CELLULAR AUTOMATA BASED MODELING OF THE

FORMATION AND EVOLUTION OF SOCIAL NETWORKS

A Case in Dentistry

Rubens A. Zimbres

, Eliane P. Z. Brito

and Pedro P. B. de Oliveira

Universidade Presbiteriana Mackenzie

Centro de Ciências Sociais e Aplicadas /

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

 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:

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

) between every (i

) individual and the pivot,

according to the individual’s factor values at issue,

by means of the expression , ,

where X

is the n

factor X of the i

-individual and

is the n

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

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 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

individual, that is, an individual that did not

change its opinions along the interactions, moved

away individuals from the 45

to the 150

, who

formed a cluster. This confirms that the information

flow comes from the individuals far from the pivot

influence. This happened because the 150

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

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.

REFERENCES

Albaum, G., 1967. Information flow and decentralised

decision making in marketing. California

Management Review, 9(4), 59-70.

Aalchian, A.A. & Demsetz, H., 1972. Production,

information costs, and economic organisation.

American Economic Review, 62.

Arrègle, J.L., Dacin, M.T., Hitt, M.A. & Borza, A., 2003.

Les modèles de sèlection des partenaires dans le cadre

d’une alliance internationale: perspectives de France et

d’Europe centrale. Management International, 7(2).

Axelrod, R., 1980. More effective choice in the Prisoner’s

Dilemma. Journal of Conflict Resolution, 24(3).

Burt, R.S., 1997. The contingent value of social capital.

Administrative Science Quarterly, 42, 339-365.

Burt, R.S., 1976. Positions in networks. Social forces,

55(1).

Carlson, W.L. & Thorne, B., 1997. Applied statistical

methods for business, economics and the social

sciences. Prentice Hall, New Jersey.

Cederman, L.E., 2003. Computational models of social

forms: Advancing generative macro theory. Paper

prepared for presentation at the 8

Annual

Methodology Meeting of the American Sociology

Association, University of Washington, Seattle.

Chwe, M.S., 1999. Structure and strategy in collective

action. The American Journal of Sociology, 105(1).

Crouse, H.J., 1991. The power of partnerships. Journal of

Business Strategy, 12(6).

Doz, Y.L. & Hamel, G., 1998. Alliance advantage: the art

of creating value through partnering. Harvard

Business School Press, Boston.

Epstein, J.M. & Axtell, R., 1996. Growing artificial

societies: Social science from the bottom up. MIT

Press, Cambridge.

Frenzen, J. & Nakamoto, K., 1993. Structure, cooperation,

and the flow of market information. Journal of

Consumer Research, 20, 360-375.

Geringer, J.M., 1991. Strategic determinants of partner

selection criteria in international joint ventures.

Journal of International Business Studies, 22(1).

Goldenberg, J., Libai, B. & Muller, E., 2001. Talk of the

network: A complex systems look at the underlying

process of word-of-mouth. Marketing Letters, 12(3).

Granovetter, M.S., 1973. The strength of weak ties.

American Journal of Sociology, 78.

Gulati, R., 1998. Alliances and networks. Strategic

Management Journal. 19(4).

Hamel, G., Doz, Y.L. & Prahalad, C.K., 1989. Collaborate

with your competitors - and win. Harvard Business

Review, 67(1).

Hegselmann, R. & Flache, A., 2004. Understanding

complex social dynamics: A plea for cellular automata

based modeling. Journal of Artificial Societies and

Social Simulation, 1(3).

Herrnstein, R.J., 1990. Behavior, reinforcement and

utility. Psychological Science, 1(4).

Kelly, M.J., Schaan, J.L. & Joncas, H., 2002. Managing

alliance relationships: Key challenges in the early

stages of collaboration. R&D Management, 32(1).

Kerlinger, F.N., Lee, H.B., 2000. Foundations of

behavioral research. 4th ed. California: Thomson

Learning.

Khanna, T., Gulati, R. & Nohria, N., 1998. The dynamics

of learning alliances: Competition, cooperation, and

relative scope. Strategic Management Journal, 19(3).

Knight, L.A., 2000. Learning to collaborate: A study of

individual and organisational learning, and

interorganisational relationships. Journal of Strategic

Marketing, 8.

Macy, M.W. & Willer, R., 2002. From factors to actors:

Computational sociology and agent-based modeling.

Annual Review of Sociology, 28.

Mitchell, M., Crutchfield, J.P. & Hraber, P.T., 1994.

Dynamics, Computation, and the 'Edge Of Chaos': A

Re-Examination. In Complexity: Metaphors, Models,

and Reality, Santa Fe Institute Studies in the Sciences

of Complexity Proceedings.

Nagpal, R., 1999. Organizing a global coordinate system

from local information on an Amorphous Computer.

MIT Artificial Intelligence Laboratory. A.I. Memo n.

1666.

Sawyer, R.K., 2004. Social explanation and computational

simulation. Philosophical Explorations, 7(3).

Schelling, T., 1978. Micromotives and Macrobehaviour.

New York, W.N. Norton & Company.

Sipper, M., 2004. Evolution of parallel cellular machines:

The cellular programming approach. Heidelberg,

Springer.

http://www.cs.bgu.ac.il/~sipper/papabs/epcm.pdf.

Tesfatsion, L., 2005. Agent-based computational

economics: A constructive approach to economic

theory. Forthcoming in Judd, K.L. Tesfatsion, L.

Handbook of Computational Economics. North-

Holland.

Wolfram, S., 2002. A new kind of science. Canada,

Wolfram Media Inc.

CELLULAR AUTOMATA BASED MODELING OF THE FORMATION AND EVOLUTION OF SOCIAL

NETWORKS: A Case in Dentistry

339