Largest nearest-neighbour link and connectivity threshold in a polytopal random sample

Material is not peer-reviewed by arXiv - the contents of arXiv submissions are wholly the responsibility of the submitter and are presented “as is” without any warranty or guarantee. By hosting works and other materials on this site, arXiv, Cornell University, and their agents do not in any way convey implied approval of the assumptions, methods, results, or conclusions of the work.Copyright © 2023 The Authors. Let $X_1,X_2, \ldots$ be independent identically distributed random points in a convex polytopal domain $A \subset \mathbb{R}^d$. Define the largest nearest neighbour link $L_n$ to be the smallest $r$ such that every point of $\mathcal X_n:=\{X_1,\ldots,X_n\}$ has another such point within distance $r$. We obtain a strong law of large numbers for $L_n$ in the large-$n$ limit. A related threshold, the connectivity threshold $M_n$, is the smallest $r$ such that the random geometric graph $G(\mathcal X_n, r)$ is connected. We show that as $n \to \infty$, almost surely $nL_n^d/\log n$ tends to a limit that depends on the geometry of $A$, and $nM_n^d/\log n$ tends to the same limit.EPSRC grant EP/T028653/1

On the complexity of color-avoiding site and bond percolation

The mathematical analysis of robustness and error-tolerance of complex networks has been in the center of research interest. On the other hand, little work has been done when the attack-tolerance of the vertices or edges are not independent but certain classes of vertices or edges share a mutual vulnerability. In this study, we consider a graph and we assign colors to the vertices or edges, where the color-classes correspond to the shared vulnerabilities. An important problem is to find robustly connected vertex sets: nodes that remain connected to each other by paths providing any type of error (i.e. erasing any vertices or edges of the given color). This is also known as color-avoiding percolation. In this paper, we study various possible modeling approaches of shared vulnerabilities, we analyze the computational complexity of finding the robustly (color-avoiding) connected components. We find that the presented approaches differ significantly regarding their complexity.Comment: 14 page

Connectivity of Soft Random Geometric Graphs Over Annuli

Nodes are randomly distributed within an annulus (and then a shell) to form a point pattern of communication terminals which are linked stochastically according to the Rayleigh fading of radio-frequency data signals. We then present analytic formulas for the connection probability of these spatially embedded graphs, describing the connectivity behaviour as a dense-network limit is approached. This extends recent work modelling ad hoc networks in non-convex domains.Comment: 12 pages, 6 figure

Linking learning with governance in networks and clusters: key issues for analysis and policy

In this paper we analyse the relationship between governance and learning in clusters and networks. In particular, we see these two key elements as interdependent, suggesting that, under particular circumstances, such interdependence may drive clusters and networks towards a dynamic development trajectory. A pure ‘governance perspective’ makes the development of any locality dependent on the system of powers which exists within the locality or across the global value chain. In parallel, a pure ‘competence-based approach’ focuses mainly on the capabilities of actors to learn and undertake activities. In contrast, we open the prospects for an interdependent relation that may change the actual competences of actors as well as the governance settings and dynamics in networks and clusters. When supported by public policies, the learning process may have the potential to modify the governance environment. Simultaneously, the learning process is intrinsically influenced by economic power, which may seriously affect the development prospects of clusters and networks. This is why an intertwined consideration of both aspects is necessary to promote specific approaches to learning and to design appropriate policies. In this paper we offer two preliminary case studies to clarify some of these dynamics: the first taken from the computers cluster in Costa Rica and the second from an Italian bio-pharmaceutical firm and its production network. The first case study refers to the software cluster that was created from scratch in Costa Rica thanks to an enlightened government policy in coordination with new local enterprises and an important foreign direct investor; while the second reflects on the ability of an individual company to create a network of relationships with large transnational companies in order to acquire new competences without falling into a subordinate position with respect to its larger partners

Foundations of Black Hole Accretion Disk Theory

This review covers the main aspects of black hole accretion disk theory. We begin with the view that one of the main goals of the theory is to better understand the nature of black holes themselves. In this light we discuss how accretion disks might reveal some of the unique signatures of strong gravity: the event horizon, the innermost stable circular orbit, and the ergosphere. We then review, from a first-principles perspective, the physical processes at play in accretion disks. This leads us to the four primary accretion disk models that we review: Polish doughnuts (thick disks), Shakura-Sunyaev (thin) disks, slim disks, and advection-dominated accretion flows (ADAFs). After presenting the models we discuss issues of stability, oscillations, and jets. Following our review of the analytic work, we take a parallel approach in reviewing numerical studies of black hole accretion disks. We finish with a few select applications that highlight particular astrophysical applications: measurements of black hole mass and spin, black hole vs. neutron star accretion disks, black hole accretion disk spectral states, and quasi-periodic oscillations (QPOs).Comment: 91 pages, 23 figures, final published version available at http://www.livingreviews.org/lrr-2013-

Exploring new physics frontiers through numerical relativity

The demand to obtain answers to highly complex problems within strong-field gravity has been met with significant progress in the numerical solution of Einstein's equations - along with some spectacular results - in various setups. We review techniques for solving Einstein's equations in generic spacetimes, focusing on fully nonlinear evolutions but also on how to benchmark those results with perturbative approaches. The results address problems in high-energy physics, holography, mathematical physics, fundamental physics, astrophysics and cosmology