Statistical distribution and distribution of power: In search for harmony in international relations

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.

Keywords

System information culture harmony international relations power law entropy military expenses

1. 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.

2. 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:

$\displaystyle I(X,R)=\Sigma p(x)\Sigma p(r/x)\log[p(r/x)/p(r)]=H(R)-H(R/X)\max,$ (1)

where $x$ – conditions of the environment (summing here and later goes over the values in parentheses, $x$ and $r$ in this case); $r$ – responses (traits, features) of the system; $p(x)$ , $p(r)$ – probabilities of $x$ and $r$ ; $p(r/x)$ – conditional probability of $r$ when $x$ occurred; $H(R)$ – unconditional entropy of responses; $H(R/X)$ – conditional entropy of responses.

This principle means that the system aspires to choose such a response $r$ which provides the maximum information about a given stimulus $x$ – or in other words, the ‘mutual information’ between the system and its environment should be maximized. The concept of mutual information $I(X,R)$ was first proposed in (Fano, 1951), and it could be illustrated by a simple example. If one has to translate a certain text from Russian into English, the stimuli $X$ are Russian words, the responses $R$ – their English equivalents, $H(X)$ – the variability of the Russian vocabulary, $H(R)$ – the variability of the interpreter’s English vocabulary, $H(R/X)$ – inexactness of the translation. The quality of the interpreter could be measured by mutual information $I(X,R)=H(R)-H(R/X)$ of the translated text.

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 $E(X,R)$ :

$\displaystyle\Sigma p(x,r)e(x,r)=E(X,R),$ (2)

where $e(x,r)$ is a resource expense in the state $(x,r)$ . The resource might be an energy (in physics), a substance (in chemistry), a number of talented persons (in cultural studies), and so on. If there are several restrictions, we write several expressions like Eq. (2).

The next logical condition is the following:

$\displaystyle\Sigma p(x,r)=1.$ (3)

Equations (1)–(3) can be combined by the Lagrange multipliers $\lambda$ and $\beta$ and presented in the form:

$\displaystyle L=I(X,R)-\lambda\Sigma p(x,r)-\beta E(X,R)=H(R)-H(R/X)-\beta E(X% ,R)\max,$ (4)

where the multiplier $\beta$ means the deficit of resources. For example, in thermodynamics, $\beta=$ 1/T, where $T$ is an absolute temperature. The less energy a system has, the bigger is $\beta$ . In other cases $\beta$ may be a deficit of space, time, power, etc. The higher is the deficit, the bigger is the weight $\beta$ and the share of corresponding resource in the total. If the deficit of the resource $E$ is absent then $\beta$ equals zero and the corresponding item in the sum is omitted.

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 $H(R)$ , and this tendency is called a search behavior;

Idealization – tendency to improve exactness of the system’s responses, to decrease the entropy $H(R/X)$ , to get a maintenance and stabilization of essential parameters providing self-identification of the system, which permits the system to survive; sometimes it is named as conservative behavior;

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 $(x,r)$ with minimum resource expense $e(x,r)$ , or by the proneness to decrease the deficit of resource $\beta$ , or to increase the resource supply.

If resource $E$ is energy and the items $H(R)$ and $H(R/X)$ are fixed, the objective Eq. (4) corresponds to the information maximum principle, or the principle of the energy conservation. This economy becomes also important when the resource deficiency $\beta$ is too big and the last item in Eq. (4) dominates over the others.

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).

3. Special cases: principle of entropy maximum restricted resources, and power law

If the response is always adequate to the reaction, then item $H(R/X)$ in expression Eq. (4) equals zero that leads to the well-known principle of the entropy (self-information) maximum. This principle reflects a tendency to increase a number and variety of a system’s responses $H(R)$ (for more detail see Golitsyn & Petrov, 1997). Originally, the principle of entropy maximum was formulated in physics, but later it has been broadly applied in many areas, such as economics, linguistics, sociology, etc. The modern usage of this principle is based on works by Jaynes (1957) who considered informational entropy as a subjective measure of lack of knowledge. The entropy maximum is a generalization of the principle of event randomization: if we know nothing about a variable $r$ , we assume that all its values have equal probabilities, $p(r)=\textit{const}$ . It can be deduced from the entropy maximum in the absence of constraints. The information theory led to using entropy in many directions (e.g., Petrov & Yablonsky, 1980; Newman, 2005), with the important role of resource limitations taken into account (Visser, 2013; Lozinsky, 1970). Nevertheless, the resource expenses were not previously combined with the entropy, while their combination was explicitly performed in the systemic-informational approach.

