TOPOLOGY AND SOCIAL BEHAVIOUR OF AGENTS

O. Hudak

Faculty of Finance, Matej Bel University, // Cesta na amfiteater 1, SK-974 15
Banska Bystrica, Slovak Republic,// e-mail<EMAIL_ADDRESS>
##  1 Abstract

We call in a social group its members as agents. On a capital market the
social group is formed from those who are buying and selling shares. In
general the social group we characterise in accordance with T. Plummer / The
Psychology of Technical Analysis, rev.ed., Probus Pub.Comp., Chicago, 1993/.
We assume that individual behaviour of agents is influenced to some degree by
the need to associate with other agents and to obtain the approval of other
agents in the group. The group is characterised by a very large nonrational
and emotional element to decisions of agents. It is due to the fact that
making decisions an individual equates own needs with those of the other
agents from the group. Any two agents from the group may interact. The
interaction consists of the exchange of information and it costs some energy.
The information is well defined, and we assume that agents interact in such a
way that they give reference to the origin of the information if asked by
other agents. Thus the agent may verify obtained information. It is natural
then that there exits a subgroup of interacting agents the interaction of
which has the following property: it is nonreducible in the sense that in this
subgroup there exists the interaction of a given agent with another one, this
another one agent again interacts with another agent, and because it has no
sense to exchange the information with the first one to verify the
information, this last mentioned agent is different from the first one. This
third agent can thus always verify the information from two sources either
interacting with the first one, either he/she is interacting with the fourth
agent. In the first case we have a minimal subgroup of interacting agents,
which form a closed subroup in which every agent verifies information from two
sources - agents. In the second case the fourth agent either interacts with
the first one and verifies exchanged information, either with the second one
and verifies information. He of course can interact with the third one agent,
however to verifiy information he needs again to interact with different agent
from that interacting with which he obtained the new information. This is the
reason why he interacts either with the first one agent, either with the
second one. In the first case we have a closed subgroup of four agents in
which every agent interacts with two agents. In the second case the closed
subgroup reduces to the group of three interacting agents /last three agents
/, verify information interacting with the fifth agent. Then by the same
procedure as in previous case we may obtain nonreducibile subgroups of three,
of four and of five agents, or the fifth agent interacts with the sixth one.
And so on. Thus we obtain closed subgroups of interacting agents in which
every agent interacts with two other agents, and in which the process of
verification of information leads to closed linear structure. If any of agents
from such an subgroup exchange and thus verifies information with any third
one agent from the subroup, the structure of the subgroup changes, the
subgroup becomes reducibile to two new nonreducibile subgroups. This is the
process of differentiation of the first nonreducibile subgroup to two new
ones. This is an example of an elementary transformation between two
configurations of nonreducibile subroups. An vice versa: if the interaction
between two agents which interact with three agents in the configuration of
two nonreducibile subgroups vanishes, a new one nonreducibile subgroup creates
and information is still verifiable. When the configuration in which an agent
was interacting with three agents which were not interacting between
themselves transform to a configuration in which two of the mentioned three
agents interact, then we may observe the process of mitosis: a new
nonreducibile group appears.Thus the transformation between this two
configurations is reversibile. A cell is such a configuration of a given
number of nonreducible subgroups in which every two interacting agents belongs
to two nonreducibile subgroups /subgroups are connected in this sense / and
which is closed . Such a cell may disappear and may be created, may change
number of nonreducible subroups in a reversible way. Becasue the structure,
configuration of interactions between agents in the group, forms a macroscopic
structure, we say that it is a microreversible process any process within a
nonreducibile subgroup and within a cell. Statistical equilibrium of the whole
group is characterized by a set of different subroups of the type mentioned
above, and by a probability that such a subgroup occurs. Thus we have
probability distribution which characterizes the group. Moreover there exists
an equation of state which enables to compare different macroscopic states of
the group.The statistical equilibrium due to microreversibility is
characterised by the maximum of entropy and by the minumum of energy /costs of
information exchange/. There exist constrains, such as a fixed number V of
agents in the group, a number E of interactions within the group, a number F
of subgroups which are nonreducibile, and a number C of cells. Thus we have a
structure which is equivalent to random cellular networks. Such networks and
their evolution were described by N. Rivier /Physica 23D (1986) p. 129/. He
applied methods of statistical mechanics to study these structures. We will
use methods described by Rivier to study social behaviour of agents, mainly
the presence of topological structure of interactions between agents and its
changes, which is the most important property of the group of agents. The area
of a nonreducibile group which belongs to those nonreducibile groups which
form the cell may be formed again for example by a sum of areas of agents
characteristic areas. Note that area of the nonreducibile group may be also
some other characteristics of the group of agents depending on studied social
relations between agents. Thus we are able to study topology properties of
interactions of agents. Their social behaviour is discussed. It can be shown
that the equilibrium number of agents with which a given agent interacts is
three for a group without cellls /the group forms a single cell/.

