Information theory and possible mathematical descriptions of economical and social systems based on real physical phenomena
Recent approaches in informatics to model large complex systems are considered following the ideas from real phenomena explained by physical tools. The econo-physics and sociophysics are considered. In particular, Master Equation approach and Markov chains approaches are discussed. Also the partial...
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Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України
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irk-123456789-138672010-12-07T17:00:11Z Information theory and possible mathematical descriptions of economical and social systems based on real physical phenomena Gaeta, M. Iovane, G. Makarenko, A. Проблеми прийняття рішень і управління в економічних, технічних, екологічних і соціальних системах Recent approaches in informatics to model large complex systems are considered following the ideas from real phenomena explained by physical tools. The econo-physics and sociophysics are considered. In particular, Master Equation approach and Markov chains approaches are discussed. Also the partial differential equations as the tool for modeling economical and social systems are represented. New approaches for modeling systems with memory and with accounting internal properties of system elements are considered and some new research problems are proposed. Рассматриваются современные подходы в информатике к моделированию больших сложных систем, аналогичные используемым в физике. Обсуждаются эконофизика и социофизика. Представлены дифференциальные уравнения в частных производных как инструмент для моделирования экономических и общественных систем. Предложены новые подходы к моделированию систем моделирования с памятью и учетом внутренних свойств элементов системы, а также новые проблемы для исследования. Розглядаються сучасні підходи в інформатиці до моделювання великих складних систем, аналогічні тим, що використовуються у фізиці. Обговорюються еконофізика і соціофізика. Наведено диференційні рівняння у частинних похідних як інструмент для моделювання економічних та суспільних систем. Запропоновано нові підходи до моделювання систем із пам’яттю та з урахуванням внутрішніх властивостей елементів системи, а також нові проблеми для досліджень. 2005 Article Information theory and possible mathematical descriptions of economical and social systems based on real physical phenomena / M. Gaeta, G. Iovane, A. Makarenko // Систем. дослідж. та інформ. технології. — 2005. — № 4. — С. 52-64. — Бібліогр.: 67 назв. — англ. 1681–6048 http://dspace.nbuv.gov.ua/handle/123456789/13867 519.5 en Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України |
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Проблеми прийняття рішень і управління в економічних, технічних, екологічних і соціальних системах Проблеми прийняття рішень і управління в економічних, технічних, екологічних і соціальних системах |
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Проблеми прийняття рішень і управління в економічних, технічних, екологічних і соціальних системах Проблеми прийняття рішень і управління в економічних, технічних, екологічних і соціальних системах Gaeta, M. Iovane, G. Makarenko, A. Information theory and possible mathematical descriptions of economical and social systems based on real physical phenomena |
| description |
Recent approaches in informatics to model large complex systems are considered following the ideas from real phenomena explained by physical tools. The econo-physics and sociophysics are considered. In particular, Master Equation approach and Markov chains approaches are discussed. Also the partial differential equations as the tool for modeling economical and social systems are represented. New approaches for modeling systems with memory and with accounting internal properties of system elements are considered and some new research problems are proposed. |
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Article |
| author |
Gaeta, M. Iovane, G. Makarenko, A. |
| author_facet |
Gaeta, M. Iovane, G. Makarenko, A. |
| author_sort |
Gaeta, M. |
| title |
Information theory and possible mathematical descriptions of economical and social systems based on real physical phenomena |
| title_short |
Information theory and possible mathematical descriptions of economical and social systems based on real physical phenomena |
| title_full |
Information theory and possible mathematical descriptions of economical and social systems based on real physical phenomena |
| title_fullStr |
Information theory and possible mathematical descriptions of economical and social systems based on real physical phenomena |
