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Immanent haziness of social system: mathematical de-
scription and control.
Thar A. Miklashevich
Laboratory of Ethnical Studying, Institute of Mathematics and Cybernetics and
Department of Theoretical Mechanics, National Technical University
Minsk, Belarus
65 Skaryny Av., Minsk, 220027, Belarus.
Tel.+375(17)2399-252, fax +375 (17) 247-24-30
e-mail: miklashevich@yahoo.com
The hierarchical multilevel system theory is used to the social system evolution
description. The development of the social system as an evolution of systems with inside
non-determination is studied on the base of non-standard analysis. The evolution of
social systems is represented as a movement in the space of states. "Human" states
space is the reduction of full space which is happening by the filtration. The global and
local optimal trajectories from point of view of mathematics are determined.
Key words: system, hazy, information, control, filter, hierarchy
Introduction: Variables set and hierarchical hazy.
The classical ‘‘descriptive" approach does not satisfy the main idea of quantita-
tive description of social systems. This approach only analyzes the past situation. But
now we need another description, which can give us the prediction, the control parame-
ters and characteristic of the critical points [1].
Nowadays, the multilevel hierarchical system theory (MHST) representation of
our world is very popular [2, 3]. The schematic images of three - level hierarchical sys-
tem presented on the fig.1. This representation contains the most important elements of
the MHST. They are the subsystems, situated on the same level, the higher level system
(sway), the control influence (solid lines), information exchange (dashed lines). Taking
into account the investigated problem, the levels have the different meaning. The three
level hierarchical system is interesting because in the first approximation this kind of
systems can represent real multilevel system. The lower level, external for our system,
acts on the higher level (on our system) and change his structure (elementary unit). The
same our level 1 “accumulate” some detail of activity of whole system. But in any case
the haziness is presented in the our system (the gray band on the level 1, 2) and have
influence on the our level. From the point of view of the general system theory the liv-
ing systems are the coordinator of a physical and a chemical systems and simultane-
ously are controlled by the intelligent
HAZY
Figure 1. Representation of hierarchical systems. The transition between subsystems of
one level generated by the operator of multiplication p", between subsystems of differ-
ent levels generated by the operator of composition p*.
systems. The intelligent systems are the systems aware of the own activity. The evolu-
tion of the system situated on any hierarchical level can be investigated as a movement
along the trajectory in the specific generalized space of states [4, 5].
MHST states that the problem of social system description can be reduced to de-
scription of interactions between subsystems and levels. Though a problem so formu-
lated is most general, it has been shown that these interactions can be adequately de-
scribed from cybernetic standpoint. However, a major difference from classical cyber-
netics is observed in this case: It is essential to introduce freedom of choice for elements
of the social system (SS).
As atule most of social systems are systems with uncertainty.
Definition: social system
In the widen meaning under the phrase "social system" we imply the objects
which are connected with the collective actions of living organism, objects, which do
not exist out of borders of the activity of the collective. «
Our aim is researching more narrow social systems and systems connected with
the activity of the human society.
Social systems which are connected with acts of human form only the part of
possible social systems, but the complicates one. The main difficulties of forecasting
and creation the adequate models of social systems connected with the acts of people
are raised by the next:
A.) This social systems are systems with uncertainty because the units of social
systems (people) have mind and freedom of will.
B.) Systems are non-determined and the onset of chaos regimes or regimes of
self- organization in them is possible.
Our interpretation of the social system is rather simple but it helps to widen all
the diversity of manifestation the activity of the living organism - from self-organization
of structures in communities of microorganisms to chaos processes in economics. By
the mathematical point of view each system which satisfies the determination of social
system generates the specific generalized space of states [6]. This full space is denoted
A. We believe that this space is continuous because the set of attitude is full in the
closed region of possible meaning. Mapping the space A‘ on the space which corre-
sponds the description of "human" social system is the limited space H® which appears
by the filtration in mean of non-standard analysis [7].
