxic interface, which coincided with a maximum abundanceof planctomycete-specific 16S rRNA gene sequences. The sequences detected at the oxic/anoxic interface were affiliatedwith the Isosphaera group, while sequences present in the anoxic peat layers were related to an uncultured planctomycete group.Pyrosequencing-based analysis identified Planctomycetes as the major bacterial group at the oxic/anoxic interface at the Obukhovskoyepeat (54% of total 16S rRNA gene sequence reads), followed by Acidobacteria (19% reads), while in the Saxnäs Mossepeat, Acidobacteria were dominant (46%), and Planctomycetes contributed to 6% of the total reads. The detection of abundantTMO lipids in planctomycetes isolated from peat bogs and the lack of TMO production by cultures of acidobacteria suggest thatplanctomycetes are the producers of TMOs in peat bogs. The higher accumulation of TMOs at the oxic/anoxic interface and thechange in the planctomycete community with depth suggest that these IPLs could be synthesized as a response to changing redoxconditions at the oxic/anoxic interface

Statistical and Dynamical Study of Disease Propagation in a Small World Network

We study numerically statistical properties and dynamical disease propagation using a percolation model on a one dimensional small world network. The parameters chosen correspond to a realistic network of school age children. We found that percolation threshold decreases as a power law as the short cut fluctuations increase. We found also the number of infected sites grows exponentially with time and its rate depends logarithmically on the density of susceptibles. This behavior provides an interesting way to estimate the serology for a given population from the measurement of the disease growing rate during an epidemic phase. We have also examined the case in which the infection probability of nearest neighbors is different from that of short cuts. We found a double diffusion behavior with a slower diffusion between the characteristic times.Comment: 12 pages LaTex, 10 eps figures, Phys.Rev.E Vol. 64, 056115 (2001

Doppler-Free Spectroscopy of Weak Transitions: An Analytical Model Applied to Formaldehyde

Experimental observation of Doppler-free signals for weak transitions can be greatly facilitated by an estimate for their expected amplitudes. We derive an analytical model which allows the Doppler-free amplitude to be estimated for small Doppler-free signals. Application of this model to formaldehyde allows the amplitude of experimentally observed Doppler-free signals to be reproduced to within the experimental error.Comment: 7 pages, 7 figures, 1 table, v2: many small improvements + corrected line assignmen

Percolation and epidemics in a two-dimensional small world

Percolation on two-dimensional small-world networks has been proposed as a model for the spread of plant diseases. In this paper we give an analytic solution of this model using a combination of generating function methods and high-order series expansion. Our solution gives accurate predictions for quantities such as the position of the percolation threshold and the typical size of disease outbreaks as a function of the density of "shortcuts" in the small-world network. Our results agree with scaling hypotheses and numerical simulations for the same model.Comment: 7 pages, 3 figures, 2 table

Universality in percolation of arbitrary Uncorrelated Nested Subgraphs

The study of percolation in so-called {\em nested subgraphs} implies a generalization of the concept of percolation since the results are not linked to specific graph process. Here the behavior of such graphs at criticallity is studied for the case where the nesting operation is performed in an uncorrelated way. Specifically, I provide an analyitic derivation for the percolation inequality showing that the cluster size distribution under a generalized process of uncorrelated nesting at criticality follows a power law with universal exponent $\gamma=3/2$. The relevance of the result comes from the wide variety of processes responsible for the emergence of the giant component that fall within the category of nesting operations, whose outcome is a family of nested subgraphs.Comment: 5 pages, no figures. Mistakes found in early manuscript have been remove

Correlation effects in a simple model of small-world network

We analyze the effect of correlations in a simple model of small world network by obtaining exact analytical expressions for the distribution of shortest paths in the network. We enter correlations into a simple model with a distinguished site, by taking the random connections to this site from an Ising distribution. Our method shows how the transfer matrix technique can be used in the new context of small world networks.Comment: 10 pages, 3 figure

The spread of epidemic disease on networks

The study of social networks, and in particular the spread of disease on networks, has attracted considerable recent attention in the physics community. In this paper, we show that a large class of standard epidemiological models, the so-called susceptible/infective/removed (SIR) models can be solved exactly on a wide variety of networks. In addition to the standard but unrealistic case of fixed infectiveness time and fixed and uncorrelated probability of transmission between all pairs of individuals, we solve cases in which times and probabilities are non-uniform and correlated. We also consider one simple case of an epidemic in a structured population, that of a sexually transmitted disease in a population divided into men and women. We confirm the correctness of our exact solutions with numerical simulations of SIR epidemics on networks.Comment: 12 pages, 3 figure

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