LATE DEVONIAN-TOURNAISIAN CONODONTS FROM THE EASTERN KHYBER REGION, NORTH-WEST PAKISTAN

Conodonts (62 species and subspecies) from acid-leaching of 226 samples from four sections through the Ali Masjid Formation west of Misri Khel in the former South Khyber Agency, north-west Pakistan, are documented by illustrations and distributional data. These indicate that most of this unit, at least in that area, spans the interval Middle crepida Zone (low in the Famennian) to at least Early crenulata Zone (mid-Tournaisian), though a fauna from low in one of the sections produced conodonts indicative of the Late falsiovalis Zone (early Frasnian). Two major hiatuses are inferred: between the Late falsiovalis and Middle crepida zones, and between the Late expansa and Early duplicata zones. Coherence of the conodont biostratigraphy accords with lithologic alignments between the sections. &nbsp

Self-avoiding walks on scale-free networks

Several kinds of walks on complex networks are currently used to analyze search and navigation in different systems. Many analytical and computational results are known for random walks on such networks. Self-avoiding walks (SAWs) are expected to be more suitable than unrestricted random walks to explore various kinds of real-life networks. Here we study long-range properties of random SAWs on scale-free networks, characterized by a degree distribution $P(k) \sim k^{-\gamma}$. In the limit of large networks (system size $N \to \infty$), the average number $s_n$ of SAWs starting from a generic site increases as $\mu^n$, with $\mu = / - 1$. For finite $N$, $s_n$ is reduced due to the presence of loops in the network, which causes the emergence of attrition of the paths. For kinetic growth walks, the average maximum length, $$, increases as a power of the system size: $\sim N^{\alpha}$, with an exponent $\alpha$ increasing as the parameter $\gamma$ is raised. We discuss the dependence of $\alpha$ on the minimum allowed degree in the network. A similar power-law dependence is found for the mean self-intersection length of non-reversal random walks. Simulation results support our approximate analytical calculations.Comment: 9 pages, 7 figure

EARLIEST TRIASSIC CONODONTS FROM CHITRAL, NORTHERNMOST PAKISTAN

Extensive tracts of very shallow water carbonates in the valleys of the Yarkhun and Mastuj rivers of Chitral (northernmost Pakistan) previously though to be Permian (or Cretaceous) are shown by conodonts from two horizons in sequences 110 km apart—near Torman Gol (Mastuj valley) and near Sakirmul (upper Yarkhun valley)—to include earliest Triassic (Scythian—Induan) horizons. Both faunas have Isarcicella staeschei Dai & Zhang, Is. lobata Perri, Is. turgida (Kozur et al.) and Hindeodus parvus (Kozur & Pjatakova), whereas Is. Isarcica (Huckriede) has been recognised only in the Torman Gol occurrence. The presence, respectively, of Is. staeschei in the Sakirmul and Is. isarcica in the Torman Gol occurrences, allows discrimination of the staeschei and isarcica zones respectively the third and the fourth conodont biozones of the Early Triassic conodont biozonation of Perri (in Perri & Farabegoli 2003). Such faunas, consisting mainly of isarcicellids and hindeodids but lacking gondolellids, are characteristic of restricted sea environments across the Permian–Triassic boundary and in the earliest Triassic in other Tethyan areas. The conodont faunas from these two occurrences are remarkably similar, nearly contemporaneous, and indicate shallow water biofacies. They are inferred to equate with the Ailak Dolomite, a sequence of Late Permian–?Late Triassic dolostones discriminated farther up the Yarkhun valley and extending eastwards into the upper Hunza region of northernmost Pakistan. The Zait Limestone and Sakirmul carbonate sequence are consistent with extension of the previously inferred Triassic carbonate platform at least 110 km farther to the SW than previously supposed

Synchronization in Scale Free networks: The role of finite size effects

Synchronization problems in complex networks are very often studied by researchers due to its many applications to various fields such as neurobiology, e-commerce and completion of tasks. In particular, Scale Free networks with degree distribution $P(k)\sim k^{-\lambda}$, are widely used in research since they are ubiquitous in nature and other real systems. In this paper we focus on the surface relaxation growth model in Scale Free networks with $2.5< \lambda <3$, and study the scaling behavior of the fluctuations, in the steady state, with the system size $N$. We find a novel behavior of the fluctuations characterized by a crossover between two regimes at a value of $N=N^*$ that depends on $\lambda$: a logarithmic regime, found in previous research, and a constant regime. We propose a function that describes this crossover, which is in very good agreement with the simulations. We also find that, for a system size above $N^{*}$, the fluctuations decrease with $\lambda$, which means that the synchronization of the system improves as $\lambda$ increases. We explain this crossover analyzing the role of the network's heterogeneity produced by the system size $N$ and the exponent of the degree distribution.Comment: 9 pages and 5 figures. Accepted in Europhysics Letter

Percolation in invariant Poisson graphs with i.i.d. degrees

Let each point of a homogeneous Poisson process in R^d independently be equipped with a random number of stubs (half-edges) according to a given probability distribution mu on the positive integers. We consider translation-invariant schemes for perfectly matching the stubs to obtain a simple graph with degree distribution mu. Leaving aside degenerate cases, we prove that for any mu there exist schemes that give only finite components as well as schemes that give infinite components. For a particular matching scheme that is a natural extension of Gale-Shapley stable marriage, we give sufficient conditions on mu for the absence and presence of infinite components

Random Networks with Tunable Degree Distribution and Clustering

We present an algorithm for generating random networks with arbitrary degree distribution and Clustering (frequency of triadic closure). We use this algorithm to generate networks with exponential, power law, and poisson degree distributions with variable levels of clustering. Such networks may be used as models of social networks and as a testable null hypothesis about network structure. Finally, we explore the effects of clustering on the point of the phase transition where a giant component forms in a random network, and on the size of the giant component. Some analysis of these effects is presented.Comment: 9 pages, 13 figures corrected typos, added two references, reorganized reference

Percolation in Hierarchical Scale-Free Nets

We study the percolation phase transition in hierarchical scale-free nets. Depending on the method of construction, the nets can be fractal or small-world (the diameter grows either algebraically or logarithmically with the net size), assortative or disassortative (a measure of the tendency of like-degree nodes to be connected to one another), or possess various degrees of clustering. The percolation phase transition can be analyzed exactly in all these cases, due to the self-similar structure of the hierarchical nets. We find different types of criticality, illustrating the crucial effect of other structural properties besides the scale-free degree distribution of the nets.Comment: 9 Pages, 11 figures. References added and minor corrections to manuscript. In pres

Dynamical Scaling Behavior of Percolation Clusters in Scale-free Networks

In this work we investigate the spectra of Laplacian matrices that determine many dynamic properties of scale-free networks below and at the percolation threshold. We use a replica formalism to develop analytically, based on an integral equation, a systematic way to determine the ensemble averaged eigenvalue spectrum for a general type of tree-like networks. Close to the percolation threshold we find characteristic scaling functions for the density of states rho(lambda) of scale-free networks. rho(lambda) shows characteristic power laws rho(lambda) ~ lambda^alpha_1 or rho(lambda) ~ lambda^alpha_2 for small lambda, where alpha_1 holds below and alpha_2 at the percolation threshold. In the range where the spectra are accessible from a numerical diagonalization procedure the two methods lead to very similar results.Comment: 9 pages, 6 figure