| title_full_unstemmed |
Information theory and possible mathematical descriptions of economical and social systems based on real physical phenomena |
| title_sort |
information theory and possible mathematical descriptions of economical and social systems based on real physical phenomena |
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Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України |
| publishDate |
2005 |
| topic_facet |
Проблеми прийняття рішень і управління в економічних, технічних, екологічних і соціальних системах |
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http://dspace.nbuv.gov.ua/handle/123456789/13867 |
| citation_txt |
Information theory and possible mathematical descriptions of economical and social systems based on real physical phenomena / M. Gaeta, G. Iovane, A. Makarenko // Систем. дослідж. та інформ. технології. — 2005. — № 4. — С. 52-64. — Бібліогр.: 67 назв. — англ. |
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2025-07-02T15:40:54Z |
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| fulltext |
M. Gaeta, G. Iovane, A. Makarenko, 2005
52 ISSN 1681–6048 System Research & Information Technologies, 2005, № 4
519.5
INFORMATION THEORY AND POSSIBLE MATHEMATICAL
DESCRIPTIONS OF ECONOMICAL AND SOCIAL SYSTEMS
BASED ON REAL PHYSICAL PHENOMENA
M. GAETA, G. IOVANE, A. MAKARENKO
Recent approaches in informatics to model large complex systems are considered
following the ideas from real phenomena explained by physical tools. The econo-
physics and sociophysics are considered. In particular, Master Equation approach
and Markov chains approaches are discussed. Also the partial differential equations
as the tool for modeling economical and social systems are represented. New ap-
proaches for modeling systems with memory and with accounting internal properties
of system elements are considered and some new research problems are proposed.
INTRODUCTION
Now the wide development of informational technologies requires the application
of mathematical modeling for qualitative and quantitative understanding of sys-
tem behavior for predicting and forecasting and for the technical design of system
structures. This clearly appear for social, economical and political systems. In
fact, the recent systems had become very complex in their structure. The society
transformation factors are besides the informational technologies, which follows
to the necessity of mentality accounting. The electronical web may serves as the
bright example.
Taking into account the increasing of complexity inside the considered sys-
tems, new adequate approaches need to make model better and better. Fortunately
in the past some physicists considered systems with many interacting particles,
which we can use also to supply corresponding models and investigation metho-
dologies in other knowledge fields. The process for introducing physical concepts
into informatics is rather new and will spread. But just now there are a great num-
ber of results which require the understanding and interpretation. Also searching
of further ways of developments and implementations of physical ideas to model-
ing is necessary. Let us remark that one of the sources of fruitful concept is syn-
ergetic. So at present work we make the system analysis and evaluation of exist-
ing approaches related to physical concepts. We consider master equation, partial
differential equations, ordinary differential equations, Ising-type models, artificial
neural network and others.
1. MASTER EQUATION APPROACH
One of the rather general approach to considering social and economical systems
had followed from theoretical physics (mainly from the physics of many-particle
systems) [1–3]. In accordance with physical approaches in such case it is neces-
sary to consider different scales of process and interactions. The basic are the mi-
Information theory and possible mathematical descriptions of economical and social systems ...
Системні дослідження та інформаційні технології, 2005, № 4 53
cro level of local interactions between elements and macro level of aggregated
macro parameters. Main mechanism in such approach is stochastic transitions
between element states. These transitions are rooled by the set of transition
probabilities. Usual description is the string of element’s states ( =)(ts
{ })(,...),(),( 21 tststs N= ). Then the basic equation is the so called master equation
for the probability distribution of the general parameters of the system. Here we
display an illustrative simple example [1] for reaching the goal of better under-
standing of model structure. The general master equation form is described in [1,
Chapter. 4, equation (4.7)] and is involved and lengthy.
