Random graphs with clustering

We offer a solution to a long-standing problem in the physics of networks, the creation of a plausible, solvable model of a network that displays clustering or transitivity -- the propensity for two neighbors of a network node also to be neighbors of one another. We show how standard random graph models can be generalized to incorporate clustering and give exact solutions for various properties of the resulting networks, including sizes of network components, size of the giant component if there is one, position of the phase transition at which the giant component forms, and position of the phase transition for percolation on the network.Comment: 5 pages, 2 figure

Equine Welfare: A Study of Dermatophilosis and the Management of Data Relevant to the Health and Wellbeing of Horses

This thesis considers aspects of equine welfare which have received little attention in the U.K. Skin disease, particularly bacterial skin disease, was highlighted as an area giving rise to concern with respect to equine welfare. Dermatophilosis was examined in detail as one of the commoner bacterial skin conditions responsible for animal suffering, and one for which management is often difficult. Essential fatty acids (EFAs) were evaluated as a dietary supplement in an alternative approach to the management of equine dermatophilosis. The pharmacokinetics of EFAs in the horse were investigated, with EFAs supplemented as evening primrose oil (EPO), containing linoleic acid (LA) and gamma-linolenic acid (GLA). A very slow conversion of LA to its active metabolites was found in the horse compared to other species. A daily dose regime of 20g of 80% EPO and 20% fish oil and vitamin E was adopted for the consequent treatment and prophylactic studies. In a placebo-controlled, double blind treatment study no significant effect was seen on severity or extent of distribution of lesions of dermatophilosis when horses received EFAs orally. When EFAs were supplemented over the traditional autumn high dermatophilosis risk period in a controlled prophylactic study, they did not prevent development of lesions or reduce incidence of infection. No significant improvement was afforded by EFAs on the condition of the coat, mane, tail or hooves, nor on general body condition. EFAs were not harmful and exerted no effect, adverse or beneficial, on haematological or biochemical parameters. The characteristics of D. congolensis were examined in relation to the site and severity of lesions of dermatophilosis, but no correlation was found. All isolates were different when examined by differential bacteriological growth characteristics and sodium dodecyl sulphate polyacrylamide gel electrophoresis (SDS-PAGE). Proteolytic enzyme production by D. congolensis was investigated with regard to the virulence of the organism, and several isolates demonstrated extracellular protease activity. The clinical and haematological consequences of bleeding horses at regular intervals were monitored in a group of animals maintained for commercial blood production. No adverse effect was recorded on clinical, protein or haematological profiles when 8 litres of blood were removed every three weeks. Thoroughbred animals supported regular bleeding better than non-Thoroughbred animals. A relational database system was created as a management tool for the manager of the horse herd. The information contained within the system, regarding horse details, bloodroom records and farm laboratory records, could be constantly updated. Rapid detection of poor performers or anaemic animals could permit prompt instigation of corrective action, avoiding undue animal distress. It is hoped that some of the work within this thesis has made a worthwhile contribution to the extension of knowledge concerning the welfare of horses in the U.K

Competing epidemics on complex networks

Human diseases spread over networks of contacts between individuals and a substantial body of recent research has focused on the dynamics of the spreading process. Here we examine a model of two competing diseases spreading over the same network at the same time, where infection with either disease gives an individual subsequent immunity to both. Using a combination of analytic and numerical methods, we derive the phase diagram of the system and estimates of the expected final numbers of individuals infected with each disease. The system shows an unusual dynamical transition between dominance of one disease and dominance of the other as a function of their relative rates of growth. Close to this transition the final outcomes show strong dependence on stochastic fluctuations in the early stages of growth, dependence that decreases with increasing network size, but does so sufficiently slowly as still to be easily visible in systems with millions or billions of individuals. In most regions of the phase diagram we find that one disease eventually dominates while the other reaches only a vanishing fraction of the network, but the system also displays a significant coexistence regime in which both diseases reach epidemic proportions and infect an extensive fraction of the network.Comment: 14 pages, 5 figure

Threshold effects for two pathogens spreading on a network

Diseases spread through host populations over the networks of contacts between individuals, and a number of results about this process have been derived in recent years by exploiting connections between epidemic processes and bond percolation on networks. Here we investigate the case of two pathogens in a single population, which has been the subject of recent interest among epidemiologists. We demonstrate that two pathogens competing for the same hosts can both spread through a population only for intermediate values of the bond occupation probability that lie above the classic epidemic threshold and below a second higher value, which we call the coexistence threshold, corresponding to a distinct topological phase transition in networked systems.Comment: 5 pages, 2 figure

