3. Vector entropy model for the effective functioning of systems

Consider the problem of developing an integral indicator that characterizes the effective functioning of the system.

Multicriteriality of complex systems functioning, including territorial, and the diversity of their elements functioning, makes the development of universal formal indicators difficult which characterizes the effectiveness of systems as a whole.

It is known that entropy is a fundamental property in any systems with probabilistic behavior [16]. The concept of entropy is flexible and allows interpretation in terms of the branch of science, where it is applied. Therefore, entropy modeling is one of the promising lines of research of complex stochastic systems [17–20].

However, the frequent use of entropy for modeling of open systems, in contrast to thermodynamics, is insufficiently formalized and has generally qualitative and private character; there are no rather simple and adequate mathematical models that allow associating entropy with the actual characteristics of conditions of multidimensional systems. Common in these works is the use of Shannon's information entropy [21]. But, as it is noted in [15], the information entropy allows developing adequate entropy models only for particular problems.

However, in the same work [21], Shannon heuristically offered a formal analog of a concept of information entropy for the m-dimensional continuous random vector of Y with a probability density:

$$H(\mathbf{Y}) = -\int\_{-\infty}^{+\infty} \dots \int\_{-\infty}^{+\infty} p\_{\mathbf{Y}}(\mathbf{y}\_1, \mathbf{y}\_2, \dots, \mathbf{y}\_m) \ln p\_{\mathbf{Y}}(\mathbf{y}) d\mathbf{y}\_1 d\mathbf{y}\_2 \dots d\mathbf{y}\_m. \tag{1}$$

This value Kolmogorov together with Gelfand and Yaglom was called subsequently differential entropy [22].

The differential entropy, being the functional given on the set of the probability density of a random vector of Y, represents a number. Therefore it cannot be an adequate mathematical model of a multidimensional system. However, the practical use of entropy (1) is complicated by the need to know the distribution law of a multidimensional random value of Y.

In [23] it was offered to use a differential entropy (further, an entropy) for modeling multidimensional stochastic systems. It is proven [23] that entropy in Eq. (1) can be represented as a sum of two components:

$$H(\mathbf{Y}) = H(\mathbf{Y})\_V + H(\mathbf{Y})\_R,\tag{2}$$

$$H(\mathbf{Y})\_V = \sum\_{i=1}^m H(Y\_i) = \sum\_{i=1}^m \ln \sigma\_{Y\_i} + \sum\_{i=1}^m \kappa\_i \text{---randomness entropy},$$

$$H(\mathbf{Y})\_R = \frac{1}{2} \sum\_{k=2}^m \ln \left( \mathbf{1} - R\_{Y\_k / Y\_1 Y\_2 \dots Y\_{k-1}}^2 \right) - \text{self-organization entropy},$$

which are irreversible and by all means are followed by the transition of a part of the energy of ordered processes (kinetic energy of a moving body, energy of electric current, etc.) into the energy of the disordered processes and eventually in warmth. The second part is caused by the exchange of energy and substance between a

Probabilistic-Entropic Concept of Sustainable Development of the Example of Territories

where dS is the total change in entropy of an open system, dSin is the change in

However, the question of the practical application of this theory for research of real systems has not been disclosed. Let us express the change of the total entropy of ΔH (Y) through the changes of entropies of randomness and self-organization:

Let us try to give an interpretation of Eq. (5) according to Eq. (6). First, it is apparent that dS ffi ΔH Yð Þ. The sign of the conditional equality "ffi" is used in view of the fact that in [24], change of a thermodynamic entropy of dS was

Let us consider the influence on the entropy of processes of exchange with the environment. From the environment multidimensional open system takes or gives

energy, which can be treated as a change of mean square deviations σYi

happening in a system is a change of self-organization entropy:

change of self-organization entropy.

self-organization entropy will decrease.

