Is Prediction Possible? Chaotic Behavior of Multiple Equilibria Regulation Model in Cellular Automata Topology
    IOANNIS D. KATERELOS AND ANDREAS G. KOULOURIS
    Psychology Department, Panteion University, 17671, Athens, Greece
    
    Received April 5, 2004; revised June 10 and September 4, 2004; accepted September 4, 2004
    
    In this article, we present the Multiple Equilibria Regulation (MER) Model in cellular automata topology. As argued in previous explorations of the model, for certain parameter values, the behavior of the system exhibits transient chaos (namely, the system is unpredictable but ends in a final steady state). In order to approach empirical reality, we introduce a cellular automata topology. Examining the outcome of the simulations leads us to conclude that for certain parameter values tested, the system yields chaotic behavior. Thus, cellular automata contribution has proven crucial, because the introduced topology converts the behavior of the system from transient chaos to “pure” chaos, i.e., the system is not only unpredictable on the long run but, in addition, it will never rest in a final steady state. According to these findings, authors argue the theoretical hypothesis that the urge for “prediction” in social sciences should be reconsidered in terms of “predictability horizon”. © 2004 Wiley Periodicals, Inc. Complexity 10: 23–36, 2004 Key Words: MER model; chaos; simulation; cellular automata; prediction; opinion dynamics; predictability horizon
    
    1. INTRODUCTION
    ositivism has exercised enormous influence over social sciences’ understanding of theirs explanatory task [1, 2]. According to this point of view, prediction is only possible if the appropriate predictors can be determined. Following this rationale, prediction error depends on the accuracy with which the predictors and the predicted be-
    
    P
    
    Correspondence to: Ioannis D. Katerelos, E-mail: iokat@ panteoin.gr
    
    havior are measured. Hence, our ability to predict future events should depend almost entirely on measurement error. However, socio-behavioral sciences seem to have certain limitations in their capacity to accumulate knowledge in the traditional sense [3, 4]. Patterns of human conduct are subject to continuous alteration across time. Therefore, many complex environmental systems are not as stable as linear statistical models might suggest. On the other hand, the ability to predict future events and to react accordingly is an important survival skill of humans. In many real life situa-
    
    © 2004 Wiley Periodicals, Inc., Vol. 10, No. 1 DOI 10.1002/cplx.20052
    
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    tions such as social interactions, the anticipation of our own and others’ behavior is vital.
    
    2. PREDICTION IN SOCIAL SCIENCES: WHENCE AND WHITHER
    No doubt, social sciences were meant to be predictive sciences. This was a key component of their constitution as sciences, in contrast to religious mythology and metaphysics [5]. According to Wright Mills, prediction had a preeminent position in the social sciences even as late as in the 1950s. He does not hesitate to claim that “the purpose of social science is the prediction and control of human behaviour” [6]. The positivist vision of social sciences is now characterised as a rather “naıve” project: we all know that ¨ there are profound problems about prediction (for example, see [7, 8]). Although prediction should not be seen as something confined to the caste of scientists,1 as Giddens [10, p. 248] claims, “orthodox” social scientists had an “oversimplified revelatory model of social science” in which they produced startling insights that supposedly left common people in awe. However, the debate about prediction in social sciences is, more or less, based on their perceived “utility” in society2: if they cannot predict, what are they good for? On the other hand, if they can make valuable predictions, social sciences can become some kind of a common foreteller, useful for individuals as well as for societies. Let us not forget that during the 1950s and 1960s, psychology as well as all other social sciences was perceived like “intimidating magic.” It was believed to offer explanation for human behavior in terms of predicting and controlling even in an “unconscious” way.3 What is demanded once again is unmistakably a “renovated” positivistic conception of social sciences. Implicit in this conception is, always, the assumption that “scientific” knowledge of social conditions can contribute, positively, to interventions that assist the course of human development. It is therefore a conception of social science that entails a particular relationship between the acquisition of knowl-
    
