# It is Useful to Analyze Correlation Graphs

A.N. Gorban<EMAIL_ADDRESS>T.A. Tyukina<EMAIL_ADDRESS>L.I. Pokidysheva
<EMAIL_ADDRESS>E.V. Smirnova<EMAIL_ADDRESS>Department of
Mathematics, University of Leicester, Leicester, UK  Lobachevsky University,
Nizhni Novgorod, Russia  Siberian Federal University, Krasnoyarsk, Russia

###### Abstract

In 1987, we analyzed the changes in correlation graphs between various
features of the organism during stress and adaptation. After 33 years of
research of many authors, discoveries and rediscoveries, we can say with
complete confidence: It is useful to analyze correlation graphs. In addition,
we should add that the concept of adaptability (‘adaptation energy’)
introduced by Selye is useful, especially if it is supplemented by ‘adaptation
entropy’ and free energy, as well as an analysis of limiting factors. Our
review of these topics, “Dynamic and Thermodynamic Adaptation Models” (Phys
Life Rev, 2021, arXiv:2103.01959 [q-bio.OT]), attracted many comments from
leading experts, with new ideas and new problems, from the dynamics of aging
and the training of athletes to single-cell omics. Methodological backgrounds,
like free energy analysis, were also discussed in depth. In this article, we
provide an analytical overview of twelve commenting papers and some related
publications.

###### keywords:

correlation graph, network biology, adaptation energy, critical transitions,
training, limiting factor

What can be considered as the main result of a long research program? Let us
take a look at great examples. According to Arnold [1], Newton’s fundamental
discovery, the one which he considered necessary to keep secret and published
only in the form of an anagram, in contemporary mathematical language means:
“It is useful to solve differential equations”. The discovery and elaboration
of a new tool, with a demonstration of its wide applicability, is a good main
result when supplemented by many smaller advances that demonstrate the
usefulness of the developed tool and should be important in their own right.

In 1987, we analyzed the changes in correlation graphs between various
features of the organism during stress and adaptation [2]. Results of 33 years
of work of many researchers and groups were summarized in the review paper
[3]. In this review, we aimed to demonstrate that it is useful to analyze
correlation graphs. In addition, we should add that the concept of
adaptability (‘adaptation energy’) introduced by Selye in 1938 [4] is useful,
especially if it is supplemented by recently invented ‘adaptation entropy’ and
free energy, as well as by an analysis of limiting factors.

However, the proper extended review and outlook were finalized when the twelve
additional comment papers were published together with our article [5, 6, 7,
8, 9, 10, 11, 12, 13, 14, 15, 16]. Our ‘reply to comments’ is an introduction
to these twelve supplementary materials. There are several aspects in which
commentators have enriched our review:

  * 1.

Presentations of materials on changes in correlation networks under stress,
illness and adaptation, which analyze various situations and use additional
tools for data preprocessing and analysis of correlation graphs. These
extensions are important for proof that analysis of correlation graphs is
useful in analysis of adaptation.

  * 2.

Suggestions to extend methods onto different phenomena and problems, from
aging and single cell omics to molecular networks and other intriguing
systems.

  * 3.

Discussion of alternative or deeper backgrounds of the effects described in
our review, and description of the conceptual problems that the proposed
methodology will face in the further development and expansion of application
areas.

Of course, each comment contributed to several aspects of the content. Below,
we briefly discuss these enrichments.

##  1 Additional materials on the main area of the review

### Dynamical network biomarkers

A treasure box of “dynamical network biomarkers” was opened for readers in
[14]. This concept was developed to characterize the critical state during the
disease progression. We agree that these ideas are consistent with our
correlation graph approach. Moreover, taking into account the publication
dates, we are proud to mention that the dynamical network biomarkers present
the further development of our approach. Our series of publications started in
1987 (see, for example, [2, 17, 18, 19, 20, 22]). The approach was later
applied to many systems and elaborated in various directions by many authors,
with references to our work [23, 24, 25, 26, 27, 29, 30, 28, 31, 32, 33, 34,
35] or with rediscovery of some ideas [36, 37, 38] (for the more detailed
review we refer to [3]). Of course, each paper was not just an application or
rediscovery of previously known ideas. Most groups of authors proposed their
own versions of methods suitable to their material and developed some
theoretical backgrounds for their findings.

