CELLULAR AUTOMATA BASED MODELING OF THE
FORMATION AND EVOLUTION OF SOCIAL NETWORKS
A Case in Dentistry
Rubens A. Zimbres
1
, Eliane P. Z. Brito
1
and Pedro P. B. de Oliveira
2
Universidade Presbiteriana Mackenzie
1
Centro de Ciências Sociais e Aplicadas /
2
Faculdade de Computação e Informática
Rua da Consolação 896, Consolação – 01302-907 São Paulo, SP – Brazil
Keywords: Cellular automaton, Dentistry, business network, social network, agent-based model, dynamics, evolution.
Abstract: The stability and evolution of networks is a research area that is not well explored. There is an inadequate
focus to the partner characteristics and to the environment influence over the evolution process. This study
analysed social network formation in its dynamic aspect through agent-based modeling, using cellular
automata. Relationships and decisions were modeled. Along the interactions the emergence of consensus in
the network could be observed and the results show that the more impulsive the individuals in a network,
the stronger will be the ties among them. Convergence of partner selection criteria could also be noticed.
Additionally, a structural hole could be shown to have a local influence on how it moves an agent away
from the network. This work waves positively towards using cellular automata in social (in the case,
business) networks modeling, in spite of their well-known limitations for these kinds of problems.
1 INTRODUCTION
The main objective of this study was to analyze the
use of cellular automata (CAs) in modeling the
dynamic process of partner selection in business
network formation in the dentistry context, starting
from a set of characteristics of potential partners; a
secondary goal was to verify the influence of
interactions among the dentists in the network
dynamics. Specifically, first we look at which
variables are involved in how dentists get together
with colleagues of distinct specialties so as to form a
social network that allows, for instance, the
redirection of a patient with special needs, beyond a
particular dentist’s expertise, to another dentist of
the network, so that the patient can be better served.
At the same we analyze how dentists make their
choices in building up a social network. Then, a CA-
based model is developed to explain some aspects of
the network creation and overall phenomenology.
Dissemination and use of information in a social
system can be compared to a complex adaptive
system with a large number of individuals that
interact with each other, in a non trivial way,
generating a visible collective behavior
(Goldenberg, Libai & Muller, 2001; Granovetter,
1976; Hegselmann & Flache, 1998; Macy & Willer,
2002; Nagpal, 1999; Tesfatsion, 2005). Agent-based
modeling (ABM) is a computational method that
allows the creation, analysis and experiments with
artificial societies composed by agents that interact
in a local and non trivial way, constituting their own
environment in an emergent fashion (Cederman,
2003; Epstein & Axtell, 1996; Macy & Willer, 2002;
Mitchell, 1994; Nagpal, 1999).
The traditional approach based on
phenomenological laws explains the combination
between previous conditions and the result of the
phenomena. In the last decade, social scientists
began to develop a different approach to the
explanation, based on causal mechanisms instead of
laws (Cederman, 2003; Epstein & Axtell, 1996;
Macy & Willer, 2002; Sawyer, 2004). With this,
models of emergent computation, such as the one
explored herein, has had an increasing role.
Macy & Willer (2002) argues that some
sociologists do not completely appreciate
computational methods and relational modeling as
tools for theoretical research. For the authors, ABM
basically differs from prior use of computation in
sociology because it considers interactions, instead
333
A. Zimbres R., P. Z. Brito E. and P. B. de Oliveira P. (2008).
CELLULAR AUTOMATA BASED MODELING OF THE FORMATION AND EVOLUTION OF SOCIAL NETWORKS - A Case in Dentistry.
In Proceedings of the Tenth International Conference on Enterprise Information Systems - AIDSS, pages 333-339
DOI: 10.5220/0001712003330339
Copyright
c
SciTePress
of simply proposing algorithms and equations to
represent behavioral processes.
ABMs or artificial societies are based upon four
premises. Agents are autonomous, interdependent,
follow simple rules, are adaptive and consider the
past (Sawyer, 2004). ABM relies on a set of
computational agents (with internal states, behavior
rules and parallel operation) and their environmental
specification; the communication among them is
made by rules, specifying a network of connectivity
that is activated through the agents’ interaction and
observing the emergent macro behaviors (Epstein &
Axtell, 1996).