    edge and its potential use. Hence Comte’s famous pronouncement that “from science comes prevision and from prevision comes control” can be argued in the sense that the outcome of social sciences had to be, in the best case, a political argument. Hence, the central task of social sciences is to “discover” the knowledge required by politicians in order to enact the necessary “rational” interventions. However, the fact that social science and policy are not simply or contingently related can be shown by looking at the nature of the relationship between causal explanation and prediction. According to Fay [12], causal explanation necessarily implies prediction and, by extension, the desire to control, because they share a “structural identity.” For Mills [6], accepting prediction as the central element for social sciences coincides with an anti-democratic (“bureaucratic”) ethos. In most cases, scientific innovation would be mutilated in a modern “cline of Procroustes” where a “grand theory with predictive power” should grant ideological legitimacy to political choices while abstracted empiricism supplies them with technology of social control [13]. Even so, recognizing the limits of prediction and abandoning prediction as the “grand project” does not mean that no predictions should be attempted [5]. Katerelos and Koulouris [14] recently presented a new model of opinion dynamics called the Multiple Equilibria Regulation (MER) Model, which seems to correspond better in the aforementioned framework: under certain parameter values, the system becomes unpredictable. In the following paragraph, we give a brief outline of the MER Model as can be found in [14] and define a cellular automata topology (CAT) in order to examine the concept of “prediction” in a chaotic system that has the ambition to represent social reality in a more realistic scope.
    
    3. MULTIPLE EQUILIBRIA REGULATION MODEL IN CAT 3.1. The Use of Topology in Social Simulations Modeling
    One weakness4 of MER model is that every agent is potentially in position to communicate with all other agents if and only if, their opinions differ less or equal than the Bound of Confidence, . This functional postulate assumes that there is not any geographical or any other kind of obstacle excluding the magnitude of the Bound of Confidence. Nevertheless, in real life, two (or more) individuals may never communicate just because they will never meet each other. Consequently, in order to approach empirical reality, we should consider first a topology of agents and then take into account their opinions. A suitable tool, according to our view, is cellular automata (CA). The main feature of CA is locality. This means that
    
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    In some cases, naive subjects can predict chaotic number series better than linear statistical models [9]. 2 Thus, the concept of “public opinion,” which is unstable in such manner, pushed many experts to accuse it of being useless, because of its instability and lack of prediction: as Converse [11, p. 77] once put it, public opinion is “impalpable,” “amorphous,” and “mercurial.” 3 The subliminal influence (advertisement) was recognized like a threat to common wealth (see rigorous legislation). A dreadful perspective indeed: someone can influence me to do things without even knowing it. Nevertheless, the experiments conducted under this title were strongly questioned in terms of replicability.
    
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    Inherited by the Bounded Confidence Model [15].
    
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    FIGURE 1
    
    3.3 A Brief Description of MER Model in CAT
    Suppose that there is a social issue that is at stake and a double opinion formation process takes place: this means that every agent holds two opinions relevant to the given issue. Thus, all agents are assigned randomly with two numbers6 that belong to the interval [0,1]. The first number denotes the agent’s opinion 1 and the second the agent’s opinion 2. Each agent, for instance agent (4,3), changes his/her former opinion 1 and replaces it with the average of the opinion 1 of the agents who fulfill two criteria. First,7 they lie on the agent’s Moore Neighborhood 3 3, as can be seen in Figure 1. This means that agent (4,3) is aware only of the opinions of these 9 agents (included him/herself). Second, their opinion 1 should differ than agent’s (4,3) opinion 1 less than the Bound of Confidence . The same is followed for opinion 2 and the whole process takes place simultaneously for all agents, i.e., they all update their opinions synchronously. Up to this point, all agents try to reach a social equilibrium, alias to aggregate via a social compliance procedure. Nevertheless, these new opinions are temporal because a structure is defined, which regulates internally, regarding each agent, opinion 1 and 2. In this simplified form of two opinions, opinion 1 goes the other direction of opinion 2 (do not “go together,” [18]). Thus, each agent, pushed by a strong motive concerning internal equilibrium, needs to look inside him/herself and see if both these changes are consistent. Because opinion 1 and 2 are opposing each other (by definition), an agent cannot stand to change synchronously both of them toward the same direction. By focusing on the opinion which suf-
    
    A Cellular automaton 10 10 and three examples of 3 neighborhoods (in green and yellow color).
    
    3 Moore
    
    all interactions take place only within well-defined spatial neighborhoods. Low dimensional, especially two-dimensional, CA is believed to be a promising modeling approach for understanding social dynamics [16]. Hence, we suppose that unlimited communication between agents is being compromised due to a proximity-locality criterion. This means those two agents, even if they find themselves having similar opinions (within the Bound of Confidence ), they do not exchange opinions unless the criterion of “proximity” is fulfilled. The idea is not new: applying a topology in the form of CA or neural networks on a set of “entities” usually signifies that we impose one more specific rule of interaction between them. Thus, one can hypothesise that (in simple terms because we do not refer to restrained groups of experts but rather to large groups of individuals or entire “artificial societies”) a geographical pattern of interaction, alias a form of social geometry, between agents must be taken into account [17].
    
