Shortest paths on systems with power-law distributed long-range connections

We discuss shortest-path lengths $\ell(r)$ on periodic rings of size L supplemented with an average of pL randomly located long-range links whose lengths are distributed according to P_l \sim l^{-\xpn}. Using rescaling arguments and numerical simulation on systems of up to $10^7$ sites, we show that a characteristic length $\xi$ exists such that $\ell(r) \sim r$ for $r>\xi$. For small p we find that the shortest-path length satisfies the scaling relation \ell(r,\xpn,p)/\xi = f(\xpn,r/\xi). Three regions with different asymptotic behaviors are found, respectively: a) \xpn>2 where $\theta_s=1$, b) 1<\xpn<2 where 0<\theta_s(\xpn)<1/2 and, c) \xpn<1 where $\ell(r)$ behaves logarithmically, i.e. $\theta_s=0$. The characteristic length $\xi$ is of the form $\xi \sim p^{-\nu}$ with \nu=1/(2-\xpn) in region b), but depends on L as well in region c). A directed model of shortest-paths is solved and compared with numerical results.Comment: 10 pages, 10 figures, revtex4. Submitted to PR

Foundations and Tools for End-User Architecting

Abstract. Within an increasing number of domains an important emerging need is the ability for technically naïve users to compose computational elements into novel configurations. Examples include astronomers who create new analysis pipelines to process telescopic data, intelligence analysts who must process diverse sources of unstructured text to discover socio-technical trends, and medical researchers who have to process brain image data in new ways to understand disease pathways. Creating such compositions today typically requires low-level technical expertise, limiting the use of computational methods and increasing the cost of using them. In this paper we describe an approach — which we term end-user architecting — that exploits the similarity between such compositional activities and those of software architects. Drawing on the rich heritage of software architecture languages, methods, and tools, we show how those techniques can be adapted to support end users in composing rich computational systems through domain-specific compositional paradigms and component repositories, without requiring that they have knowledge of the low-level implementation details of the components or the compositional infrastructure. Further, we outline a set of open research challenges that the area of end-user architecting raises

Analytical Solutions for Multicomponent, Two-Phase Flow in Porous Media with Double Contact Discontinuities

This paper presents the first instance of a double contact discontinuity in analytical solutions for multicomponent, two-phase flow in porous media. We use a three-component system with constant equilibrium ratios and fixed injection and initial conditions, to demonstrate this structure. This wave structure occurs for two-phase injection compositions. Such conditions were not considered previously in the development of analytical solutions for compositional flows. We demonstrate the stability of the double contact discontinuity in terms of the Liu entropy condition and also show that the resulting solution is continuously dependent on initial data. Extensions to four-component and systems with adsorption are presented, demonstrating the more widespread occurrence of this wave structure in multicomponent, two-phase flow systems. The developments in this paper provide the building blocks for the development of a complete Riemann solver for general initial and injection conditions