Network 'small-world-ness': a quantitative method for determining canonical network equivalence

Background: Many technological, biological, social, and information networks fall into the broad class of 'small-world' networks: they have tightly interconnected clusters of nodes, and a shortest mean path length that is similar to a matched random graph (same number of nodes and edges). This semi-quantitative definition leads to a categorical distinction ('small/not-small') rather than a quantitative, continuous grading of networks, and can lead to uncertainty about a network's small-world status. Moreover, systems described by small-world networks are often studied using an equivalent canonical network model-the Watts-Strogatz (WS) model. However, the process of establishing an equivalent WS model is imprecise and there is a pressing need to discover ways in which this equivalence may be quantified. Methodology/Principal Findings: We defined a precise measure of 'small-world-ness' S based on the trade off between high local clustering and short path length. A network is now deemed a 'small-world' if S. 1-an assertion which may be tested statistically. We then examined the behavior of S on a large data-set of real-world systems. We found that all these systems were linked by a linear relationship between their S values and the network size n. Moreover, we show a method for assigning a unique Watts-Strogatz (WS) model to any real-world network, and show analytically that the WS models associated with our sample of networks also show linearity between S and n. Linearity between S and n is not, however, inevitable, and neither is S maximal for an arbitrary network of given size. Linearity may, however, be explained by a common limiting growth process. Conclusions/Significance: We have shown how the notion of a small-world network may be quantified. Several key properties of the metric are described and the use of WS canonical models is placed on a more secure footing

Evaluation of vaccination strategies for SIR epidemics on random networks incorporating household structure

This paper is concerned with the analysis of vaccination strategies in a stochastic SIR (susceptible → infected → removed) model for the spread of an epidemic amongst a population of individuals with a random network of social contacts that is also partitioned into households. Under various vaccine action models, we consider both household-based vaccination schemes, in which the way in which individuals are chosen for vaccination depends on the size of the households in which they reside, and acquaintance vaccination, which targets individuals of high degree in the social network. For both types of vaccination scheme, assuming a large population with few initial infectives, we derive a threshold parameter which determines whether or not a large outbreak can occur and also the probability and fraction of the population infected by such an outbreak. The performance of these schemes is studied numerically, focusing on the influence of the household size distribution and the degree distribution of the social network. We find that acquaintance vaccination can significantly outperform the best household-based scheme if the degree distribution of the social network is heavy-tailed. For household-based schemes, when the vaccine coverage is insufficient to prevent a major outbreak and the vaccine is imperfect, we find situations in which both the probability and size of a major outbreak under the scheme which minimises the threshold parameter are \emph{larger} than in the scheme which maximises the threshold parameter

Low pH immobilizes and kills human leukocytes and prevents transmission of cell-associated HIV in a mouse model

BACKGROUND: Both cell-associated and cell-free HIV virions are present in semen and cervical secretions of HIV-infected individuals. Thus, topical microbicides may need to inactivate both cell-associated and cell-free HIV to prevent sexual transmission of HIV/AIDS. To determine if the mild acidity of the healthy vagina and acid buffering microbicides would prevent transmission by HIV-infected leukocytes, we measured the effect of pH on leukocyte motility, viability and intracellular pH and tested the ability of an acidic buffering microbicide (BufferGel(®)) to prevent the transmission of cell-associated HIV in a HuPBL-SCID mouse model. METHODS: Human lymphocyte, monocyte, and macrophage motilities were measured as a function of time and pH using various acidifying agents. Lymphocyte and macrophage motilities were measured using video microscopy. Monocyte motility was measured using video microscopy and chemotactic chambers. Peripheral blood mononuclear cell (PBMC) viability and intracellular pH were determined as a function of time and pH using fluorescent dyes. HuPBL-SCID mice were pretreated with BufferGel, saline, or a control gel and challenged with HIV-1-infected human PBMCs. RESULTS: Progressive motility was completely abolished in all cell types between pH 5.5 and 6.0. Concomitantly, at and below pH 5.5, the intracellular pH of PBMCs dropped precipitously to match the extracellular medium and did not recover. After acidification with hydrochloric acid to pH 4.5 for 60 min, although completely immotile, 58% of PBMCs excluded ethidium homodimer-1 (dead-cell dye). In contrast, when acidified to this pH with BufferGel, a microbicide designed to maintain vaginal acidity in the presence of semen, only 4% excluded dye at 10 min and none excluded dye after 30 min. BufferGel significantly reduced transmission of HIV-1 in HuPBL-SCID mice (1 of 12 infected) compared to saline (12 of 12 infected) and a control gel (5 of 7 infected). CONCLUSION: These results suggest that physiologic or microbicide-induced acid immobilization and killing of infected white blood cells may be effective in preventing sexual transmission of cell-associated HIV