    3.2. A Cellular Automata Topology
    Suppose that we have 100 agents5 distributed (each one on a different cell) in a square cellular automaton 10 10. Each agent is assigned a pair of integers (i,j) that denote the coordinates of his/her position on the CA. For example, as we can see in Figure 1, agent (4,3) is posed on the cell that lies on the fourth line and on the third column of the cellular automaton. Each agent is aware of the opinions only of the agents that are on his 3 3 Moore neighborhood. Namely, each agent is aware of the opinions of either 9 [i.e., agent (4,3)] or 6 [i.e., agent (10,6)] or 4 others [i.e., agent (1,10)] in yellow color, depending on his/hers position on the cellular automaton. Obviously, each agent is aware of his own opinion.
    
    Besides the use of a random initial profile, there is not any other stochastic procedure in MER Model. 7 In terms of programming, the sequence of “events” is purely indifferent but how can one verify that another has a similar opinion with him if it is impossible to meet him?
    
    6
    
    TABLE 1
    Rescale Process: Scaling Factor According to the Range of Values of Opinions 1 and 2 Value’s Range max max max max 1 1 1 1 and and and and min min min min 0 0 0 0 Scaling Factor max min max 1 min 1
    
    5
    
    In this article, we fix the number of agents to 100. The investigation of the influence of the number of agents to the model will be investigated in the future.
    
    As presented in Ref. 14.
    
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    FIGURE 2
    
    MER model in CAT (for
    
    0.5 combined with
    
    0.1,
    
    0.2, and
    
    0.3).
    
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    FIGURE 3
    
    MER model in CAT (for
    
    1 combined with
    
    0.1,
    
    0.2, and
    
    0.3).
    
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    FIGURE 4
    
    Opinions 1 and 2 after 500,000 iterations (
    
    0.1,
    
    1).
    
    fered the maximum change (lets say that this was opinion 1), s/he accepts new opinion 1 and s/he changes his/hers other opinion (opinion 2) on the opposite direction. The magnitude of the change equals the change suffered in opinion 1 multiplied with the intra-regulation factor . Obviously, can take values between zero (where the tendency to intra-individual equilibrium is absent) and infinity. Nevertheless, is considered to be limited theoretically because values above 2 seem to be rather “unrealistic” [14]. This means that adding or subtracting the double of the difference found in one opinion to the other can already 1, be characterised as “over-reaction.” For instance, if the magnitude of maximal difference is added or subtracted 2, the dissonant opinion is to the dissonant opinion. If 0.5 adjusted by doubling the maximal difference and if the dissonant opinion is corrected by adding or subtracting half of the maximal difference. Because, in MER Model, opinions escape the predefined interval [0,1], in order to keep them into the original interval and, at the same time, not to change the dynamical behavior of the system, a procedure called “rescale” is applied. If, for some iteration, in case that even one opinion of an agent escapes the interval [0,1], we rescale all opinions as follow: we calculate the maximum max and the minimum min of opinions 1 and 2 of all agents. If min is negative, then we translate all opinions upwards by subtracting min (min is negative), so in fact we augment all opinions and make them positive or zero. Then we divide all opinions by a scaling factor, which is the range of their values. All options of rescale are described in Table 1.
    
    4. SIMULATIONS: DISTURBING PERIODICITY LEADS TO “PURE” CHAOS
    As we can see in Figure 2 ( 0.5 and 0.1, 0.2, 0.3), the fact of imposing a cellular automata topology in MER Model does not affect significantly (compare with Table 3 in [14]) the dynamics of the system. Nevertheless, we observe a more fragmented repartition, alias a configurational change, due to the topological restrictions in communication between agents. 1 (regardless of ), MER Model As shown in [14], for (without CAT) presented periodicity as final state. This kind of dynamics signifies that the system is fully “balanced” but not motionless. In this setting of parameters, the system attains equilibrium in the form of periodicity, which is rather “improbable” regarding “social reality”. On the contrary, in MER Model with CAT, as can be seen in Figure 3. 1 and 0.1, 0.2, 0.3), periodicity is trans( formed into chaos, a radical change concerning the system’s dynamic. The system “deteriorates” in an unpredictable future (regardless of ). We also note that in both opinions a group formation takes place. This means that the system exhibits self-organization. However, these groups appear to move in a chaotic8 way and group membership9
    
    Although a general definition of chaos suitable for the most interesting cases is still lacking, most researchers would agree that a dynamical system is chaotic if it is deterministic and unpredictable. Chaotic systems are characterised only by short-term predictability. An unpredictable system can be either deterministic or stochastic. MER model is clearly a
    
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    FIGURE 5
    
    MER model in CAT (for
    
    1.5 combined with
    
    0.1,
    
    0.2, and
    
    0.3).
    
