Systems analysisi: mathematical modeling and approach to structural complexity measure using Polyhedral Dynamics approach

(Part 1)

Introduction

One of the main stages in systems studying is a stage of analysis that leads to the obtaining of important information both on systems under design and real systems (natural or as a manmade). Analysis process dwells upon the observer’s «level of understanding» of a system as a whole, his ability to distinguish without destruction the integrity properties of a system using a priori knowledge about an object.

Intense growth of interest in problems arising in the field of large complex systems has led to the application of profound mathematical methods for initiating a systematic inquiry into structural analysis of systems and measures of their complexity. The notions of «complexity» and «structure» are used in systems science (and elsewhere) in various ways, and this paper examines and discusses from the systematic point of view some modifications of the holist approach proposed by R.H.Atkin for analysis of systems structures both at the global level (system as a whole) and at the local level (level of elements that are connected to each other to form a structure), as well as for estimation of structural complexity of systems based on the results of such analysis. This approach known as Q-analysis (or, Polyhedral Dynamics), uses the ideas of C.Dowker as a mathematical background. The origins of the idea itself are in the publications Analysis Situs and Complement a l’Analysis Situs A.Poincare, who brought into life the «idea of computing with topological objects». Within the scope of this approach a structure of the system is used with a purpose of getting its geometric and algebraic representation as a simplicial complex formed by simplices of particular dimensionalities, the last depending on initial information about the system and the level of its mathematical description. Analysis of system’s model is performed at each dimensional level through studying the chains of connectivity, which link simplices together and lead to the appearance of connectivity components. A measure of structural complexity of simplicial complex deals with the results of performed analysis.

A concept of complex system (or, complexity in general) is many-sided and rich, and because of that we distinguish only structural features which could bring a valuable contribution to systems studying. Classification of systems as simple or complex normally takes into account several factors, among which a variety of elements and interactions (connections) are of importance. Preliminary conclusions on complexity of system are drawn on the basis of observation of its behavior,which depends upon a system’s organization. In general, organization is a dynamic component, but it includes a fixed (constant) part – namely, it is structure. What we mean in this paper are aspects of hypothetical complexity which appear in a system’s structure and «arise through connectivity and the inter-relationships of a system’s constituent elements>>.

Background

Algebraic topology concepts

We shall now turn to some useful basic concepts from algebraic topology used in the Polyhedral Dynamics approach. A simplicial complex K is a collection of finite non-empty sets, called simplice s of K, such that if simplex s an element of K, so is every non-empty subset, called face of, of

complex K. The vertex set V of complex K is a union of one-point sets of K. A simplex formed by exactly (d+1) vertices has a dimension d (one less than the number of its vertices), and is called d-simplex. We can say that an 0-dimensional simplex is a point, a 1-dimensional simplex is a straight line segment, a two-dimensional simplex is a triangle (including the plane region, which it bounds), etc. Any simplices of K meet, if at all, strictly

in the common face. The union of simplices of K (a result of «gluing them together» along their faces) is called a polyhedron. The dimension of complex K is the largest dimensionality of its simplices, and it is often denoted dimK.

In general outline, the Polyhedral Dynamics approach is based on studying the way simplices are connected to each other by means of chains of connectivity (the word «chain» corresponds to a sequence of intermediate simplices of particular dimensionalities, which connects two examined simplices). Finally, these chains specify the internal multidimensional structure of a system under modeling. From the geometrical point of view,analysis of complex K appears to be difficult, especially because of simplices dimensionalities, but it opens the way to the application of algebraic methods.