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

A weighted configuration model and inhomogeneous epidemics

A random graph model with prescribed degree distribution and degree dependent edge weights is introduced. Each vertex is independently equipped with a random number of half-edges and each half-edge is assigned an integer valued weight according to a distribution that is allowed to depend on the degree of its vertex. Half-edges with the same weight are then paired randomly to create edges. An expression for the threshold for the appearance of a giant component in the resulting graph is derived using results on multi-type branching processes. The same technique also gives an expression for the basic reproduction number for an epidemic on the graph where the probability that a certain edge is used for transmission is a function of the edge weight. It is demonstrated that, if vertices with large degree tend to have large (small) weights on their edges and if the transmission probability increases with the edge weight, then it is easier (harder) for the epidemic to take off compared to a randomized epidemic with the same degree and weight distribution. A recipe for calculating the probability of a large outbreak in the epidemic and the size of such an outbreak is also given. Finally, the model is fitted to three empirical weighted networks of importance for the spread of contagious diseases and it is shown that $R_0$ can be substantially over- or underestimated if the correlation between degree and weight is not taken into account

Routine monitoring and assessment of adults living with HIV: results of the British HIV Association (BHIVA) national audit 2015

BACKGROUND: The clinical care of people living with HIV changed fundamentally as a result of the development of effective antiretroviral therapy (ART). HIV infection is now a long-term treatable condition. We report a national audit to assess adherence to British HIV Association guidelines for the routine investigation and monitoring of adult HIV-1-infected individuals. METHODS: All UK sites known as providers of adult HIV outpatient services were invited to complete a case-note review and a brief survey of local clinic practices. Participating sites were asked to randomly select 50-100 adults, who attended for specialist HIV care during 2014 and/or 2015. Each site collected data electronically using a self-audit spreadsheet tool. This included demographic details (gender, ethnicity, HIV exposure, and age) and whether 22 standardised and pre-defined clinical audited outcomes had been recorded. RESULTS: Data were collected on 8258 adults from 123 sites, representing approximately 10% of people living with HIV reported in public health surveillance as attending UK HIV services. Sexual health screening was provided within 96.4% of HIV services, cervical cytology and influenza vaccination within 71.4% of HIV services. There was wide variation in resistance testing across sites. Only 44.9% of patients on ART had a documented 10-year CVD risk within the past three years and fracture risk had been assessed within the past three years for only 16.7% patients aged over 50 years. CONCLUSIONS: There was high participation in the national audit and good practice was identified in some areas. However improvements can be made in monitoring of cardiovascular risk, bone and sexual health

Routine monitoring and assessment of adults living with HIV: results of the British HIV Association (BHIVA) national audit 2015.

BACKGROUND: The clinical care of people living with HIV changed fundamentally as a result of the development of effective antiretroviral therapy (ART). HIV infection is now a long-term treatable condition. We report a national audit to assess adherence to British HIV Association guidelines for the routine investigation and monitoring of adult HIV-1-infected individuals. METHODS: All UK sites known as providers of adult HIV outpatient services were invited to complete a case-note review and a brief survey of local clinic practices. Participating sites were asked to randomly select 50-100 adults, who attended for specialist HIV care during 2014 and/or 2015. Each site collected data electronically using a self-audit spreadsheet tool. This included demographic details (gender, ethnicity, HIV exposure, and age) and whether 22 standardised and pre-defined clinical audited outcomes had been recorded. RESULTS: Data were collected on 8258 adults from 123 sites, representing approximately 10% of people living with HIV reported in public health surveillance as attending UK HIV services. Sexual health screening was provided within 96.4% of HIV services, cervical cytology and influenza vaccination within 71.4% of HIV services. There was wide variation in resistance testing across sites. Only 44.9% of patients on ART had a documented 10-year CVD risk within the past three years and fracture risk had been assessed within the past three years for only 16.7% patients aged over 50 years. CONCLUSIONS: There was high participation in the national audit and good practice was identified in some areas. However improvements can be made in monitoring of cardiovascular risk, bone and sexual health