Let us consider the role of resources available for a system. If we know restrictions for a variable $r$ (such as its mean, variance, etc.) it is possible to obtain a more exact probability distribution for it by adding these restrictions to the entropy maximization. A typical form of this restriction occurs when the total or mean quantity of resources is fixed, so Eq. (2) reduces to the following relation:

$\displaystyle\Sigma p(r)e(r)=E(R).$ (5)

The entropy maximization can be described as follows: if $r$ is a random variable, probabilities $p(r)$ can be obtained from the condition:

$\displaystyle H(R)=-\Sigma p(r)\log p(r)\max.$ (6)

Also, the total probability equals 1, so Eq. (3) reduces to:

$\displaystyle\Sigma p(r)=1.$ (7)

The objective Eq. (6) with two constraints Eqs (5) and (7) combined by Lagrange multipliers $\lambda$ and $\beta$ , corresponds to the objective Eq. (4) reduced to:

$\displaystyle L=H(R)-\lambda\Sigma p(r)-\beta E(R)=\max.$ (8)

To maximize the objective Eq. (8), the partial derivative of $L$ with respect to probability $p(r)$ is put to zero, which yields:

$\displaystyle\partial L/\partial p(r)=\partial[-\Sigma p(r)\log p(r)-\lambda% \Sigma p(r)-\beta\Sigma p(r)e(r)]/\partial p(r)=-\log p(r)-1-\lambda-\beta e(r% )=0.$ (9)

From Eq. (9) we get:

$\displaystyle\log p(r)=-1-\lambda-\beta e(r).$ (10)

Then the optimum probability distribution $p(r)$ equals:

$\displaystyle p(r)=\exp[-1-\lambda-\beta e(r)]=C\exp[-\beta e(r)],$ (11)

where $C=\exp(-1-\lambda)$ is the normalizing constant.

Let $e(r)$ be a resource expense needed to achieve the state $r$ . Then Eq. (11) shows that the higher is the resource expense the lower is the probability of the corresponding state $r$ . If the resource expense is growing then $p(r)$ is diminishing to zero, i.e., the state $r$ becomes unattainable. Consider the term $\beta$ which characterizes the deficit of resource. For a bigger $\beta$ the distribution $p(r)$ becomes more uneven and concentrates around zero that corresponds to the lower entropy of responses $H(R)$ because the entropy is maximal when the probability distribution is uniform. In other words, a deficit of resources leads to a decreasing variety in the responses of a system. The inverse value $1/\beta$ can be interpreted as activeness of the system. For example, the temperature plays a role of a measure of activeness in physics. The higher is the deficit of resources, the less is the activeness.

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 $\beta$ . The link between the increment of the resource expense $d e$ by the total resource $r$ is defined in (Golitsyn & Petrov, 1997) as follows:

$\displaystyle de=dr/r,$ (12)

Integral of this expression yields the equation:

$\displaystyle\log p(r)=\log C-\beta\log r,$ (13)

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 $\bm{\beta}$ . Two behavioral properties of such curves are the following: the more resource expense $e(r)$ is needed to the function at a given level, the less is the probability $p(r)$ at this level; the more is the deficit of resource $\beta$ , the steeper is the slope of the curve $p(r)$ . It is worth mentioning that the value of this coefficient was used as an indicator of inter-personal differences in various sociological and psychological measurements (Golitsyn & Petrov, 1997; Petrov & Mazhul, 2014).

Figure 1.

Dependence of the probability of response $p(r)$ on resource expense $e(r)$ and the deficit of resource $\beta$ .

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.

4. Quantitative view on the structure of unipolar world

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.

a)
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 $N$ was rapidly growing together with the number of subscribers $n$ as the number of pairs among $n$ elements.

$\displaystyle N=n(n-1)/2.$ (14)

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

$\displaystyle N=n.$ (15)
b)
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.
c)
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.

Table 1
Military expenses of top 15 countries of the world in 2016

Rank Country Expenses, bill. $

1 USA 611.0

2 China 215.0

3 Russia 69.2

4 Saudi Arabia 63.7

5 India 55.9

6 France 55.7

7 Great Britain 48.3

8 Japan 46.1

9 Germany 41.1

10 Republic of Korea 36.8

11 Italy 27.9

12 Australia 24.6

13 Brazil 23.7

14 Arab Emirates 22.8

15 Israel 18.0

The order of these countries by their ranks is shown in Fig. 2 where the logarithmic scales are used by both axes.

