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Journal ArticleDOI

Tensor Analysis of Networks. By Gabriel Kron. Pp. xxix, 635. 70s. 1965. (Macdonald)

01 Feb 1967-The Mathematical Gazette (Cambridge University Press (CUP))-Vol. 51, Iss: 375, pp 85-86
About: This article is published in The Mathematical Gazette.The article was published on 1967-02-01. It has received 402 citations till now.
Citations
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Journal ArticleDOI
TL;DR: In this article, the authors derive global patterns of global relations from a detailed social network, within which classes of equivalently positioned individuals are delineated by a "functorial" mapping of the original pattern.
Abstract: The aim of this paper is to understand the interrelations among relations within concrete social groups. Social structure is sought, not ideal types, although the latter are relevant to interrelations among relations. From a detailed social network, patterns of global relations can be extracted, within which classes of equivalently positioned individuals are delineated. The global patterns are derived algebraically through a ‘functorial’ mapping of the original pattern. Such a mapping (essentially a generalized homomorphism) allows systematically for concatenation of effects through the network. The notion of functorial mapping is of central importance in the ‘theory of categories,’ a branch of modern algebra with numerous applications to algebra, topology, logic. The paper contains analyses of two social networks, exemplifying this approach.

1,488 citations


Cites background from "Tensor Analysis of Networks. By Gab..."

  • ...The theory of electric circuits comes to mind as an analogue, particularly in the elegant formulation of Kron (1939), ably simplified and restated by LeCorbeiller (1950), or in the formulation of Slepian (1968)....

    [...]

  • ...The theory of electric circuits comes to mind as an analogue, particularly in the elegant formulation of Kron (1939), ably simplified and restated by LeCorbeiller (1950), or in the formulation of Slepian (1968). There are three crucial defects in the analogy, exploration of which can guide further treatment of social ties....

    [...]

Posted Content
TL;DR: In this paper, a general graph-theoretic framework for the Kron reduction of a weighted and undirected graph with self-loops and its corresponding Laplacian matrix is proposed.
Abstract: Consider a weighted and undirected graph, possibly with self-loops, and its corresponding Laplacian matrix, possibly augmented with additional diagonal elements corresponding to the self-loops. The Kron reduction of this graph is again a graph whose Laplacian matrix is obtained by the Schur complement of the original Laplacian matrix with respect to a subset of nodes. The Kron reduction process is ubiquitous in classic circuit theory and in related disciplines such as electrical impedance tomography, smart grid monitoring, transient stability assessment in power networks, or analysis and simulation of induction motors and power electronics. More general applications of Kron reduction occur in sparse matrix algorithms, multi-grid solvers, finite--element analysis, and Markov chains. The Schur complement of a Laplacian matrix and related concepts have also been studied under different names and as purely theoretic problems in the literature on linear algebra. In this paper we propose a general graph-theoretic framework for Kron reduction that leads to novel and deep insights both on the mathematical and the physical side. We show the applicability of our framework to various practical problem setups arising in engineering applications and computation. Furthermore, we provide a comprehensive and detailed graph-theoretic analysis of the Kron reduction process encompassing topological, algebraic, spectral, resistive, and sensitivity analyses. Throughout our theoretic elaborations we especially emphasize the practical applicability of our results.

480 citations

Journal ArticleDOI
TL;DR: The tensorial approach suggests that the ‘covariant analysis’ and ‘contravariant synthesis’, via metric tensor, may be a general principle of the organization of the CNS.

361 citations

Journal ArticleDOI
TL;DR: RICE focuses specifically on the passive interconnect problem by applying the moment-matching technique of Asymptotic Waveform Evaluation (AWE) and application-specific circuit analysis techniques to yield large gains in run-time efficiency over circuit simulation without sacrificing accuracy.
Abstract: This paper describes the Rapid Interconnect Circuit Evaluator (RICE) software developed specifically to analyze RC and RLC interconnect circuit models of virtually any size and complexity RICE focuses specifically on the passive interconnect problem by applying the moment-matching technique of Asymptotic Waveform Evaluation (AWE) and application-specific circuit analysis techniques to yield large gains in run-time efficiency over circuit simulation without sacrificing accuracy Moreover, this focus of AWE on passive interconnect problems permits the use of moment-matching techniques that produce stable, pre-characterized, reduced-order models for RC and RLC interconnects RICE is demonstrated to be as accurate as a transient circuit simulation with hundreds or thousands of times the efficiency The use of RICE is demonstrated on several VLSI interconnect and off-chip microstrip models >

215 citations

Journal ArticleDOI
25 Apr 2018
TL;DR: This paper surveys some fundamental and historic as well as recent results on how algebraic graph theory informs electrical network analysis, dynamics, and design, and reviews the algebraic and spectral properties of graph adjacency, Laplacian, incidence, and resistance matrices.
Abstract: Algebraic graph theory is a cornerstone in the study of electrical networks ranging from miniature integrated circuits to continental-scale power systems. Conversely, many fundamental results of algebraic graph theory were laid out by early electrical circuit analysts. In this paper, we survey some fundamental and historic as well as recent results on how algebraic graph theory informs electrical network analysis, dynamics, and design. In particular, we review the algebraic and spectral properties of graph adjacency, Laplacian, incidence, and resistance matrices and how they relate to the analysis, network reduction, and dynamics of certain classes of electrical networks. We study these relations for models of increasing complexity ranging from static resistive direct current (dc) circuits, over dynamic resistor..inductor..capacitor (RLC) circuits, to nonlinear alternating current (ac) power flow. We conclude this paper by presenting a set of fundamental open questions at the intersection of algebraic graph theory and electrical networks.