The next session reviews some cellular automata
concepts, which is followed by a presentation of the
research methods employed. Subsequently, the results
obtained are discussed, and then conclusions are drawn.
2 BACKGROUND: CELLULAR
AUTOMATA
Cellular automata (CAs) are fully discrete, complex
systems that possess both a dynamic and a
computational nature. They consist of a grid-like
regular lattice of cells, and a state transition rule
(Wolfram, 2002). The cells in the lattice have an
identical pattern of local connections to other cells,
and are subjected to some boundary condition,
usually periodic. Each cell can take on one of a
discrete set of possible states, and the
neighbourhood of a cell is defined as the cell,
together with the others that are connected to it.
The state transition rule yields the next state for
each cell, as a function of its neighbourhood, and, at
each time step, all cells synchronously have their
states updated. In computational terms, a cellular
automaton is, therefore, an array of finite automata,
where the state of each automaton depends on the
state of its neighbours.
Figure 1: Rule 234 and the state transitions that define it.
For elementary cellular automata (ECA), that is,
CAs where the neighbours are located only at the
right and left of the centre cell, the size m of the
neighbourhood is usually written as m=2r+1, where
r is called the radius of the automaton. In the case of
binary-state CAs, the transition rule is given by a
state transition table, which lists each possible
neighbourhood together with its output bit, that is,
the updated value for the state of the central cell in
the neighbourhood. Figure 1 gives an example, with
rule 234 of the elementary space – the set of one-
dimensional cellular automata rules with 2 states per
cell and radius 1 – in which black cells represent
state 1, and white cells represent 0. The rule
denomination as 234 comes from the decimal
number corresponding to the binary number that is
formed from its rule table, from neighbourhood
111…1 on the left-hand side, as shown in the figure;
in fact, such a naming scheme is widely used in the
literature, for any radius, so that here we preserve it.
3 MODELING
3.1 Variables Considered
A number of concepts discussed in the literature (for
instance, Granovetter, 1973; Frenzen & Nakamoto,
1993; Khana, Gulati & Nohria, 1998; Herrnstein,
1990) led us to elaborate the hypotheses of the
present study:
H1
A structural hole in the social network has
local influence. Structural hole (Burt, 1997) is an
imperfection in an network structure, due to a lack
of information transmission to some of the agents of
the network.
H2 The existence of an individual that acts as a
structural hole in the network increases this
individual distance to the network.
H3 The greater the convergence of partner
selection criteria, the stronger will be the ties among
the individuals.
H4 The larger the distance between individual
and network rationale the larger the tendency to the
individual to move away from the network.
H5 The stronger the tie between two actors, the
greater is the impulsiveness in their decision
making.
H6 The lesser the impulsiveness of an actor, the
further away from the network the actor will be kept.
In pursuing the latter, the strength of ties among
partners was measured herein through the product of
the number of clients indicated to the partner during
a week and for how long these indications occurred
added by the number of clients received in a week
and for how long the clients have been received.
This way of measuring the construct is coherent with
Granovetter (1973).
The business scope was measured herein by
summing the amount of specialties offered, number
of employees, number of work days in the week, and
number of clients treated in a day. Additionally,
ICEIS 2008 - International Conference on Enterprise Information Systems
334
network scope was measured by the amount of
specialties involved in the relationship, in both
directions.
In respect to partner selection, firms tend to make
partner selection based on easily observable
technical criteria, ignoring or underestimating
personal compatibility criteria, important in initial
stages of network formation and stabilisation. Some
criteria were chosen to be tested in our network
modeling according to some authors, listed below
with their corresponding choice criteria:
Partner’s reputation choice was based, for
instance, on Geringer (1991). This construct
was measured following the idea of
characteristics that define the reputation, which
are: strategy quality, products/services quality,
management quality, market orientation,
innovation and financial strength.
Partner’s proximity choice was based on
Geringer (1991).