The presence of such "freedom of choice" is equivalent to the presence of inde-
terminate behavior within the system, or hierarchical haze in the system description.
Likewise objects have been intensively researched in all branches of science lately.
Physics regards dynamic systems (DS), that is, systems whose state depends on time, to
be an instance of such objects. In general case SS is not equal to DS. In first approxima-
tion the primitive tribe in the stable natural condition has the stable existence and can be
interpreted as a non-dynamical SS because the averaging characteristics of the society
have any dynamics in really observed time. (The good example at present is the small
tribe in Amazon’s river basin). But in global this tribe is the SS with slowly dynamics.
The contacts with extemal systems can change dynamic of SS evolution.
Our deliberate activity on this model is thought to consist in a series of logical
operations directed to choose an appropriate state function and transition functions in
SS regarded as DS. The number of such operations is, in principle, infinite provided we
admit of the existence of free will. We suppose that indeterminate dynamics of an SS
resulting from free choice of state and transition functions can be described in terms of a
DS theory with nonstandard time. Such structured are mathematically considered using
nonstandard analysis. So, choice in SS is viewed as "standardization procedure" for cor-
responding DS objects.
There is a term regarded as a key factor by nonstandard analysis, called a monad
[8]. Besides, any finite number, being an SS characteristic, has a corresponding part of
the monad located at the point R on the number axis. Phase space can be introduced in a
DS, where a phase point mirrors each element (state) of the system. In this case inde-
termination of an SS is characterized as a halo n(G ), that is, we have an area of possi-
ble states for a system instead of a completely deterministic description. This corre-
sponds to the impossibility of sociological parameters to be determined exactly, with
one exception for those exact by nature, like, say, the exact number of voters.
Any SS is both a sway for its subservient system and a subservient system for its
own sway. As an SS situated on a certain level evolves, it attempts to gain knowledge of
the laws of its sway and "to hide in the hierarchical haze" its influence on the levels that
nomnally control it. Meanwhile, any interaction between systems includes exchange of
partial information about the structure and functioning of the interacting systems. If this
information increases the system's "understanding", the hierarchical haze diminishes
and the system thus grows more competent. If, in turn, the competence increases suffi-
ciently, the system can increase its rank and move to a new structural level.
In terms of the above mentioned concepts, the social structure of countries with
democratic structures, transparent legislation and determined boundaries for all sub-
structures' activities is intuitively understood to function more efficiently. Mathemati-
cally speaking, the halo of the model occupies a smaller area in the phase space, so the
phase volume of the hierarchical haze is small in this case, too.
The social systems strive to correct its evolution to obtain the optimal evolution
trajectory because they have enough information about the sway activity. This amend-
ments are made basing as on the main information, which presents in the system, so on
the base of the anticipatory information. In the formalism of hierarchical mathematics
[8] the prognostication is possible in case the system situated on the hierarchical level é
receives the information (knowledge) about the level 4, where 4 is the arising time
(level). The classical example of system with the corrections and the anticipations is the
business corporation. Without the activity of managers (sway for the system) the corpo-
ration can’t have a success. The effectively of the corporation depends on the successful
anticipating.
The reception of such information relatively to any level is possible because any
system contains the hierarchical hazy in its own structure [9, 10]. This hierarchical hazy
is closely connected with the indeterminacy of social system. Any system which is situ-
ated on any level translates its image to the hierarchical hazy too and his corresponding
interpretation (the reception of the mathematical expression) causes the collapse of the
hazy and induces the representation of knowledge.
Il. Space of states and hierarchical hazy.