25

region) development dynamics in 1992–2017.

appearance of new properties, states can also occur from the outside, from the environment. Therefore, the change of distribution type, and therefore entropy indicators, is also due to the process of exchange of the system with the environment, that is, it is possible to consider that the change of entropy during processes of exchange with the environment represents a change of randomness entropy:

System elements in the process of functioning can strengthen or weaken the interaction between them due to the increase or decrease of the narrowness of correlation communication. Therefore, the change of entropy during the processes

On the basis of Eqs. (7) and (8), it is possible to make the following hypothesis. Hypothesis 1. The total change of entropy of an open system consists of the sum of two items. The first item characterizes the impact of the interaction of a system with the external environment and represents a change of randomness entropy. The second item characterizes the processes occurring within a system and represents a

Example 1. Entropy analysis of Yekaterinburg (regional center of Sverdlovsk

The effective functioning of the megalopolis as a complex system according to the vector entropy model Eq. (4) consists in the simultaneous growth of diversity, opportunities for all elements of this system, and the presence of a close interrelation between these elements. This is manifested in the fact that with the development of a megalopolis, its randomness entropy should gradually increase, and the

entropy during the processes occurring in a system, and dSout is the change of

entropy during the processes of exchange with the environment.

dS ¼ dSin þ dSout, (5)

ΔHð Þ¼ Y ΔHð Þ Y <sup>V</sup> þ ΔHð Þ Y <sup>R</sup> (6)

dSout ffi ΔH Yð ÞV: (7)

dSin ffi ΔH Yð ÞR: (8)

. Besides the

system and a surrounding medium:

DOI: http://dx.doi.org/10.5772/intechopen.89287

considered.

where σ<sup>2</sup> Yi —dispersion, κ<sup>i</sup> ¼ H Yi=σYi ð Þ—entropy indicator shows a type of random value distribution law Yi, i = 1,2,...,m; R<sup>2</sup> Yk=Y1Y2…Yk�<sup>1</sup> - coefficient of determination of regression dependencies of random vector Y, k = 2,3,…, m.

In particular, for multidimensional normally distributed random variable Y.

$$H(\mathbf{Y})\_V = \sum\_{i=1}^m \ln \sigma\_{Y\_i} + m \ln \sqrt{2\pi\epsilon}, H(Y)\_R = \frac{1}{2} \ln \left( |R| \right), \tag{3}$$

where R is the correlation matrix of random vector Y.

The formula (2) does not always explain the behavior of the system. The addition of the component H(Y)<sup>V</sup> and H(Y)<sup>R</sup> in terms of systems analysis is incorrect since they characterize various regularities of the complex systems: H(Y)<sup>V</sup> is additivity, and H(Y)<sup>R</sup> is an integrity of the system.

The practical use of the relation (2) showed that there are situations when systems with different functional states have approximately the same general entropies of H(Y), but the corresponding values of entropies of randomness H(Y)<sup>V</sup> and self-organization H(Y)<sup>R</sup> have significant differences. It schematically looks as follows. There are two the same systems Y(1) and Y(2) with different states. At the same time, <sup>H</sup> <sup>Y</sup>ð Þ<sup>1</sup> � � <sup>¼</sup> 0, <sup>H</sup> <sup>Y</sup>ð Þ<sup>1</sup> � � <sup>V</sup> <sup>¼</sup> 1, <sup>H</sup> <sup>Y</sup>ð Þ<sup>1</sup> � � <sup>R</sup> ¼ �1 and <sup>H</sup> <sup>Y</sup>ð Þ<sup>2</sup> � � <sup>¼</sup> 0, H Yð Þ<sup>2</sup> � � <sup>V</sup> <sup>¼</sup> 10, <sup>H</sup> <sup>Y</sup>ð Þ<sup>2</sup> � � <sup>R</sup> ¼ �10.

Complex systems, including territorial ones, are open, and their entropy can both increase and decrease. Moreover, the directions of change in the entropies of randomness H(Y)<sup>V</sup> and self-organization H(Y)<sup>R</sup> of systems may be different. To build adequate models and investigate multidimensional stochastic systems, differential entropy should be considered not in scalar, but in vector form as two components—the entropies of randomness and self-organization as [15]:

$$h(\mathbf{Y}) = (h\_V; \ h\_R) = \left( H(\mathbf{Y})\_V; \ H(\mathbf{Y})\_R \right). \tag{4}$$

In specific situations, the direction and values of the entropy vector Eq. (4) should be set on the basis of the features of the studied system. In other words, complex systems should have a balance between the entropies of randomness and self-organization.