In illustrative simple case the state of social (economical and so on) systems
at the moment t is described by cells representation ( ) { ...,,...,,...,, 1
11
1
P
C nnnn =θ
P
C
P
C
P
Cn θθθθ ,...,,...,,...,;,... 1
11
1 , where n , θ are integers, which represents the
number of elements in i-th cell (numbers of individuals with i-th opinion) and θ
trend parameter (propensity to opinion) in i-th cell of individuals. For simplicity
in the case of two sates of opinion ( + ) or (–) the master equation has the form
(equation (5.20) in [1])
[ −+++−−= +−−+ ),,1(),1(),,1(),1(),,( tnPnwtnPnw
td
tnPd
θθθθ
θ
] +−− +−−+ ),,(),(),,(),( tnPnwtnPnw θθθθ
[ −++−−−+
↓↑
),1,()1,(),1,()1,( tnPnqtnPnq θθθθ
]),,(),(),,(),( tnPnqtnPnq θθθθ
↓↑
−−
where ),( θ
n ),,( tnP θ is the probability for system to be at state ),( θ
n at mo-
ment t .
This equation corresponds to the probabilistic transition processes between
cells in number of individuals which take opinion ( + ) or (–). The coefficients
↓↑+−−+ qqww ,,, account the probability of transitions and balance of such transi-
tions. The previous equation is the basis for many investigations and conse-
quences. We remark that the possibility of deriving the equations for mean values
from master equation is very important. Maybe the main advantage of this ap-
proach is that in principle it can bring strong background for models (and another
approaches). The advantage is accompanied with some drawbacks. First of all it is
very difficult to collect the values of a lot of parameters for real problems (for
example the transition probabilities). So, some simple analytical formulas usually
are exploited. But the main problems still is the proof of applicability of theoreti-
cal concepts in social and economical systems. Moreover, usually the applications
of master equations use some kind of regularity of space elements, which is poor
adjusted to real geography, social structure and so on. Besides it is very difficult
to include in master equation the mental factors (mentality, beliefs, decision-
making, consciousness and so on). So, other approaches may be also useful and
they may take into account more adequate other ‘human’ aspects. However the
master equation may be used for aggregated (average) models. One of the devel-
M. Gaeta, G. Iovane, A. Makarenko
ISSN 1681–6048 System Research & Information Technologies, 2005, № 4 54
oped models of such kind for example has the form of ordinary differential equa-
tions [4, 5], where the model for opinion formation on nuclear power problem is
zykx
dt
dx
+−= γ ,
zxyy
dt
dy εββ +−= 0 ,
xz
td
zd
δα +−= ,
where )(tx is the difference between the numbers of supporter and opponents on
some opinion at time t ; )(ty is the difference between the number of journalists
writing about nuclear energy topics and those not writing about nuclear energy
topics; )(tz is the difference between the number of scientist doing their research
on innovations in nuclear energy; δαεβγ ,,,,, k — some coefficients. It is inter-
esting and very illustrative that these equations may be obtained from the master
equation by some averaging procedure. Also the probabilities of opinion change
may be accounting explicitly by introducing the transition probabilities [4, 5]. It is
very useful that in this problem some levels of description had been exploited and
strict background for o.d.e. in principle may be received.
2. ECONOPHYSICS
Recently some other physical concepts became the key issues in modeling of non-
physical systems. Strong support for applications of physics was the needs in eco-
nomical applications, especially in financial problems. Such field of applications
has a special name ‘econophysics’. Initially such term had been coined mainly for
the phenomenon of fluctuations and critical phenomena in economical data sets
and processes. But such title is useful also for more wide field of physical con-
cepts for the economics. So here we describe some of such concepts which are or
may be essential to understanding and modeling in economics.
2.1. Mean field approach. One of the useful idea is that in many-particles
systems the dynamics of each particle is determined by the influence of all anoth-
er particles (or in some surrounding region) [6, 7]. It is well known the fact that in
economical and social systems the behavior of elements (participants) of such
system is in many cases determined by surrounding influences following to possi-
ble analogies with physical concepts. Such idea had been already used explicitly
or implicitly. One of the first explicit applications of such idea was the investiga-
tions of market [8, 9]. In fact in the cellular automata approach some restricted
surrounding vicinity is exploited.