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    is not stable any more: occasionally agents seem to quit their group and join another. In order to find out whether this system stays restless indeed or reaches a final steady state, we run the simulation10 for 500,000 iterations. In Figure 4, we see that, even after 500,000 iterations the system is still moving, which strongly indicates that there will be no final steady state. Nevertheless, we cannot state that the system is chaotic unless we calculate the Lyapunov Exponent (see §5). 1.5 and 0.1, 0.2, 0.3 (Figure 5), the For alteration caused by the cellular automata topology is significant regarding the distribution of groups in the final steady state (compared with results obtained without CAT [14]). Although the system’s dynamics is obviously similar to the outcome of MER Model, for the same parameter setting (transient chaos), we observe that the agent’s corpus is highly fragmented in both opinions. In [14], the system presented noteworthy differences in the configurations of the two opinions: one of the opinions was highly fragmented, whereas the other ended either in consensus or in fragmentation of a much lower order (significantly fewer groups). 1.5 and 0.2, 0.3, It is noticeable, however, for that a “crazy” agent appears to be capable to create turbu0.2, an agent lence in the whole system’s dynamic. For seems to oscillate in a continuously larger width until s/he “coordinates” with all the others and let the whole system rest in a final steady state, after about 10,000 iterations. For 0.3, another agent follows a highly differentiated trajectory for a number of iterations and, then, when s/he reaches the others, s/he torments gravely their trajectories and causes a “mini crisis” (see Figure 5). In both cases, it seems that a single agent characterized by a rather “abnormal” behavior can seriously alter the dynamics of the whole system. This incident can only happen in a system whose dynamic is characterized by the well-known “butterfly effect.”
    
    FIGURE 6
    
    Running average of the Lyapunov Exponent for 1.5 with 0.1.
    
    0.5,
    
    1, and
    
    5. COMPUTATION OF LYAPUNOV EXPONENT
    In order to examine whether the system is unpredictable and thus chaotic, we examine if it exhibits sensitivity to initial conditions. The phenomenon of sensitivity means that even the slightest “error” in the initial conditions will be magnified after a number of iterations. Nevertheless, sensi-
    
    deterministic system and the trajectories of each individual as well as of the whole system is unpredictable, as we will explain in paragraph 5. 9 In social simulations bibliography, the word “group” generally is defined as a set of individuals (“agents”) between which absolute consensus reigns [15]. 10 Only for 0.1 due to the enormous computational power needed.
    
    tivity does not automatically lead to unpredictability. Indeed, there are some sensitive systems that are predictable, ,n N . e.g., all linear transformations xn 1 c xn, c In this case, an initial error is magnified after we iterate, but the error remains significantly smaller than xn [19, p. 512]. In a chaotic system, given the slightest deviation in initial conditions, the error increases exponentially with time and after a certain number of iterations, it becomes of the same order of magnitude as the correct values. In order to quantify error propagation and examine if MER model in CAT exhibits strong sensitivity to initial conditions and thus is unpredictable, we compute the Largest Lyapunov Exponent. The Lyapunov numbers are the average per-step divergence rate of nearby points along the orbit and the Lyapunov Exponents are the natural logarithm of the Lyapunov numbers [20]. Thus, Lyapunov Exponent characterizes the average logarithmic growth of the relative error per iteration. A small error in an initial point will be scaled by the factor of e (on the average) in each iteration. Lyapunov Exponents provide the most useful indication of chaos [21, p. 58]. When the system is Lyapunov-positive, the perturbation grows and predictability is lost. A chaotic orbit is one that forever continues to experience the unstable behavior; it never manages to find a sink to be attracted to. At any point of such an orbit, there are points arbitrarily near that will move away from the point during further iteration. Lyapunov Numbers and Lyapunov Exponents quantify this sustained irregularity. For a more detailed explanation about the calculation of the Lyapunov Exponent, see the Appendix. The results of the calculations of Lyapunov Exponents 0.5, 1 and 1.5 with 0.1 are presented in for Figure 6. In all cases, we present a running average of the exponent as an indication of whether the values have settled to a unique number and test the reliability of the calculation.
    