##  2 Introduction

We call in a social group its members as agents. In general the social group
is characterised in accordance with T. Plummer [1] . We assume that individual
behaviour of agents is influenced to some degree by the need to associate with
other agents and to obtain the approval of other agents in the group. The
group is characterised by a very large nonrational and emotional element to
decisions of agents. It is due to the fact that making decisions an individual
equates own needs with those of the other agents from the group. Any two
agents from the group may interact. The interaction consists of the exchange
of information and it costs some energy. The information is well defined, and
we assume that agents interact in such a way that they give reference to the
origin of the information if asked by other agents. Thus the agent may verify
obtained information. It is natural then that there exits a subroup of
interacting agents the interaction of which has the following property: it is
nonreducible in the sense that in this subgroup there exists the interaction
of a given agent with another one, this another one agent again interacts with
another agent, and because it has no sense to exchange the information with
the first one to verify the information, this last mentioned agent is
different from the first one. This third agent can thus always verify the
information from two sources either interacting with the first one, either
he/she is interacting with the fourth agent. In the first case we have a
minimal subgroup of interacting agents, which form a closed subroup in which
every agent verifies information from two sources - agents. In the second case
the fourth agent either interacts with the first one and verifies exchanged
information, either with the second one and verifies information. He of course
can interact with the third one agent, however to verifiy information he needs
again to interact with different agent from that interacting with which he
obtained the new information. This is the reason why he interacts either with
the first one agent, either with the second one. In the first case we have a
closed subgroup of four agents in which every agent interacts with two agents.
In the second case the closed subgroup reduces to the group of three
interacting agents /last three agents /. This process can continue futher,
because the fourth agent may also verify information interacting with the
fifth agent. Then by the same procedure as in previous case we may obtain
nonreducibile subgroups of three, of four and of five agents, or the fifth
agent interacts with the sixth one. And so on. Thus we obtain closed subroups
of interacting agents in which every agent interacts with two other agents,
and in which the process of verification of information leads to closed linear
structure. If any of agents from such an subgroup exchange and thus verifies
information with any third one agent from the subroup, the structure of the
subgroup changes, the subgroup becomes reducibile to two new nonreducibile
subgroups. This is the process of differentiation of the first nonreducibile
subgroup to two new ones. This is an example of an elementary transformation
between two configurations of nonreducibile subroups. An vice versa: if the
interaction between two agents which interact with three agents in the
configuration of two nonreducibile subgroups vanishes, a new one nonreducibile
subgroup creates and information is still verifiable. When the configuration
in which an agent was interacting with three agents which were not interacting
between themselves transform to a configuration in which two of the mentioned
three agents interact, then we may observe the process of mitosis: a new
nonreducibile group appears.Thus the transformation between this two
configurations is reversibile. A cell is such a configuration of a given
number of nonreducible subroups in which every two interacting agents belongs
to two nonreducibile subgroups /subgroups are connected in this sense / and
which is closed . Such a cell may disappear and may be created, may change
number of nonreducible subroups in a reversible way. Becasue the structure,
configuration of interactions between agents in the group, forms a macroscopic
structure, we say that it is a microreversible process any process within a
nonreducibile subgroup and within a cell. Statistical equilibrium of the whole
group is characterized by a set of different subroups of the type mentioned
above, and by a probability that such a subroup occurs. Thus we have
probability distribution which characterizes the group. Moreover there exists
an equation of state which enables to compare different macroscopic states of
the group.The statistical equilibrium due to microreversibility is
characterised by the maximum of entropy and by the minumum of energy /costs of
information exchange/. We will use methods of statistical physics to study
social behaviour of agents, mainly the presence of topological structure of
interactions between agents and its changes, which is the most important
property of the group of agents. There are three empiricaly observed
dependencies of personal radius which enabled us to characterize the
quantities of cells, faces, vertices and bonds [2]. There exist constrains,
such as a fixed number V of agents in the group, a number E of interactions
within the group, a number F of subgroups which are nonreducibile, and a
number C of cells. Thus we have a structure which is equivalent to random
cellular networks. Such networks and their evolution were described by [3],
[4] N. Rivier /Physica 23D (1986) p. 129/. He applied methods of statistical
mechanics to study these structures. We will use methods described by Rivier
to study social behaviour of agents, mainly the presence of topological
structure of interactions between agents and its changes, which is the most
important property of the group of agents. The area of a nonreducibile group
which belongs to those nonreducibile groups which form the cell may be formed
again for example by a sum of areas of agents characteristic areas. Note that
area of the nonreducibile group may be also some other characteristics of the
group of agents depending on studied social relations between agents. Thus we
are able to study topology properties of interactions of agents. Their social
behaviour is discussed in. It can be shown that the equilibrium number of
agents with which a given agent interacts is three for a group without cellls
/the group forms a single cell/.

