On the Nature of Structure and Its Identification

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).