2.2. Ferromagnetics models. Important source for many concepts, which
had lead as to the self-organization theory as particularly to social and economical
modeling were ferromagnetic systems, that the system which consist from a large
number of small interacting magnets [6, 10]. Main properties in such systems
theory are mean field, phase transition, critical indexes, renormalization groups,
domens (clusters), fluctuations caused by temperature influence, potential energy
surfaces, order and disorder and spin glasses.
Information theory and possible mathematical descriptions of economical and social systems ...
Системні дослідження та інформаційні технології, 2005, № 4 55
The most known before had been Ising model of ferromagnetics. Now we
posed sketch of such model because of its importance for analogies with con-
densed physics phenomena. In classical Ising model N elements had been consi-
dered with two possible states 1+ and 1− (directions of spins). The stats of the
system (set of element’s state) tend to the minimum of the system’s energy
∑
≠
−=
ji
jiij ssJE ,
where ijJ denote the interactions between the elements. In physics ijJ are random
and symmetrical jiij JJ = and some transition probability for state change exist
for non-zero temperature [10, 11].
The dynamics of the models for variables )(ts { })(,...),(),( 21 tststs N= (‘pat-
tern’ of situation) follows to special dynamical laws and the variables tend to one
of the minima of the energy. The graphic of the ‘energy’ E in the space NR has
the title ‘landscape’ (or ‘potential landscape’) and has very involved character
with many local minima in case of random bounds ijJ . Of course all of these
properties should be reconsidered for the case of social and economical applica-
tions because of peculiarities of non-physical systems.
2.3. Critical indexes. A large number of papers had been devoted to applica-
tions of physical concepts to the description on the behavior of economical and
closely related to economical variables. The examples are numerous: size of
firms, probability of price change, correlations and fluctuations in time series of
parameters [12–15]. Remark that one of the typical problems is searching the laws
of correlation functions in dependence on distance (time intervals). Frequently the
experimental data is reconstructed as power law. Many different approaches had
been exploited for such goals. For example in [15] the statistical model by Sher-
rington–Kirkpatric had been used for imitationing the correlations laws which has
stock market. Also the models remembering spin glass had been developed for
investigating the market fluctuations [16, 17]. Some applications of Ising model
are in [18]. The results had been interpreted in terms of transitions (‘flip-flop’) of
spins. Numerical investigations of quasi-equilibrium process followed by abrupt
change (so-called ‘punctuated equilibrium’, which had been found in biology
[17], are especially important.
2.4. Complex networks and related problems. Recently the problems from
subsections 2.2–2.3 follow to new theoretical and practical results which have
been connected with complex networks investigating by theoretical physics
methods. Many definitions of complex networks exist now but intuitively many
systems had been recognized as complex. The examples are electrical power net-
works, trophycal chains, ecological systems, world-wide web, phone-call net-
works, economics, language, ontological models of knowledge, social networks,
communicational networks and many others [13, 19–21]. The complex network is
the object which consists from many nodes (elements) and connections between
elements; moreover, different elements can have different numbers of connec-
tions. We remark that frequently many problems had been reformulated as the
graph theory problems.
M. Gaeta, G. Iovane, A. Makarenko
ISSN 1681–6048 System Research & Information Technologies, 2005, № 4 56
An example of typical problem is the evaluation of )(kP — the probability
that a randomly selected node will have k connections with the rest elements. This
problem had been intensively investigated since Erdosh and Reniy in the graph
theory and now one of the results is the power law for the probability )(kP .
Another important problem is the percolation problem [13, 22]. In such case the
connected subpart of network is investigated (more frequently for regular net-
works). One of the most important facts is the existence of a thresholds for k. If k
is less then some critical *k then the network consists of isolated clusters (con-
nected subparts of whole network). If *kk > than network has one large cluster.
