The Air Transportation System as a Subsystem of Modern Communication Space: Analysis Based on Transfer Entropy Graphs

Abstract

The processes of information exchange and the movement of material flows form a communication space that reflects the relationship of complex intersystem interactions in various spheres of our life within the framework of the concepts of information-theoretical theory. One of these concepts, reflecting the mutual influence between processes at a qualitative level, is the transfer of entropy. The direction and intensity of these flows reflect the main social and economic processes. As it is known, air transport is one of the most reliable and high-speed modes of transport, influencing the processes of socio-cultural interaction between different regions. This indirectly affects the development of industrial relations, the development of technology and intercultural exchange. New technologies in aviation improve the flight performance of airliners and reduce the costs of transporting passengers. The size and range of modern airliners are increasing, and ticket prices are being optimized. The processes of the liberalization of developing air transportation markets, the emergence of low-cost air carriers, open skies agreements, and the reduction in restrictions on the nomenclature of carriers and routes have led to the growth and diversity of air transport links. This article considers air transport as a complex system that takes into account the interconnectedness of the elements of the transportation system and the influence of some subsystems on others, which are not always obvious. The object of the study was the communication space formed on the basis of air transportation between regions of the world. To assess the dynamic properties of the world communication space, ICAO data for the period of 1970–2021 were used. The subject of the analysis was a time series reflecting the flows of passengers and cargo over the considered time horizon. The entropy transfer algorithm was used as an analysis tool. In the course of the research, the features of dynamic changes in the properties of the communication space were revealed. The analysis showed that the flows of entropy transfer between regions of the world change depending on political, economic, social, and technological factors. Examples of the application of the proposed approach are considered: an analysis of the cognitive model of the air transport flow structure, an analysis of the regional communication space, and an analysis of changes in the global communication field. The results of the analysis can be useful for assessing the development of the communication field of various regions, which will allow us to solve the problems of forming forecasts and effective scenarios for the development of transport flows at different hierarchical levels of economic management.

1. Introduction

The concept of communication space is used in sociology and reflects a view of the development of society [1]. It is implied that without interaction (communication), society could not arise and exist. At present, modern means of information exchange are used for communication: the Internet, telephones, etc. Communication, according to Luhmann’s theory, is a key type of operation of social systems; it constitutes systems and ensures their differentiation and reproduction. Social systems and society are communications; systems consist of interconnected communications; society is a special form of social system, including all possible communications [2]. Everything that is not communication is excluded from the operational area of social systems; if there is communication, there is society. If it is possible to identify the signs of communication, then we can talk about the existence of society. If there are no communications in the system, then it is impossible to talk about the existence of a social system. The communication space is a set of communication links of various natures that are generated and perceived by communication agents. Communication agents are understood as separate individuals or groups of individuals, as well as cultural, social, or political institutions. It should be taken into account that each society under consideration forms a unique communication space with its own characteristics and features [3].

In the 21st century, the world communication space is usually associated with the media, the Internet, and the virtual information field [4,5]. However, we should not forget about the real communication processes associated with the movement of a large number of people in a short time. These processes are provided exclusively by means of transport systems. For the socio-cultural space, it is not so much the geographical location of the centers of social and cultural attraction that plays an important role, but their transport accessibility. This allows for the formation of a much more complex system of interconnected peripheral and central nodes. On the one hand, transport systems and transport processes are an effective tool for the dense internal aggregation of different spatial segments within the world as a whole. On the other hand, transport facilitates the expansion of individual civilizations into the outside world [6,7]. In this case, not only the ability to reach a certain point in space in a certain time is important, but also the dependence of the economic and cultural components of a given geographic point on transport communications. Despite the fact that individual regions of the world are developing in different directions, the general trend in the development of global society is interconnectedness. This point of view is reflected in one of the modern concepts of world development, namely connectography. This social hypothesis reflects a new developing trend, which is characterized by active human mobility, both from the position of tourism and from the position of migration. Also, within the framework of this hypothesis, an active global exchange of material and intellectual resources is implied. Connectography substantiates the transition from traditional political geography to functional geography. In other words, within the framework of this concept, the emphasis shifts from the official division of the world into countries and regions towards the infrastructure that ensures global socio-cultural and economic ties [8].

Globalization and high levels of technology contribute to a radical change in the modern socio-cultural and economic space, as well as the system of relationships in it. Transport communications have always played a key role in intercultural exchange between regions, both within individual countries and for the world as a whole. The peculiarities of the formation of transport communications are associated with differences in the economic and geographical position of regions of the world and individual states [9].