##  3 Human groupings have hierarchical structure.

Human groupings have hierarchical structure, for an introduction see [1]. The
civilisation consists of societes, societes consist of groups, and groups
consist of individuals. A social group has power to rganise individuals and se
them for its own purposes. An agent we call a member of a social group, we
assume that his/her special properties are defined.

Agent - its behaviour is influenced to some degree by the need to associate
with other agents and to obtain the approval of other agents in the group [1].
A group is chararcterized by a very large nonrational and emotional elements
in decisions of agents, and agent equates his/her needs with those of the
other agents from the group [1]. Equates here means interacts, interaction
costs some energy and consists of the exchange of information about his/her
needs, the information is well defined. Specification of agents here: agents
interact in such a way that they give reference to the origin of the
information if asked by the other agent. Thus an agent may verify obtained
information.

Nonreducible subgroup of the group of agents is a group which defined in the
following way. There exists interaction of a given agent with another one.
This another one agent interacts with another one. it has no sense for him/her
to exchange information with the first one to verify information /to obtain
approval of at least another one agent/: thus the agent interacts with and
agent diferent from the first one. This the third agent can verify the
information from two sources: either interacting with the first one, either
interacting with the fourth one. The first case leads to formation of a
nonreducibile subgroup of three agents in which every agent verifies
information from two different sources /we asuem that information is verified
if an agent verifies information from at least two different sources/. The
second case: the fourth agent interacts either with the first one, and we
ontain a nonreducibile subgroup of four agents, either this agent interacts
with the second agent, and we obtain a nonreducibile subgroup of three agents,
this process may continue further: the fourth agent may verifiy information
interacting with the first ones, the process just describes now may lead to a
nonreducibile subgroup with three, four or five agents, etc. Thus in the group
of interacting agents there exist nonreducibile subgroups of agents which are
closed as concerning exchange of an information and in which every agent
interacts with two and only two other agents, thus the structure of
interaction of agents is "linear-circular".

##  4 Nonreducibile subgroups of the group of agents may transform.

Reduction of a nonreducibile subgroup to two nonreducibile subgroups: if any
agent from a given nonreducibile subgroup interacts as concerning exchange of
an information with a third one agent from the nonreducibile subgroup. And
vice versa: if the interaction as concerning exchange of an information
between two agents in the configuration of two nonreducibile subgroups
vanishes, a new one nonreducibile subgroup appears. Mitosis is a process in
which an agent which was interacting with three agents which were not
interacting between themselves transforms to a configuration in which two of
the mentioned three agents interact, then a new nonredcibile subgroup appears,
this process is reversibile.

Cell is a configuration of a given number of nonreducibile subgroups in which
everz two interacting agents belong to two nonredcibile subgroups of a closed
subgroup formed from nonreducibile subgroups. Cella of the group of agents may
reversible transform, they may disappear, may be created, may change a number
of nonreducibile subgroups.

