Method of Synthesis of a Stable Multi-Connected Cognitive Map of a Complex System

Abstract

In the article the more effective than analogues method of synthesis of a stable multi-connected cognitive map of a complex system on the basis of a basic stable cognitive map of a smaller order is considered. The peculiarity of this method is that the higher-order cognitive map is designed with a pseudo-triangular adjacency matrix, which is more interconnected than the adjacency matrices used in other similar methods. Such a pseudo-triangular contiguity matrix is designed taking into account the following rules: 1) all elements of the basic cognitive map are elements of the higher order cognitive map, i.e. the contiguity matrix for this higher order cognitive map is built "around" the adjacent cognitive maps were part of the diagonal elements of the higher order cognitive map; 2) the elements of the adjacency matrix of the cognitive map "around" the base cognitive map above or below the main diagonal should be 0, i.e. have the form of a triangular matrix (if you ignore the elements of the base cognitive map, so for such a matrix or rearranging columns and rows.

The paper proves that cognitive maps with pseudo-triangular adjacency matrices that satisfy the above rules will be guaranteed to be stable. The proof is made by using the rules of calculating the determinant of the matrix as the sum of algebraic complements of any column or row and taking into account that the weight of cognitive maps does not exceed 1, by definition.

To prove the efficiency of the proposed method of synthesis of a stable multi-connected cognitive map, the example of successively located sections of the river on the example of the Southern Bug near Vinnytsia was considered. In this example, the peaks of the cognitive map are the concentration of biochemical oxygen consumption in each section of the river, which is a characteristic of the concentration of organic matter in water, and the target peak is the concentration of dissolved oxygen in the closing section of the river affected by pollution. To automate calculations, the Python program was developed as a public laptop based on the famous Kaggle platform. Studies and computer experiments have shown the effectiveness and efficiency of the proposed method.