Generalized percolation in random directed networks

We develop a general theory for percolation in directed random networks with arbitrary two point correlations and bidirectional edges, that is, edges pointing in both directions simultaneously. These two ingredients alter the previously known scenario and open new views and perspectives on percolation phenomena. Equations for the percolation threshold and the sizes of the giant components are derived in the most general case. We also present simulation results for a particular example of uncorrelated network with bidirectional edges confirming the theoretical predictions

Giant strongly connected component of directed networks

We describe how to calculate the sizes of all giant connected components of a directed graph, including the {\em strongly} connected one. Just to the class of directed networks, in particular, belongs the World Wide Web. The results are obtained for graphs with statistically uncorrelated vertices and an arbitrary joint in,out-degree distribution $P(k_i,k_o)$. We show that if $P(k_i,k_o)$ does not factorize, the relative size of the giant strongly connected component deviates from the product of the relative sizes of the giant in- and out-components. The calculations of the relative sizes of all the giant components are demonstrated using the simplest examples. We explain that the giant strongly connected component may be less resilient to random damage than the giant weakly connected one.Comment: 4 pages revtex, 4 figure

Evolution equation for a model of surface relaxation in complex networks

In this paper we derive analytically the evolution equation of the interface for a model of surface growth with relaxation to the minimum (SRM) in complex networks. We were inspired by the disagreement between the scaling results of the steady state of the fluctuations between the discrete SRM model and the Edward-Wilkinson process found in scale-free networks with degree distribution $P(k) \sim k^{-\lambda}$ for $\lambda <3$ [Pastore y Piontti {\it et al.}, Phys. Rev. E {\bf 76}, 046117 (2007)]. Even though for Euclidean lattices the evolution equation is linear, we find that in complex heterogeneous networks non-linear terms appear due to the heterogeneity and the lack of symmetry of the network; they produce a logarithmic divergency of the saturation roughness with the system size as found by Pastore y Piontti {\it et al.} for $\lambda <3$.Comment: 9 pages, 2 figure

Ageing and menopause considerations for women with HIV in the UK

OBJECTIVES: Treatment rollout has dramatically improved life expectancy for people with HIV and AIDS. Women represent a substantial proportion of patients in the UK (approximately one-third of patients in care are female according to the HIV Annual Report 2014). This study examines psychosocial and biomedical issues for women diagnosed with HIV in the UK, comparing those above and below 45 years of age to examine menopause and ageing issues. METHODS: Consecutive clinic attenders in a large outpatient London HIV clinic were invited to participate in the study. Data were available for 170 (68%) women. In 57 women above the age of 45 data were available regarding menopause detailed insights. RESULTS: Compared with women aged under 45, women >45 years old were significantly less likely to be in a relationship (P=0.01), had higher anxiety scores (P=0.002), more likely to be classified as moderate to severe (25.9% vs 9.1%; χ(2)=6.1, P=0.01). There were no differences in terms of suicidal ideation, which was high for both groups of women (56.6%). Older women had higher psychological symptoms on the MSAS scale form and significantly higher PHQ-9 depression levels. A higher proportion of older women scored above the cut-off point for moderate to severe depression (9.2% vs 21.8%; χ(2)=3.7, P=0.048). Fewer older women had no mental health challenges (26.1% vs 42.4%) and more had multiple comorbidities (P=0.07). CONCLUSIONS: The vast majority of women reported experiencing a variety of physical and psychological menopause-related symptoms and there was a high suicide ideation rate in both groups of women. Over half of the group of menopausal women recorded distressing symptoms such as hot flushes, sweating, decreased sexual desire, back pain, night sweats, avoiding intimacy, involuntary urination and skin changes, yet few sought help. Age-specific, psychosexual and menopause services should be routinely available for women with HIV

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

Non-equilibrium mean-field theories on scale-free networks

Many non-equilibrium processes on scale-free networks present anomalous critical behavior that is not explained by standard mean-field theories. We propose a systematic method to derive stochastic equations for mean-field order parameters that implicitly account for the degree heterogeneity. The method is used to correctly predict the dynamical critical behavior of some binary spin models and reaction-diffusion processes. The validity of our non-equilibrium theory is furtherly supported by showing its relation with the generalized Landau theory of equilibrium critical phenomena on networks.Comment: 4 pages, no figures, major changes in the structure of the pape

The Floor of the Arctic Ocean: Geographic Names

A table listing 54 major features of the ocean floor in the Arctic, gives the final suggested name, approx location, and the status of the name with the US Board on Geographic Names and the Intl. Hydrographic Bureau. Recommendations are based on decisions made at a meeting called by the US Navy Electronics Laboratory, San Diego, Calif, Jan 1966. The criteria for decisions were: consistency with Undersea terms and definitions (US Board on Geographic Names, 1964) and Limits of oceans and seas (Intl Hydrographic Bureau 1953), common usage, priority of discovery or naming, association with established geographic features, and minimizing ambiguity. It is suggested that a straight line across the narrowest constriction of Bering Strait should mark the southern boundary of Chukchi Sea, rather than the Arctic Circle, as recommended by the I.H.B., and that, in the absence of any hydrographic or physiographic features designating a unique region, the name Beaufort Sea should be dropped. The opinions ar those of the writers personally

Laplacian spectra of complex networks and random walks on them: Are scale-free architectures really important?

We study the Laplacian operator of an uncorrelated random network and, as an application, consider hopping processes (diffusion, random walks, signal propagation, etc.) on networks. We develop a strict approach to these problems. We derive an exact closed set of integral equations, which provide the averages of the Laplacian operator's resolvent. This enables us to describe the propagation of a signal and random walks on the network. We show that the determining parameter in this problem is the minimum degree $q_m$ of vertices in the network and that the high-degree part of the degree distribution is not that essential. The position of the lower edge of the Laplacian spectrum $\lambda_c$ appears to be the same as in the regular Bethe lattice with the coordination number $q_m$. Namely, $\lambda_c>0$ if $q_m>2$, and $\lambda_c=0$ if $q_m\leq2$. In both these cases the density of eigenvalues $\rho(\lambda)\to0$ as $\lambda\to\lambda_c+0$, but the limiting behaviors near $\lambda_c$ are very different. In terms of a distance from a starting vertex, the hopping propagator is a steady moving Gaussian, broadening with time. This picture qualitatively coincides with that for a regular Bethe lattice. Our analytical results include the spectral density $\rho(\lambda)$ near $\lambda_c$ and the long-time asymptotics of the autocorrelator and the propagator.Comment: 25 pages, 4 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

Generation of uncorrelated random scale-free networks

Uncorrelated random scale-free networks are useful null models to check the accuracy an the analytical solutions of dynamical processes defined on complex networks. We propose and analyze a model capable to generate random uncorrelated scale-free networks with no multiple and self-connections. The model is based on the classical configuration model, with an additional restriction on the maximum possible degree of the vertices. We check numerically that the proposed model indeed generates scale-free networks with no two and three vertex correlations, as measured by the average degree of the nearest neighbors and the clustering coefficient of the vertices of degree $k$, respectively