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    FIGURE 7
    
    System’s dynamics with (simulation 2) or without (simulation 1) a small error e
    
    10
    
    10
    
    in the initial opinion 1 of agent (1,1).
    
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    For 0.5 the running average of the Lyapunov Exponent is negative for a few number of iterations (error decreases and trajectories converge) and, finally, tends to zero (the system reaches a finally steady state). 1 the running average of the Lyapunov Exponent For is clearly positive and converges11 to the number 0.126. Because Lyapunov Exponent equals 0.126, a tiny mistake in the initial conditions will increase by e0.126 1.13 times on the average for each iteration. Thus, the system exhibits strong sensitivity to initial conditions and it is chaotic. 1.5 the running average of the Lyapunov ExpoFor nent is positive for a large number of iterations and tends to zero. This means that Lyapunov Exponent equals zero. For the first few iterations, trajectories of agents diverge exponentially but finally the system stabilise in a final state, so trajectories neither converge nor diverge. This phenomenon can be found in the bibliography as final state sensitivity [19] or transient chaos [22] and is the same as presented and analysed in [14].
    
    FIGURE 8
    
    Opinion 1 of agent (1,1) in simulations 1 and 2.
    
    6. SENSITIVITY TO INITIAL CONDITIONS: BUTTERFLY EFFECT ON (SIMULATED) SOCIAL DATA
    Assuming that MER Model can depict social reality in a certain degree, we re-approach our results in a more “empirical” way, trying to show the effect of sensitivity to initial conditions in analyzing social data.
    
    simulations in the left column of Figure 7 are the same with those presented in Figures 3 and 5. Both agents’ trajectories and final configurations are different due to sensitivity to initial conditions. The observed dissimilarities are noteworthy14: this demonstration shows clearly that the effects of sensitivity to initial conditions are decisive concerning the outcome of the system in a macro view.
    
    6.2. A Micro View
    Although personal histories are rather indifferent for the “typical” social researcher, it seems appealing to locate the aforementioned sensitivity in a micro (agent) level (Figure 8). The slightest perturbation in initial conditions causes a completely different outcome. The difference of their opinion in these two simulations is given in Figure 9. Sensitivity to initial conditions is not limited to one agent who experienced the slight difference. As previously shown, in such a system, each component of the system is highly interrelated with all others, i.e., if we shift slightly even one of them, all the others will be affected. For example, in
    
    6.1. A Macro View
    We re-run the simulation that we can see in the first row of Figure 3 (referred as simulation 1) with the same parameter values (100 agents on a 10 10 cellular automaton, a 3 3 Moore Neighborhood) but with “almost” the same initial profile. This means that all agents have exactly the same initial opinion 1 and 2 except agent (1,1) whose opinion 1 in simulation 1 is 0.7055475 and in simulation 2 is 0.7055475001. This slight difference (error) of e 10 10 is magnified in simulation 2 as can be seen in Figure 7 (for 1.5112 combined with 0.113). Note that the 1 and
    
    14
    
    For
    
    1.5 a “crazy” agent appeared in simulation 2!
    
    We have run the simulation up to 1,000,000 iterations to test the convergence of the running average of the Lyapunov Exponent. We have also tested that this number is independent from the initial opinion profile and the direction of the initial error at the first iteration. 12 We do not present the simulations concerning 0.5 because in these cases, the initial error becomes even smaller or disappears. 13 Because the type of system’s dynamic seems to be indifferent to 0.1. This argument the variation of , we have chosen concerns the type of dynamic only and not the outcome, which can be more or less fragmented according to the magnitude of .
    
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    FIGURE 9
    
    Differences of opinion 1 of agent (1,1) in simulations 1 and 2.
    
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    FIGURE 10
    
    “Personal” history of agents (2,4) and (10,10) according to two slightly different versions of the same model.
    
    Figure 10, we give the trajectories of two other agents in the two simulations [agent (2,4) and agent (10,10)]. Agent (10,10) [who is posed on the bottom right cell of the cellular automaton, the most “distant” agent from agent (1,1)] had exactly the same initial opinion 1 and 2 in both simulations but we changed slightly the initial opinion of agent (1,1). Table 2 shows agent (10,10)’s opinion 1 for a number of iterations. In the first six iterations, the perturbation of the system caused by the change in the initial opinion 1 of agent (1,1) has not yet touched agent (10,10). However, when this “disturbance” reaches agent (10,10) at iteration 7, it continu-
    
    ously increases and reaches the same magnitude as the opinions themselves. We also present agent’s (2,4) trajectory, because s/he suffers one of the biggest changes between the two simulations.
    