Expected partner’s quality was measured by the
technical knowledge, physical facilities and
employees appearance, equipments, attention
provided to the client, employees courtesy and
availability to helping the client (Arrègle et al.,
2003; Geringer, 1991).
Financial conditions offered by the partner to
the client were defined as financial options and
charges (Geringer, 1991).
Resources complementary choice was based on
Geringer (1991) and Hamel, Doz & Prahalad
(1989).
Impulsiveness is considered a moderating
variable, since it intervenes in the decision making
process. The variables used to measure the construct
– namely, the time spent to analyze the situation,
level of emotion in the decision, degree of risk
aversion, planning time and degree of qualitative
and quantitative analysis of the benefits generated by
the network participation – were all derived, for
instance, from Doz & Hamel (1998).
Leadership is a variable that has influence over
the network profile and probably moderates the
individuals decision making process. It was included
in the research but only the transformational
characteristics were included. It was measured
through the perception on the leadership
characteristics of the partner.
The structural hole and its local effect will be
measured evaluating the distances among the
network participants, i.e., the differences among
their partner selection characteristics along each one
of the interactions.
3.2 Data Collection
The sample was composed by dental offices located
in São Paulo city, Brazil, where, according to the
Regional Council of Dentistry database (CROSP)
there are 17,571 dentists.
The sampling process followed a snow-ball type,
i.e., starting with CROSP’s database and then using
the respondents indications. More specifically,
invitations (for questionnaire answering) were
initially sent to a randomly selected sample from the
database, composed by 2200 letters and 960 emails.
After that, the respondents were thanked by writing,
where the opportunity was used to request from
them the indication of potential new respondents.
This procedure was repeated until a useful sample
was reached, in the case 313 dentists; from these,
240 came from those directly contacted and 73 from
the respondents’ indications. Four questionnaires
were excluded due to an excess of missing values, as
they might compromise the modeling and statistical
analysis. Eighteen questionnaires had a small
amount of missing values, and were simply filled in
by the valid sample average. The option for
accepting incomplete questionnaires was
appropriate, as it prevented the respondents from
giving up the answering. Furthermore, four outliers
were identified and removed from the modeling. In
order to check whether their absence was relevant or
not, the model was also run with them.
As an instrument to data collection, the
questionnaire was hosted in a website. Also, a Likert
scale from 1 to 6 was used, so as to avoid a neutral
positioning of the respondent (Kerlinger & Lee,
2000). Before the actual research a pre-test was
carried out with 30 individuals, which helped to
improve the questionnaire, but the pre-test
respondents were not considered in the final sample.
3.3 Data Processing
After preparing the data, a factorial analysis, limited
to 10 factors and Varimax rotation, was run and the
principal components extracted.
The factors obtained from factorial analysis were
orthogonal and had a Kaiser-Meyer Olkin sample
adequacy measure of 0.587, and the result for the
Bartlett Sfericity Test of 1865.52 significant at
0.000, meaning that factorial analysis was viable.
Only variables with loads greater than 0.400 were
selected. The outcome of the factor analysis served
as variables to the modeling phase; they are listed
below, together with the attributes they comprise:
1. Leadership: partner interest in the relationship,
optimism of the partner regarding the future,
partner that searches for alternatives, individual
CELLULAR AUTOMATA BASED MODELING OF THE FORMATION AND EVOLUTION OF SOCIAL
NETWORKS: A Case in Dentistry
335
attention of the partner in the relationship and
the perception that the network increased the
business strength.
2. Quality: equipments, employees appearance,
facilities, customisation of service and
technology.
3. Reputation: ability, honesty and innovation.
4. Customer care: proximity, flexibility, financial
conditions.
5. Propensity to collaborate: network scope,
market saturation, market competition and risk
aversion.
6. Impulsiveness: indication by a strong tie,
planning horizon, emotion in the decision,
persistence and inversely related to invested
time in the decision making.
7. Network utility: calculation of benefits,
contribution equivalence, dependence and
inversely related to conflict and learning.
8. Degree of external segregation: importance
given by a weak tie indication, elitism and
hardness of the partner.