A region of the world is determined by the unity of the social, economic, political, and cultural spaces of the countries included in it. The spatial and communication factors are taken into account when studying the history and prospects for the development of society, allowing us to identify key points and time periods based on observed population movements.

Transport systems, regardless of their nature, be it land, air, or water transport, initially have a communication nature. At the present stage, transport is associated primarily with tourism, urbanization, population migration, and globalization [10]. When moving to a new stage of development, regions strive for the planned and long-term development of transport infrastructure [11]. In this case, one should focus on global trends in socio-economic development.

Modern air transport is undoubtedly one of the most reliable and is high-tech and high-speed. Air transport occupies a leading position in regional and global passenger transportation and transports a significant portion of important cargo [12]. Unlike automobile or rail transport, the air transport system allows for the reconstruction of existing routes and the creation of new ones quite easily. The total length of the network of the world air routes is approaching 10 million kilometers (Figure 1 [13]). The number of airports in the world is constantly changing due to the construction of new ones and the closure of old airports. However, according to the Federal Aviation Administration (FAA), there are currently about 19,000 airports in the world [14]. The emergence and development of aviation have significantly changed society, both regionally and globally. Air transport has accelerated the processes of socio-cultural interaction and influenced the increase in the number and diversity of communication processes. This has affected people’s daily lives. An example is the express delivery of mail and goods. The cultural aspect is affected by the opportunity to attend theatres, concerts, exhibitions, and sporting events in other cities and sometimes in other countries.

Qualitative leaps in the development of air transport have been associated with the development of new types of jet aircraft, including wide-body aircraft. New technologies used in the creation of aircraft and their components and units, including jet engines, have made it possible to significantly improve the performance characteristics of aircraft and, consequently, reduce the life cycle costs of aircraft and aircraft engines [15,16]. This made it possible to increase the efficiency of transportation by increasing the size and range of airliners, reduce ticket prices, and increase the intensity of economic and socio-cultural exchange between Asia and the rest of the world in the late 1980s and in 2010 [9].

Other important factors changing the air transport landscape include the emergence of low-cost carriers (LCCs), the deregulation of air travel since the late 1970s, and the liberalization of emerging air travel markets such as China, Indonesia, India, and Brazil [17]. Open skies agreements have played an important role in the development of air transport. Thanks to the signing and implementation of such agreements, the number of restrictions on the nomenclature of carriers and possible routes within the airspace specified in the agreements between countries has been sharply reduced. The liberalization of air travel in general (a process that includes deregulation and privatization) has led to the growth and diversity of air transport links and increased the adaptability of the global air transport network [18].

In our study, air transport is considered a complex organizational and technical system interacting with various other complex economic and social systems. When defining a system as an object of study, particular attention is paid to the interconnectedness of the system’s elements [19]. The formed relationships determine the behavior of the system as a whole. In this case, it is not always possible to explicitly determine the influence of some subsystems on others [20].

In our case, to solve the problem, we considered constructing a model of a communication system based on information about passenger and freight flows, since, in general, they reflect the dynamics of social and economic processes in modern society.

Currently, various methods and algorithms for analyzing time series have been developed that allow the construction of various models based on collected statistical data [21,22].

A system model constructed on the basis of time series can be described as a system of differential equations, graphs, and other tools. In this study, we selected the apparatus of discrete mathematics: graphs, which are a popular and effective tool for analyzing complex systems reflecting the behavior of the communication space [23,24].

When analyzing complex systems, an analysis method is used, depending on the decomposition of the system into subsystems and the construction of hierarchical relationships between them [25,26]. Depending on the level of the hierarchical subsystem, such as the executive level, the coordination level, and the planning level, it is necessary to build models that reflect the properties of each of these levels. When considering the air transport system, one can identify a hierarchy of problem-solving for analyzing communication at the regional level, the level between regions, and the global level of inter-regional transportation.

In the analysis of complex systems and processes, various methods and algorithms based on the idea of entropy transfer are widely used [27,28,29]. This allows us to obtain qualitative dependencies (causality) between processes in large systems [30,31,32].

2. Research Methodology

The causality hypothesis can be briefly described as follows. If two variables Y and X are given at a given time interval, and if the behavior of variable Y can be predicted, including on the basis of variable X, using historical values of both X and Y, and not only historical values of Y. Transfer entropy is a method in information theory that can make it possible to assess the relationship and direction of causality due to its asymmetry. Transfer entropy can measure both the direction and the amount of information transmitted, which is suitable for nonlinear spatial and temporal modeling of cause-and-effect relationships of traffic flows.

