Abstract
This paper describes how systemic-informational approach could be applied to the analysis of international relations. It considers the interplay of two fundamental concepts: the shape of distribution of countries by their military might and the unipolarity of world structure. It concludes that the most harmonious version of the global distribution of forces is based on the power law, and demonstrates that contemporary military expenses of fifteen leading countries correspond to this regularity, which allows to assume that the existing distribution is optimal and stable.
Introduction
A well-known systemic-informational approach is applied to the problem of military relations between leading countries of the world. This approach has already demonstrated its effectiveness and universality in other areas; such as semantic, linguistic, creative arts (Golitsyn, 2000; Golitsyn & Petrov, 1995, 2005; Petrov, 2017) which supports the claim that this attempt could be productive. The paper considers the optimal structure of international relations, specifically in its military aspect, describing the relations most advantageous for the global system. This approach provides some new results in area, which could be tentatively called “statistical analysis of global conflicts”, being developed for the long time, since (Richardson, 1960) to the latest conclusions of the Club of Rome on the necessity to create a new worldview (von Weizsaeker & Wijkman, 2018).
The paper is organized as follows: Section 2 describes theoretical foundation of the systemic-informational approach, Section 3 considers relation between entropy maximization and power law distribution, Section 4 applies developed techniques to empirical data on international relations, Section 5 considers perspectives of the multi-polar world, and Section 6 summarizes.
Theoretical foundation: The principle of information maximum
The principle of the information maximum was formulated in (Golitsyn & Petrov, 1995; 1997). It permits to explain many facts and regularities of behavior and evolution of various kinds of adaptive systems and could be presented as follows:
where
This principle means that the system aspires to choose such a response
The system maximizes the mutual information subject to restrictions which prevent the system from reaching the absolute (unconditional) maximum of information. The system has to be satisfied with the conditional maximum. A very general and typical condition is the restriction of the average resource
where
The next logical condition is the following:
Equations (1)–(3) can be combined by the Lagrange multipliers
where the multiplier
As far as the Eq. (4) contains three items, the principle of the information maximum describes three principal tendencies inherent to the behavior and development of any system. Those are:
Expansion – aspiration to increase a number and variety of the system’s responses
Idealization – tendency to improve exactness of the system’s responses, to decrease the entropy
Economy of resources – when the third item in Eq. (4) contains the terms defined in Eq. (2), they influence either by a choice of situations
If resource
The systemic-informational approach is congenial to the idea of famous Leonhard Euler to use purely logical (mathematical) tools to explain all physical and mental processes from the only principle of optimality. Some attempts to reach this aim seem to be promising (see Petrov, 2017, for application of this idea to physics and mental processes, and cultural phenomena).
If the response is always adequate to the reaction, then item
Let us consider the role of resources available for a system. If we know restrictions for a variable
The entropy maximization can be described as follows: if
Also, the total probability equals 1, so Eq. (3) reduces to:
The objective Eq. (6) with two constraints Eqs (5) and (7) combined by Lagrange multipliers
To maximize the objective Eq. (8), the partial derivative of
From Eq. (9) we get:
Then the optimum probability distribution
where
Let
One of the results of the principle of entropy maximum is the well-known power law (Zipf, 1949; Newman, 2005). Versions of this regularity are known under names of Pareto, Lotka, Zipf, Mandelbrot, and others, with the important findings summarized by Newman (2005) (see also Petrov & Yablonsky, 1980, 2013). Let us consider connection of the power distribution with the degree of the deficit of resource
Integral of this expression yields the equation:
which is a linear function with a negative slope. This dependence is discussed below.
Figure 1 shows two curves corresponding to this regularity, with different values of
Dependence of the probability of response 
Thus, the power law is the optimal distribution of the system’s states from the point of view of the entropy maximization under condition Eq. (12). It describes a degree of diversity of distribution of the system’s states caused by the resource restrictions.
Concerning the global world system, we may assume that it could correspond to the power law, and it is possible to check this hypothesis by a statistical analysis of the distribution of resources spent by the countries for their various needs.
As it was shown in (Golitsyn & Petrov, 1995), the development of numerous multi-element systems comes sooner or later to a certain centralized state where the elements are connected with each other via a core element – several examples can illustrate it as follows.