Figure 2.
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 $r$ -th rank can be expressed as follows (see Petrov & Yablonsky, 1980, 2013):

$\displaystyle I_{r}=I_{1}r^{-\alpha}.$ (16)

In logarithmic coordinates it becomes a straight line:

$\displaystyle\log I_{r}=\log I_{1}-\alpha\log r,$ (17)

where $I_{1}$ is the intensity of functioning of the first element ( $r=$ 1), and $\alpha$ is the slope of linear approximation. The correlation between this line and empirical data is very high, with its square $R^{2}=$ 0.94. The slope of this line $\alpha=$ 1.14 is close to such values in similar processes (Newman, 2005; Martindale, 1995).

Table 2
Dynamics of military expenses (US dollars in billion) some leading countries belonging to NATO and BRICS

Country 2010 2011 2012 2013 2014 2015 2016

Members of NATO:

USA 698. 2 711. 3 684. 8 639. 7 609. 9 596. 0 611. 0

Great Britain 58. 1 60. 3 58. 5 56. 9 59. 2 55. 5 48. 3

France 61. 8 64. 6 60. 0 62. 4 63. 6 50. 9 55. 7

Germany 46. 3 48. 1 46. 5 45. 9 46. 1 39. 4 41. 1

Italy 36. 0 38. 1 33. 7 33. 9 31. 6 23. 8 27. 9

NATO in total 1013. 7 1044. 5 996. 6 968. 5 942. 4 892. 1 918. 3

NATO without USA 315. 5 333. 2 347. 8 328. 8 332. 5 296. 1 307. 3

Members of BRICS:

China 115. 7 138. 0 157. 5 177. 9 199. 7 214. 8 215. 0

Russia 58. 7 70. 2 61. 5 88. 4 84. 7 66. 4 69. 2

India 46. 1 49. 6 47. 2 47. 4 50. 9 51. 3 55. 9

Brasilia 34. 0 36. 9 34. 0 32. 9 32. 7 24. 6 23. 7

South Africa 4. 2 4. 6 4. 5 4. 1 3. 9 3. 5 3. 5

BRICS in total 258. 7 299. 3 304. 7 350. 7 371. 9 360. 6 377. 3

Figure 3.
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 $R^{2}=$ 0.96 and the slope $\alpha=$ 1.71. It is higher than the slope for the entire world system, which allows to suppose that the structure of NATO has more rational features of organization than the self-organized global system.

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.
5. Conclusion

Rank	Country	Expenses, bill. $
1	USA	611.0
2	China	215.0
3	Russia	69.2
4	Saudi Arabia	63.7
5	India	55.9
6	France	55.7
7	Great Britain	48.3
8	Japan	46.1
9	Germany	41.1
10	Republic of Korea	36.8
11	Italy	27.9
12	Australia	24.6
13	Brazil	23.7
14	Arab Emirates	22.8
15	Israel	18.0

Country	2010	2011	2012	2013	2014	2015	2016
Members of NATO:
USA	698.	2	711.	3	684.	8	639.	7	609.	9	596.	0	611.	0
Great Britain	58.	1	60.	3	58.	5	56.	9	59.	2	55.	5	48.	3
France	61.	8	64.	6	60.	0	62.	4	63.	6	50.	9	55.	7
Germany	46.	3	48.	1	46.	5	45.	9	46.	1	39.	4	41.	1
Italy	36.	0	38.	1	33.	7	33.	9	31.	6	23.	8	27.	9
NATO in total	1013.	7	1044.	5	996.	6	968.	5	942.	4	892.	1	918.	3
NATO without USA	315.	5	333.	2	347.	8	328.	8	332.	5	296.	1	307.	3
Members of BRICS:
China	115.	7	138.	0	157.	5	177.	9	199.	7	214.	8	215.	0
Russia	58.	7	70.	2	61.	5	88.	4	84.	7	66.	4	69.	2
India	46.	1	49.	6	47.	2	47.	4	50.	9	51.	3	55.	9
Brasilia	34.	0	36.	9	34.	0	32.	9	32.	7	24.	6	23.	7
South Africa	4.	2	4.	6	4.	5	4.	1	3.	9	3.	5	3.	5
BRICS in total	258.	7	299.	3	304.	7	350.	7	371.	9	360.	6	377.	3

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.

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