The research proposes a dynamic causality analysis methodology including several main stages. Firstly, transfer entropy is applied to detect the air transportation flows’ causality from the given air transportation flows’ time series data for the given air transportation network to calculate the transfer entropy matrix, which can denote the dynamic nonlinear causal relationships of the air transportation system’s states among different regions’ transportation flows. Secondly, the matrix of causal correlation coefficients used to design the transfer entropy graph of the system’s interactions is calculated on the basis of the transfer entropy matrix. This obtained graph can be used for various tasks, for example, the search for maximum flow, analysis of the network connectivity, graph-based entropy determination, etc. This may allow us to identify and analyze the dynamic properties of the air transportation system, reflecting hidden socio-economic processes.

Statistical data reflecting the quantitative characteristics of passenger and freight flows at selected time intervals were used as a source of information for analyzing the characteristics of the communication space. Statistical data were obtained from open sources (reports and databases) and are presented in the form of time series [33,34,35,36,37].

Figure 2 shows the life cycle of the entropy transfer-based model’s formation. The first stage involves collecting data from the available sources and forming the raw data. The second stage involves data processing (ETL process): removing outliers, filling gaps, and normalizing and preparing the dataset. The third stage involves forming the data arrays used to determine the entropy transfer values and forming an entropy transfer graph based on the data obtained.

When analyzing time series, trends should be taken into account, including seasonal trends. For the first two examples, models were built, in which time series were represented by a small (available) dataset. The entropy transfer analysis algorithm requires more data. Therefore, averaged results were obtained for these examples. In the third case (an example for regions of the world), there are more available data, which made it possible to show the dynamics of interactions (trends).

When formalizing the problem statement, mathematical objects are used: time series, matrices, the concept of entropy, entropy transfer, graphs, and entropy transfer graphs. Next, we consider the elements of the analysis used to construct entropy transfer graphs, on the basis of which we can draw a conclusion about the properties of the communication space under consideration.

2.1. Problem Formulation

We consider a system K_s, based on a set of time series S

S = [s₁, s₂, …, s_n],

(1)

where n is the number of time series used in the analysis of the system K_s.

It is necessary to construct a model of the communication system K_s using a time series set and present it as a transfer entropy graph.

2.2. Procedure for Constructing an Entropy Transfer Graph

When one analyzes the influence of one process over time on another process, a measure of influence called transfer entropy (TE) is used [27]. This measure refers to information-theoretical characteristics and is based on Shannon’s information entropy.

To analyze the system of time series K_s, we need to construct a mapping (algorithm) that would allow us to move from a set of time series to a matrix whose elements would reflect the mutual influence of selected pairs of time series over a given time interval.

Thus, we need to determine a set of pairs of numbers over a certain time interval, which allows us, on the one hand, to evaluate the influence of time series in the context of their direction of influence on each other and, on the other hand, to obtain a pair of quantitative characteristics when constructing an information transfer matrix.

The matrix elements reflect the systemic connections in a set of time series.

When constructing a matrix whose elements are pairs of numbers, we incorporate the Ks system into a real matrix Ms.

The matrix Ms now can be associated with the adjacency matrix of the graph G_TE.

By applying the matrix Ms, the entropy transfer graph G_TE can be constructed. The vertices of this matrix are the names of the time series, and the edges determine the presence of a connection between the vertices; the weights of the edges reflect the strength of the influence of one process in time on another process.

The structure of this graph reflects the entropy of the system. The value of entropy of this graph depends on the adopted measure.

In the simplest case, the entropy of a matrix is determined on the basis of its structural properties. For example, it is the power of the set of the weights |V| of matrix M_TE, where V is the set of edges of the matrix M_TE.

One of the graph analysis algorithms that can be useful in our case is the analysis of the maximum flow on graphs [38]. The maximum flow route can be used in the analysis of the properties of various transport networks. It allows one to estimate the throughput of the selected route on graphs.

2.3. The Method of Analyzing the Communication Properties of Air Transportation

Next, the problem of analyzing communications at various levels of territorial organization is considered: regional, federal, and world regions. The subject of analysis is datasets presented in the form of time series reflecting the flows of transported passengers and cargo transportation in certain periods of time. The main research tool is an information-theoretical measure based on the assessment of the transfer of entropy between the elements of time series.

It should be noted that if the data are presented in the form of multivariate time series, the analysis is carried out on the basis of paired comparisons at specified time intervals, which allows us to construct a matrix of paired comparisons. This matrix is then used as a connectivity matrix when constructing an entropy transfer graph.