Network robustness and fragility: Percolation on random graphs

Recent work on the internet, social networks, and the power grid has addressed the resilience of these networks to either random or targeted deletion of network nodes. Such deletions include, for example, the failure of internet routers or power transmission lines. Percolation models on random graphs provide a simple representation of this process, but have typically been limited to graphs with Poisson degree distribution at their vertices. Such graphs are quite unlike real world networks, which often possess power-law or other highly skewed degree distributions. In this paper we study percolation on graphs with completely general degree distribution, giving exact solutions for a variety of cases, including site percolation, bond percolation, and models in which occupation probabilities depend on vertex degree. We discuss the application of our theory to the understanding of network resilience.Comment: 4 pages, 2 figure

A statistical network analysis of the HIV/AIDS epidemics in Cuba

The Cuban contact-tracing detection system set up in 1986 allowed the reconstruction and analysis of the sexual network underlying the epidemic (5,389 vertices and 4,073 edges, giant component of 2,386 nodes and 3,168 edges), shedding light onto the spread of HIV and the role of contact-tracing. Clustering based on modularity optimization provides a better visualization and understanding of the network, in combination with the study of covariates. The graph has a globally low but heterogeneous density, with clusters of high intraconnectivity but low interconnectivity. Though descriptive, our results pave the way for incorporating structure when studying stochastic SIR epidemics spreading on social networks

Random graphs with arbitrary degree distributions and their applications

Recent work on the structure of social networks and the internet has focussed attention on graphs with distributions of vertex degree that are significantly different from the Poisson degree distributions that have been widely studied in the past. In this paper we develop in detail the theory of random graphs with arbitrary degree distributions. In addition to simple undirected, unipartite graphs, we examine the properties of directed and bipartite graphs. Among other results, we derive exact expressions for the position of the phase transition at which a giant component first forms, the mean component size, the size of the giant component if there is one, the mean number of vertices a certain distance away from a randomly chosen vertex, and the average vertex-vertex distance within a graph. We apply our theory to some real-world graphs, including the world-wide web and collaboration graphs of scientists and Fortune 1000 company directors. We demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world, while in others there is a measurable discrepancy between theory and reality, perhaps indicating the presence of additional social structure in the network that is not captured by the random graph.Comment: 19 pages, 11 figures, some new material added in this version along with minor updates and correction

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

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

Analysis of lesion localisation at colonoscopy: outcomes from a multi-centre U.K. study

Background: Colonoscopy is currently the gold standard for detection of colorectal lesions, but may be limited in anatomically localising lesions. This audit aimed to determine the accuracy of colonoscopy lesion localisation, any subsequent changes in surgical management and any potentially influencing factors. Methods: Patients undergoing colonoscopy prior to elective curative surgery for colorectal lesion/s were included from 8 registered U.K. sites (2012–2014). Three sets of data were recorded: patient factors (age, sex, BMI, screener vs. symptomatic, previous abdominal surgery); colonoscopy factors (caecal intubation, scope guide used, colonoscopist accreditation) and imaging modality. Lesion localisation was standardised with intra-operative location taken as the gold standard. Changes to surgical management were recorded. Results: 364 cases were included; majority of lesions were colonic, solitary, malignant and in symptomatic referrals. 82% patients had their lesion/s correctly located at colonoscopy. Pre-operative CT visualised lesion/s in only 73% of cases with a reduction in screening patients (64 vs. 77%; p = 0.008). 5.2% incorrectly located cases at colonoscopy underwent altered surgical management, including conversion to open. Univariate analysis found colonoscopy accreditation, scope guide use, incomplete colonoscopy and previous abdominal surgery significantly influenced lesion localisation. On multi-variate analysis, caecal intubation and scope guide use remained significant (HR 0.35, 0.20–0.60 95% CI and 0.47; 0.25–0.88, respectively). Conclusion: Lesion localisation at colonoscopy is incorrect in 18% of cases leading to potentially significant surgical management alterations. As part of accreditation, colonoscopists need lesion localisation training and awareness of when inaccuracies can occur