Abstract
When working on systems of the real world, abstractions in the form of graphs have proven a superior modeling and representation approach. This paper is on the analysis of such graphs. Based on the paradigm that a graph of a system contains information about the system’s structure, the paper contributes within the following respects: 1. It introduces a new and lucid structure measure, the so-called weighted partial connectivity, Λ, whose maximization defines a graph’s structure (Section 2). 2. It presents a fast algorithm that approximates a graph’s optimum Λ-value (Section 3).
Moreover, the proposed structure definition is compared to existing clustering approaches (Section 4), resulting in a new splitting theorem concerning the well-known minimum cut splitting measure. A key concept of the proposed structure definition is its implicit determination of an optimum number of clusters.
Different applications, which illustrate the usability of the measure and the algorithm, round off the paper (Section 5).
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
T. Bailey and J. Cowles. Cluster definition by the optimization of simple measures. IEEE Transactions on Pattern Analysis and Machine Intelligence, September 1983. 128
L. A. R. Eli, B. Messinger and R. R. Henry. A divide-and-conquer algorithm for the automatic layout of large directed graphs. IEEE Transactions on Systems, Man, and Cybernetics, January/February 1991. 131, 131
B. S. Everitt. Cluster analysis. Edward Arnold, a division of Hodder & Stoughton, 1992. 129, 129
K. Florek, J. Lukaszewiez, J. Perkal, H. Steinhaus and S. Zubrzchi. Sur la liason et la division des points d’un ensemble fini. Colloqium Mathematicum, 1951. 128, 129
T. Hesse and B. Stein. Hybrid Diagnosis in the Fluidic Domain. Proc. EIS 98, International ICSC Symposium on Engineering of Intelligent Systems, University of La Laguna, Tenerife, Spain, Feb. 1998. 130
S. C. Johnson. Hierarchical clustering schemes. Psychometrika 32, 1967. 128, 129
D. Jungnickel. Graphen, Netzwerke und Algorithmen. BI Wissenschaftsverlag, 1990. 124
B. Kernighan and S. Lin. Partitioning graphs. Bell Laboratories Record, January 1970. 128
T. Kohonen. Self Organizing and Associate Memory. Springer-Verlag, 1990. 128
T. Lengauer. Combinatorical algorithms for integrated circuit layout. Applicable Theory in Computer Science. Teubner-Wiley, 1990. 124, 127, 128
P. MacNaughton-Smith, W.T. Williams, M.B. Dale and L.G. Mockett. Dissimilarity analysis. Nature 202, 1964. 128
O. Niggemann, B. Stein, and M. Suermann. On Resource-based Configuration — Rendering Component-Property Graphs. In J. Sauer and B. Stein, editors, 12. Workshop “Planen und Konfigurieren”, tr-ri-98-193, Paderborn, Apr. 1998. University of Paderborn, Department of Mathematics and Computer Science. 131
T. Roxborough and Arunabha. Graph Clustering using Multiway Ratio Cut. In S. North, editor, Graph Drawing, Lecture Notes in Computer Science, Springer Verlag, 1996. 128, 128
R. Sablowski and A. Frick. Automatic Graph Clustering. In S. North, editor, Graph Drawing, Lecture Notes in Computer Science, Springer Verlag, 1996. 128, 131
G. Sander. Graph Layout through the VCG Tool. Technical Report A/03/94, 1994. 131
P. H. A. Sneath. The application of computers to taxonomy. J. Gen. Microbiol. 17, 1957. 128, 129
B. Stein. Functional Models in Configuration Systems. Dissertation, University of Paderborn, Department of Mathematics and Computer Science, 1995. 131
B. Stein and E. Vier. Computer-aided Control Systems Design for Hydraulic Drives. Proc. CACSD 97, Gent, Apr. 1997. 130
K. Sugiyama, S. Tagawa, and M. Toda. Methods for Visual Understandig of Hierarchical System Structures. IEEE Transactions on Systems, Man, and Cybernectics, Vol. SMC-11, No. 2, 1981. 131
Z. Wu and R. Leahy. An optimal graph theoretic approach to data clustering: Theory and its application to image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, November 1993. 127
J.-T. Yan and P.-Y. Hsiao. A fuzzy clustering algorithm for graph bisection. Information Processing Letters 52, 1994. 128
C. T. Zahn. Graph-Theoretical Methods for Detecting and Describing Gestalt Clusters. IEEE Transactions on computers Vol. C-20, No. 1, 1971. 128
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Stein, B., Niggemann, O. (1999). On the Nature of Structure and Its Identification. In: Widmayer, P., Neyer, G., Eidenbenz, S. (eds) Graph-Theoretic Concepts in Computer Science. WG 1999. Lecture Notes in Computer Science, vol 1665. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46784-X_13
Download citation
DOI: https://doi.org/10.1007/3-540-46784-X_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-66731-5
Online ISBN: 978-3-540-46784-7
eBook Packages: Springer Book Archive