The resulting matrices for different time intervals are considered as adjacency matrices for entropy transfer graphs, which allow us to study the properties of the system represented by time series on the basis of various graph analysis methods.

Figure 3 shows the main stages of the formation of the entropy transfer graph G_TE based on the time series system S_n, where S_n is a time series, D represents the values of the time series for the selected time interval t_m, p is a set of paired estimates p^m_i,j, M_TE is a matrix composed of the values of paired estimates of the transfer of entropy, and G_TE is a graph (model) of the system of communication interactions of processes.

3. Case Studies

In Section 3.1, we will consider the analysis of the communication space for passenger transportation in the system of one region. In Section 3.2, we will consider the problem of analyzing the communication space using the example of inter-regional transportation and the center. In Section 3.3, we will consider the problem of analyzing changes in the properties of the communication space over a long time horizon using the example of world regions’ passenger transportation.

3.1. Regional Transportation

The airport network reflects the economic and social characteristics of the region and is an integral part of the transport system for the carriage of passengers and cargo by air. Airports provide reliable logistics links within a region or country, and with other regions of the world. Airport congestion, and thus passenger traffic, reflects the geographical, social, and economic characteristics of the region, directly affecting the characteristics of the communication space. Airport capacity statistics reflect the overall traffic of aircraft, the directions of air transport movements, and the intensity of the flow of passengers and cargo.

Let us further consider an analysis of the activities of the airport network of one region over a certain period of time using the example of Great Britain. Statistical data were collected by the Civil Aviation Authority [33]. The statistics reflect the operating characteristics of more than 60 airports in Great Britain from 1981 to 2015. Figure 4 shows curves reflecting the main characteristics of the intensity of traffic flows: total traffic density, air transport traffic density, and cargo traffic density.

Figure 5 shows the corresponding change curve of the flow for registered passengers over the same period of time.

Using the previously presented method for analyzing the mutual influence of the processes S_n (n = 1…4), we determined the values of the set of paired estimates of transfer entropy p and formed the matrix M_TE. The results of calculation are presented in

Table 1. Transfer entropy from Y to X based on movements, passengers, and freight statistics.

Y	X
	Total Movements	Air transport Movements	Terminal Passengers	Freight Tonnes
Total movements	0	0.13	0	0.13
Air transport movements	0.25	0	0	0.13
Terminal passengers	0	0.13	0	0
Freight tonnes	0	0	0	0

.

Next, we present the data from Table 1 as a directed graph G_TE = (N, V), where N is the set of vertices describing processes in time, and V is the set of edges whose weights reflect the information measure of entropy transfer between processes.

Figure 6a shows the graph G_TE, where Node 1 is total movements, Node 2 is air transportation movements, Node 3 is terminal passengers, and Node 4 is freight tonnes.

Figure 6b shows the approximation of the graph G_TE in the form of the graph G_TES, in which edges with small weights (c < 0.095) are removed. The graph G_TES reflects the cause-and-effect relationship between the processes of organizing transportation.

This model in the form of the graph G_TES can be used as a knowledge representation model for forming a data structure in a graph database, which can be used to solve various problems of analyzing the activities of the country’s airport network and assessing the performance indicators of the passenger and cargo transportation system, which in turn will allow assessing the properties of the communication space.

3.2. Analysis of the Communication Space of the Center and Regions

The regional transport system is a complex of subsystems interacting with regional systems of air transport, land transport, and water transport. The organization of these systems has a hierarchical structure, requiring the solution of problems of coordination and synchronization of the participants in transportation [11].

A large role in the airport system is played by aviation hubs, allowing the solution of problems of synchronization and coordination in the transportation system.

Let us consider an example of how we can analyze the communication space of the center and regions.

The airports of Moscow were chosen as air hubs: Vnukovo Airport, Domodedovo Airport, and Sheremetyevo Airport. The airports of Ufa City, Yekaterinburg City, and Chelyabinsk City were chosen as regional airports. These airports are used for the transportation of goods and passengers, including the movement of labor capital and tourist flows, influencing the social landscape of the regions. They usually have strong connections with large aviation hubs.

The analysis of the communication space was carried out on the basis of statistics of passenger and cargo flows for all selected airports (from 1 January 2018 to 31 August 2018). The statistical data are presented in the form of graphs in Figure 7 and Figure 8 [34].

Next, on the basis of the available statistical information, we construct the matrix M_TE for the G_TE graph, the weight values of which are presented in

Table 2. Transfer entropy from Y to X based on regional transportation statistics.

