Abstract

In the exploration of complex systems, a pivotal aspect involves analyzing specific numerical data to comprehend the system's functioning. This effort often extends to identifying specialized elements or subsystems within the system, discerned by their consistent response to defined 'actions' and intricate 'relations' among homoge-neous subsystems. Understanding these nuanced characteristics through rigorous mathe-matical analysis, elucidating the underlying structure of the system, is crucial, particu-larly as a foundation for conducting complex or resource-intensive statistical studies. The research explores this basic methodology to identify single-peak sequences that define components of what we call "monotonic systems," where peaks represent "kernels" and "hikes" are depicted as "stable sets." Furthermore, we extensively delve into an additional constructive methodology involving two defining sequences within monotonic systems. Through meticulous exploration, we uncover the complex relation-ship between these defining sequences, ultimately leading to the formulation of the duality theorem. This theorem not only serves as a cornerstone in our understanding but also provides a systematic approach for limiting the search area for kernels and stable sets. In light of this, we present an algorithm designed specifically for the identification of extremal subsystems, namely kernels and stable subsets, within a monotonic system, encapsulated by a certain dual scheme. Keywords: monotonic; system; matrix; graph; cluster

Extremal Subsystems of Monotonic Systems, I (pdf)

A general theoretical method is described which is intended for the initial analysis of systems of interrelated elements. Within the framework of the model a specially postulated monotonicity property for systems guarantees the existence of a special kind of subsystems called kernels. A number of extremal properties and the structure of the kernels are found. The language of description of Monotonic Systems of interrelated elements is described in general set-theoretic terms and leads to a constructive system of notions in case of system with finite number of elements. A series of properties of special finite sequences of elements are studied whereby kernels in Monotonic System are classified.

Extremal Subsystems of Monotonic Systems, II (pdf)

A constructive procedure is considered for obtaining singular determining sequence of elements of Monotonic systems studied in I. The relationship between two determining sequences   a⁺   and   a^-   is also examined, and the obtained result is formulated as a duality theorem. This theorem is used for describing a procedure of restricting the domain of search for extremal subsystems (or kernels of a Monotonic System): the corresponding search scheme is also presented.