The structure of classical telephone nets. At the early stages of their development, subscribers were connected with each other directly. However, the number of inter-element links
The solution of the problem turned out to be easy: the central telephone exchange was introduced within each net, where each subscriber was connected with other subscribers only through the exchange. The number of links sharply diminished to
The system of commerce evolved towards introducing one central currency element (commonly, it was gold), instead of the direct barter exchange of one product for another. Fast processes (avalanche, crises) develop when a system has a certain element which is stronger than others, and this element becomes the center of exchange followed by its fast development (Petrov, 2013, pp. 237-254).
In general, evolution of almost any complex system can be described by the power law combined with the centralization. Let us apply these principles for the description of the world system.
We studied military expenses of 15 leading countries in 2016 (by the data from Tian et al., 2017): USA, China, Russia, Saudi Arabia, India, France, Great Britain, Japan, Germany, Republic of Korea, Italy, Australia, Brazil, Arab Emirates, and Israel – see Table 1.
Military expenses of top 15 countries of the world in 2016
The order of these countries by their ranks is shown in Fig. 2 where the logarithmic scales are used by both axes.
Distribution of 15 leading countries of the world, over their ranks on defense budget in logarithmic scales by both axes. The sequence of countries: USA, China, Russia, Saudi Arabia, India, France, Great Britain, Japan, Germany, Republic of Korea, Italy, Australia, Brazil, Arab Emirates, Israel – from the left to right dots.
We see an approximately linear dependence which is in full agreement with the power law distribution in its rank form: the intensity of the element possessing an
In logarithmic coordinates it becomes a straight line:
where
Dynamics of military expenses (US dollars in billion) some leading countries belonging to NATO and BRICS
Dynamics of military expenses of BRICS, NATO, and USA, % in relation to the level of 2010.
This result does not contradict to the above described concept of unipolar structure, i.e., a structure possessing only one center of the whole system. The contemporary world is controlled militarily by the ‘central element’ which is the USA with 36% of the world military expenses. The American tax-payers do actually support ‘harmony and stability’ of international relations making this system more or less harmonious, which is advantageous for the entire world stability. Note, however, that in this case the central element of the system (in the sense of the examples a)–c) above) is replaced by something else – the most important player, ranked as the biggest by the specific indicator (military expenses). It makes the whole structure analogous to that described earlier as a centralized one; nevertheless, this analogy is quite telling.
The dependence of military expenses of 15 largest countries in NATO in 2016 (USA, Great Britain, France, Germany, Italy, Turkey, Canada, Spain, Poland, Netherlands, Norway, Greece, Belgium, Portuguese, and Denmark) was also built as the function of these countries’ ranks. The linear relation on logarithmic scales was observed as well, with a very high coefficient of determination
Besides the quantitative arguments on the unipolar structure of the contemporary world system in its military aspect, the problems with other resources, like economic and cultural, should become the subject of a specialized study. Consider a possibility of transformation of the unipolar structure into a bi-polar or multi-polar one. Table 2 presents the dynamics of military expenses of NATO (excluding USA), the USA separately, and five countries united in the so-called BRICS (Brazil, Russia, India, China, and South Africa), by the data of 2010–2016 years.
Figure 3 illustrates the same data graphically. We see a continual growth of BRICS expenses, signaling inclination of its members to become independent of the other world system players.
Sure enough, the defense budget of BRICS is no more than about 40–45% of the USA, but dynamics like that, if preserved, could alternate this proportion noticeably. We may expect some changes which could be not so good for the previously achieved harmonious or optimal structure of the world. A simple extrapolation of the data on the dynamics presented in Table 2 indicates a possibility for destruction of unipolarity – mainly because of growth of military expenses of China which by the year 2025 might become equal with those of the USA. Of course, such a situation requires a separate research.
The described systemic-informational approach to international problems needs further development especially in such fields as diplomacy, social and cultural politics. Sure enough, the author recognizes that such a universal view of very complex processes is an extreme simplification, although it may help to grasp some important features. The idea that relations between nations can be revealed as a result of certain self-organized system in its optimization is a fascinating possibility worthwhile more research. International relations, however, cannot be reduced to conflicts and military matters only, the economic and inter-cultural influences play a huge role in the national and global developments (Uspensky, 2002; Petrov, 2013), but this is a problem for future studies.
Footnotes
Acknowledgments
I express my deep gratitude to the late friend, Prof. Colin Martindale, as well as to Prof. Robert Hogenraad, Dr. Lidia A. Mazhul, and Dr. Peter A. Kulichkin – for cheerful discussions; and to anonymous reviewers for multiple comments, which significantly helped to improve the article. My special thanks to Dr. Igor Mandel for valuable advices concerning the content and style of this paper.