Y	X
	Yekaterinburg	Chelyabinsk	Ufa	Moscow (Vnukovo)	Moscow (Domodedovo)	Moscow (Sheremetyevo)
Yekaterinburg	0	0	0.4/0	0	0	0
Chelyabinsk	0	0	0.1/0	0	0	0
Ufa	0/0.2	0/0.3	0	0.3/0.53	0.3/0.56	0.5/0.2
Moscow (Vnukovo)	0	0	0	0	0	0
Moscow (Domodedovo)	0	0	0	0	0	0
Moscow (Sheremetyevo)	0	0	0	0	0	0

. The vertices G_TE are numbered in accordance with the row numbers in Table 2, i.e., 1—Yekaterinburg, 2—Chelyabinsk, 3—Ufa, 4—Moscow (Vnukovo), 5—Moscow (Domodedovo), and 6—Moscow (Sheremetyevo).

An illustration of the obtained results in the form of a graph model is presented in Figure 9. From an analysis of the communication space based on the obtained results, it follows that there is some heterogeneity in the communication spaces for regional air transport hubs and the center, which requires an increase in the efficiency of solving the problems of managing the coordination of regions and the center.

3.3. Air Transportation Between World Regions and Communication Space

When analyzing time series that describe processes in a system, which in our case, is a communication space based on air transportation, the task of analyzing a set of regional subsystems as a single whole arises, which reflects the properties of the global communication space.

In this example, an analysis of passenger transport flows between different regions of the world was performed, which allowed us to evaluate the dynamic properties of the world communication space. The following statistical regions of the world according to ICAO were selected: Eastern and Southern Africa, Western and Central Africa, the Arab world, Central Europe and the Baltic, East Asia and the Pacific, Europe and Central Asia, the European Union, Latin America and the Caribbean, Middle East and North Africa, North America, South Asia, and Sub-Saharan Africa [35]. The integrating indicator was the total registered indicator of global passenger transportation volumes. Figure 10 shows the geography of these regions.

Figure 11 shows curves describing the change in the flow of air passenger traffic within regions and between regions of the world for the period 1970–2021. The data were divided into decades, which allowed us to assess the changes in the influence of regions on communication directions over these years.

Table 3. Transfer entropy values for 1970–2021.

1970–1921	1	2	3	4	5	6	7	8	9	10	11	12	13
	Africa Eastern and Southern	Africa Western and Central	Arab World	Central Europe and the Baltics	East Asia and Pacific	Europe and Central Asia	European Union	Latin America and Caribbean	Middle East and North Africa	North America	South Asia	Sub-Saharan Africa	World
Africa Eastern and Southern	0	0	0	0	0	0	0	0	0	0	0.18	0	0
Africa Western and Central	0	0	0	0	0	0	0	0	0	0	0.18	0	0
Arab World	0	0	0	0	0	0	0	0	0	0	0.18	0	0
Central Europe and the Baltics	0	0	0	0	0	0	0	0	0	0	0.18	0	0
East Asia and Pacific	0	0	0	0	0	0	0	0	0	0	0.18	0	0
Europe and Central Asia	0	0	0	0	0	0	0	0	0	0	0.18	0	0
European Union	0	0	0	0	0	0	0	0	0	0	0.18	0	0
Latin America and Caribbean	0	0	0	0	0	0	0	0	0	0	0.18	0	0
Middle East and North Africa	0	0	0	0	0	0	0	0	0	0	0.18	0	0
North America	0	0	0	0	0	0	0	0	0	0	0.18	0	0
South Asia	0.04	0.04	0.04	0.04	0	0.04	0	0.04	0.04	0.04	0	0.04	0
Sub-Saharan Africa	0	0	0	0	0	0	0	0	0	0	0.18	0	0
World	0	0	0	0	0	0	0	0	0	0	0.18	0	0

,

Table 4. Transfer entropy values for 1970–2080.

