Mathematical Aspects of the Concept of Influence in Cognitive Simulations

The mathematical aspects of algorithms for assessing the influence of external factors in the cognitive simulation of complex systems are considered. The systems are represented as directed graphs, to the nodes and edges of which certain weights are assigned. In the conventional approach, the influence of external factors (input nodes) on the state (weight) of system elements (internal nodes) is computed successively on the set of all paths using the method of additive convolution of the input nodes and the corresponding edges. In the new approach proposed here, such influences are defined for each internal node as a partial derivative of a functional dependence that corresponds to the graph. Formulas for calculating the influence coefficient are derived that take into account the graph structure, partitions into cycles, and paths between corresponding nodes. The results provided by the proposed method and the conventional one are compared using as an example a simple cognitive model that assesses the impact of a certain viral disease on the population and production and measures used to counteract it. The introduced definition of the concept of influence is also valid for more complex, nonlinear formulas or fuzzy characteristics of nodes and edges.