Traceroute sampling makes random graphs appear to have power law degree distributions

The topology of the Internet has typically been measured by sampling traceroutes, which are roughly shortest paths from sources to destinations. The resulting measurements have been used to infer that the Internet's degree distribution is scale-free; however, many of these measurements have relied on sampling traceroutes from a small number of sources. It was recently argued that sampling in this way can introduce a fundamental bias in the degree distribution, for instance, causing random (Erdos-Renyi) graphs to appear to have power law degree distributions. We explain this phenomenon analytically using differential equations to model the growth of a breadth-first tree in a random graph G(n,p=c/n) of average degree c, and show that sampling from a single source gives an apparent power law degree distribution P(k) ~ 1/k for k < c

An algorithm for counting circuits: application to real-world and random graphs

We introduce an algorithm which estimates the number of circuits in a graph as a function of their length. This approach provides analytical results for the typical entropy of circuits in sparse random graphs. When applied to real-world networks, it allows to estimate exponentially large numbers of circuits in polynomial time. We illustrate the method by studying a graph of the Internet structure.Comment: 7 pages, 3 figures, minor corrections, accepted versio

Morphometric analysis of the submarine arc volcano Monowai (Tofua – Kermadec Arc) to decipher tectono-magmatic interactions

Morphometric analysis of multibeam bathymetry and backscatter data is applied to Monowai, a submarine volcano of the active Tofua–Kermadec Arc to map and document the structure and evolution of the volcanic centre. Low rates of erosion and sedimentation, and pervasive tectonic and magmatic processes, allow quantification through detailed structural analysis and measurement of deformation. The Slope, Aspect, Curvature, Rugosity, and Hydrology (flow) tools of ArcGIS provide a robust structural interpretation and the development of a model of Monowai evolution.A nested caldera structure with a volume of ~ 31 km3 and a stratovolcano of ~ 18 km3 dominate the magmatic constructs. The outer caldera is elongate along 125°, and the inner caldera along 135°. Numerous parasitic cones and fissure ridges are also observed, oriented at 039° and 041°, respectively. Northeast trending faults (with a regional average strike of 031°) are widespread within this part of the backarc, forming a nascent rift graben to the west of the Monowai caldera complex. The distribution of throw varies spatially, reaching a maximum total along-rift of 320 m and across rift of 120 m, with greater throw values measured in the west.Elongation directions of the two nested calderas are near-perpendicular to the trends of faults and fissure ridges. The inner caldera is more orthogonal to the magmatic constructs (fissure ridges and aligned vent cones) and the outer caldera is approximately orthogonal to the regional fault fabric, suggesting a strong interaction between magmatic and tectonic processes, and the directions of the horizontal principal stress. We present a detailed morphometric analysis of these relationships and the data are used to interpret the spatial and temporal evolution of the tectono-magmatic system at Monowai, and classify the type of rifting as transtensional. Similar analysis is possible elsewhere in the Kermadec backarc and within other regions of submarine volcanism

A variable amplitude fretting fatigue life estimation technique: formulation and experimental validation

The aims of the research work summarised in this paper are twofold. The first goal is to make available a large number of new experimental results generated by testing specimens of grey cast iron under both constant and variable amplitude fretting fatigue loading. The second goal is to formulate an advanced fretting fatigue design approach based on the combined use of the Modified Wӧhler Curve Method, the Theory of Critical Distances and the Shear Stress-Maximum Variance Method. The validation exercise based on the experimental results being produced demonstrates that the proposed methodology is a powerful tool suitable for designing mechanical assemblies against fretting fatigue

Quilting sutures for nasal septum

Suturing of the nasal septum after septal surgery is a commonly performed procedure designed to prevent complications such as septal haematoma and bleeding. It is also useful for closing any inadvertent tears of the septal mucosa and providing additional support for the cartilage pieces retained in septoplasty. In addition, the suture can be placed through the middle turbinates, stabilising them during the healing process. Placing knots for interrupted sutures in the posterior and middle part of the nasal septum can be technically difficult. We describe a continuous suturing technique for approximating the mucosal flaps following septal surgery.C Hari, C Marnane and P J Wormal

High degree graphs contain large-star factors

We show that any finite simple graph with minimum degree $d$ contains a spanning star forest in which every connected component is of size at least $\Omega((d/\log d)^{1/3})$. This settles a problem of J. Kratochvil

On Bootstrap Percolation in Living Neural Networks

Recent experimental studies of living neural networks reveal that their global activation induced by electrical stimulation can be explained using the concept of bootstrap percolation on a directed random network. The experiment consists in activating externally an initial random fraction of the neurons and observe the process of firing until its equilibrium. The final portion of neurons that are active depends in a non linear way on the initial fraction. The main result of this paper is a theorem which enables us to find the asymptotic of final proportion of the fired neurons in the case of random directed graphs with given node degrees as the model for interacting network. This gives a rigorous mathematical proof of a phenomena observed by physicists in neural networks

Finding long cycles in graphs

We analyze the problem of discovering long cycles inside a graph. We propose and test two algorithms for this task. The first one is based on recent advances in statistical mechanics and relies on a message passing procedure. The second follows a more standard Monte Carlo Markov Chain strategy. Special attention is devoted to Hamiltonian cycles of (non-regular) random graphs of minimal connectivity equal to three

Self-organization of collaboration networks

We study collaboration networks in terms of evolving, self-organizing bipartite graph models. We propose a model of a growing network, which combines preferential edge attachment with the bipartite structure, generic for collaboration networks. The model depends exclusively on basic properties of the network, such as the total number of collaborators and acts of collaboration, the mean size of collaborations, etc. The simplest model defined within this framework already allows us to describe many of the main topological characteristics (degree distribution, clustering coefficient, etc.) of one-mode projections of several real collaboration networks, without parameter fitting. We explain the observed dependence of the local clustering on degree and the degree--degree correlations in terms of the ``aging'' of collaborators and their physical impossibility to participate in an unlimited number of collaborations.Comment: 10 pages, 8 figure

Ising Model on Networks with an Arbitrary Distribution of Connections

We find the exact critical temperature $T_c$ of the nearest-neighbor ferromagnetic Ising model on an `equilibrium' random graph with an arbitrary degree distribution $P(k)$. We observe an anomalous behavior of the magnetization, magnetic susceptibility and specific heat, when $P(k)$ is fat-tailed, or, loosely speaking, when the fourth moment of the distribution diverges in infinite networks. When the second moment becomes divergent, $T_c$ approaches infinity, the phase transition is of infinite order, and size effect is anomalously strong.Comment: 5 page