1970–1980	1	2	3	4	5	6	7	8	9	10	11	12	13
	Africa Eastern and Southern	Africa Western and Central	Arab World	Central Europe and the Baltics	East Asia and Pacific	Europe and Central Asia	European Union	Latin America and Caribbean	Middle East and North Africa	North America	South Asia	Sub-Saharan Africa	World
Africa Eastern and Southern	0	0	0	0	0.2	0	0	0	0	0	0.45	0	0
Africa Western and Central	0	0	0	0	0.2	0	0	0	0	0	0.45	0	0
Arab World	0	0	0	0	0.2	0	0	0	0	0	0.45	0	0
Central Europe and the Baltics	0	0	0	0	0.2	0	0	0	0	0	0.45	0	0
East Asia and Pacific	0	0	0	0	0	0	0	0	0	0	0.45	0	0
Europe and Central Asia	0	0	0	0	0.2	0	0	0	0	0	0.45	0	0
European Union	0	0	0	0	0.2	0	0	0	0	0	0.45	0	0
Latin America and Caribbean	0	0	0	0	0.2	0	0	0	0	0	0.45	0	0
Middle East and North Africa	0	0	0	0	0.2	0	0	0	0	0	0.45	0	0
North America	0	0	0	0	0.2	0	0	0	0	0	0.45	0	0
South Asia	0	0	0	0	0.2	0	0	0	0	0	0	0	0
Sub-Saharan Africa	0	0	0	0	0.2	0	0	0	0	0	0.45	0	0
World	0	0	0	0	0.2	0	0	0	0	0	0.45	0	0

,

Table 5. Transfer entropy values for 1981–1990.

1981–1990	1	2	3	4	5	6	7	8	9	10	11	12	13
	Africa Eastern and Southern	Africa Western and Central	Arab World	Central Europe and the Baltics	East Asia and Pacific	Europe and Central Asia	European Union	Latin America and Caribbean	Middle East and North Africa	North America	South Asia	Sub-Saharan Africa	World
Africa Eastern and Southern	0	0.02	0	0.02	0	0	0	0	0	0	0	0	0
Africa Western and Central	0	0	0	0	0	0	0	0	0	0	0	0	0
Arab World	0.23	0.02	0	0.48	0.5	0	0	0	0	0	0	0.023	0
Central Europe and the Baltics	0.23	0.02	0	0	0	0	0	0	0	0	0	0.023	0
East Asia and Pacific	0.23	0.02	0	0	0	0	0	0	0	0	0	0.023	0
Europe and Central Asia	0.23	0.02	0	0.48	0.5	0	0	0	0	0	0	0.023	0
European Union	0.23	0.02	0	0.48	0.5	0	0	0	0	0	0	0.023	0
Latin America and Caribbean	0.23	0.02	0	0.48	0.5	0	0	0	0	0	0	0.023	0
Middle East and North Africa	0.23	0.02	0	0.48	0.5	0	0	0	0	0	0	0.023	0
North America	0.23	0.02	0	0.48	0.5	0	0	0	0	0	0	0.023	0
South Asia	0.23	0.02	0	0.48	0.5	0	0	0	0	0	0	0.023	0
Sub-Saharan Africa	0	0.02	0	0.48	0	0	0	0	0	0	0	0	0
World	0.23	0.02	0	0.48	0.5	0	0	0	0	0	0	0.023	0

,

Table 6. Transfer entropy values for 1991–2000.

1991–2000	1	2	3	4	5	6	7	8	9	10	11	12	13
	Africa Eastern and Southern	Africa Western and Central	Arab World	Central Europe and the Baltics	East Asia and Pacific	Europe and Central Asia	European Union	Latin America and Caribbean	Middle East and North Africa	North America	South Asia	Sub-Saharan Africa	World
Africa Eastern and Southern	0	0.48	0	0	0	0	0	0	0	0	0.483	0	0
Africa Western and Central	0.02	0	0	0	0	0	0	0	0	0	0.023	0.023	0
Arab World	0.48	0.02	0	0	0	0	0	0	0	0	0.023	0.483	0
Central Europe and the Baltics	0.48	0.02	0	0	0	0	0	0	0	0	0.023	0.483	0
East Asia and Pacific	0.48	0.02	0	0	0	0	0	0	0	0	0.023	0.483	0
Europe and Central Asia	0.48	0.02	0	0	0	0	0	0	0	0	0.023	0.483	0
European Union	0.48	0.02	0	0	0	0	0	0	0	0	0.023	0.483	0
Latin America and Caribbean	0.48	0.02	0	0	0	0	0	0	0	0	0.023	0.483	0
Middle East and North Africa	0.48	0.02	0	0	0	0	0	0	0	0	0.023	0.483	0
North America	0.48	0.02	0	0	0	0	0	0	0	0	0.023	0.483	0
South Asia	0.02	0	0	0	0	0	0	0	0	0	0	0.023	0
Sub-Saharan Africa	0	0.48	0	0	0	0	0	0	0	0	0.483	0	0
World	0.48	0.02	0	0	0	0	0	0	0	0	0.023	0.483	0

,

Table 7. Transfer entropy values for 2001–2010.

