Benchmarks for testing community detection algorithms on directed and weighted graphs with overlapping communities

Many complex networks display a mesoscopic structure with groups of nodes sharing many links with the other nodes in their group and comparatively few with nodes of different groups. This feature is known as community structure and encodes precious information about the organization and the function of the nodes. Many algorithms have been proposed but it is not yet clear how they should be tested. Recently we have proposed a general class of undirected and unweighted benchmark graphs, with heterogenous distributions of node degree and community size. An increasing attention has been recently devoted to develop algorithms able to consider the direction and the weight of the links, which require suitable benchmark graphs for testing. In this paper we extend the basic ideas behind our previous benchmark to generate directed and weighted networks with built-in community structure. We also consider the possibility that nodes belong to more communities, a feature occurring in real systems, like, e. g., social networks. As a practical application, we show how modularity optimization performs on our new benchmark.Comment: 9 pages, 13 figures. Final version published in Physical Review E. The code to create the benchmark graphs can be freely downloaded from http://santo.fortunato.googlepages.com/inthepress

Spin Glass Phase Transition on Scale-Free Networks

We study the Ising spin glass model on scale-free networks generated by the static model using the replica method. Based on the replica-symmetric solution, we derive the phase diagram consisting of the paramagnetic (P), ferromagnetic (F), and spin glass (SG) phases as well as the Almeida-Thouless line as functions of the degree exponent $\lambda$, the mean degree $K$, and the fraction of ferromagnetic interactions $r$. To reflect the inhomogeneity of vertices, we modify the magnetization $m$ and the spin glass order parameter $q$ with vertex-weights. The transition temperature $T_c$ ($T_g$) between the P-F (P-SG) phases and the critical behaviors of the order parameters are found analytically. When $2 < \lambda < 3$, $T_c$ and $T_g$ are infinite, and the system is in the F phase or the mixed phase for $r > 1/2$, while it is in the SG phase at $r=1/2$. $m$ and $q$ decay as power-laws with increasing temperature with different $\lambda$-dependent exponents. When $\lambda > 3$, the $T_c$ and $T_g$ are finite and related to the percolation threshold. The critical exponents associated with $m$ and $q$ depend on $\lambda$ for $3 < \lambda < 5$ ($3 < \lambda < 4$) at the P-F (P-SG) boundary.Comment: Phys. Rev. E in pres

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

Effectiveness of a social support intervention on infant feeding practices : randomised controlled trial

Background: To assess whether monthly home visits from trained volunteers could improve infant feeding practices at age 12 months, a randomised controlled trial was carried out in two disadvantaged inner city London boroughs. Methods: Women attending baby clinics with their infants (312) were randomised to receive monthly home visits from trained volunteers over a 9-month period (intervention group) or standard professional care only (control group). The primary outcome was vitamin C intakes from fruit. Secondary outcomes included selected macro and micro-nutrients, infant feeding habits, supine length and weight. Data were collected at baseline when infants were aged approximately 10 weeks, and subsequently when the child was 12 and 18 months old. Results: Two-hundred and twelve women (68%) completed the trial. At both follow-up points no significant differences were found between the groups for vitamin C intakes from fruit or other nutrients. At first follow-up, however, infants in the intervention group were significantly less likely to be given goats’ or soya milks, and were more likely to have three solid meals per day. At the second follow-up, intervention group children were significantly less likely to be still using a bottle. At both follow-up points, intervention group children also consumed significantly more specific fruit and vegetables. Conclusions: Home visits from trained volunteers had no significant effect on nutrient intakes but did promote some other recommended infant feeding practices

Vertex similarity in networks

We consider methods for quantifying the similarity of vertices in networks. We propose a measure of similarity based on the concept that two vertices are similar if their immediate neighbors in the network are themselves similar. This leads to a self-consistent matrix formulation of similarity that can be evaluated iteratively using only a knowledge of the adjacency matrix of the network. We test our similarity measure on computer-generated networks for which the expected results are known, and on a number of real-world networks

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

Statistical ensemble of scale-free random graphs

A thorough discussion of the statistical ensemble of scale-free connected random tree graphs is presented. Methods borrowed from field theory are used to define the ensemble and to study analytically its properties. The ensemble is characterized by two global parameters, the fractal and the spectral dimensions, which are explicitly calculated. It is discussed in detail how the geometry of the graphs varies when the weights of the nodes are modified. The stability of the scale-free regime is also considered: when it breaks down, either a scale is spontaneously generated or else, a "singular" node appears and the graphs become crumpled. A new computer algorithm to generate these random graphs is proposed. Possible generalizations are also discussed. In particular, more general ensembles are defined along the same lines and the computer algorithm is extended to arbitrary (degenerate) scale-free random graphs.Comment: 10 pages, 6 eps figures, 2-column revtex format, minor correction