    6.3. Stability of “Traditional” Statistical Measures
    On the other hand, tracing personal histories for all agents, although interesting, does not comply with the usual practices concerning applied social research. From this point of view, habitually, we take into account statistical measures of central tendency and variation that should indicate the existence of some kind of law (more specifically, a causal
    
    TABLE 2
    Differentiated Opinions of Agent (10, 10) According to Two Simulations with Slightly Different Initial Opinion of Agent (1,1) Agent (10, 10), Opinion 1, Simulation 1 0.390292 0.390574 0.434567 0.469148 0.51461 0.543467 0.546105 0.554347 0.542563 0.721104 0.761712 0.737392 Agent (10, 10), Opinion 1, Simulation 2 0.390292 0.390574 0.434567 0.469148 0.51461 0.543467 0.546105 0.554347 0.542563 0.721104 0.740134 0.443623 Opinion’s Difference in the Simulations 0 0 0 0 0 0 0 8.1E-14 1.4E-13 1.7E-12 0.021577 0.293769
    
    Iteration 0 1 2 3 4 5 6 7 8 100 300 771
    
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    FIGURE 11
    
    FIGURE 12
    
    Mean value of opinion 1 in simulation 1 and 2 (
    
    0.1,
    
    1).
    
    Standard deviation of Opinion 1 in simulations 1 and 2 ( 1).
    
    0.1,
    
    inference law [23]. Thus, attesting variations concerning one and only person, among others, is considered to be of little value and the focus is always on evaluating more “social” or “collective” measures, which should depict the stability,15 occulted by “tormented” personal histories: means and standard deviations. As an example, in Figures 11 and 12, we can see the mean and standard deviation concerning opin0.1 and 1. ion 1 of the system for 10,000 iterations, Sensitivity to initial conditions is present also in mean and standard deviation. The picture is analogous for all other cases shown in Figure 3 (regarding different values of and opinion 2). The slight difference e 10 10 of initial opinion 1 of agent (1,1) is capable of differentiating both mean and standard deviation concerning opinions in simulations 1 and 2. These results are very fruitful: if we wanted to “simulate” a typical social researcher’s point of view, we would have several sequential measurements of means and standard deviations. These “snapshots” have a very limited meaning by themselves: a static description in a specific time. On the contrary, all dynamical aspects of the system are occluded: the predictive power of all these measures seems to be limited according to a specific time perspective.
    
    that each iteration is in analogy with a time-like unit not yet defined as “minute,” “hour,” or “day,” the Characteristic Time of a system is the time in which the error is magnified 10 times [24]. In this system the Characteristic Time T is
    
    1.13
    
    T
    
    10 N T
    
    log 10 log 1.13
    
    1 log 1.13
    
    18.84
    
    6.4. Is Prediction Possible? Defining a “Predictability Horizon”
    Because the system for 1 is deterministic and exhibits sensitive dependence on initial conditions (positive Lyapunov Exponent), it is chaotic. In an earlier paragraph, we mentioned that no prediction could be made if we have not taken into account a “predictability horizon.” Considering
    
    This means that the Characteristic Time for our system (for these parameter values) is about 18 iterations. We could say that the predictability horizon for our system is 18 iterations, meaning that if we have a measurement that is more or less accurate of the two agents’ opinions at a specific iteration, then by simulating we could predict their trajectories after about 18 iterations, knowing that an initial tiny error will become “only” 10 times larger. This is the “Predictability Horizon”16 of our system with the aforementioned parameter values. The Predictability Horizon represents the short time period during which above-chance prediction can occur in a chaotic system. For the example, less than or equal to 18 time periods ahead. Hence, the question of prediction shifts from “controlling accurate values” to “controlling the error propagation of inaccurate values.”
    
    7. DISCUSSION
    Much of cognitive social psychology has been concerned to reveal the “rules of social thinking.” For some social psychologists, this involves the attempt to discover the deontic rules that should be followed, if thinking is to proceed
    
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    Thus, measuring one individual’s attitudes does not give us valuable information compared to the information acquired from a group of individuals: we assume implicitly that group properties should “absorb” the individual variations considered, more or less, as “annoying” noise.
    
