Elements of Complexity Theory

Formation of the Concept of Complexity. Range of Difficulties

All sciences accumulate a variety of information about the world and about humans as its element. However, the science of systems pays special attention to the interactions of people with their environment, that is, the cognition and transformation of reality by humans. In the study of human activity to change the environment in accordance with their goals, a special place is occupied by clarifying the conditions for successfully achieving the goal, determining the causes of difficulty (complexity!) in achieving the goal, as well as finding ways to overcome, bypass, or at least reducing complexity. These problems are the central subject of applied systems analysis.[1]

The transition from the Age of Machines to the Age of Systems in the activities of mankind (see Chapter 1) takes place in the form of developing management practices in a continuously changing environment, with increasingly encountered difficulties in achieving the goals. During the research regarding ways to overcome these difficulties, the experience of “difficult” situations was accumulated and generalized; gradually, the understanding of the essence of the phenomenon of complexity in human activity in the surrounding reality increased. Understandably, the explanation of the incomprehensible is reduced to the representation of the accumulated information in the form of three basic models: models of the “black box”, composition, and structure of the considered system. All the difficulties in our practice arise from the “difficulties” lurking in the resulting models, which we further use in the development, adoption, and execution of our management decisions. We now understand that the source of all difficulties lies in the fact that our models, that is, ideas about the reality that we want to transfer into the desired state, are somewhat different from the reality; and that these differences can relate to the different elements of any of the models, as a result of which complexity can have different nature, vary in degree of complexity, as well as in quality characteristics.

Classification of the Complexity Types

The simplest model of diversity is classification (see Section 3.5, Part II). Like all models, classification has a purpose: different classifications are built for different purposes. Consider the most common classification of types of complexity designed to “localize” the reasons for this type of complexity (finding “lever points”), followed by the definition of ways to reduce the complexity of this type (fully or to the minimum level possible in this case).

Classification According to the Degree of Objective Complexity in the Behavior of the Controlled Object

Study of systems dynamics has revealed the discreteness of the types of behavior of such systems, and it was found to be subject to completely deterministic differential equations of the second order: their trajectories in the phase space converge to one of the four possible configurations (point, cyclic, toroidal, and strange attractors, see the section “Attractors” in Section 4.4.3). The real sensation was the discovery of random trajectories inside toroidal and strange attractors. This makes the remote future of the system unpredictable, making it very difficult to manage.

The discovery that nature itself limits the filling of the phase space with trajectories of the dynamic system only within a limited area of the attractor, and the set of attractors is discrete and finite, with the attractors themselves differing in a growing degree of chaotic trajectories, was one of the three great achievements of the 20th century in the knowledge of nature (the first two include the theory of relativity and quantum mechanics). The feeling that this is not only a physical regularity of striking beauty but a manifestation of the universal law, which is fair to all forms of organization of the material world, was confirmed by the consideration of chemical, biological, and social systems: everywhere there are similar features of the dynamics of the state of the system.

In particular, in the theory of management of social systems, there are four types of systems: simple, complicated, complex, and chaotic, in order of increasing complexity of management (due to the growing uncertainty of the future). For the first three types of complexity, methods were developed to overcome the difficulties (control algorithms). Naturally, the algorithm for overcoming a particular difficulty is associated with the possibility of neutralizing a specific cause of complexity — the lack of some resources (substance, energy, information) necessary for successfully achieving the goal.

Different types of control correspond to different reasons of complexity (see Section 4.3 in Part I):

  • • simple system management (all necessary resources are available) —program management;
  • • management of a complex system (with lack of information about the managed system) — trial and error (Note that here the term “complex” is used in a narrow sense when the reason for the complexity is only a lack of awareness about the managed system);
  • • parameter control, or regulation (for small, compensable differences between the model and the real state of the environment);
  • • management structure, reorganization (in case of large deviations, non- compensable due to the lack of material resources);
  • target management (if the current target is unattainable);
  • • management of large systems (with lack of time to find the optimal solution). (Note that here the terms “big” and “small” also have a special meaning which differs from everyday);
  • • management of the uncertainty in the ultimate goal (if information is unavailable about the ultimate goal, but you have confidence in the existence of the best state) — heuristic (revolutionary) and empirical (evolutionary) approaches.

However, in relation to chaotic (objectively stochastic) systems, the question of control is different: the laws of nature are not subject to us, and we cannot change the natural randomness of events. It remains only to adapt to the unexpected changes taking place around, like fishermen in a storm, or pilots caught in a zone of turbulence, or a surfer, maneuvering on his board on the slope of a steep wave. As D. Meadows puts it, chaotic systems cannot be controlled, but they can be danced with [13].

System thinking offers a set of techniques for managing social systems that have entered a chaotic phase of their lifecycle. These techniques are based on the use of information about the objective laws of natural development, including frac- tality, change of attractors, self-organization, etc. (see Chapter 3). However, experts warn [14] that due to the presence of people’s consciousness and free will, the development of global information technology and other factors, chaos in social systems is much more complicated than physical chaos, which requires caution in the practical use of the results of the mathematical theory of chaos.

Complexity Classification of Types of Models of the Managed System

Targeted impact on reality is planned on the basis of a working model of the system undergoing transformation. Which of the three basic models (black box, composition model, structure model) or a combination of them should serve as a working model in a particular case depends on the characteristics of the problem and the problem situation. However, when constructing any particular model, errors may creep into it (see Section 2.1, Part I), or the constructed model may be inconvenient to use. All this leads to difficulties and creates complexities in work. Again, we are faced with different types of complexities that require different approaches. Let’s consider the classification of difficulties that may arise in each of the basic models.

The Complexity Caused by the Large Dimension of the Composition Model

In some cases, developing a successful solution to the problem requires information about all elements of the problem situation, that is, a fairly complete model of the composition of the system. The importance of this requirement is related to one of the basic laws of cybernetics — the Law of Requisite Variety proposed by Ashby. The variety of the system is the number of its possible states. Ashby’s law claims that problem-free management is possible only if the diversity of the controlling system is not less than the diversity of the controlled system. Often complexity is due to the lack of diversity of the system composition model. There are systems consisting of a very large number of elements, and working with their composition model becomes extremely time-consuming and difficult. The reason for the complexity in such cases is the lack of available modeling resources to complete the processing of all information at the required time. In computational mathematics, this difficulty is called the “curse of dimension”; in control practice, such systems are called “large”.

A good example of this situation is the delay of three to four years of compilation by the USSR State Planning Committee of the annual input-output balance among the millions of produced and consumed products. This alone (in addition to other difficulties) sharply reduced the effectiveness of rigidly centralized operational management of Soviet Union’s national economy.

There are two options for controlling a large system (see Section 4.3, Part I): demanding physical acceleration of simulation (increasing computing power, parallel simulation techniques), and speeding up the simulation by simplifying the models (intentional reduction in the number of considered variables, linear approximation of nonlinear dependencies, etc.), that is, conscious low-quality, but timely decisions.

As a measure of the complexity of large systems, A. N. Kolmogorov proposed to use the length (the number of bits in the program) of the algorithm that completely describes the composition of the system.

  • [1] Note that since in English the meaning of the word analysis is reduced to reductionistn, that is,consideration of only the composition (sometimes, structure) of the system, and the Russian termapplied systems analysis includes a synthetic consideration of the system (i.e., the black box model), inEnglish system terminology, it corresponds to the terms systems thinking and design thinking (projectthinking).
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