The examples of percolation problems supply the electrical networks or water
channel systems, capillary networks in porous media. Spread of epidemic or resis-
tance to the attacks (which imitates by some part of nodes removal) also is very
interesting. Physical approaches, physical analogies are very useful for investiga-
tions of phenomena above. Remark here only the master equation approach for
probability that the node added at the moment t has k connections [13, 23]. Also
the models for number of connections for i-th node of o.d.e. type may be applied.
Recently new sub branch of network investigations had been created namely
small-world networks [13, 19, 24]. This term is related to the network with inter-
mediate extent of regularity between regular network with small number of con-
nections and random graph with a great number of connections between nodes.
But now it is recognized that a lot of natural, social, technical systems have the
formal characteristics of small worlds. One of the basic models is the small-world
network on the ring [13, 19]. In this case each element is connected to k2 neigh-
bors (usually 2/Nk < ). Also some part of bounds randomly relates far elements.
One of the questions is the next: How many bounds need to reach one element
from another? If N is the number of nodes, k — number of neighbor connected
nodes and p — probability to randomly rewire each node, then average path
length evaluation is
)(~),( αNkpf
k
NpNP ,
>>
<<
=
.1if/)(ln
,1ifconst
)(
uuu
u
uf
Maybe the most intriguing property is that if kNp /2≥ the l begin increases
and the transition level depends on the system size. Remark that such kind of re-
sults had been received by statistical physics methods. As another special type of
growing network behavior, the power law distribution γβ −kmkP /12~)( had been
proved for asymptotical case ( ∞→t ), where k is the degree of node (number of
connections), 11
+=
β
γ . Continuum theory and master equations are the tools for
such derivations.
Another problem closely connected to networks is the attack tolerance (that
is to the disconnecting of network after removing some edges (or nodes)). This
phenomenon is also named as defragmentation of network. From the physical
Information theory and possible mathematical descriptions of economical and social systems ...
Системні дослідження та інформаційні технології, 2005, № 4 57
point of view it is related to the percolation of the network in dependence of net-
work parameter. Another aspect is the robustness of networks (especially WWW).
And finally we should remark the epidemical process on network for which prop-
agation laws had been considered in dependence on parameters [13, 25]. The
most important is the threshold for epidemic spreading.
2.5. Synchronization. Important for interpretation and understanding the
behavior of considered systems are synchronization phenomena. The synchroni-
zation is collective behavior of systems elements when some elements behave in
familiar type [26, 27]. Earlier the synchronization had been considered in pure
physical systems. The classical example is the field of connected oscillators.
Whole synchronizations, clusters of synchronization, chaotic synchronization,
defragmentation are different types of such behavior. Now the point of interests if
pushed to coupled nonlinear maps. Remark that such type of models has already
some economical applications [26].
3. SOCIOPHYSICS
Applications of physical ideas to considering social systems is more rare (al-
though the term ‘sociophysics’ had been known since early 80 th of XX century).
But the way of considering is the same as for the econophysics. First class of
models is the master equation and related equations. Remember also as the exam-
ple the problem of public opinion formation model [1, 4, 5, 28]. More recently the
analogies to Ising systems had been applied occasionally. One example is again
opinion formation and one of the first papers is [29] with two state of opinion.
Further such approach has been developed in the works by S. Galam, J.A. Holist,
Stauffer D and others. Here we consider the simplest case of equation
>
≤−
=−+
0)(,0
,0)(),(2
)()1(
tf
tfts
tsts
i
ii
ii
for states of element, f some nonlinear function which corresponds to account-
ing of the mean influence. The ferromagnetic analogies also had been exploited in
the investigations of international relations [30–32].
Recently new approach to social problems follows from physics, namely
from the theory of Brownian particles. Classical theory of Brownian motion con-
siders the movement of mechanical (‘test’) particles in the gas of small particles
[33]. Now the same problem is formulated for hard particles with internal struc-
ture [34]. In most interesting case active particles had been considered. The ‘ac-
tive' particles can search its neibourhood and make some decision on the move-
ment direction in dependence of own goal. So the model consists from three
blocks of equations: for probability of individual opinion (master equation), para-
bolic equations for communicational field and Langeven equation for individual
movement [34].