The complex systems are open. Influence of entropy on the evolution of open systems was investigated by many scientists. In their publications, it is noted that the change of open systems either leads to degradation or it is self-organization process as a result of which more complex structures appear. Prigogine [24] in 1955 formulated an extended version of the second law of thermodynamics. According to this law, the total change of entropy dS of an open system must be represented in the form of two parts. The reason of the first of them serves internal processes

Probabilistic-Entropic Concept of Sustainable Development of the Example of Territories DOI: http://dx.doi.org/10.5772/intechopen.89287

which are irreversible and by all means are followed by the transition of a part of the energy of ordered processes (kinetic energy of a moving body, energy of electric current, etc.) into the energy of the disordered processes and eventually in warmth. The second part is caused by the exchange of energy and substance between a system and a surrounding medium:

$$d\mathbb{S} = d\mathbb{S}\_{\text{in}} + d\mathbb{S}\_{\text{out}},\tag{5}$$

where dS is the total change in entropy of an open system, dSin is the change in entropy during the processes occurring in a system, and dSout is the change of entropy during the processes of exchange with the environment.

However, the question of the practical application of this theory for research of real systems has not been disclosed. Let us express the change of the total entropy of ΔH (Y) through the changes of entropies of randomness and self-organization:

$$
\Delta H(\mathbf{Y}) = \Delta H(\mathbf{Y})\_V + \Delta H(\mathbf{Y})\_R \tag{6}
$$

Let us try to give an interpretation of Eq. (5) according to Eq. (6). First, it is apparent that dS ffi ΔH Yð Þ. The sign of the conditional equality "ffi" is used in view of the fact that in [24], change of a thermodynamic entropy of dS was considered.

Let us consider the influence on the entropy of processes of exchange with the environment. From the environment multidimensional open system takes or gives energy, which can be treated as a change of mean square deviations σYi . Besides the appearance of new properties, states can also occur from the outside, from the environment. Therefore, the change of distribution type, and therefore entropy indicators, is also due to the process of exchange of the system with the environment, that is, it is possible to consider that the change of entropy during processes of exchange with the environment represents a change of randomness entropy:

$$d\mathbb{S}\_{out} \cong \Delta H(Y)\_V. \tag{7}$$

System elements in the process of functioning can strengthen or weaken the interaction between them due to the increase or decrease of the narrowness of correlation communication. Therefore, the change of entropy during the processes happening in a system is a change of self-organization entropy:

$$d\mathbb{S}\_{\text{in}} \cong \Delta H(Y)\_{\mathbb{R}}.\tag{8}$$

On the basis of Eqs. (7) and (8), it is possible to make the following hypothesis. Hypothesis 1. The total change of entropy of an open system consists of the sum of two items. The first item characterizes the impact of the interaction of a system with the external environment and represents a change of randomness entropy. The second item characterizes the processes occurring within a system and represents a change of self-organization entropy.

Example 1. Entropy analysis of Yekaterinburg (regional center of Sverdlovsk region) development dynamics in 1992–2017.

The effective functioning of the megalopolis as a complex system according to the vector entropy model Eq. (4) consists in the simultaneous growth of diversity, opportunities for all elements of this system, and the presence of a close interrelation between these elements. This is manifested in the fact that with the development of a megalopolis, its randomness entropy should gradually increase, and the self-organization entropy will decrease.

In [23] it was offered to use a differential entropy (further, an entropy) for modeling multidimensional stochastic systems. It is proven [23] that entropy in

ln <sup>σ</sup>Yi <sup>þ</sup>X<sup>m</sup>

In particular, for multidimensional normally distributed random variable Y.

The formula (2) does not always explain the behavior of the system. The addition of the component H(Y)<sup>V</sup> and H(Y)<sup>R</sup> in terms of systems analysis is incorrect since they characterize various regularities of the complex systems: H(Y)<sup>V</sup> is

The practical use of the relation (2) showed that there are situations when systems with different functional states have approximately the same general entropies of H(Y), but the corresponding values of entropies of randomness H(Y)<sup>V</sup> and self-organization H(Y)<sup>R</sup> have significant differences. It schematically looks as follows. There are two the same systems Y(1) and Y(2) with different states. At the

<sup>V</sup> <sup>¼</sup> 1, <sup>H</sup> <sup>Y</sup>ð Þ<sup>1</sup> � �

components—the entropies of randomness and self-organization as [15]:

Complex systems, including territorial ones, are open, and their entropy can both increase and decrease. Moreover, the directions of change in the entropies of randomness H(Y)<sup>V</sup> and self-organization H(Y)<sup>R</sup> of systems may be different. To build adequate models and investigate multidimensional stochastic systems, differential entropy should be considered not in scalar, but in vector form as two

hð Þ¼ Y ð Þ¼ hV; hR Hð Þ Y <sup>V</sup>; Hð Þ Y <sup>R</sup>

In specific situations, the direction and values of the entropy vector Eq. (4) should be set on the basis of the features of the studied system. In other words, complex systems should have a balance between the entropies of randomness and

The complex systems are open. Influence of entropy on the evolution of open systems was investigated by many scientists. In their publications, it is noted that the change of open systems either leads to degradation or it is self-organization process as a result of which more complex structures appear. Prigogine [24] in 1955 formulated an extended version of the second law of thermodynamics. According to this law, the total change of entropy dS of an open system must be represented in the form of two parts. The reason of the first of them serves internal processes

ln <sup>σ</sup>Yi <sup>þ</sup> <sup>m</sup> ln ffiffiffiffiffiffiffi

Yk=Y1Y2…Yk�<sup>1</sup> � �

i¼1

—dispersion, κ<sup>i</sup> ¼ H Yi=σYi ð Þ—entropy indicator shows a type of ran-

<sup>2</sup>π<sup>e</sup> <sup>p</sup> , H Yð Þ<sup>R</sup> <sup>¼</sup> <sup>1</sup>

Hð Þ¼ Y Hð Þ Y <sup>V</sup> þ Hð Þ Y <sup>R</sup>, (2)

κi—randomness entropy,

—self‐organization entropy,

2

<sup>R</sup> ¼ �1 and <sup>H</sup> <sup>Y</sup>ð Þ<sup>2</sup> � � <sup>¼</sup> 0,

� �: (4)

Yk=Y1Y2…Yk�<sup>1</sup> - coefficient of determina-

ln ð Þ j j R , (3)

Eq. (1) can be represented as a sum of two components:

H Yð Þ¼<sup>i</sup>

ln 1 � <sup>R</sup><sup>2</sup>

Xm i¼1

tion of regression dependencies of random vector Y, k = 2,3,…, m.

H Yð Þ<sup>V</sup> <sup>¼</sup> <sup>X</sup><sup>m</sup>

H Yð Þ<sup>R</sup> <sup>¼</sup> <sup>1</sup>

where σ<sup>2</sup> Yi i¼1

dom value distribution law Yi, i = 1,2,...,m; R<sup>2</sup>

<sup>H</sup>ð Þ <sup>Y</sup> <sup>V</sup> <sup>¼</sup> <sup>X</sup><sup>m</sup>

i¼1

additivity, and H(Y)<sup>R</sup> is an integrity of the system.

<sup>R</sup> ¼ �10.

same time, <sup>H</sup> <sup>Y</sup>ð Þ<sup>1</sup> � � <sup>¼</sup> 0, <sup>H</sup> <sup>Y</sup>ð Þ<sup>1</sup> � �

<sup>V</sup> <sup>¼</sup> 10, <sup>H</sup> <sup>Y</sup>ð Þ<sup>2</sup> � �

H Yð Þ<sup>2</sup> � �

self-organization.

24

where R is the correlation matrix of random vector Y.

2 Xm k¼2

Sustainability Assessment at the 21st Century

The analysis will be performed according to the official data from the Russian Federal State Statistics Service (Rosstat) [24]. Of the many basic socioeconomic indicators of cities, we will form a system of signs that characterize all the main aspects of the city's infrastructure [15]:


When calculating the entropy, the estimates were performed for periods of 13 years. This period turned out to be optimal, on the one hand from the statistical smoothing point of view and on the other hand because it takes into account the dynamics of entropy change. Entropy was estimated in the vector form Eq. (4). Accounting for inflation was carried out by recalculation in 2017 prices based on consumer price indices; the different populations of cities were taken into account by the transition to relative indicators per inhabitant. Since the sample was quite small, the deviations of the empirical distributions of the considered features from the normal distribution are practically impossible to establish. Therefore, when calculating the entropies of randomness and self-organization, we use Eq. (3).

Figure 2 shows the graphs of changes in the entropies of randomness and selforganization in Yekaterinburg. Figure 3 shows the entropy dynamics.

(2012–2013). The direction of vector entropy gradually changes (at first hV

5.Functioning in terms of sanctions (since 2014). There is a practical lack of vector entropy dynamics, with the value of self-organization entropy hR fixed at the level of 2006–2007, and randomness entropy hV at the level of 2013.