Directed percolation with incubation times

We introduce a model for directed percolation with a long-range temporal diffusion, while the spatial diffusion is kept short ranged. In an interpretation of directed percolation as an epidemic process, this non-Markovian modification can be understood as incubation times, which are distributed accordingly to a Levy distribution. We argue that the best approach to find the effective action for this problem is through a generalization of the Cardy-Sugar method, adding the non-Markovian features into the geometrical properties of the lattice. We formulate a field theory for this problem and renormalize it up to one loop in a perturbative expansion. We solve the various technical difficulties that the integrations possess by means of an asymptotic analysis of the divergences. We show the absence of field renormalization at one-loop order, and we argue that this would be the case to all orders in perturbation theory. Consequently, in addition to the characteristic scaling relations of directed percolation, we find a scaling relation valid for the critical exponents of this theory. In this universality class, the critical exponents vary continuously with the Levy parameter.Comment: 17 pages, 7 figures. v.2: minor correction

Random graphs containing arbitrary distributions of subgraphs

Traditional random graph models of networks generate networks that are locally tree-like, meaning that all local neighborhoods take the form of trees. In this respect such models are highly unrealistic, most real networks having strongly non-tree-like neighborhoods that contain short loops, cliques, or other biconnected subgraphs. In this paper we propose and analyze a new class of random graph models that incorporates general subgraphs, allowing for non-tree-like neighborhoods while still remaining solvable for many fundamental network properties. Among other things we give solutions for the size of the giant component, the position of the phase transition at which the giant component appears, and percolation properties for both site and bond percolation on networks generated by the model.Comment: 12 pages, 6 figures, 1 tabl

A Technique for the Quantitative Estimation of Soil Micro-organisms

RESP-269

Chaotic synchronizations of spatially extended systems as non-equilibrium phase transitions

Two replicas of spatially extended chaotic systems synchronize to a common spatio-temporal chaotic state when coupled above a critical strength. As a prototype of each single spatio-temporal chaotic system a lattice of maps interacting via power-law coupling is considered. The synchronization transition is studied as a non-equilibrium phase transition, and its critical properties are analyzed at varying the spatial interaction range as well as the nonlinearity of the dynamical units composing each system. In particular, continuous and discontinuous local maps are considered. In both cases the transitions are of the second order with critical indexes varying with the exponent characterizing the interaction range. For discontinuous maps it is numerically shown that the transition belongs to the {\it anomalous directed percolation} (ADP) family of universality classes, previously identified for L{\'e}vy-flight spreading of epidemic processes. For continuous maps, the critical exponents are different from those characterizing ADP, but apart from the nearest-neighbor case, the identification of the corresponding universality classes remains an open problem. Finally, to test the influence of deterministic correlations for the studied synchronization transitions, the chaotic dynamical evolutions are substituted by suitable stochastic models. In this framework and for the discontinuous case, it is possible to derive an effective Langevin description that corresponds to that proposed for ADP.Comment: 12 pages, 5 figures Comments are welcom

A Comparison of a Direct- and a Plate-counting Technique for the Quantitative Estimation of Soil Micro-organisms

RESP-308

Validation and Calibration of Models for Reaction-Diffusion Systems

Space and time scales are not independent in diffusion. In fact, numerical simulations show that different patterns are obtained when space and time steps ($\Delta x$ and $\Delta t$) are varied independently. On the other hand, anisotropy effects due to the symmetries of the discretization lattice prevent the quantitative calibration of models. We introduce a new class of explicit difference methods for numerical integration of diffusion and reaction-diffusion equations, where the dependence on space and time scales occurs naturally. Numerical solutions approach the exact solution of the continuous diffusion equation for finite $\Delta x$ and $\Delta t$, if the parameter $\gamma_N=D \Delta t/(\Delta x)^2$ assumes a fixed constant value, where $N$ is an odd positive integer parametrizing the alghorithm. The error between the solutions of the discrete and the continuous equations goes to zero as $(\Delta x)^{2(N+2)}$ and the values of $\gamma_N$ are dimension independent. With these new integration methods, anisotropy effects resulting from the finite differences are minimized, defining a standard for validation and calibration of numerical solutions of diffusion and reaction-diffusion equations. Comparison between numerical and analytical solutions of reaction-diffusion equations give global discretization errors of the order of $10^{-6}$ in the sup norm. Circular patterns of travelling waves have a maximum relative random deviation from the spherical symmetry of the order of 0.2%, and the standard deviation of the fluctuations around the mean circular wave front is of the order of $10^{-3}$.Comment: 33 pages, 8 figures, to appear in Int. J. Bifurcation and Chao