In particular, the important new tools for selection of the relevant dominant
or leading subnetworks in the large networks of attributes near crisis were
developed by comparison between the internal (within the group) and external
correlations. A brief review of the theory and practice of dynamical network
biomarkers was presented in the comment [14]. A different approach for
selection of the most relevant subnetworks of attributes is based on principal
component analysis [26, 27]. In this approach, at the first stage, the most
important principal components for disease recognition should be found (the
components with the greatest significance for the classification task), and
then the attributes that dominate in these principal components should be
selected for further correlation and variance analysis (for more details, we
refer to the original works).

The series of work presented in [14] systematically uses the concept of
attractor bifurcations as a hidden cause of the observing transformation of
the network. The general theory of bifurcation of attractors and critical
delays in general dynamic systems (with and without noise) was developed in
1978-1980 in the PhD thesis of one of the authors [39, 40]. (An abridged
version of the thesis [39] was published later in English [41], see also a
brief review [42] with further references.) However, we found that a
systematic theory of the transformation of attribute networks under adaptation
and stress can benefit from the concepts developed by Selye and his school.
Below, we discuss this question and the non-trivial relations between
attractor bifurcations and the concept adaptability in more detail using also
other comments that considered bifurcations of attractors as one of the main
mechanisms of the observed effects [6].

### Someone’s noise is another one’s signal

K. Selvarajoo in his comment [12] discussed high-throughput gene expressions
data analyzed to interpret and predict the complex immune, cancer and biofilm
responses. In general, he supported our findings but attracted more attention
to biological noise. This noise can be independent from the technical noise
like instrumental effects and can be a signal of approaching critical
transitions. We completely agree with this comment. Moreover, the correlation
graph approach discussed in our review with analysis of correlations and
variances is a tool for study of sensible noise in the networks of attributes.
These methods are widely used and modified for different purposes. For
example, correlations and variances both in the organism-to-organism [2, 19,
29] (or cell-to-cell [12]) variations and in the fluctuations of the
attributes of one organism in time [32] can provide important information
about adaptation and crisis. The further development of this methodology
provides inspiring examples even from applied humanities, such as the analysis
of fluctuations in time of physiological parameters (cardiac activity and skin
conductance) in a ‘psychotherapist-patient’ pair, which measures the level of
empathy and may predict the success of psychotherapy [43], or the analysis of
correlations between variations of public fears for different regions of the
country that can predict political crises [34].

This topic, analysis of noise and individual behavior of various structural
components (both material, like cells, tissues, etc., and virtual like
principal components, biomarkers, and subnetworks) was stressed by many
commentators. C. Franceschi stated that “each cell has a specific signature
and present subtle differences with respect to the other cells derived from
the same source,” – and these differences increase with age [5]. In this
comment, like in [12], our approach is friendly criticized for insufficient
attention to this heterogeneity of life system: “A major usually non explicit
assumption of the approach adopted by Gorban et al. is that the elements of
each specific cellular network are basically similar each other’s. This basic
assumption is likely not true.” We fully accept the fundamental heterogeneity
of many life systems at all levels but cannot accept that their homogeneity is
our ‘major assumption’.

However, the source of the impression that our systems are “basically similar”
is very clear (to us): To apply the correlation graph approach to a group of
biological objects (organisms, cells, etc.), no one needs their homogeneity.
In fact, ‘meta-uniformity’ is desirable instead: the attributes selected for
analysis must be measurable for all objects. Of course, some palliative
solutions can be used when attributes are immeasurable for some objects, for
example, a special value can be entered meaning ‘impossibility of
measurement’. However, the correlation of the two attributes can only be
calculated for groups (or subgroups) where both attributes are measurable.
Technically, one can consider a ‘crazy kilt’ of such subgroups with a
different set of objects for different pairs of attributes, but meta-
homogeneity with the same attributes for all objects makes data analysis much
easier and more transparent.

