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

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

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

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

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

RESP-308

Contact process with long-range interactions: a study in the ensemble of constant particle number

We analyze the properties of the contact process with long-range interactions by the use of a kinetic ensemble in which the total number of particles is strictly conserved. In this ensemble, both annihilation and creation processes are replaced by an unique process in which a particle of the system chosen at random leaves its place and jumps to an active site. The present approach is particularly useful for determining the transition point and the nature of the transition, whether continuous or discontinuous, by evaluating the fractal dimension of the cluster at the emergence of the phase transition. We also present another criterion appropriate to identify the phase transition that consists of studying the system in the supercritical regime, where the presence of a "loop" characterizes the first-order transition. All results obtained by the present approach are in full agreement with those obtained by using the constant rate ensemble, supporting that, in the thermodynamic limit the results from distinct ensembles are equivalent

Non-equilibrium Phase Transitions with Long-Range Interactions

This review article gives an overview of recent progress in the field of non-equilibrium phase transitions into absorbing states with long-range interactions. It focuses on two possible types of long-range interactions. The first one is to replace nearest-neighbor couplings by unrestricted Levy flights with a power-law distribution P(r) ~ r^(-d-sigma) controlled by an exponent sigma. Similarly, the temporal evolution can be modified by introducing waiting times Dt between subsequent moves which are distributed algebraically as P(Dt)~ (Dt)^(-1-kappa). It turns out that such systems with Levy-distributed long-range interactions still exhibit a continuous phase transition with critical exponents varying continuously with sigma and/or kappa in certain ranges of the parameter space. In a field-theoretical framework such algebraically distributed long-range interactions can be accounted for by replacing the differential operators nabla^2 and d/dt with fractional derivatives nabla^sigma and (d/dt)^kappa. As another possibility, one may introduce algebraically decaying long-range interactions which cannot exceed the actual distance to the nearest particle. Such interactions are motivated by studies of non-equilibrium growth processes and may be interpreted as Levy flights cut off at the actual distance to the nearest particle. In the continuum limit such truncated Levy flights can be described to leading order by terms involving fractional powers of the density field while the differential operators remain short-ranged.Comment: LaTeX, 39 pages, 13 figures, minor revision

Electrophysiological correlates of high-level perception during spatial navigation

We studied the electrophysiological basis of object recognition by recording scalp\ud electroencephalograms while participants played a virtual-reality taxi driver game.\ud Participants searched for passengers and stores during virtual navigation in simulated\ud towns. We compared oscillatory brain activity in response to store views that were targets or\ud nontargets (during store search) or neutral (during passenger search). Even though store\ud category was solely defined by task context (rather than by sensory cues), frontal ...\ud \u

Edinburgh Wave Power Project: Fourth Year Report Volume 1 Of 3

The results of measurements carried out at the National Maritime Institute, Feltham in February 1977 confirmed that scaling l aws operate well over the range from 1/150 to 1/15 and so we believe that it is safe to continue work in small tanks. We have tested ducks on a variety of mountings in the narrow tank over the entire range of sea conditions found at OWS India. The results p rovide sufficient input data for the full-scale power take-off design . We have developed techniques for generating very steep waves and have tested ducks in conditions likely to induce slamming. We are satisfied with duck behaviour in these conditions and do not regard slamming as our most serious problem. Photographs of the tests are contained in Volume 2 of this report. We have explored duck behaviour on mountings of variable compliance, and have discovered some striking effects. We find that there are two regions of efficient operation, one of which requires no restraint in heave . Ducks on mountings 1 with the right compliance can work better than those on fixed mountings and the right compliance can easily be achieved at full scale. Most of our effort has gone into making a wide tank with control of directional characteristics of random seas. Its design may prove of interest to other groups. It was ready for use in January 1978 and the cost e stimates proved accurate . Descriptions of the design and performance will be found in Volume 3 of this report. Results from our first month of experiments on free-floating backbones without ducks show that bending moments fall in the central sections of very long backbones and that static beam theory is difficult to apply. The full-time engineering strength of the team has risen to four with the arrival of Glenn Keller but I am sorry to report that we shall be losing two welcome visitors. Rick Jefferies, who has been working on non-linear problems for his Cambridge doctorate, will be going to CEGB, Marchwood. Ian Young is leaving us to r ead computer science here at Edinburgh. I would like to draw the attention of WESC to r eports by my colleague David Mccomb on polymer additions for hydraulic power transmission, by Graham Dixon on the anomalous heave force behaviour of horizontal cylinders and by Rick J efferies on theoretica l frequency and time domain models for ducks