Topic

# Comparison of topologies

About: Comparison of topologies is a research topic. Over the lifetime, 879 publications have been published within this topic receiving 18460 citations.

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31 Oct 1993

TL;DR: In this paper, the Attouch-Wets and Hausdorff Metric Topologies for Convex Functions are presented. But they do not consider the relationship between the two topologies.

Abstract: Preface. 1. Preliminaries. 2. Weak Topologies determined by Distance Functionals. 3. The Attouch--Wets and Hausdorff Metric Topologies. 4. Gap and Excess Functionals and Weak Topologies. 5. The Fell Topology and Kuratowski--Painleve Convergence. 6. Multifunctions - the Rudiments. 7. The Attouch--Wets Topology for Convex Functions. 8. The Slice Topology for Convex Functions. Notes and References. Bibliography. Symbols and Notation. Subject Index.

906 citations

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TL;DR: In this paper, Affirmative and refutative assertions are made for the point logic and spectral algebraic locales, and the definitions of the topology of the spectral lattice are discussed.

Abstract: 1. Introduction 2. Affirmative and refutative assertions 3. Frames 4. Frames as algebras 5. Topology: the definitions 6. New topologies for old 7. Point logic 8. Compactness 9. Spectral algebraic locales 10. Domain theory 11. Power domains 12. Spectra of rings Bibliography.

559 citations

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TL;DR: An approach to visualizing flow topology that is based on the physics and mathematics underlying the physical phenomenon is presented and can be displayed as a set of points and tangent curves or as a graph.

Abstract: The visualization of physical processes in general and of vector fields in particular is discussed. An approach to visualizing flow topology that is based on the physics and mathematics underlying the physical phenomenon is presented. It involves determining critical points in the flow where the velocity vector vanishes. The critical points, connected by principal lines or planes, determine the topology of the flow. The complexity of the data is reduced without sacrificing the quantitative nature of the data set. By reducing the original vector field to a set of critical points and their connections, a representation of the topology of a two-dimensional vector field is much smaller than the original data set but retains with full precision the information pertinent to the flow topology is obtained. This representation can be displayed as a set of points and tangent curves or as a graph. Analysis (including algorithms), display, interaction, and implementation aspects are discussed. >

486 citations

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11 Aug 2006

TL;DR: This work presents a new, systematic approach for analyzing network topologies, introducing the dK-series of probability distributions specifying all degree correlations within d-sized subgraphs of a given graph G, and demonstrates that these graphs reproduce, with increasing accuracy, important properties of measured and modeled Internet topologies.

Abstract: Researchers have proposed a variety of metrics to measure important graph properties, for instance, in social, biological, and computer networks. Values for a particular graph metric may capture a graph's resilience to failure or its routing efficiency. Knowledge of appropriate metric values may influence the engineering of future topologies, repair strategies in the face of failure, and understanding of fundamental properties of existing networks. Unfortunately, there are typically no algorithms to generate graphs matching one or more proposed metrics and there is little understanding of the relationships among individual metrics or their applicability to different settings. We present a new, systematic approach for analyzing network topologies. We first introduce the dK-series of probability distributions specifying all degree correlations within d-sized subgraphs of a given graph G. Increasing values of d capture progressively more properties of G at the cost of more complex representation of the probability distribution. Using this series, we can quantitatively measure the distance between two graphs and construct random graphs that accurately reproduce virtually all metrics proposed in the literature. The nature of the dK-series implies that it will also capture any future metrics that may be proposed. Using our approach, we construct graphs for d=0, 1, 2, 3 and demonstrate that these graphs reproduce, with increasing accuracy, important properties of measured and modeled Internet topologies. We find that the d=2 case is sufficient for most practical purposes, while d=3 essentially reconstructs the Internet AS-and router-level topologies exactly. We hope that a systematic method to analyze and synthesize topologies offers a significant improvement to the set of tools available to network topology and protocol researchers.

481 citations