2001–2010	1	2	3	4	5	6	7	8	9	10	11	12	13
	Africa Eastern and Southern	Africa Western and Central	Arab World	Central Europe and the Baltics	East Asia and Pacific	Europe and Central Asia	European Union	Latin America and Caribbean	Middle East and North Africa	North America	South Asia	Sub-Saharan Africa	World
Africa Eastern and Southern	0	0	0.02	0	0	0	0	0.48	0.02	0	0	0	0
Africa Western and Central	0	0	0.02	0	0	0	0	0.48	0.02	0	0	0	0
Arab World	0.02	0.02	0	0	0	0.02	0	0.02	0	0.02	0.023	0.023	0
Central Europe and the Baltics	0.48	0.48	0.02	0	0	0.48	0	0.02	0.02	0.48	0.483	0.483	0.5
East Asia and Pacific	0.48	0.48	0.02	0	0	0.48	0	0.02	0.02	0.48	0.483	0.483	0.5
Europe and Central Asia	0	0	0.02	0	0	0	0	0.48	0.02	0	0	0	0
European Union	0.48	0.48	0.02	0	0	0.48	0	0.02	0.02	0.48	0.483	0.483	0.5
Latin America and Caribbean	0.02	0.02	0.02	0	0	0.02	0	0	0.02	0.02	0.023	0.023	0
Middle East and North Africa	0.02	0.02	0	0	0	0.02	0	0.02	0	0.02	0.023	0.023	0
North America	0	0	0.02	0	0	0	0	0.48	0.02	0	0	0	0
South Asia	0	0	0.02	0	0	0	0	0.48	0.02	0	0	0	0
Sub-Saharan Africa	0	0	0.02	0	0	0	0	0.48	0.02	0	0	0	0
World	0	0	0.02	0	0	0	0	0.48	0.02	0	0	0	0

,

Table 8. Transfer entropy values for 2011–2021.

2011–2021	1	2	3	4	5	6	7	8	9	10	11	12	13
	Africa Eastern and Southern	Africa Western and Central	Arab World	Central Europe and the Baltics	East Asia and Pacific	Europe and Central Asia	European Union	Latin America and Caribbean	Middle East and North Africa	North America	South Asia	Sub-Saharan Africa	World
Africa Eastern and Southern	0	0	0	0	0.5	0	0	0	0	0	0	0	0
Africa Western and Central	0	0	0	0	0.5	0	0	0	0	0	0	0	0
Arab World	0	0	0	0	0.5	0	0	0	0	0	0	0	0
Central Europe and the Baltics	0	0	0	0	0.5	0	0	0	0	0	0	0	0
East Asia and Pacific	0	0	0	0	0	0	0	0	0	0	0	0	0
Europe and Central Asia	0	0	0	0	0.5	0	0	0	0	0	0	0	0
European Union	0	0	0	0	0.5	0	0	0	0	0	0	0	0
Latin America and Caribbean	0	0	0	0	0.5	0	0	0	0	0	0	0	0
Middle East and North Africa	0	0	0	0	0.5	0	0	0	0	0	0	0	0
North America	0	0	0	0	0.5	0	0	0	0	0	0	0	0
South Asia	0	0	0	0	0.5	0	0	0	0	0	0	0	0
Sub-Saharan Africa	0	0	0	0	0.5	0	0	0	0	0	0	0	0
World	0	0	0	0	0.5	0	0	0	0	0	0	0	0

and

Table 9. Transfer entropy values for 2005–2016.

2005–2016	1	2	3	4	5	6	7	8	9	10	11	12	13
	Africa Eastern and Southern	Africa Western and Central	Arab World	Central Europe and the Baltics	East Asia and Pacific	Europe and Central Asia	European Union	Latin America and Caribbean	Middle East and North Africa	North America	South Asia	Sub-Saharan Africa	World
Africa Eastern and Southern	0	0	0	0	0	0	0	0	0	0.02	0	0	0
Africa Western and Central	0	0	0	0	0	0	0	0	0	0.02	0	0	0
Arab World	0	0	0	0	0	0	0	0	0	0.02	0	0	0
Central Europe and the Baltics	0	0	0	0	0	0	0	0	0	0.02	0	0	0
East Asia and Pacific	0	0	0	0	0	0	0	0	0	0.02	0	0	0
Europe and Central Asia	0	0	0	0	0	0	0	0	0	0.02	0	0	0
European Union	0	0	0	0	0	0	0	0	0	0.02	0	0	0
Latin America and Caribbean	0	0	0	0	0	0	0	0	0	0.02	0	0	0
Middle East and North Africa	0	0	0	0	0	0	0	0	0	0.02	0	0	0
North America	0	0	0	0	0	0	0	0	0	0	0	0	0
South Asia	0	0	0	0	0	0	0	0	0	0.02	0	0	0
Sub-Saharan Africa	0	0	0	0	0	0	0	0	0	0.02	0	0	0
World	0	0	0	0	0	0	0	0	0	0.02	0	0	0

show the results of calculating the transfer entropy by decades.