##  5 The structure of the group is macroscopic

The group has a given number of agents, of nonreducibile subgroups and of
cells as concerning exchange of information. Statistical equilibrium of the
group is characterized by different macroscopic structure as concerning
exchange of an information and by a probability that a nonreducible group
appears, this probability characterizes the group. Equation of state
comparison of different macroscopic states of the group. Statistical
equilibrium exist in the group due to reversibility, it is characterized by
the maximum of entropy and by the minimum of energy /costs of information
exchange/. Constraints are given by a fixed number V of agents in the group, a
number E of interactions, a number F of subgroups which are nonreducibile, a
number C of cells, note that conservation law holds [4]:

$-C+FE+V=0$ (1)

Note that structural stability exists [4]: only agents with interaction with
four agents are structuraly stable in 3d, and only agents with interaction
with three agents are structuraly stable in 2d. This structure is equivalent
to random cellular networks, such networks and their evolution were described
by N. RIVIER [3], [4] by methods statistical mechanics.

We will use these methods to study social behaviour of agents, and changes of
this behaviour, mainly the presence of the topological structure of
interactions between agents and its changes, the most important property of
the group of agents which coresponds to maximum of informational entropy.

One can define an area of nonreducibile subgroup and volume of the cell. We
asume that there is homogeneity, and no costs of information.

##  6 Personal area and social behaviour of agents of the group

The average area A(n) of an n-sided cell is [3], [4] / if A is the total area
in which group is localised/:

$A(n)=\frac{A}{F}\lambda(n-(6-\frac{1}{\lambda}))$ (2)

Asume that this area corresponds to a personal area of agents forming an
n-sised nonreducibile subgroup, due to homogeneity and equilibrium every agent
contributes $\frac{1}{3}$ personal area A(n), thus $\lambda=\frac{1}{6}$ and

$A(n)=\frac{A}{F}\frac{n}{6}$ (3)

One agent contributes $\frac{1}{3}$ of his personal area to the area A(n), if
we denote r the radius of the agent’s personal area then:

$\frac{\pi r^{2}}{3}=\frac{A}{6F}$ (4)

As we see personal radius increases as a square root with increased

total area A per a nonreducibile group. As we see personal radius decreases as
an inverse square root with increasing number of nonreducibile groups, in
towns personal area is observed to be smaller than in villages [2], this
corresponds to smaller $\frac{A}{F}$ ratio in towns than in villages according
to our equation above, the ratio $\frac{A}{F}$ is more-less constant in area A
in these cases, however it is increasing with the density of agents in the
group (town, village) and thus number of nonreducibile groups is increasing in
our equation for personal area qualitatively, personal radius is increasing
with decreasing risk which a person expects [2]: intimous - smaller radius is
0.15 m and less for intimous contacts /love, security, … /, intimous - larger
radius is 0.15 m to 0.45 m for less intimous contacts /relatives/, personal -
smaller radius is 0.45 m to 0.90 m for personal contacts with close friends
and relatives, personal - larger radius is 0.90 m to 1.20 m for personal
contacts with friend, bussines people, neoghbours. Thus we see from our
equation that F may be associated with risk: for constant area A the larger F
the smaller r and smaller acceptable risk (*). Personal radius is large for
inhabitants [2] of New Zeeland, Australia and white North-Americans, it is
middle for inhabitants of Great Britain, Switzerland, Sverige, Germany,
Austria and it is small for inhabitants of Arab countries, Japans, South-
Americans, inhabitants of countries around Middleterrenien sea /Italy, France,
Greece, … / and black North-Americans: one can say qualitatively that
(probably due to temperament) the first group is characterised by low risk
activities and by their preference, the second group is characterised by
middle risk activities, and the last group is characterised by high risk
activities. Thus first group has F lower than the second group, and second
group lower than the third group: this is consistent with the statement (*).

##  7 Conclusion

We conclude that the three mentioned empiricaly observed dependencies of
personal radius dependence on some factors enabled us to characterize the
quantity F as the quantity which characteriyes verification of information,
the smaller personal radius the larger process of verification of verbal and
non-verbal information about the other person in other to accept the risk that
this person will be closer to us as a person. We expect that this conclusion
is a general conclusion.

##  8 Acknowledement

The paper represents a part of results of the VEGA project 1/0495/03.

## References

  * [1] T. PLUMMER, THE PSYCHOLOGY OF TECHNICAL ANALYSIS, REV.ED., PROBUS PUB.COMP., CHICAGO-CAMBRIDGE, 1993
  * [2] D. LEWIS: THE SECRET LANGUAGE OF SUCCESS, CAROLL AND GRAF PUB.INC., USA, 1989
  * [3] N. RIVIER, JOURNAL DE PHYSIQUE, C9 N12 T46 (1985) 155
  * [4] N. RIVIER, PHYSICA, 23D (1986) 129