6.The total entropy H(Y) in the period under consideration has changed slightly.

Example 2. The modeling of a system that characterizes the safety of the

increases, and hR decreases, but gradually this trend fades).

4.Announcement of sanctions (2014). A steep increase in hR.

The change in self-organization and randomness entropies in Yekaterinburg.

Probabilistic-Entropic Concept of Sustainable Development of the Example of Territories

DOI: http://dx.doi.org/10.5772/intechopen.89287

production.

27

Figure 3.

Entropy dynamics in Yekaterinburg.

Figure 2.

Analysis of graphs in Figures 2 and 3 allows us to make the following conclusions:


Probabilistic-Entropic Concept of Sustainable Development of the Example of Territories DOI: http://dx.doi.org/10.5772/intechopen.89287

(2012–2013). The direction of vector entropy gradually changes (at first hV increases, and hR decreases, but gradually this trend fades).


Example 2. The modeling of a system that characterizes the safety of the production.

The analysis will be performed according to the official data from the Russian Federal State Statistics Service (Rosstat) [24]. Of the many basic socioeconomic indicators of cities, we will form a system of signs that characterize all the main

3.Average monthly nominal accrued wages (in 2017 prices), thousand rubles.

5.The total area of residential premises per one urban resident (at the end of

6.Number of pupils in preschool educational organizations, thousand people.

9.The volume of work performed under construction contracts (in 2017 prices),

4.The share of retirees registered with the social security authorities, %.

aspects of the city's infrastructure [15]:

Sustainability Assessment at the 21st Century

.

ths. rub. for one person.

the year), m<sup>2</sup>

we use Eq. (3).

conclusions:

26

1.Natural increase, decrease () per 1000 population.

7.The number of doctors per 1000 population, people.

8.The number of registered crimes per thousand people.

10.Retail trade turnover (in 2017 prices), ths. rub. for 1 person.

11. Investments in fixed assets (in 2017 prices), ths. rub. for one person.

13 years. This period turned out to be optimal, on the one hand from the statistical smoothing point of view and on the other hand because it takes into account the dynamics of entropy change. Entropy was estimated in the vector form Eq. (4). Accounting for inflation was carried out by recalculation in 2017 prices based on consumer price indices; the different populations of cities were taken into account by the transition to relative indicators per inhabitant. Since the sample was quite small, the deviations of the empirical distributions of the considered features from the normal distribution are practically impossible to establish. Therefore, when calculating the entropies of randomness and self-organization,

organization in Yekaterinburg. Figure 3 shows the entropy dynamics. Analysis of graphs in Figures 2 and 3 allows us to make the following

hV increases, and the self-organization entropy hR decreases.

direction of vector entropy to the opposite.

When calculating the entropy, the estimates were performed for periods of

Figure 2 shows the graphs of changes in the entropies of randomness and self-

1.The period of stabilization of operation (until 2008). The randomness entropy

2.The global financial crisis of 2008–2009. Short-term sharp change in the

3.The period of economic recovery after the financial crisis (2009–2011), followed by a decrease in the growth rate of gross domestic product

2.The share of the working population in organizations, %.

Seventeen coal mining enterprises were investigated [23]. On the basis of two generalized factors (Y1 is the factor characterizing the organization of safe production; Y2 is the factor reflecting the professionalism of the staff), all enterprises were divided into two groups: (1) enterprises with a low level of injury; (2) enterprises with a high level of injury. For the first and second groups of mines, respectively, we have.

subsystems can adapt to others, compensating shortcomings their functioning

Probabilistic-Entropic Concept of Sustainable Development of the Example of Territories

Consider the task of ensuring the system sustainable functioning. Here you first need to concretize the "sustainability" concept. Typically the stability of the system functioning is interpreted in terms of its safety. Security issues are resolved with the help of risk analysis [25]. Some authors note that growth rates of damage considerably exceed growth rates of the economy [26, 27]. This can be explained with a constant increase of risk in the conditions of a scientific and technical revolution and the forced development of a technosphere [28]. Therefore, we will assume that the functioning stability of the territorial system is intimately connected with risk; the lower the risk level, the more stable the system state. Thus, the diagnosis of system sustainability can be made on the basis of monitoring its risk. This requires