Cortisol levels and history of depression in acute coronary syndrome patients

Background Depressed mood following an acute coronary syndrome (ACS) is a risk factor for future cardiac morbidity. Hypothalamic-pituitary-adrenal (HPA) axis dysregulation is associated with depression, and may be a process through which depressive symptoms influence later cardiac health. Additionally, a history of depression predicts depressive symptoms in the weeks following ACS. The purpose of this study was to determine whether a history of depression and/or current depression are associated with the HPA axis dysregulation following ACS. Method A total of 152 cardiac patients completed a structured diagnostic interview, a standardized depression questionnaire and a cortisol profile over the day, 3 weeks after an ACS. Cortisol was analysed using: the cortisol awakening response (CAR), total cortisol output estimated using the area under the curve method, and the slope of cortisol decline over the day. Results Total cortisol output was positively associated with history of depression, after adjustment for age, gender, marital status, ethnicity, smoking status, body mass index (BMI), Global Registry of Acute Cardiac Events (GRACE) risk score, days in hospital, medication with statins and antiplatelet compounds, and current depression score. Men with clinically diagnosed depression after ACS showed a blunted CAR, but the CAR was not related to a history of depression. Conclusions Patients with a history of depression showed increased total cortisol output, but this is unlikely to be responsible for associations between depression after ACS and later cardiac morbidity. However, the blunted CAR in patients with severe depression following ACS indicates that HPA dysregulation is presen

Does the McNeill Alexander model accurately predict maximum walking speed in novice and experienced race walkers?

Background: Mathematical models propose leg length as a limiting factor in determining the maximum walking velocity. This study evaluated the effectiveness of a leg length-based model in predicting maximum walking velocity in an applied race walking situation, by comparing experienced and novice race walkers during conditions where strictly no flight time (FT) was permitted and in simulated competition conditions (i.e., FT ≤ 40 ms). Methods: Thirty-four participants (18 experienced and 16 novice race walkers) were recruited for this investigation. An Optojump Next system (8 m) was used to determine walking velocity, step frequency, step length, ground contact time, and FT during race walking over a range of velocities. Comparisons were made between novice and experienced participants in predicted maximum velocity and actual velocities achieved with no flight and velocities with FT ≤ 40 ms. The technical effectiveness of the participants was assessed using the ratio of maximum velocity to predicted velocity. Results: In novices, no significant difference was found between predicted and maximum walking speeds without flight time but there was a small 5.8% gain in maximum speed when FT ≤ 40 ms. In experienced race walkers, there was a significant reduction in maximum walking speed compared with predicted maximum (p < 0.01) and a 11.7% gain in maximum walking speed with FT ≤ 40 ms. Conclusion: The analysis showed that leg length was a good predictor of maximal walking velocity in novice walkers but not a good predictor of maximum walking speed in well-trained walkers who appear to have optimised their walking technique to make use of non-visible flight periods of less than 40 ms. The gain in velocity above predicted maximum may be a useful index of race walking proficiency

Emotional triggering and low socio-economic status as determinants of depression following acute coronary syndrome

Background The determinants of depression following acute coronary syndrome (ACS) are poorly understood. Triggering of ACS by emotional stress and low socio-economic status (SES) are predictors of adverse outcomes. We therefore investigated whether emotional triggering and low SES predict depression and anxiety following ACS. Method This prospective observational clinical cohort study involved 298 patients with clinically verified ACS. Emotional stress was assessed for the 2 h before symptom onset and compared with the equivalent period 24 h earlier using case-crossover methods. SES was defined by household income and education. Depression was measured with the Beck Depression Inventory and the Hamilton Rating Scale for Depression and anxiety with the Hospital Anxiety and Depression Scale 3 weeks after ACS and again at 6 and 12 months. Age, gender, ethnicity, marital status, the Global Registry of Acute Coronary Events risk score, duration of hospital stay and history of depression were included as covariates. Results Emotional stress during the 2-h hazard period was associated with increased risk of ACS (odds ratio 1.88, 95% confidence interval 1.01-3.61). Both low income and emotional triggering predicted depression and anxiety at 3 weeks and 6/12 months independently of covariates. The two factors interacted, with the greatest depression and anxiety in lower income patients who experienced acute emotional stress. Education was not related to depression. Conclusions Patients who experience acute emotional stress during their ACS and are lower SES as defined by current affluence and access to resources are particularly vulnerable to subsequent depression and anxiet