The sets of graphs G_TE obtained on the basis of the analysis of transfer entropy for different time periods presented in Table 3, Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9 are shown in Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17 and Figure 18.

Let us further consider the results of the analysis of changes in the directions of air passenger flows over time, as an illustration of the dynamics of the communication space.

The analysis of the changes in the communication space based on air transport between regions of the world shows the dynamic features of changes in the properties of this space, which corresponds to the concepts of complex nonequilibrium dynamic systems, or the process of self-organization. Communication flows change dynamically depending on the economic, political, social, and technological factors. An analysis of the communication space allows us to assess the state of the world system from the standpoint of its sustainable development.

3.4. TE Graph Max Flows Analysis

Next, we consider an example of constructing maximum entropy transfer flows based on the algorithm for determining the maximum flow in the graph (Figure 19) [38].

As an example, we consider the transfer entropy graph for the period from 2001 to 2010 (Figure 15).

Figure 20a–d show the results of calculating the maximum flows between the selected nodes of the graph G_TE for 2001–2010. When designating the graph’s vertices, the abbreviations of statistical regions according to the ICAO were used.

The presented results of the analysis can be useful in assessing the development of the communication field between regions over a given period of time.

For the following statistical regions of the world according to ICAO [35] that were selected, the following abbreviations were applied in Figure 20: AE&S, Eastern and Southern Africa; AW&C, Western and Central Africa; AW, Arab world; CE&B, Central Europe and the Baltics; EA&P, East Asia and the Pacific; E&CA, Europe and Central Asia; EU, European Union; LA&C, Latin America and the Caribbean; ME&NA, Middle East and North Africa; NA, North America; SA, South Asia; SSA, Sub-Saharan Africa; W, World.

So, we can conclude that the results of the analysis showed that transfer entropy flows between world regions change dynamically depending on the political, economic, social, and technological factors that manifest themselves over a historical period of time.

4. Conclusions

The concept of communicative space is used in sociology and reflects a view of the development of society. It is implied that without interaction (communication), society could not arise and exist. At present, modern means of information exchange are used for communication: the Internet, telephones, etc. In our case, the influence of air transport in this context was studied. The main result of the research is that the communication space has changed dynamically along with changes in society and it reflects both economic and social factors. For the study, the method of analyzing the mutual influence of time series based on the entropy transfer algorithm was used.

The article considered the problem of analyzing the communication space at various levels of its organization, using air transport as an example. The features of passenger transportation processes and material flows in various regions that form the communication space reflecting the relationship of complex intersystem interactions in various spheres of life in the thesaurus of concepts of theoretical and information theory were considered. It is noted that the development of the communication space is the basis for the sustainable development of modern society.

The object of the analysis was data presented in the form of time series, reflecting the characteristics of passenger and cargo flows with a given time horizon. A method for analyzing the communication space based on the construction and analysis of the entropy transfer graph was proposed. A procedure for constructing an entropy transfer graph was suggested, including the transition from time series to a matrix of paired comparisons, which was used to construct an entropy transfer graph. The resulting model could be used to analyze the structure of the regional communication space. The research revealed the features of dynamic changes in the properties of the communication space of the regions under consideration.

An analysis of the cognitive model of air transport, a graph model of the regional communication space, and a graph model of changes in the global communication field was conducted. The analysis showed that entropy transfer flows between regions of the world change depending on the political, economic, social, and technological factors reflecting the historical landscape. The developed approach, the results of the analysis of the structure, and the dynamic properties of the communication space can be useful in studying the characteristics of the communication field of various regions. This will allow the solution of problems of forming forecasts and effective scenarios for the development of transport systems at different hierarchical levels of regional economic management systems.

As a key point of the research results, it should be noted that interactions based on the analysis of the dynamics of transport flows allow us to study the hidden processes of relationships in society and the economy of regions.

In our opinion, the target audience of this study includes researchers, practitioners, students, and graduate students in various applied areas. The approach under consideration can be used in various areas where data are available, presented in the form of a set of time series reflecting the dynamic properties of processes in complex systems.