Article

Article title CLUSTERING OF ORIENTED WEIGHTED GRAPHS BASED ON THE FUNCTIONAL OF POTENTIAL ENERGY OF ELASTIC DEFORMATION WITH THE USE OF COGNITIVE MODELS
Authors A. N. Tselykh, V. S. Vasilev, L. A. Tselykh
Section SECTION I. METHODS AND ALGORITHMS OF INFORMATION PROCESSING
Month, Year 03, 2018 @en
Index UDC 004.891.2
DOI
Abstract A new approach to clustering based on the model of division of vertices into groups based on higher potential energy of edges within groups than between the groups themselves is proposed. This method is implemented by minimizing the elastic strain potential energy functional for oriented weighted signed graphs. Existing structures in the adjacency matrix of a graph are expressed in its structure. The search for the best ordering of the graph vertices is carried out by means of the optimization process in the space of real numbers, which is inevitably displayed on the set of permutations of vertex indices. The method is developed for networks representing interrelations in economic systems. These relations are represented by heterogeneous factors and cause-effect relationships between them. Systems of objects and their relations are presented in the form of a fuzzy cognitive map, which, in fact, is a weighted oriented sign graph. Clustering methods are applied to this graph. The novelty of the approach is that the solution of the clustering problem is found as a solution to the optimization problem of the function of many variables (functionals). The proposed method uses a mechanical analogy, which is a special case of metric representations, which provides the convexity of the problem. This approach allows us to design various functionalities with a clear interpretation and predictability of the minimization process. The work of the algorithm is to find the numbering of the vertices of the graph, which reaches the lowest value of the functional. In this numbering, gradient components of the functional are used to determine the boundaries of the clusters. The criterion for classifying vertices as a single community is the proximity of the indices and the same gradient signs. These criteria are clearly formalized. The proposed algorithm classifies the vertices of a particular cluster and at the same time determines the order of the nodes in the cluster. This order reflects the degree of vertex distribution in relation to the intracluster energy to the energy of the extracluster bond. The hierarchical structure of the graph is revealed by recursive application of the proposed algorithm in each cluster without taking into account inter-cluster connections. The potential energy of the elastic deformation functional used in the clustering algorithm reflects the causal nature of the relationship of factors in the socio-economic system. Functional minimization is monotonous and requires no user intervention. The algorithm is computationally efficient.

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Keywords Clustering; elastic deformation functional; oriented weighted graph; optimization methods; cognitive maps.
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