9. Decision importance: involvement and
quantification of risks prior to the partnering.
10. Decision value: egotism, complementarity of
resources, costs and risks of Dentistry
practicing.
3.4 Implementation
Once we had the 10 factors, data was classified
following the leadership factor, so that the individual
that identifies the partner the most as a leader
became the pivot, that is, the reference in the
network organisation. The remaining individuals
were placed to the right of the pivot, in decreasing
order of the value of their leadership factor, so that
the individual that identifies itself the least with the
leader ends up at the left-hand side of the pivot,
since the model has periodic boundary.
The different scales of factor values were then
transformed into a single scale, from 0 to 255, and,
for every factor, a distance measure was computed
(D
i0
) between every (i
th
) individual and the pivot,
according to the individual’s factor values at issue,
by means of the expression , ,
where X
ni
is the n
th
factor X of the i
th
-individual and
X
n0
is the n
th
factor X of the pivot.
Then, a threshold scale was used to transform the
decimal factor values into binary representation. The
threshold scale was composed of classes and their
number was defined by Sturge’s rule
C=1+3.332*Log
10
N , where N is the sample size,
namely, 305 individuals. Sturge’s rule application
led to 9 classes, defined by intervals of 28.333.
Each factor value was then converted to 9-bit-long
binary number, following the threshold scale. Each
binary number corresponding to a given factor value
was created by associating bit 1 to the position
corresponding to the interval containing the value,
and 0 to the other positions, so that the resulting
binary number would have just a single 1-bit at the
position corresponding to the interval containing the
value at issue. For example, a factor value of 43.3
would have a value 1 in the second bit and 0 in all
others. So, for each factor, nine ECAs are then
created, each one of them consisting of a 305-bit-long
binary numbers. The initial state of the ECAs were
then obtained and the state of each ECA was
independently updated. The dimension chosen for the
neighbours, one at each side, was selected to preserve
information quality. With such an approach a society
connected through strong ties has thus been created,
in accordance with Chwe (1999), who argues that
strong ties are better in the creation of common
knowledge, an essential feature to collective action.
In order to select the most appropriate rule to run
the model, a rule was chosen so as to allow more
than 10 iterations, convergence to a fixed point and
not a cyclic attractor in less than 308 iterations, and
such that its transient would not be too long. These
conditions were determined due to the
computational limitations and in order to make data
analysis possible. Rules that converged too rapidly
(in less than 10 iterations) were eliminated, as well
as the fixed point and chaotic rules. We considered
fixed point as the point in which all cellular
automata converged to a steady state.
The selection criterion also considered the
possibility of the rule admitting a rationale, in that
the rule could be translated into a coherent action
with the theory. In tune with this idea, the chosen
rule was rule 234 (displayed below), out of the 256
possible ECA rules; in addition to meeting all
requirements, it embeds the rationale that an
individual’s opinion follows the local majority, but
if the individual does not consider the specific
partner selection criteria as relevant the same way
the most influent neighbour (i.e., the one in the
leftmost position) does, that is, both of them have
value 0 to that partner selection criteria, the
individual changes its original opinion and turns it to
the same as the least influent neighbour; Fig. 1
illustrates the rationale. The change from 0 to 1
means that the individual’s opinion change and the
individual start considering the partner selection
criteria as relevant. If 0 remains 0 it means that the
individual does not change its opinion and does not
consider that partner selection criteria as relevant in
its choice.
)(
0
10
1
0 n
n
nii
XXD
=
=
ICEIS 2008 - International Conference on Enterprise Information Systems
336
The stop criterion of the ECAs associated to each
factor was determined when all corresponding 9
ECAs had reached a fixed point, or, otherwise, after
308 iterations. At that moment each column of the
lattice, i.e., individual values, was converted back to
a decimal number following the average class
intervals that contained 1-bits; more precisely, every
cell containing a 1 was given the average value of
that specific class, and then all values of that
individual were summed up. Then, the final
distances relative to the pivot were measured just
like initially.
4 RESULTS
Professionals working in small dentistry
organisations containing 3 to 5 employees answered
the research questionnaire and the major percentage
of answers (36%) came from general practitioners.