    We can not give the same definition for 1.5, because there the average of Local Lyapunov numbers for the first iterations is not stable and therefore we cannot be sure for a certain number of iterations that the error propagation will not exceed any threshold we would like to pose.
    
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    successfully. Other social psychologists are not so much concerned with the rules, which ought to be followed [25], but with those that naıve people actually do follow [26, 27]. ¨ It will be suggested that there is something missing in both these cognitive accounts of thinking. It is neither that social psychologists are relying on theories that make erroneous predictions, nor that their experimental results have arisen from faulty methodological procedures. What is missing is a feel of the contentious and dynamical nature of social thinking. Although, both approaches accept conventional theory testing in statistical terms, they both fall short to present an understandable aspect of social action in a combined micro and macro way. In our simulated social reality, the initial position of the agents is extremely important in a precision, which makes the applied errors useless in socio-psychological view: it is important however to say that such microscopic errors in opinion assessment can magnify themselves in a disproportional degree. The sensitivity to initial conditions is rather a characteristic of the system itself than a characteristic of the applied measurement tool. The system derived by MER Model in CAT is complex, self-organizing, and disorganizing with emerging dynamics. Cellular Automata contribution has been proven crucial, because the introduced topology converts the behavior of the system in a noteworthy way: from periodicity to “pure” chaos [22]. The imposed cellular automata topology seems to create micro-communicative anomalies in the social tissue of the agents’ macro-world, which result in selective group formatting first, and chaotic group trajectories afterwards. This is to say that the system is not only unpredictable on the long run but, in addition, it will never rest in a final steady state: prediction is only possible for a relatively short predictability horizon. Always on the move, “snapshots” seem meaningless, because they cannot capture the dynamics of the system. The use of “Predictability Horizon” (in terms of system’s Characteristic Time) could permit an attenuated prediction (a “soft” or “relative” prediction) of the system’s future states and can be employed by the social scientist for conclusions without surrendering himself in a deontic logic of compliance and social control: as Benzecri [28] claims, “the model must fit the data and not viceversa.”17 On the contrary, by presenting these simulated results and despite the lack of field-operationalization, we want to underline the necessity of focusing more in a dynamic than static aspect of social phenomena18: usually,
    
    “social dynamics” is treated as a taken-for-granted concept that is not in need of any explicit investigation and discussion.
    
    APPENDIX: LYAPUNOV EXPONENTS
    The main idea for the calculation of the Lyapunov Exponent is this: By an opinion profile x we mean a vector with 200 components, each one on the interval [0,1], the first 100 denoting the opinions 1 of the 100 agents and the rest the opinions 2. We consider two initial profiles, x1 and another one x2 that carries an error 0 10 10 in connection with x1 [the Euclidean distance of x1 and x2 equals 0]. We iterate both x1 and x2 separately and we get y1 and y2, respectively, L with their Euclidean distance being 1. The ratio 1/ 0 quantifies how the error is amplified (the error increases if L 1 and decreases if L 1) and is named Local Lyapunov Number (also relative error per iteration or error amplification factor) and its logarithm Local Lyapunov Exponent. We record this ratio. In order to continue the procedure, we keep opinion profile y1 and move opinion y2 so that it 19 has distance 0 with y1, without changing its direction. We iterate opinions y1 and y2 and go on with the same procedure. We accumulate Local Lyapunov Exponents and take their average for a large number of iterations. This is the (global) Lyapunov Exponent (from now on, we will call it Lyapunov Exponent). 0) signifies chaos, A positive Lyapunov Exponent ( meaning that the growth of the relative error per iteration 1) and therefore nearby orbits becomes greater than 1 (e move away, whereas a negative exponent means that nearby orbits are attracted [19, p. 709 –710]. For a fixed point or a periodic movement as a final state, the Lyapunov Exponent equals zero. For a more detailed account on Lyapunov Exponent see [19, 20, 22] and for a program in basic for the calculation of the Lyapunov Exponent see [30].
    
    REFERENCES
    1. Halfpenny, P. Positivism and Sociology: Explaining Social Life; George Allen and Unwin: London, 1982. 2. Halfpenny, P. Situating simulation in sociology. Sociological Research Online 1997, 2(3), http://www.socresonline.org.uk/socresonline/2/3/ 9.html. 3. Gergen, K.J. Social psychology as history. J Personality Social Psychol 1973, 26, 309 –20. 4. Gergen, K.J. Toward Transformation in Social Knowledge; SpringerVerlag: New York, 1982. 5. Aldrigde, A. Prediction in sociology: Prospects for a devalued activity. Sociological Research Online 1999, 4(3), http://www.socresonline.org.uk/socresonline/4/3/aldridge.html. 6. Mills, C.W. The Sociological Imagination; Harmondsworth: Penguin, 1970.
    