4. DIFFERENTIAL EQUATIONS AS MODELS
In the previous sections we consider some models which are very sophisticated
and sometimes non-usual for scientific community. Vice versa the models based
M. Gaeta, G. Iovane, A. Makarenko
ISSN 1681–6048 System Research & Information Technologies, 2005, № 4 58
on ordinary or partial differential equations are more familiar. Such type of equa-
tion had been introduced phenomenologically. But now the possibility for deriv-
ing them by physical analogies exists. The first possibilities is in exploiting con-
servation laws. Economical and social sciences take from physics as model
equations as ideas for leading concepts. Also synergetics is the source of many
ideas in interpretation. Especially fruitful is the idea of dynamical chaos. So here
we consider the realization of such ideas at the level of differential equations.
4.1. Lorents system and dynamical chaos. Maybe the most known model
with the complex behavior is the so-called Lorents system [35, 36]
)( xy
dt
dx
−=σ ,
yxrzx
dt
dy
−+−= ,
zbyx
td
zd
−= .
The investigation of its solutions follows to the concept of dynamical chaos -
the behavior which remembers realization of stochastic processes and has the sen-
sitive dependence to disturbances [10, 36]. Next very important concept now is
attractive sets (attractor) especially strange. Just understanding of attractor’s pos-
sibilities in social and economical systems may be important in the case of lack of
information and models.
Further as the models as the idea of dynamical chaos had been applied to
econonomics [37–39], sociology [40, 41]; especially it should be stressed finan-
cial applications [38]. Lorents system and many others are representative of con-
centrated systems. Of course the space and time aspects are important.
4.2. Models for distributed systems. At the present some applications of
partial differential equations to socio-economical systems already exist; frequent-
ly we find models for spreading, diffusion of parabolic type (diffusive equation).
For illustrating that we pose here one of the examples in [37, Chapter 3]
( )pqpp
p
q
t
p
γ−++
∇∇=
∂
∂ 1 ,
where ),( txp is the distribution of economical parameter. Remark that in [37]
also applications of wave equations to finance had been described. Another class
of models are considered for describing the concurrence in a time process. The
source of all such field of investigations are the so-called Lottka–Volterra equa-
tions for the concurrency of species
NiNN
dt
Nd N
j
jjiii
i ,...,2,1,
1
=
−= ∑
=
γε ,
where iji γε , are coefficients iN number of representatives of i-th species
(commodities, firms and so on) [28, 42–44].
In the case of a continual second independent variable τ and continual ana-
log equation the model has the form [43]
Information theory and possible mathematical descriptions of economical and social systems ...
Системні дослідження та інформаційні технології, 2005, № 4 59
),()(
),(),(
ττ
τ
ττ
txd
tx
t
tx
ii
ii −=
∂
∂
+
∂
∂
,
∫
∞
=
0
),()(),( τττο txbdtx iii .
We also remark the work by [45] where such models had been applied to
some industrial processes and just chaotic solutions had been considered. Closely
related to models in this subsection are integral and integro-differential equations.
Some other models for demography belong to considered classes of models
[46, 47] o.d.e.:
22
1 )( τ+−
=
TT
C
td
Nd
or to partial differential equations [48].
It should be stressed in socio-economical interpretations the notion of blow-
up solutions (or solutions with collapses) [49, 50]. In the blow-up regimes the so-
lutions of the correspondent models can tends to the infinity by the finite time. In
demography blow-up regimes had been interpreted as acceleration in the popula-
tion growth of the Earth [46]. Also blow-up regimes sometimes relate to the fi-
nancial crashes. At the considered level of models (o.d.e. and p.d.e.) frequently
the account of external disturbances should be investigated. One of the analytical
tools for such research is found in partial differential equations with impulsive
right hand [51, 52] which are reformulated as operator equations.
