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

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

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

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

Network growth for enhanced natural selection

Natural selection and random drift are competing phenomena for explaining the evolution of populations. Combining a highly fit mutant with a population structure that improves the odds that the mutant spreads through the whole population tips the balance in favor of natural selection. The probability that the spread occurs, known as the fixation probability, depends heavily on how the population is structured. Certain topologies, albeit highly artificially contrived, have been shown to exist that favor fixation. We introduce a randomized mechanism for network growth that is loosely inspired in some of these topologies' key properties and demonstrate, through simulations, that it is capable of giving rise to structured populations for which the fixation probability significantly surpasses that of an unstructured population. This discovery provides important support to the notion that natural selection can be enhanced over random drift in naturally occurring population structures

Circuits in random graphs: from local trees to global loops

We compute the number of circuits and of loops with multiple crossings in random regular graphs. We discuss the importance of this issue for the validity of the cavity approach. On the one side we obtain analytic results for the infinite volume limit in agreement with existing exact results. On the other side we implement a counting algorithm, enumerate circuits at finite N and draw some general conclusions about the finite N behavior of the circuits.Comment: submitted to JSTA

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

Efficacy of Repeat Selective Laser Trabeculoplasty in Medication-Naïve Open Angle Glaucoma and Ocular Hypertension during the LiGHT Trial

PURPOSE: To determine the efficacy of repeat selective laser trabeculoplasty (SLT) in medication-naïve open angle glaucoma (OAG) and ocular hypertensive (OHT) patients requiring repeat treatment for early to medium-term failure during the Laser in Glaucoma and Ocular Hypertension (LiGHT) trial. // DESIGN: Post-hoc analysis of SLT treatment arm of a multicentre prospective randomised-controlled-trial. // PARTICIPANTS: Treatment-naïve OAG or OHT requiring repeat 360-degree SLT within 18 months. Re-treatment was triggered by pre-defined IOP and disease-progression criteria (using objective individualised target IOPs) // METHODS: After SLT at baseline, patients were followed for a minimum of 18 months after second (‘repeat’) SLT. A mixed model analysis was performed with the eye as the unit of analysis, with crossed random-effects to adjust for correlation between fellow eyes and repeated measures within eyes. Kaplan-Meier curves plot the duration of effect. // OUTCOME MEASURES: Initial (‘early’) IOP lowering at 2-months and duration of effect following initial and Repeat SLT. // RESULTS: 115 eyes of 90 patients received Repeat SLT during first 18 months of the trial. Pre-treatment IOP prior to Initial SLT was significantly higher than that prior to pre-retreatment IOP of Repeat SLT (mean difference: 3.4, 95% confidence interval (CI) 2.6 to 4.3, mmHg; p<0.001). Absolute IOP reduction at 2-months was greater following Initial, compared to Repeat, SLT (mean difference: 1.0, 95% CI 0.2 to 1.8, mmHg; p=0.02). Adjusted absolute IOP reduction at 2-months (adjusting for IOP prior to initial or repeat laser) was greater following Repeat SLT (adjusted mean difference: -1.1, 95% CI -1.7 to -0.5, mmHg; p=0.001). 34 eyes were ‘early failures’ (retreated 2-months after Initial SLT) vs 81 ‘later failures’ (retreatment beyond 2-months following Initial SLT). No significant difference in early absolute IOP reduction at 2-months following Repeat SLT was noted between ‘early’ vs ‘later’ failures’ (mean difference: 0.3, 95% CI, -1.1 to 1.8,mmHg; p=0.655). Repeat SLT maintained drop-free IOP control in 67% of 115 eyes at 18 months, with no clinically-relevant adverse events. // CONCLUSION: These exploratory analyses demonstrate Repeat SLT can maintain IOP at or below Target IOP in medication-naive OAG and OHT eyes requiring retreatment with atleast an equivalent duration of effect to initial laser