Analyzing the responses it is possible to notice
that only few variables were not highly considered
important as partner selection criteria. Among them
are strength of ties, perception of partner elitism, and
price charged by the partner. Respondents have a
medium to short term perspective in their decision
making process, and see their partners as leaders.
In the modeling phase each rule generated a
different special positioning, thus confirming
hypothesis H4
,
because according to the existing
rationale in the network, entailed from the rule, the
individual attributes and distances change.
Initial distances had great variation, in the range
from -250 to 250, which represents the individual
differences regarding different selection criteria. The
reduction of distances after rule 234 application to a
range of 0.08 shows that individuals get closer to
each other, due to tie strengthening and an increase
in the degree of similarity. There was an opinion
convergence, represented by the reduction of
distances among individuals, but at the micro-level
they did not become exactly the same. Results are
compatible with the Matching Law, since similar
individuals strengthen the tie between them and are
susceptible to constant behavior reinforcements,
therefore supporting hypothesis H3.
Final distances showed that individual 249 was
the one who modified the least its positioning,
keeping himself away from the network. This
individual was the least influenced by the network
and increased its factor values 9 fold, while the
network increased 9.3 times. Analyzing the initial
variables, we noticed that individual 249 does not
consider partner indication by a strong tie as
important, had considered more the benefits prior to
entering the network, perceives that the contribution
of its partner in the relationship is not equivalent,
and takes more time than the average to decide. So
he is a less impulsive individual, very rational and
with low cooperation propensity. He wants to
maximize its utility, perceives little value in the
decision of participating in the network and is
refractory to strong ties reinforcement. So,
participating in a network has a small weight in its
decision, thus making him a weak tie. Hence, such a
finding supports hypothesis H6.
The increase in tie strength caused an increase in
impulsiveness and a greater cooperation propensity,
because there was the same rationality in the
network, so that individuals were deciding
accordingly. Network utility increased, as well as the
outcome value and its importance, because
according to the Matching Law, individuals are
subject to behaviour reinforcements. These facts
confirm hypothesis H5.
Initial distances had great variation, in the range
from -250 to 250, which represents the individual
differences regarding different selection criteria. The
reduction of distances after rule 234 application to a
range of 0.08 shows that individuals get closer to
each other due to tie strengthening and an increase in
the degree of similarity. that is, everyone perceives
that artificial network formation increases each
business strength. The reputation factor was the only
one for which there was a convergence in two
clusters. The smaller one was composed in its
extremity by individuals 33 e 173 and its size was
85% smaller than the other. In order to know who
was the individual that attracted the others to the
second cluster, both were analysed.
Individual 33 does not see its partner as a leader,
differently from individual 173. So, individual 33 is
farther from the pivot. The larger discrepancy in the
factor values, considering the rest of the network,
happened for individual 33, suggesting that he is the
one that changes its opinion the most. We concluded
that he is the source of influence that moves away
the adjacent individuals, up to individual 173. Those
adjacent to 33 and 173 do not receive the
information because 33 alters the quality of the
information transported. However, individuals from
173 and beyond converge to zero because the CA
has periodic boundary, that is, the pivot interacts
with distant network layers. The periodic boundary
shows the importance of the manager to be in
contact with employees in order to spread its
influence to the different layers of the organisation
and of the network.
CELLULAR AUTOMATA BASED MODELING OF THE FORMATION AND EVOLUTION OF SOCIAL
NETWORKS: A Case in Dentistry
337
In order to verify the influence of outliers that
were removed from the model, we ran the model
with them. The only noticed alteration was the
increase in time convergence, that turned from 199
to 288 iterations. This result is coherent, since the
total number of individuals is much larger relatively
to the number of outliers, and as consequence the
latter were not capable of altering the overall system
configuration.
The insertion of a structural hole using rule 204 in
the 150
th
individual, that is, an individual that did not
change its opinions along the interactions, moved
away individuals from the 45
th
to the 150
th
, who
formed a cluster. This confirms that the information
flow comes from the individuals far from the pivot
influence. This happened because the 150
th
individual
interrupted information flow. However, information
flow is compensated by transitivity. That is the reason
why there was only a local influence, therefore
supporting hypotheses H1
and H2.