4.3. Dissipative structures and self-organization. One of the most impor-
tant concepts from physics is self-organization and types of self-organized solu-
tions. The examples of self-organization objects are dissipative structures, auto
waves, collapses, sinchronisation and so on [1, 2, 28, 42, 43, 47, 53, 54]. At the
first such examples had been recognized and investigated primarily in physics and
biology. Now the fields of applications naturally spread on socio-economical sys-
tems where the next qualitative and quantitative problems are important: origin of
structures, complexity of the structures, bifurcations of the solutions, and stability
and transformation linguistics of system and many others. The examples of appli-
cations are city development processes, demography, epidemiology [52, 53].
Usually such problems had been considered on the background of diffusive sys-
tems.
5. ASSOCIATIVE MEMORY APPROACH WITH MENTALITY ACCOUNTING
The approaches and models, described before in this paper including Ising model,
had its origin in physics and biology. It leads to many advantages of exploiting
more developed existing tools from physics. But in such case some implicit draw-
backs exist. Namely, physical concepts pose some frame of consideration; socio-
economical systems as it also stressed in references should have specific features
in its description and behavior. So if the investigations were start from the origi-
nal system following the physical tools then we were received new intrinsic mod-
els and knowledge. Here we describe some of such interesting new features in
M. Gaeta, G. Iovane, A. Makarenko
ISSN 1681–6048 System Research & Information Technologies, 2005, № 4 60
modeling approach. The examples of such peculiarities are: evolutionary nature of
society, evident hierarchical nature and the most important — the presence of
human individuals in the system. The last theme assume the mentality accounting
of different forms, types in different circumstances. Namely such accounting fol-
lows to surprisingly new properties of models. These properties are prospective
for understanding of such system behavior, decision-making, and management of
social systems. Moreover it may be useful for traditional humanitarian sciences:
sociology, politology, social psychology etc. Some of such considerations and
concepts had been developed and formalized since the beginning of 90-th (see
publications [55–59]). Very shortly, the approach is based on the associative
memory analogies of social systems.
5.1. System’s description. Let us consider a society consisting of 1>>N
individuals and each individual characterising by vector of state
},...,,,...,{ 11
i
M
i
k
i
k
i
i iii
ssssS + i
l
i
i
l MlMs ,...,1, =∈ , where l
iM is a set of possible
values is . There are many possibilities to compose the elements in blocks and
levels in such models. In sufficiently developed society, individuals have many
complex connections. Let us formalise this. We assume that there are connections
between the individuals i and j . Let qp
ijJ be the connection between p compo-
nents of element i and q component of element j . Thus the set
( )NjiJsQ pq
iji ,...,1,},{},{ == characterises state of society. Analysis of recent
models for media from sets of elements and bonds shows the resemblance of such
society models to neural network models.
In reality the society is an evolutionary system with dynamical changes on
time. Further for simplicity we will consider only discrete time models with mo-
ments of time: ...,,...,2,1,0 n . Following evolutionary nature of the considered
systems, it is natural to consider as input of system at the moment n the values of
parameters in n-th time moment and as output the values at next )1( +n time
moment (for ...,2,1,0=n ). In the simplest case the model takes the form of well-
known Hopfield model [60], and dynamical equations have the form:
)(sign)1( ii htS =+ ,
where ∑
≠
=
N
ij
iiji ssJh and }01,01{)(sign =−>+= WifWifW .
Surprisingly the models are familiar with models of brain activity — the
neuronets [55–59]. It is well known that Hopfield model is derived from the unc-
tional called 'energy’ of the form:
∑
≠
=
N
ji
jiij ssJE .
