Neighborhood structures and metastructural identification

N. M. Mishachev, A. M. Shmyrin

Lipetsk State Technical University

Abstract: In the article, the concept of metastructural identification of a modeled system is formalized as the construction of a pair consisting of a neighborhood structure (graph) and the type of interactions between the nodes of this structure. In the language of metagraphs, two types of interactions are defined: vertex type, when the equations of the model correspond to the nodes of the structure, and the relational type, when the equations correspond to the edges of the structure.

The relations between vertex-type and relational-type models are discussed. Structural identification of the modeled system, as a rule, can be divided into two stages. At the first stage we specify the nodes of the model, the connections between them and the sets of variables corresponding to these nodes and connections. On the second, we define the model equations with unknown parameters that are subject to further parametric identification. In this article, we propose to call the first stage a metastructural identification and define such identification as the construction of a neighborhood structure (graph), the choice of the type of interactions between the nodes of this structure and the indication of the corresponding variables. Our experience in modeling complex systems shows that in many cases it makes sense to distinguish between two types of such interactions: vertex-type, when the equations of the model correspond to the nodes of the structure, and the relational-type (edge-type) when the equations of the model correspond to the edges of the structure. The main purpose of this article is to create a system of definitions to describe these two situations and to clarify the relationships between them. These two types of models are convenient to define using the language of metagrafics. In order to describe the relationships between vertex-type and relational-type models, we are define the notions of clustering and declustering of neighborhood structures, and show that each relational-type structure can be uniquely declustered down to a vertex-type. This (fairly simple) result does not mean that we need to exclude the relational-type models, since declustering of the relational-type model often loses its visibility. We also discuss the inverse problem of clustering the vertex-type structures into more compact relational ones.

Keywords: neighborhood structure, neighborhood system, metastructural identification, metagraph, vertex system, relational system.

UDC: 519.71

MSC: 93B30