According to the comment [6], the crucial importance of fluctuations in living
systems are caused by the “small integer numbers of key and lock molecules” of
many types. “The key issue is that randomness is a central feature of the
microscopic level, and it is therefore unlikely that microscopic details could
help in understanding the logic behind living organisms.” Therefore, the
special mesoscale approach is needed, that does not rely on the central limit
theorem and mass action law but, on the other hand, is not as noisy as the
microscopic level is [6]. Despite of that, another comment [10] is focused on
the single cell level and discussed the data collections that are produced at
that ‘microscopic’ level by single cell omics as a promising area of
applications of the correlation graph approaches and their further
development. The comment [8] goes even deeper, to more ‘microscopic’ and noisy
molecular networks. The fluctuating world of many personalized objects can be
represented as a supernetwork, that is a network of networks. Promising
approaches for constructing of such supernetworks of potentially infinitely
depth hierarchy were discussed in the comment [7]. This hierarchy starts from
the graphs of binary correlations (or their close analogues) and then goes
deeper to more complex structures.

Separation of real life high-dimensional and heterogeneous data into signal
and noise is extremely problematic and can be never done for sure. The
important data mining ‘axiom’, Someone’s noise is another one’s signal, should
be included into undergraduate textbooks, and biological networks can provide
good examples of transforming apparent noise into important signal.

##  2 Extending of methods to new phenomena and problems

### Dynamics of aging

According to the comment of C. Franceschi, [5], the approach presented in our
work “will help not only in disentangling the complexity of aging
(particularly in humans) but also in establishing a new ‘aging medicine’, and
above all in identifying new tools capable of early predicting power regarding
human health trajectories.” In this comment, the phenomena of aging that can
be covered by our approach were analyzed and some open problems were
formulated.

The concept of adaptation energy and optimization of its spending discussed in
the paper “fits very well with what I learned from the study of human aging
and longevity,” wrote one of the world’s leading experts on aging. This
opinion supports our claim that the concept of adaptability (‘adaptation
energy’) is useful. It could be worth to mention that a very recent critical
review of adaptation energy and its applications to exercise science [44]
states that this concept is applicable to the phenomenon of aging rather than
to sports.

C. Franceschi drafted a research program for application of dynamic and
thermodynamic models of adaptation to aging phenomena. As per his comment, the
main and urgent challenge to our approach, is “the fundamental heterogeneity
and uniqueness of humans at all biological level (genetic, epigenetic,
metabolic, among others) which emerges whenever the results of a study are
analyzed at the individual level,” and an enormous amount of work is expecting
to meet this challenge. We have already begun work on analysis large
heterogeneous systems and have created open access software tools for
identifying development trajectories in big data [45, 46, 47, 48]. These tools
can help in predicting “human health trajectories”, as requested by the
commentator. Nevertheless, there remain more inspiring questions than answers
in the theory and practice of optimal aging and longevity [5].

### Exercises and sport

Exercise and sports data provide a wealth of material for the study of
adaptation. Any theory of adaptation is incomplete without assimilation of
these materials. We were deeply concerned that Selye’s classical concept of
adaptation energy did not seem to be able to describe the training phenomena
that are well known in the science of exercise and beyond to anyone with a
sporting background. Similar concern was formulated by several authors, see a
review [44]. Our commentators, [9] are experts in exercise science. They
analyzed the concepts of adaptation from the “exercise science perspective”.
We are grateful for an accurate understanding of our modification of the
concept of energy of adaptation and for examples that demonstrate the need for
modification. Here, we would like to add one more comment. To capture the
apparent training phenomena we had to introduce “free energy of adaptation”.
It included two components: adaptation energy and adaptation entropy. The
entropy of adaptation is necessary for the formalization of adaptive changes
in the scale of stressors. After training, the organism can withstand larger
loads. The same load can be ‘small’ for a trained body and ‘large’ (or even
impossible) for an untrained one. This rescaling of the stressors can be
measures by a logarithm of the rescaling coefficient (logarithm is needed for
additivity). This is a typical entropic measure – logarithm of available
‘volume’. The dynamics of the load scale described by our free energy models
[3] are qualitatively consistent with the phenomena discussed in the comment
[9], but additional efforts are required for detailed modeling and analysis to
verify the quantitative agreement.

