Networks of coupled dynamical systems have been used to model biological oscillators, Josephson junction arrays, excitable media, neural networks, spatial games, genetic control networks and many other self-organizing systems. Ordinarily, the connection topology is assumed to be either completely regular or completely random. But many biological, technological and social networks lie somewhere between these two extremes. Here we explore simple models of networks that can be tuned through this middle ground: regular networks 'rewired' to introduce increasing amounts of disorder. We find that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs. We call them 'small-world' networks, by analogy with the small-world phenomenon (popularly known as six degrees of separation. The neural network of the worm Caenorhabditis elegans, the power grid of the western United States, and the collaboration graph of film actors are shown to be small-world networks. Models of dynamical systems with small-world coupling display enhanced signal-propagation speed, computational power, and synchronizability. In particular, infectious diseases spread more easily in small-world networks than in regular lattices.

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Collective dynamics of small-world networks

https://arxiv.org/pdf/astro-ph/9907117.pdf

Catastrophic Disruptions Revisited

The Cambridge Geotechnical Centrifuge routinely tests models made of up to 0·2 m3 of soil, at accelerations up to 125g. Self-weight effects in models and in corresponding prototypes are similar within a few per cent. Seepage and diffusion of pore water into the plane section of an unlined tunnel serve to illustrate scaling relationships. The stress/strain behaviour of soil alters with change of effective pressure p′and specific volume v. A theory of plasticity explains cohsolidation and yielding in shear at states above critical, but failures below critical state involve rupture or fracture. The range of values of a new equivalent liquidity LI5 = LI+0·5 log (p′/5)associated with yielding is 1·9<LI5<1, with rupture is l<LI5<0·5, and with fracture is 0·5<LI5. One example of centrifuge modelling is for tunnels in soft ground near critical states (LI5 = 1), in which plastic collapse is observed closely to match upper and lower bound calculations of theory of plasticity. In contrast, another example of centrif...

/pdf/cambridge-geotechnical-centrifuge-operations-4e1983txhq.pdf

Cambridge Geotechnical Centrifuge Operations

Deep Impact collided with comet Tempel 1, excavating a crater controlled by gravity. The comet's outer layer is composed of 1- to 100-micrometer fine particles with negligible strength ( 1000 kelvins). A large increase in organic material occurred during and after the event, with smaller changes in carbon dioxide relative to water. On approach, the spacecraft observed frequent natural outbursts, a mean radius of 3.0 ± 0.1 kilometers, smooth and rough terrain, scarps, and impact craters. A thermal map indicates a surface in equilibrium with sunlight.

https://science.sciencemag.org/content/sci/310/5746/258.full.pdf

Deep Impact: Excavating Comet Tempel 1

Current approaches to scaling laws for planetary impact processes are reviewed. Only the simplest concepts and first-order theories that are the most determined are presented. Experimental, analytical, and calculative approaches to determining scaling laws are examined, including the advantages and deficiencies of each.

The scaling of impact processes in planetary sciences

Some recent advances in the scaling of impact and explosion cratering

Self-consistent scaling laws are developed for meteoroid impact crater ejecta. Attention is given to the ejection velocity of material as a function of the impact point, the volume of ejecta with a threshold velocity, and the thickness of ejecta deposit in terms of the distance from the impact. Use is made of recently developed equations for energy and momentum coupling in cratering events. Consideration is given to scaling of laboratory trials up to real-world events and formulations are developed for calculating the ejection velocities and ejecta blanket profiles in the gravity and strength regimes of crater formation. It is concluded that, in the gravity regime, the thickness of an ejecta blanket is the same in all directions if the thickness and range are expressed in terms of the crater radius. In the strength regime, however, the ejecta velocities are independent of crater size, thereby allowing for asymmetric ejecta blankets. Controlled experiments are recommended for the gravity/strength transition.

Crater ejecta scaling laws - Fundamental forms based on dimensional analysis

Holsapple and Schmidt (1980) previously addressed the problem of the scaling of explosive cratering. Their analysis included results which show under which conditions the scaling can be bounded between quarter-root and cube-root rules. The present investigation is an extension of the earlier analysis and approaches the case of impact cratering. More restrictive bounds are found for impact cratering than for the explosive case. These stronger results come from considering the role of the impactor momentum as an independent variable for impact cratering. Attention is given to impact cratering variables, general scaling rules, the bounds on scaling rules, a generalization to more variables, and previous scaling rules and results.

On the Scaling of Crater Dimensions 2. Impact Processes

The use of a point source of an impactor energy and momentum to replace the effects of the impactor is examined. The general framework and notation of the impact cratering problems are described; it is determined that the cratering phenomena are governed by Froude, Cauchy, and Reynolds numbers. The coupling parameter concept is defined mathematically as the measure that governs limit point source solutions. Examples of cases where coupling parameters are used are presented. The relationships of the coupling parameter concept with steady flow and the Z-model of cratering of Maxwell (1973, 1977) are studied. Crater size, ejecta distributions, growth histories, time of formation, melt volume, and shock decay for various scale factors for impact cratering mechanics are calculated, and the applicability of the coupling parameter to the study of cratering mechanics is revealed.

Robert M. Schmidt

Papers

Some recent advances in the scaling of impact and explosion cratering

Crater ejecta scaling laws - Fundamental forms based on dimensional analysis

On the Scaling of Crater Dimensions 2. Impact Processes

Point source solutions and coupling parameters in cratering mechanics

Laboratory simulations of large scale fragmentation events