Ioannis D Katerelos and Andreas G Koulouris (2004)
Seeking Equilibrium Leads to Chaos: Multiple Equilibria Regulation Model
Journal of Artificial Societies and Social Simulation
vol. 7, no. 2
<http://jasss.soc.surrey.ac.uk/7/2/4.html>
To cite articles published in the Journal of Artificial Societies and Social Simulation, reference the above information and include paragraph numbers if necessary
Received: 16-Nov-2003 Accepted: 24-Feb-2004 Published: 31-Mar-2004
Table 1: Rescale process: scaling factor according to the range of values of opinions 1 and 2 | |
Value's range | Scaling factor |
max>=1 and min<0 | max-min |
max>1 and min>=0 | max |
max<=1 and min<0 | 1-min |
max<=1 and min>=0 | 1 |
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Table 2. Simulation of MER Model for Ψ = 0.5 and three values of ε |
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Table 3. Simulation of MER Model for Ψ = 1 and three values of ε |
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Table 4. Simulation of MER Model for Ψ = 1.5 and three values of ε |
Figure 1. Opinion 1 of agent 1 in the two simulations (Ψ = 1.5 and ε = 0.1) |
Figure 2. Differences of opinion 1 of agent (1) in simulations 1 and 2 |
Figure 3. A minimal change on Agent (1) signifies a major change of Agent (95) -"Personal" history of Agent (95) according to two slightly different (in Agent (1), opinion 1) simulations of the MER model |
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Table 5. System's dynamics with (simulation 2) or without (simulation 1) a small error (e = 10^{-10}) in the initial opinion 1 of agent 1. Both agents' trajectories and final configurations are different due to sensitivity to initial conditions. |
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Table 6. Examining system's nonlinear characteristics: Lyapunov Exponents and Information Entropy |
where #I(i, k, t) denotes the number of elements of the set I(i, k, t).
where
ε | Bound of confidence |
Ψ | Intra-regulation factor |
Opinion (i, p) | Opinion p, p ∈ {1, 2} of agent i at the previous iteration. |
TemporaryOpinion (i, p) | The mean of the Opinions of agents j that their Opinion differ from agent i less than the Bound of Confidence ε. |
NewOpinion (i, p) | Opinion p of agent i at the next iteration. |
Abs | Absolute Value |
For all opinions p For all agents i Sum=0: Counter=0 For all agents j Distance (i, j, p) = Abs (Opinion (i, p) – Opinion (j, p)) If Distance (i, j, p) <= ε Then Counter = Counter + 1 Sum = Sum + Opinion (j, p) End If TemporaryOpinion (i, p) = Sum / Counter For all agents i For all opinions p Change (p)= TemporaryOpinion (i, p) – Opinion (i, p) Max=Maximum { Abs (Change (1)), Abs (Change (2)) } If Max= Abs (Change (1)) Then NewOpinion (i, 2) = TemporaryOpinion (i, 2) - Ψ * Max If Max= Abs (Change (2)) Then NewOpinion (i, 1) = TemporaryOpinion (i, 1) - Ψ * Max For all opinions p For all agents i Rescale NewOpinion (i, p) into [0,1] Calculate Information Entropy Calculate Lyapunov Exponent
^{2} This conflict is defined as a psycho-logical concept and not on a pure logical basis.
^{3} These inter-relations among cognitive elements (CE) can be of three kinds: consonant, dissonant and irrelevant (Festinger 1957).
^{4} In our paradigm, opinions have a sentence-like form: they can be defined as cognitive elements (CE) of the same cognitive space: i.e. if CE A goes up then CE B must go down or vice-versa.
^{5} Maybe the word "conforming" is more suitable in our case (Sherif 1936; Asch 1951, 1956).
^{6} Of course, as for social equilibrium, a state of achieved "Intra-individual" equilibrium is purely theoretical (see Introduction on Intra-individual equilibrium). Therefore, to be more exact, a person rather "tries" to attain the equilibrium than "maintaining" an already achieved equilibrium.
^{7} Just like rhinoceros, one is considered "myopic" in static values but accurate in changes. A rhinoceros is very deficient in seeing static things but he distinguishes motion easily.
^{8} For the moment, we consider that values above 2 are rather "unrealistic". This means that adding or subtracting the double of the difference found in one opinion to the other can already be characterised as "over-reaction".
^{9} This procedure is like every time some opinion escapes the interval [0,1] we "photocopy" the opinions by reducing their size proportionally and so we put them into [0,1] again. We have tested for a large number of parameter values that the dynamical behavior of the system is unaltered and thus we obtain similar configurations.
^{10} Given the expected measurement error produced in every data collection technique concerning social phenomena, the magnitude of "error" can be extremely high (or even "gross") in terms of exact sciences' mathematical analysis. This means that an error of the order 10^{-10} is simply senseless in socio-psychological research and one considers that social sciences' measurement cannot be "exact", so these sciences could never enter the "exact sciences club".
^{11} We do not present simulation 2 for Ψ = 0.5 and Ψ = 1, since in these cases the initial error becomes even smaller or disappears. The system gives exactly the same picture for simulation 2, as we can see for simulation 1 in table 2 and apparently does not exhibit sensitivity to initial conditions.
^{12} Sensitivity, however, does not automatically lead to unpredictability. Indeed, there are sensitive systems that are predictable, e.g. the linear transformations x_{n+1} = c · x _{n}, c ∈ R, n ∈ N^{*}. An initial error is magnified after we iterate, but the error remains significantly smaller than x_{n} (Peitgen et al 1992, pg.512). In a chaotic system, given the slightest deviation in initial conditions, after a certain number of iterations, it will become as large as the true signal itself. In other words, the error will be of the same order of magnitude as the correct values.
^{13} For every pair of values for Ψ and ε the system has 200 Lyapunov Exponents since it is defined in R^{200}. In all cases, we compute only the Larger Lyapunov Exponent, which is the best indication for the predictability or unpredictability of the system.
^{14} We have also tested that the values of the Lyapunov Exponents calculated above are independent of the magnitude of error (when error tends to zero) and independent of different initial conditions.
^{15}"Final configuration" stands for the system position beyond which there is no further change of values.
^{16} The simplest way of defining methodological individualism is the thesis in which every proposition about a group is, implicitly or explicitly, formulated in terms of behavior or interaction of the individuals constituting the group (Schumpeter 1908).
^{17} Our model does not include any stochastic process; therefore, the trajectories of our agents are predetermined by means of interaction's algorithms.
^{18}"Zeitgeist" means a trend of a culture and taste characteristic of an era.
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