    17 18
    
    This is the second out of three principles stated in his book. It has become a commonplace to see the major disputes of contemporary sociology as organized around a dualism of “static”— expressed by “structure”—and “dynamic”— usually referred as “action” [29].
    
    19
    
    We rescale 0 at each iteration as we have done with opinions 1 and 2 and the Bound of Confidence .
    
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    7. Fiske, D.W.; Sweder, R.A. Metatheory in Social Science; The University of Chicago Press: Chicago, 1986. 8. Gilbert, N. Simulation: An Emergent Perspective. Lecture given at the conference on New Technologies in the Social Sciences, 27–29th October 1995, Bournemouth, UK and then at LAFORIA, Paris, 22nd January 1996. http://www.soc.surrey.ac.uk/research/cress/resources/emergent.html. 9. Heath, R.A. Can people predict chaotic sequences? Nonlinear Dynamics, Psychology and Life Sciences 2002, 6(1), 37–54. 10. Giddens, A. The Constitution of Society; Polity Press: Cambridge, 1984. 11. Converse, P.E. Public opinion and voting behavior. In: Handbook of Political Science; Greenstein, F.I.; Polsby, N.W., Eds.; Addison-Wesley: Reading, MA, 1975; pp 75–168. 12. Fay, B. Social Theory and Political Practice; George Allen and Unwin: London, 1975. 13. Hodgkinson, P. Who wants to be a social engineer? A commentary on David Blunkett’s Speech to the ESRC. Sociological Research Online 2000, 5(1), http://www.socresonline.org.uk/5/1/hodgkinson.html. 14. Katerelos, I.; Koulouris, A. Seeking equilibrium leads to chaos: Multiple equilibria regulation model. J Artif Soc Soc Simul 2004, http:// jasss.soc.surrey.ac.uk/7/2/4.html. 15. Hegselmann, R.; Krause, U. Opinion dynamics and bounded confidence models: Analysis and simulation. J Artif Soc Soc Simul 2002, 5(3), http://jasss.soc.surrey.ac.uk/5/3/2/2.pdf. 16. Hegselmann, R.; Flache, A. Understanding complex social dynamics: A plea for cellular automata based modelling. J Artif Soc Soc Simul 1998, 1(3), http://jasss.soc.surrey.ac.uk/1/3/1.html.
    
    17. Klu J. The evolution of social geometry. Complexity 2004, 9(1), 13–22. ¨ver, 18. Flament, C. L’ Analyse de Similitude: Une Technique pour les Recherches sur les Representations Sociales. Cahiers de Psychologie ´ Cognitive 1981, 4, 357–396. 19. Peitgen, H.-O.; Jurgens, H.; Saupe, D. Chaos and Fractals, New Frontiers of Science. Springer-Verlag: New York, 1992. 20. Alligood, K.T.; Sauer, T.D.; Yorke, J.A. Chaos, an Introduction to Dynamical Systems; Springer-Verlag: New York, 2000. 21. Kiel, L.D.; Elliott, E. Chaos Theory in the Social Sciences; The University of Michigan Press: 1997/2000. 22. Sprott, J.C. Chaos and Time-Series Analysis; Oxford University Press: Oxford, 2003. 23. McKim, V.; Turner, S. Causality in Crisis? University of Notre Dame Press: South Bend, IN, 1997. 24. Ekeland, I. Le Chaos. Flammarion: Paris, 1995. 25. Campbell, D.T. Common fate, similarity and other indices of the status of aggregates of persons as social entities. Behav Sci 1958, 3, 14 –25. 26. Guimelli, C. La pensee Sociale; Presses Universitaires de France: ´ Paris, 1999. 27. Papastamou, S., Ed. Introduction to Social Psychology; Tome I, Vol. 1, Ellinika Grammata: Athens, GA, 2001. 28. Benzecri, J.P. Correspondence Analysis Handbook; Marcel Dekker: New York, 1992. 29. Lopez, J.; Scott, J. Social Structure; Open University Press: Philadelphia, PA, 2000. 30. Sprott, J.C. Numerical Calculation of Largest Lyapunov Exponent; 1998. http://sprott.physics.wisc.edu/chaos/lyapexp.htm.
    
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