Let S be some multidimensional stochastic system. Let us consider an adequate representation of this system as a random vector X = (X1, X2, ..., Xl) with a certain probability density pX(x). The development of a complex system and an increase in the efficiency of its functioning are an inevitable cause of increasing risks. Therefore, it is necessary to assess the risks of such systems. Consider the risk model of

Instead of the generally conventional selection of concrete dangerous situations, we will define the geometric area D of adverse outcomes. Formally this area can

The concept of dangerous states as larger and improbable deviations of a conception of dangerous conditions as large and improbable deviations of random variables from some best provision Θ is mostly distributed. In this case, D repre-

Setting the function of consequences from dangerous situations (risk function)

ð

g xð ÞpXð Þ x dx: (9)

in the form of g(x), we will receive a model for the quantitative assessment of

R<sup>m</sup>

r Xð Þ¼ P Xð Þ ∈ D , that is, the risk is estimated as a probability of an unfavorable

g(x) function, then Eq. (2) becomes an assessment of P(D) and is a convenient

the studied system depending on values of risk factors. It demands to carry out separate research. Let us note that values of the function g(x) are given in the nominal units. But they are usually quite simply interpreted in the respective subject area. The essence of the function g(x) is as follows. It accepts the least nonnegative (e.g., zero) value in a point of Θ or in its neighborhood of U(Θ). Further in each direction during removal from U(Θ) the g(x) function has to increase monotonously. For scaling on each risk factor, we will set some limit values, at which consequences become dangerous (or irreversible). Let us set values g(x) at each

If at an early stage of system analysis is difficult to describe enough precisely the

To define a function g(x) requires a quantitative assessment of consequences for

r Xð Þ¼ ð ð …

If in Eq. (9) to accept g xð Þ¼ 1 ∀x∈ D and g xð Þ¼ 0 ∀x ∉ D, that

4. Multidimensional risk model of complex systems

multidimensional stochastic systems proposed in [29].

look arbitrarily depending on a specific objective.

sents an external area of an m-axis ellipsoid.

initial approximation of risk model.

(substitution effect).

DOI: http://dx.doi.org/10.5772/intechopen.89287

adequate risk models.

risk [30]:

outcome.

29

$$\left(h\_V^{(1)} \mathcal{H}\_{\mathbb{R}}^{(1)}\right) = (2, 42; \ -0, 31), H\left(Y^{(1)}\right) = 2, 11,$$

$$\left(h\_V^{(2)} \mathcal{H}\_{\mathbb{R}}^{(2)}\right) = (3, 74; \ -0, 70), H\left(Y^{(2)}\right) = 3, 04.$$

In this case, the direction of the entropy change vector will differ from Example 1: at the enterprises with a high level of injury, the randomness entropy needs to be reduced, and the self-organization entropy needs to be increased.

For example, this can be accomplished as follows: to bring the state of the second group of mines to the state of the first group, it is necessary to reduce the dispersion of the factor characterizing the organization of safe production and reduce interrelation with the factor reflecting the professionalism of the staff. This means a more specific and accurate organization of production. The organization of safe production should not depend on the degree of professionalism and competence of staff strongly.

Example 3. We investigate the possibilities of entropy modeling on the example of the population analysis in terms of prevention of chronic noninfectious diseases (CNID) by biological risk factors [23]. For carrying out the analysis of change of population entropy depending on the health status, two equal age groups were formed: 18–26 years and 27–35 years. Four risk factors were identified: "total cholesterol," "systolic blood pressure," "body mass index," and "glucose level." The results of the analysis are given in Table 1.

As the health of the population deteriorates, the total population entropy and the randomness entropy increase. This can be explained by the fact that the additional damaging influence of CNID, in general, is added to the pathological influence of risk factors on a human body separately and on all population.

Conversely, the self-organization entropy as the deterioration of the health status of the population decreases which corresponds to the strengthening of the narrowness of the interrelations between subsystems. This can be explained by the fact that the development of diseases in the organism happens in many respects and it is interdependent. On the other hand, at the development of diseases, some


#### Table 1.

Entropy levels in different groups of people.

Probabilistic-Entropic Concept of Sustainable Development of the Example of Territories DOI: http://dx.doi.org/10.5772/intechopen.89287

subsystems can adapt to others, compensating shortcomings their functioning (substitution effect).