### Single cell omics

We dealt [3] with whole organisms (mice, people, plants) or “super-organismic”
objects (stock market or social systems). A. Zinovyev proposed to change the
focus to a significantly smaller scale, to a single cell [10]. This is not
just an interesting and important question asked by many commentators. The
comment [10] proposed to work with single-cell omics. This is a new technology
that gives information about genomes, gene expression, and cell heterogeneity
for thousands of individual cells from a single organism, simultaneously. This
revolutionary technology brings a huge amount of new data we have never seen
before. It is a challenge to biology and medicine to assimilate these
datasets. Several specific difficulties of single cell omics data analysis
were discussed in the comment [10]. In particular, individual cells are
usually destroyed when measured, so we can track the population of cells over
time, but the state of an individual cell can only be observed once. It is
very desirable to join individual cells (represented by points in a
multidimensional space) into trajectories of their development. These
trajectories can branch and the whole picture is expected to be a bunch of
branching cell development and differentiation trajectories. Extraction of
this bunch from a single cell data cloud is a special data mining problem and
several tools are developed (for example, the method of topological grammars
and principal graphs [45, 49]). However, this is not the end of the story. The
motion of cells in the feature space may include also oscillation in cell
cycle that is different from the development and differentiation. This is
especially important for growing cell populations like cancer. Special
techniques are needed to separate these two types of motion: cell cycle and
development with differentiation. In a recent work [50] a motion separation
approach based on independent component analysis was developed. Universal
mathematical [51] and computational [52] models of cell cycle are useful for
such separation. The developed methods for processing of single cell omics
data open the door for detailed analysis of adaptation and stress at the
single cell level. A research program of this analysis is drafted in the
comment [10] and we will be happy to see the results and participate in this
new fascinating intellectual travel.

### Molecular networks

Can we go deeper than the single cell level? The comment [8] demonstrates one
more step: from cells to molecular networks. Adaptation at the level of
protein networks is possible. Recent works reveal that protein networks adapt
to mutations. They develop new allosteric networks. For us, this is a
completely new angle of view, and we are more than happy that the integrated
dynamic and topological approach works at that level and helps in analysis of
adaptation at the level of protein networks. As a result, we can imagine the
future construction of adaptation science, from the very bottom level
(molecular networks) to the single cell adaptation, then the tissue and
organismic levels, and finally, the level of super-organismic entities. In
this context, the concept of “network of networks” [7, 53] seems to be very
useful.

##  3 Alternative or deeper backgrounds of the observed effects

### Bifurcations versus adaptability dynamics

In two comments, [6, 14] the bifurcations of attractors is considered as a
main mechanism of the transformation of correlation graphs near the ‘tipping
points’. In our review [3], we discussed three proposed mechanisms of the
observed effects: dynamics of adaptability described by the factors–resource
models, approaching bifurcations of hidden dynamic systems, and approaching
the border of survival (in the framework of Fokker–Planck models). All these
mechanisms describe some experimental data, at least, qualitatively. Moreover,
even if some data are not described, there remains a hope the the model can be
modified accordingly to meet better the real phenomena.

It is difficult to select the best type of models because all the models refer
to hidden phenomena that are not observable directly. For example, the
adaptation energy is not a physical quantity and cannot be measured directly.
Instead, adaptation energy (or free energy) should be considered as an
internal coordinate on the dominant path in the model of adaptation [22]. Many
questions can be asked: What is the dynamical system that has this dominant
path, what is its phase space, why the dominant path appears? (This can be the
result of either averaging, or the existence of a slow manifold, or a
combination of these reasons.)

Very similar questions can be asked for the bifurcation model. Nobody has
observed the underlying dynamical systems and its ‘network of attractors’.
Therefore, all this reasoning remains a useful metaphor. The situation is very
similar to the ‘self-organized criticality’ (SOC) that provides us by a
metaphor and simple nice examples like a sandpile but does not create
quantitative models. “One of the main misconceptions about SOC is to think
that it is a traditional quantitative theory and that, as such, it can yield
detailed quantitative predictions once confirmed that the system exhibits SOC
dynamics [54].”

Existence of dominant path and a single parameter that describes behavior of a
complex system is usual near bifurcation and phase transitions. Perhaps, the
first famous example gave Lev Landau in his work about stability loss of
laminar flows [55]. In this work, he found the equation for the deviations
from the steady laminar flow without explicit use of the Navier-Stokes
equations. The closeness to the bifurcation was the main assumption. In that
sense, two approaches, bifurcation and a special dominant path with
adaptability as a main coordinate, are not in principal contradictions.
Fluctuations and critical delays become very important near bifurcations, here
we agree with our commentators. (See also our detailed analysis of attractor
bifurcations and critical delays in dynamical systems, [39, 40, 41, 42]. In
particular, we analyzed effect of fluctuations modeled by
$\varepsilon$-trajectories.)

