K-method of cognitive mapping analysis

📝 Abstract

Introduced a new calculation method (K-method) for cognitive maps. K - method consists of two consecutive steps. In the first stage, allocated subgraph composed of all paths from one selected node (concept) to another node (concept) from the cognitive map (directed weighted graph) . In the second stage, after the transition to an undirected graph (symmetrization adjacency matrix) the influence of one node to another calculated with Kirchhoff method. In the proposed method, there is no problem inherent in the impulse method. In addition to “pair” influence of one node to another, the average characteristics are introduced, allowing to calculate the impact of the selected node to all other nodes and the influence of all on the one selected. For impulse method similar to the average characteristics in the case where the pulse method “works” are introduced and compared with the K-method.

💡 Analysis

Introduced a new calculation method (K-method) for cognitive maps. K - method consists of two consecutive steps. In the first stage, allocated subgraph composed of all paths from one selected node (concept) to another node (concept) from the cognitive map (directed weighted graph) . In the second stage, after the transition to an undirected graph (symmetrization adjacency matrix) the influence of one node to another calculated with Kirchhoff method. In the proposed method, there is no problem inherent in the impulse method. In addition to “pair” influence of one node to another, the average characteristics are introduced, allowing to calculate the impact of the selected node to all other nodes and the influence of all on the one selected. For impulse method similar to the average characteristics in the case where the pulse method “works” are introduced and compared with the K-method.

📄 Content

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K-method of cognitive mapping analysis

A.A.Snarskiia,b, D.I.Zorinetsa,1, D.V.Landea,b, A.V.Levchenkoc

a NTUU “Kyiv Polytechnic Institute”, Kiev, Ukraine b Institute for Information Recording NASU, Kiev, Ukraine c ROSEN Europe B.V., Kiev, Ukraine

Abstract

Introduced a new calculation method (K-method) for cognitive maps. K - method consists of two consecutive steps. In the first stage, allocated subgraph composed of all paths from one selected node (concept) to another node (concept) from the cognitive map (directed weighted graph) . In the second stage, after the transition to an undirected graph (symmetrization adjacency matrix) the influence of one node to another calculated with Kirchhoff method. In the proposed method, there is no problem inherent in the impulse method. In addition to “pair” influence of one node to another, the average characteristics are introduced, allowing to calculate the impact of the selected node to all other nodes and the influence of all on the one selected. For impulse method similar to the average characteristics in the case where the pulse method “works” are introduced and compared with the K-method

1. INTRODUCTION One of the directions in modeling of complicated network structures is the creation of cognitive maps, their description and analysis [1]. A cognitive map is an orientated graph to the edges of which weights may be collated. As any graph a cognitive map is defined by adjacency matrix, the elements of which correspond to the weights of the edges between the vertices. Some concepts correspond to the vertices of a cognitive map and cause-effect (casual) relations between concepts correspond to the edges (relations). The weight of the cognitive map edge has a positive value if the increasing of the weight of the concept-cause leads to the increasing of the weight of the concept. The negative weight of the edge means that the increasing of the weight of the concept- cause leads to the decreasing of the weight of the concept-effect. Thus, a cognitive map is an digraph to each concept of which the weight is assigned and some concept corresponds to each concept. Fundamental models of cognitive maps are sign digraphs, weight digraphs and so-called fuzzy cognitive maps. The peculiarity of cognitive maps, being investigated now, is their weak structuralization which is so characteristic to social, economic, organizing, political and many other networks. In this paper weight digraphs are regarded as cognitive maps, and sign digraphs are interpreted as weight ones with weight value ±1.

Thus, a cognitive map is the digraph to each relation of which weight is assigned and concept of a cognitive map corresponds to each concept In the first point the most frequently utilized method of quantitative description of cognitive maps –impulse method and the principal drawbacks so characteristic to it are very briefly considered. In the second point a new method of quantitative description of cognitive maps is described, we call it K-method for short. In point 3 some examples of calculations of cognitive

1 Corresponding author at: Kyiv Polytechnic Institute, Kyiv, Ukraine. D.I. Zorinets. deniszorinets@gmail.com

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maps with the help of K-method are given. In point 4 two characteristics of the obtained K-matrix, analogs of characteristics in HITS – algorithm proposed in [2], which are called “pressure” and “consequence” have been introduced and analyzed. In point 5 the cognitive maps in which the series of adjacency matrix diverge have been considered as a result this map cannot be investigated in the frames of impulse method. In the supplement for method characteristics which are analogous to the introduced for K-method “pressure” and “consequence” have been represented. The comparison of these characteristics has been made in case when the calculation of cognitive map in pulse method is possible.

2. THE IMPULSE METHOD According to the most frequently utilized method of quantitative description of cognitive maps - the impulse method, to each concept 𝑖 – some primordial value 𝑣𝑖(initial) is assigned. The problem is to define the final value of concept 𝑣𝑖(𝑡→∞), or in some cases the rate of change in time.
   For definition 𝑣𝑖(𝑡) it is necessary to set the law of value change of the concept depending on its initial value, values of related neighboring to it concepts, weights of relations. The basic procedure of cognitive mapping analysis is iterational method, described in [1] in detail. According to this method the values of each concept 𝑣𝑖(𝑛) at the moment of discrete time 𝑛 (𝑛= 1,2 … ) is defined by the following rule 𝐯(𝑛+ 1) = 𝐯(𝑛) + 𝐖𝐖(n), 𝑛= 0,1,… , (1) where 𝐯(𝑛) –vector-column of net concepts of cognitive map 𝐖 – adjacency matrix of digraph of relations with weight, and

𝐩(𝑛) = 𝐯(𝑛) −𝐯(𝑛−1), 𝑛= 1,2, … . (2) At the initial moment

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