In Hopfield’s like neuroronet the system tends to one of the few stable states
(attractors) with minimum of ‘energy’ (energetical landscape). Many of possible
initial condition lead to a little number of such minimal ‘energy’ states called at-
tractors.
Information theory and possible mathematical descriptions of economical and social systems ...
Системні дослідження та інформаційні технології, 2005, № 4 61
5.2. Mentality accounting. The mentality accounting requires considera-
tions about the inner structures and incorporating them in global hierarchical
models. The most natural way for implementing this task is to consider as model
for internal structure also neuronet models. The simplest way consists in repre-
senting image of World in the individual’s brain or in a model as collection of
elements and bonds between elements.
5.3. Anticipatory property. Let us name the pattern of society )()1( tQ in
the section above as ‘image of real world’ in discrete moment of time t . We also
introduce the )(wish tQ — ‘desirable image of world in moment t by the first
individual’ as the set of element states and bonds which are desired by the first
individual at the moment t .
( ))}(),({)( wishwish)1(
wish tJtstQ iji= .
Then we assume that the change of elements state depends on difference be-
tween real and desirable image of the world. In the simplest variant the equations
of model have the form (for discrete time) [57, 58]:
{ }( )}{,)(,...),2(),1(),()1( JKtstststsFtsi +++=+
, Ni ,...,2,1=
with N — number of elements in system; }{J the connections matrix between
elements; K horizon of anticipation; { })(,...),(),()( 21 tstststs N=
the state of
system at moment t ; F – nonlinear function of sigmoid type (see [60] for de-
scription of traditional neural network).
The main peculiarity is essentially the anticipated nature.
5.4. Multivaluedness and choice. The main consequence of anticipation is
the possible multiplicity of solutions [57, 58]. To make a choice it need to have
the set of possibilities (that is many variants). First and the most important source
of multivaluedness is the anticipatory property as had been considered in [61] (see
also discussion in [57]) and some possible applications in the scenario in [58].
Many possibilities follow from the necessity of making choice of unique variant
in such circumstances. Some recipe, for example in using the theory of
catastrophes, or implicit choice in numerical computations of solutions, is in [61].
But in considering the system which involves the humans, the problems of choice
are very important. In such case we may speak not on the simple choice but on the
decision-making.
Recently the quantum – mechanical analogies in society properties had been
proposed as the consequences of approach. Moreover the proposed approach
allows reconsidering other new approaches: multiagent approach [62, 63], cellular
automata [64, 65], measure - valued stochastic process [66]. The advantage of
proposed methodology is in adaptively and flexibility. Prospective is the possibili-
ty for flexible description of the systems in terms of ‘patterns’. Remark that this
may be adjust with data mining and with ontology concept in knowledge acquisi-
tion.
CONCLUSIONS
In this paper we had posed first of all a comparative analysis of physics-related
models with respect to the theory of information in the context of possible
M. Gaeta, G. Iovane, A. Makarenko
ISSN 1681–6048 System Research & Information Technologies, 2005, № 4 62
mathematical descriptions for socio-economical modeling. The analysis had lead
to the choice of some new prospective class of interdisciplinary models which
exploit the concepts from physics and biology to informatics and cybernetics.
Such approach is familiar with artificial neural networks. Such models already
had been applied to some systems: sustainable development, economics, geopolit-
ics (see references in [54–59]). Adaptive possibilities of approach may be com-
bined with approaches from sections 2–4, which will be the issues for further ap-
plications and theoretical investigations. Bonabeau
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Received 22.03.2005
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INFORMATION THEORY AND POSSIBLE MATHEMATICAL DESCRIPTIONS OF ECONOMICAL AND SOCIAL SYSTEMS BASED ON REAL PHYSICAL PHENOMENA
M. Gaeta, G. Iovane, A. Makarenko
Introduction
1. Master equation approach
2. Econophysics
3. Sociophysics
4. Differential equations as models
5. Associative memory approach with mentality accounting
Conclusions
|