Despite of some similarity, there exists a difference between just a general
‘closeness to a bifurcation’ and a specific dominant path factor-resource
models. Instead of rephrasing of our large paper, we will present a very
recent simple example of a social system, where these factor-resource
relations are obvious.

Development of world-class universities in China was analyzed in [56]. This
process was started in 1998 and recently, high positions of Chinese
universities in world ranking are widely known. At the beginning of this race
for ranking, vast investments of money have indeed earned China’s universities
‘quick wins’. In that sense, one factor dominated the accelerated development
of the universities to the relatively high level. But after that, more
individual institutional and cultural factors determine the development of the
universities, “including dominion over university-level strategic initiatives,
promotion, tenure, research directions, establishment of new academic
programs, and publication and academic content” [56]. Thus, efforts and
successes of different universities become different and the race for higher
success is not dominated just by funding. According to [56], the simple
government’s command-and-control approach to higher education institutes
became largely inconsistent with the diverse world of universities and at this
“multifactorial” stage, a more flexible approach is necessary. We cannot
comment this advices abut management, but the difference between the single-
factor development and polyphonic multifactorial motion seems to be obvious.

The comments [6, 14] include much more material. In particular, [6] presents a
very dense review of many mathematical, biological and even epistemological
problems in understanding biological adaptation with special attention to
stochastic processes and limited predictive power of many classical families
of models like ordinary differential equations of chemical kinetics. (The
comment [12] also stressed that these models are insufficient in the
stochastic world of genome activity.) High importance of fluctuations was
demonstrated also in the comment [14]. All these problems are very important,
indeed. Here we would like just to stress that relatively simple ‘middle
level’ models may work well despite the complexity and stochasticity of the
low-level detailed picture.

### Statistical physics of life and network thermodynamics

The factor-recourse models work like thermodynamics, whereas the hidden
bifurcation of unknown attractors is a metaphor of statistical physics. A.
Giuliani in his comment [13] presented an idea of ‘organized complexity’ and
‘statistical physics of life’. He proposed that the foundation of a biological
statistical mechanics “should focus on the changes in correlation patterns
marking state transitions”. At the same time, the network thermodynamics of
life is necessary to consider changes in macroparameters. For this purpose, we
utilized the typology of systems of factors and their interactions proposed by
Tilman [57]. We developed these ideas further and applied to analysis of
adaptation. In addition to the Liebig’s law of the minimum, various different
interaction of factors are possible like synergy. The rich family of “factor-
resource” models can work not only as a metaphor, but also provide
quantitative models of real systems. To overcome the quasistatic nature of
these models (‘periodic adaptation’), a family of dynamic models of individual
adaptation was proposed.

The manifesto of statistical physics of life [13] supports the main results:
It is useful to analyze correlation graphs and thermodynamic models based on
adaptability (‘adaptation energy’) concept. It also opens further
methodological perspectives.

### Networks of networks

The concept of “networks of networks” covers many ideas expressed by the
commentators. The hierarchy with molecular networks [8] at the low level, then
networks of cells and their features [10], then correlation networks of the
attributes of an organisms, and so on, presents an example of such a network
of networks. Analyzing these constructs may require special mathematical and
statistical tools.

In our paper [3], we considered the correlation graphs, where the nodes were
attributes and edges represented the correlations between the attributes. The
simplest definition of edge weight is Pearson’s correlation coefficient. In
our previous works, we tried many various characteristics and, finally, were
satisfied by the simplest version. Nevertheless, this is not a compulsory
choice and for formal description of ‘networks of networks’ various different
approaches may be proposed. Informational characteristics, like information
gain, may be useful, especially for qualitative attributes. Various regression
residuals are used in the “parenclitic” (from the Greek for “deviation”) and
synolytic (from the Greek for “ensemble”) network representations [7]. These
networks were successfully used in medical applications. Essentially, they are
also the correlation networks but with the different measures of correlations.
However, the details matter and different measures may be useful in different
situations. It will be important to compare different approaches and we hope
that the comment [7] could be extended to an analytical work about different
versions of correlation graphs and their performance analysis. Even more, the
ensemble of different correlation graphs can be used for description of a
complex biological system, and we may not need to choose the best correlation
graph, but rather use all of them together [7].