The main task in mathematical description DS at present moment is choosing
the corresponding space of states for the concrete systems. This choosing is limited by
the set of actual determining hierarchical variables. The full set of hierarchical variables
is well known [9, 10] but the transition to concrete system demands the definition of the
filter (in meaning of non-standard analysis). This filter must cut the subspace of realiza-
tion of present system from the generalized hierarchical space. It is possible to write this
filter as [10]
H® =A%/X,
where H" is the limited space, connected with the activity of studied system, A’ is the
full space of states for social system on the present level, ¥ is the filter of transforma-
tion degree (N-n). The exact structure of & depends on our understanding the distin-
guishing differences between concrete systems and unit elements of present level. In
such case the social system is determined not only by the set of variables but by the cor-
responding filter, too. The halo of the unit structure on the present level is organized
because the homogeneous system (as an example, the corporation) is situated simulta-
neously on the same level but have different structure and sway (figure 2). There gray
disks are represent the corporation image in the phase space and intersection (black) ge-
Figure2. Origin of halo in SS.
nerated the monad (the exact value) of corporation characteristics. The monad can be
interpreted as a common elements of different corporations structures. The success or
un-success of corporation activity depend on the system sway tactics. Difference of tac-
tics is realized through halo because the different firms has a distinguishes halo.
Definition: exception filter
In the case N=n the filter & is the exception filter. This filter cuts down from
the full set of the variables only the characteristics of the Homo.
The process of optimization of SS development must be realized on all hierar-
chical level, as an example in the culture, in the social systems, on the level of chemical
and physical processes because the filtration is acting on the different structure levels.
For example, the principle of maximum effectively of investment can be taken as a fil-
ter, which optimizes all the acts and the system development on the business sublevel.
But this filter is not applicable for the culture, because this filter cuts only some impor-
tant variables from A‘ and we receive Hines =H dime. It is a mistake because a lot of
variables are not actualized and power of haziness for the space H ¢ijure iS very high and
system is undetermined. The choice of the filter depends on the sway for any hierarchi-
cal system. In the simplest case of two-levels system appearing of the registrar (it is the
simplest variant of coordinator) corresponds with the knowledge creation. The knowl-
edge creation differs from the information reception and can be compared with an ele-
mentary act of self-organization [10].
Ill. Some mathematical background
Let’s investigate the interaction and influence of two business systems or busi-
ness and other social organizations situated simultaneously on the same level. This in-
teraction can be realized only through sway level. For this social systems the sway is
acting on the knowledge level [2, 3, 5, 9].
Let's introduce the coefficient of receptivity of the sway influence
Tl
a =o (1)
I, is the amount of control information production of the sway, I, is the amount of in-
formation accepted by system. Under other equal condition let
lim I, =const =1°, (2)
A>B
but the I, is the essential non-constant,
1, =I, (VED). (3)
VED is the assembly of averaging data which characterised the person belong-
ing to the social system.
The full VED set of variables depends on sorts of social system. All elements
are interacting according to the law. As we note the main problem is the correct choos-
ing the space of states. The basis vectors of this space must be the main conditions
which characterise studied social groups. The full description of the set of variables is
the object of next research.
The general task of the sway control is
lim R=max. (4)
A+B
Taking into account expression (1, 2) this gives
lim I, =max. (7)
A>B
According to [5,9] the total process of interaction between business systems and
social systems through knowledge level has the form
Ne *p ep? , (8)
Xp is the multiplying act of original state, tp is the uniting act of connection of the
ordinary units and creation of the new sway, © is the symbol of operation between two
acts. With (1, 2, 8) for the full coefficient of receptivity we will obtain
0 0
ig®_ i Gl 1 i Ty Ue
=> 3) => +6. (9)
i
ant = 2 Holl He 27,1,
j#k j#k
Equation (9) written in generalised case when i systems are situated on the same
level. The optimal strategy of the knowledge management available from equations (7),
(9). This management would be realised by the controlling of the R value.
V. Conclusion: Variables set and hierarchical hazy.
The hierarchical hazy is out of the boundary of structure of system (Symbol ? in
equation of hierarchical mathematics [2, 3, 8]). We can not know the exact law of the
hazy, but only some influence of this higher strata to our lower strata can be studied.
