Comparing Multilevel Clustering Methods on Weighted Graphs : The Case of Worldwide Air Passenger Traffic 2000-2004

Specific properties emerge from the structure of large networks, such as that of worldwide air traffic, including a highly hierarchical node structure and multi-level small world sub-groups that strongly influence future dynamics. We have developed clustering methods to understand the form of these structures, to identify structural properties, and to evaluate the effects of these properties. Graph clustering methods are often constructed from different components: a metric, a clustering index, and a modularity measure to assess the quality of a clustering method. To understand the impact of each of these components on the clustering method, we explore and compare different combinations. These different combinations are used to compare multilevel clustering methods to delineate the effects of geographical distance, hubs, network densities, and bridges on worldwide air passenger traffic. The ultimate goal of this methodological research is to demonstrate evidence of combined effects in the development of an air traffic network. In fact, the network can be divided into different levels of âeurooecohesionâeuro, which can be qualified and measured by comparative studies (Newman, 2002; Guimera et al., 2005; Sales-Pardo et al., 2007).

A topological invariant to predict the three-dimensional writhe of ideal configurations of knots and links.

We present herein a topological invariant of oriented alternating knots and links that predicts the three-dimensional (3D) writhe of the ideal geometrical configuration of the considered knot/link. The fact that we can correlate a geometrical property of a given configuration with a topological invariant supports the notion that the ideal configuration contains important information about knots and links. The importance of the concept of ideal configuration was already suggested by the good correlation between the 3D writhe of ideal knot configurations and the ensemble average of the 3D writhe of random configurations of the considered knots. The values of the new invariant are quantized: multiples of 4/7 for links with an odd number of components (including knots) and 2/7 plus multiples of 4/7 for links with an even number of components.

Properties of ideal composite knots.

The shortest tube of constant diameter that can form a given knot represents the 'ideal' form of the knot. Ideal knots provide an irreducible representation of the knot, and they have some intriguing mathematical and physical features, including a direct correspondence with the time-averaged shapes of knotted DNA molecules in solution. Here we describe the properties of ideal forms of composite knots-knots obtained by the sequential tying of two or more independent knots (called factor knots) on the same string. We find that the writhe (related to the handedness of crossing points) of composite knots is the sum of that of the ideal forms of the factor knots. By comparing ideal composite knots with simulated configurations of knotted, thermally fluctuating DNA, we conclude that the additivity of writhe applies also to randomly distorted configurations of composite knots and their corresponding factor knots. We show that composite knots with several factor knots may possess distinct structural isomers that can be interconverted only by loosening the knot.

Subknots in ideal knots, random knots, and knotted proteins.

We introduce disk matrices which encode the knotting of all subchains in circular knot configurations. The disk matrices allow us to dissect circular knots into their subknots, i.e. knot types formed by subchains of the global knot. The identification of subknots is based on the study of linear chains in which a knot type is associated to the chain by means of a spatially robust closure protocol. We characterize the sets of observed subknot types in global knots taking energy-minimized shapes such as KnotPlot configurations and ideal geometric configurations. We compare the sets of observed subknots to knot types obtained by changing crossings in the classical prime knot diagrams. Building upon this analysis, we study the sets of subknots in random configurations of corresponding knot types. In many of the knot types we analyzed, the sets of subknots from the ideal geometric configurations are found in each of the hundreds of random configurations of the same global knot type. We also compare the sets of subknots observed in open protein knots with the subknots observed in the ideal configurations of the corresponding knot type. This comparison enables us to explain the specific dispositions of subknots in the analyzed protein knots.

The evolution of helping and harming on graphs: the return of the inclusive fitness effect.

Evolutionary graph theory has been proposed as providing new fundamental rules for the evolution of co-operation and altruism. But how do these results relate to those of inclusive fitness theory? Here, we carry out a retrospective analysis of the models for the evolution of helping on graphs of Ohtsuki et al. [Nature (2006) 441, 502] and Ohtsuki & Nowak [Proc. R. Soc. Lond. Ser. B Biol. Sci (2006) 273, 2249]. We show that it is possible to translate evolutionary graph theory models into classical kin selection models without disturbing at all the mathematics describing the net effect of selection on helping. Model analysis further demonstrates that costly helping evolves on graphs through limited dispersal and overlapping generations. These two factors are well known to promote relatedness between interacting individuals in spatially structured populations. By allowing more than one individual to live at each node of the graph and by allowing interactions to vary with the distance between nodes, our inclusive fitness model allows us to consider a wider range of biological scenarios leading to the evolution of both helping and harming behaviours on graphs.