Another technique is discussed in the comment [15]. Many modern tools are
developed for learning of intellectual agents, including neural networks.
Adaptation can be modeled by learning. Evolution of a population of such
agents with small mutations of ‘genotypes’ and optimization of phenotypes by
learning can clarify the problem of instantaneous fitness discussed in our
work. These systems of intellectual agents can be used for modeling networks
of networks.

### Three (or four?) faces of free energy in adaptation and stress

Two free energies were used in our work. One of them is the classical physico-
chemical free energy used in [3, Appendix A] for demonstration of
‘thermodynamic cost of homeostasis’. Another is the adaptation free energy,
combined from the adaptation energy and the adaptation entropy introduced for
modeling of the adaptive rescaling of stressors in training. Two comments,
[11, 16] added a lot to this topic.

The main challenge, formulated in [11] is “to establish a real correspondence
between axiomatic structures of adaptation and microscopic models.” According
to this comment, the difference between homeostasis and adaptation is the
difference between two types of control: for homeostasis, this is control of
dynamic variables (‘thermodynamic coordinates’), like concentrations, and for
adaptation this is control of the conjugated variables (‘thermodynamic
forces’), like chemical potentials. Behind this definition, we found the
representation of biological systems as controlled dynamical system of non-
equilibrium thermodynamics. We fully support this basic idea. Adaptation in
dynamical systems [58] is important for analysis of biological processes, and
these dynamical systems must obey thermodynamics. However, realization of this
idea still requires a lot of work and further details, even in the basic
definitions.

The comment [11] presents Bauer’s axiomatics of adaptation [59] and explains,
why it meets the physico-chemical backgrounds of biological adaptation better
than Selye (or Selye–Goldstone) axioms. A research program “towards the
physics of adaptation” is proposed that consists of two items. We reformulate
them as follows:

  1. 1.

Develop accurate thermodynamic estimates of energy consumption in adaptation
and homeostasis (according to [11], the existing models overestimate this
consumption, especially for adapted steady states).

  2. 2.

Integrate the Selye–Goldstone phenomenology with the physico-chemical approach
to adaptation proposed by Bauer (and the modern versions of these approaches)
and create more realistic and interesting models.

It seems that the comment [16] approaches thermodynamic of adaptation from the
opposite side. The concept of variational free energy is clearly explained
from scratch, from variational Bayesian inference. The formal definition [16,
Box 1] is followed by an explanation of qualitative details, meaningful
analogies and even metaphors. According to [16], the variational free energy
evaluate the surprisal under an optimism-biased expectations from the world,
that is, it measures instantaneous distress. It can be used for deriving the
instantaneous fitness using the learning metaphor of adaptation (compare to
the remark from [15] about evolutionary learning of intellectual agents and
models of adaptation and evolution). It is impressive that the dynamics of
“Bayesian thermostats” in active inference demonstrate the behavior of
variance and correlations between intellectual agents that was observed in
many living systems under stress in adaptation process.

Finally, the concept of variational free energy meets physical thermodynamics
in the following form: in practice of modeling living systems, the variational
free energy and the storaged of thermodynamic free energy have inverse
relationships because “the minimisation of variational free energy in living
agents is associated with the build-up and maintenance of the physical
structure realizing these agents.”

The combination of comments [11, 16] gives hope that a new revolutionary
concept of variational thermodynamics of adaptation will soon appear, taking
into account both non-equilibrium thermodynamics and Bayesian learning.

A very minor answer to a minor comment from [16] is ethically important for
us. In a brief discussion of our estimate of free energy cost of a non-
equilibrium state [3, Appendix A], an earlier work of Horowitz et al. (2017)
[60] was mentioned in [16]. We were ready to cite our book (1984), which
contains these estimates (and much more) [61] (some results of this book were
later published in English [62]), but we were ashamed when found that the
first results of this type were published in 1935 by Bauer [59] (not cited in
our review). We must never forget this very talented researcher who was
arrested and killed on the peak of his creative life.

##  4 Instead of conclusion

The combination of comments creates a polyphonic effect around a very deep
problem of adaptation. In this ‘reply’ we tried to share our impressions and
enjoyment of this polyphony. We are very grateful to all commentators for new
questions, tools, hypotheses, and answers to some problems. The main expected
result of our joint efforts will be “the enlargement of the spectrum of the
phenomena amenable of a quantitative approach” [13].

## Acknowledgments

The work was supported by the Ministry of Science and Higher Education of the
Russian Federation (Project No. 075-15-2021-634). .

## References

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