Figure 2: Quality factor CA evolution for the first CA.
Initial condition is at the top, and time flows downward.
At iteration 26 there was an exponential
threshold from where a characteristic convergence
could be observed. In order to identify it properly we
looked at the CA global state at that iteration. Fig 2
shows the first 70 iterations of the rule temporal
evolution. The threshold is showed by the white
circle at the tip of the next-to-last triangle. This
threshold happens at the 26
th
iteration and
represents the moment where the network is taken
by the “rebels” (black areas), individuals that
propagate their influence and represent the majority.
It can be noticed that individuals far from the pivot
join the network later (white triangles on the right
whose size is bigger than the ones on the left).
Iteration 26 determines the network threshold, where
the majority of “rebels” take the network, a situation
that is usually found at innovation adoption events.
After 199 iterations, all the network is in consensus
(black area), thus meaning that all network starts to
consider quality as a relevant factor in partner
selection.
5 CONCLUSIONS
The selected area of the sample used herein may not
represent the general condition of the Dentistry
universe used. Therefore, conclusions may not be
general to all firms of the dental sector in São Paulo
city. Clippings and options done to measure the
constructs chosen as important to formation and
development of networks were selected following
the authors’ objectives and do not include all partner
selection decision criteria found in the literature.
Nevertheless, many interesting observations could
still be made, confirming the validity and strength of
the model
Specifically, results show that a structural hole in
the network increased the distance of adjacent
individuals and its influence decreases in distant
neighbours. Different distances also came from
different network rationale, which means that
environment influence prevailed over individual
decision criteria. This happened because individuals
with different decision criteria were attracted to the
same position through the use of the same reasoning,
suggesting that the environment exerts strong pressure
over the individual and that decisions are contingent.
Along the interactions there was a convergence of
opinions in decision criteria, which represents the
consensus being formed among dentists. The
distances decreased for all individuals, meaning that
decision criteria similarity increased along the
interactions. The increase of similarity among
decision criteria caused a greater propensity to
cooperate. There was an increase in tie strength
among network participants, what caused an
increase in their impulsiveness, compatible with the
Matching Law.
Macy & Willer (2002) observe that many agent-
based models treat social forms as behavioral
interactions, not varying the topology and actor
identities. The model explores this gap in the literature
and considers topology, once it has special relevance in
the modeling results (Sipper, 2004). The model also
considers structural attributes of the agents, the same
way Tesfatsion (2005) did, but in another
circumstances. In tune with latter, actor identity is
varied herein, insofar as the factors change along time,
since individuals do interact with their neighbours and
may acquire opinions that initially might have belonged
to another. Topology varied according to discrepancies
of the individuals’ characteristics. and was measured
by the distance among actors and a chosen individual
(the so called pivot), in the same way as Burt (1976)
and Gulati (1995).
All in all, this study contributes to theory insofar
as it allowed to verify the potential use of cellular
automata to understand formation and evolution of
social networks, in tune with the enormous attention
the topic has received recently, and in spite of well-
known limitations of cellular automata use in these
kinds of problems. Emergent phenomena was
observed, and there were environment and rationale
influences in network configuration. This fact
suggests a dynamic partner selection decision
ICEIS 2008 - International Conference on Enterprise Information Systems
338
criteria classification and allows to understand the
formation of weak ties as well as the emergence of
consensus in the network. This study also
contributes to practice once it allows to understand
the knowledge management in the network and the
direction of information transmission.
ACKNOWLEDGEMENTS
RAZ thanks CAPES, for financial support, and E.
Macedo for helpful discussions. PPBO is thankful
for research grants provided by Wolfram Research
(Mathematica Academic Grant No 1149), FAPESP
(Proc. 05/04696-3), and MackPesquisa – Fundo
Mackenzie de Pesquisa (Edital 2007). EPZB thanks
MackPesquisa and CNPq for financial support.
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