Linear relations between writhe and minimal crossing number in Conway families of ideal knots and links

We showed earlier how to predict the writhe of any rational knot or link in its ideal geometric configuration, or equivalently the average of the 3D writhe over statistical ensembles of random configurations of a given knot or link (Cerf and Stasiak 2000 Proc. Natl Acad. Sci. USA 97 3795). There is no general relation between the minimal crossing number of a knot and the writhe of its ideal geometric configuration. However, within individual families of knots linear relations between minimal crossing number and writhe were observed (Katritch et al 1996 Nature 384 142). Here we present a method that allows us to express the writhe as a linear function of the minimal crossing number within Conway families of knots and links in their ideal configuration. The slope of the lines and the shift between any two lines with the same

Ideal and reality: do countries adopt and follow recommended procedures in comprehensive multiyear planning guidelines for national immunization programmes?

BACKGROUND: Meticulous steps and procedures are proposed in planning guidelines for the development of comprehensive multiyear plans for national immunization programmes. However, we know very little about whether the real-life experience of those who adopt these guidelines involves following these procedures as expected. Are these steps and procedures followed in practice? We examined the adoption and usage of the guidelines in planning national immunization programmes and assessed whether the recommendations in these guidelines are applied as consistently as intended. METHODS: We gathered information from the national comprehensive multiyear plans developed by 77 low-income countries. For each of the 11 components, we examined how each country applied the four recommended steps of situation analysis, problem prioritization, selection of interventions, and selection of indicators. We then conducted an analysis to determine the patterns of alignment of the comprehensive multiyear plans with those four recommended planning steps. RESULTS: Within the first 3 years following publication of the guidelines, 66 (86%) countries used the tool to develop their comprehensive multiyear plans. The funding conditions attached to the use of these guidelines appeared to influence their rapid adoption and usage. Overall, only 33 (43%) countries fully applied all four recommended planning steps of the guidelines. CONCLUSIONS: Adoption and usage of the guidelines for the development of comprehensive multiyear plans for national immunization programmes were rapid. However, our findings show substantial variation between the proposed planning ideals set out in the guidelines and actual use in practice. A better understanding of factors that influence how recommendations in public health guidelines are applied in practice could contribute to improvements in guidelines design. It could also help adjust strategies used to introduce them into public health programmes, with the ultimate goal of a greater health impact.

Ideal Body Size as a Mediator for the Gender-Specific Association Between Socioeconomic Status and Body Mass Index: Evidence From an Upper-Middle-Income Country in the African Region.

While obesity continues to rise globally, the associations between body size, gender, and socioeconomic status (SES) seem to vary in different populations, and little is known on the contribution of perceived ideal body size in the social disparity of obesity in African countries. We examined the gender and socioeconomic patterns of body mass index (BMI) and perceived ideal body size in the Seychelles, a middle-income small island state in the African region. We also assessed the potential role of perceived ideal body size as a mediator for the gender-specific association between SES and BMI. A population-based survey of 1,240 adults aged 25 to 64 years conducted in December 2013. Participants' BMI was calculated based on measured weight and height; ideal body size was assessed using a nine-silhouette instrument. Three SES indicators were considered: income, education, and occupation. BMI and perceived ideal body size were both higher among men of higher versus lower SES (p< .001) but lower among women of higher versus lower SES (p< .001), irrespective of the SES indicator used. Multivariate analysis showed a strong and direct association between perceived ideal body size and BMI in both men and women (p< .001) and was consistent with a potential mediating role of perceived ideal body size in the gender-specific associations between SES and BMI. Our study emphasizes the importance of gender and socioeconomic differences in BMI and ideal body size and suggests that public health interventions that promote perception of healthy weight could help mitigate SES-related disparities in BMI.