But hazy problem is important only for some of known social systems. As an example,
at this moment we do not have full set of variables and the law of the ethnos. In this
case the ethnos as object belongs to the relatively higher strata, i.e. to hierarchical hazy
and we understand only some of its aspects. As the system of the highest strata ethnos
also includes genetic, social, and business substrata and it is the co-ordinator (sway) of
this strata. The whole ethnos is the complex system, which includes all lower strata (all
pyramid) but have more general characteristics. Because the existence of the sway de-
pends on the existence of the lower strata that is why hierarchical hazy is condensed
near the breaking point of the evolution lower strata. (The sociological example is the
war that is accenting the national spirit).
The full “hierarchical portrait” for the most part of the other systems now can be
received. Thus influence and direction of development can be optimised. In our change-
able world only correct understanding of the main principles of system evolution allows
to win in the development and rivalry.
The determination of full set of the variables and the mathematical expression of
actual system and coordinator could help to create the base for the mathematical repre-
sentation of higher strata of hierarchical model of the world. The actuality of the bio-
logical level studies is connected with direct link between the consciousness processes
and the R. Feynman’s reversible quantum mechanical computer functioning [10]. Evo-
lution of biological systems in such approach tums out to be an evolution of registers
and programs contained in living substances.
The received result could make clearer the determinacy - indeterminacy transi-
tion rules for the living and social systems. Besides this the set and the structure of ac-
tual variables of the living and social systems can be determined. The collection of re-
ceived variables could help to build the filter corresponding level for the full space of
states.
Acknowledgement
I am specially grateful to my colleague prof. S. Novikava and prof. V. Barkaline
from the Institute of Mathematics and Cybernetics for discussions on this theme and P.
Buka for a plenty important correction.
References
1. Chestnut, H. (1995) Improving International Stability and Maintaining
Peace. Preprints of the IFAC Conference on Supplementary Ways for Improving In-
ternational Stability, SWIIS'95, Vienna, Austria, 1995, p.11-14.
2. Novikava, S., et al. (1993) Hierarchical Multilevel Systems in Aed Realiza-
tion. Proceedings of 9th International Conference on Mathematical and Computer
Modelling, ICMCM'93, p.71, Berkeley, USA.
3. Novikava, S., Gancharova S., Zhybul A., et al. (1997) The Statute of Hierar-
chical Mathematics and Its Cybernatical Maintenance. Preprints of Eleventh Interna-
tional Conference on Mathematical and Computer Modelling and Scientific Computing.
Washington, DC, USA. 1997. p.149.
4. Gemet Dieter (2000), Towards a closed description of observation proc-
esses. BioSystems 54, 165-180, (2000).
5. Miklachevich, I. (2000) Modeling and optimal control in ethnosocial sys-
tems. 18" Int. System Dynamics Conference “ Sustainability in the Third Millennium”
Bergen, Norway, August 6-10.
6. Weidlich, W (1991) Physics and social science - the approach of synergetics
Phys. Rep., 204(1), pp.1-166.
7, Albeverio (1986) Albeverio S., Fenstad J.E., Hoegh-Krohn R., and
Lindstrom T. Nonstandard Methods in Stochastic Analysis and Mathematical Physics.
Orlando/San Diego, Academic Press, 1986.
8. Novikava, S., Ananich, G. and Miatliuk, K. (1990) The Structure and the
Dynamics of Information in Design Systems. Proceedings of 7th Conference on Engi-
neering Design, ICED'90, Dubrovnik, 2, pp.946-953,.
9. Ray (1994) Ray T.S. Evolution, complexity, entropy and artifical reality.
Physica D 75, 239-263.
10. Miklachevich, I. (2002) Mathematical representation of social systems.
Uncertainty and optimization of social system evolution. Submitted to Open System
and Information Dynamic
11. Barkaline (2002), Barkaline V.V.and Miklashevich I.A. Knowledge Mani-
festation in Biological Systems Dynamics International J ournal of Computing Antici-
patory Systems, to be published (2002).
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