191 citations


Cites background or methods from "Tensor Analysis of Networks. By Gab..."

  • ...This algebraic elimination procedure is termed Kron reduction after Gabriel Kron [89]....

    [...]

  • ...Indeed, the series-circuit contraction and Y-∆ transformation are special cases of the more general Kron reduction [89] that permits an elegant analysis via algebraic graph theory....

    [...]

  • ...This algebraic elimination procedure is termed Kron reduction after Gabriel Kron [89]....

    [...]

References
More filters
Journal ArticleDOI
TL;DR: In this article, the authors derive global patterns of global relations from a detailed social network, within which classes of equivalently positioned individuals are delineated by a "functorial" mapping of the original pattern.
Abstract: The aim of this paper is to understand the interrelations among relations within concrete social groups. Social structure is sought, not ideal types, although the latter are relevant to interrelations among relations. From a detailed social network, patterns of global relations can be extracted, within which classes of equivalently positioned individuals are delineated. The global patterns are derived algebraically through a ‘functorial’ mapping of the original pattern. Such a mapping (essentially a generalized homomorphism) allows systematically for concatenation of effects through the network. The notion of functorial mapping is of central importance in the ‘theory of categories,’ a branch of modern algebra with numerous applications to algebra, topology, logic. The paper contains analyses of two social networks, exemplifying this approach.

1,488 citations

Journal ArticleDOI
TL;DR: This paper provides a comprehensive and detailed graph-theoretic analysis of Kron reduction encompassing topological, algebraic, spectral, resistive, and sensitivity analyses and leads to novel insights both on the mathematical and the physical side.
Abstract: Consider a weighted undirected graph and its corresponding Laplacian matrix, possibly augmented with additional diagonal elements corresponding to self-loops. The Kron reduction of this graph is again a graph whose Laplacian matrix is obtained by the Schur complement of the original Laplacian matrix with respect to a specified subset of nodes. The Kron reduction process is ubiquitous in classic circuit theory and in related disciplines such as electrical impedance tomography, smart grid monitoring, transient stability assessment, and analysis of power electronics. Kron reduction is also relevant in other physical domains, in computational applications, and in the reduction of Markov chains. Related concepts have also been studied as purely theoretic problems in the literature on linear algebra. In this paper we analyze the Kron reduction process from the viewpoint of algebraic graph theory. Specifically, we provide a comprehensive and detailed graph-theoretic analysis of Kron reduction encompassing topological, algebraic, spectral, resistive, and sensitivity analyses. Throughout our theoretic elaborations we especially emphasize the practical applicability of our results to various problem setups arising in engineering, computation, and linear algebra. Our analysis of Kron reduction leads to novel insights both on the mathematical and the physical side.

715 citations

Journal ArticleDOI
TL;DR: The tensorial approach suggests that the ‘covariant analysis’ and ‘contravariant synthesis’, via metric tensor, may be a general principle of the organization of the CNS.

361 citations

Journal ArticleDOI
TL;DR: RICE focuses specifically on the passive interconnect problem by applying the moment-matching technique of Asymptotic Waveform Evaluation (AWE) and application-specific circuit analysis techniques to yield large gains in run-time efficiency over circuit simulation without sacrificing accuracy.
Abstract: This paper describes the Rapid Interconnect Circuit Evaluator (RICE) software developed specifically to analyze RC and RLC interconnect circuit models of virtually any size and complexity RICE focuses specifically on the passive interconnect problem by applying the moment-matching technique of Asymptotic Waveform Evaluation (AWE) and application-specific circuit analysis techniques to yield large gains in run-time efficiency over circuit simulation without sacrificing accuracy Moreover, this focus of AWE on passive interconnect problems permits the use of moment-matching techniques that produce stable, pre-characterized, reduced-order models for RC and RLC interconnects RICE is demonstrated to be as accurate as a transient circuit simulation with hundreds or thousands of times the efficiency The use of RICE is demonstrated on several VLSI interconnect and off-chip microstrip models >

215 citations

Journal ArticleDOI
25 Apr 2018
TL;DR: This paper surveys some fundamental and historic as well as recent results on how algebraic graph theory informs electrical network analysis, dynamics, and design, and reviews the algebraic and spectral properties of graph adjacency, Laplacian, incidence, and resistance matrices.
Abstract: Algebraic graph theory is a cornerstone in the study of electrical networks ranging from miniature integrated circuits to continental-scale power systems. Conversely, many fundamental results of algebraic graph theory were laid out by early electrical circuit analysts. In this paper, we survey some fundamental and historic as well as recent results on how algebraic graph theory informs electrical network analysis, dynamics, and design. In particular, we review the algebraic and spectral properties of graph adjacency, Laplacian, incidence, and resistance matrices and how they relate to the analysis, network reduction, and dynamics of certain classes of electrical networks. We study these relations for models of increasing complexity ranging from static resistive direct current (dc) circuits, over dynamic resistor..inductor..capacitor (RLC) circuits, to nonlinear alternating current (ac) power flow. We conclude this paper by presenting a set of fundamental open questions at the intersection of algebraic graph theory and electrical networks.

191 citations