Sylvie Huet and Guillaume Deffuant (2008)
Differential Equation Models Derived from an Individual-Based Model Can Help to Understand Emergent Effects
Journal of Artificial Societies and Social Simulation
vol. 11, no. 2 10
<http://jasss.soc.surrey.ac.uk/11/2/10.html>
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Received: 04-Aug-2007 Accepted: 15-Mar-2008 Published: 31-Mar-2008
Figure 1. Examples of temporal evolution of the attitude of an individual with g = 6.5. On the left, the individual receives the features in order u, U, U, u, u. Its final attitude is negative |
Figure 2. Trajectory for order of exposure u, u, u, U, U, and the final attitude is positive |
Figure 3. The 10 possible individual trajectories, for case g = 6.5, 5 features composed of 2 U and 3 u with U = -6 and u = +4. Three trajectories over ten lead to a final negative attitude |
Figure 4. Final percentage of negative individuals, averaging on 100 replicas, for g = 2.5 for three various dynamics: isolated individuals; interacting individuals transmitting only congruent features and interacting individuals transmitting any feature |
Table 1: Final sign of the global attitude for the ten different trajectories and all different values of g | |||||||
Exposure order |
g < 0 | 0 ≤ g < 0.5u | 0.5u ≤ g < u | u ≤ g < 1.5u | 1.5u ≤ g < 2u | 2u ≤ g < 3u | g ≥ 3 u |
UUuuu | - | - | - | - | - | - | + |
UuUuu | - | - | - | - | - | + | + |
UuuUu | - | - | - | - | + | + | + |
UuuuU | - | - | - | - | + | + | + |
uUUuu | - | - | - | - | - | + | + |
uUuUu | - | - | - | + | + | + | + |
uUuuU | - | - | + | + | + | + | + |
uuUUu | - | - | - | + | + | + | + |
uuUuU | - | + | + | + | + | + | + |
uuuUU | - | + | + | + | + | + | + |
Figure 5. The graph of transitions between the groups for u ≤ g < 1.5 u, defined by the set of retained features. The groups with a negative attitude are in grey |
Table 2: communicated features for each group in the case u ≤ g < 1.5 u or U | |||||||
Group | Media | {U*} | {u} | {Uu} | {uu*} | {uUU} | {uUu*} |
Communicated features | U, u | U | u | U | u | U | U |
(1) |
with:
S_{0}: proportion of individuals with a void list of retained features,
S_{u}: proportion of individuals with a list of retained features containing only u,
S_{ U*}: proportion of individuals following all trajectories beginning with U,
S_{uU}: proportion of individuals with a list of retained features containing only u at first and U at second
S_{ uu*}: proportion of individuals following all trajectories beginning with uu.
S_{uUU}: proportion of individuals following all trajectories beginning with uUU.
S_{ uUu*}: proportion of individuals following all trajectories beginning with uUu.
f: frequency of media feature communication.
We compute finally the evolution of groups at the end by calculating, for each group S_{G} with G ∈ {0, u , U*, uU, uu*,UU, uUu*}:
(3) |
Figure 6. Comparison of trajectories of each groups of aggregated and IBM model. One measure of the IBM's replicas is put all the ten measures of the aggregated model |
Figure 7. Comparison of probability of U emission due to interaction with the probability of U emission due to medium for u ≤ g < 1.5 u |
Figure 8. The graph of transitions between the groups for 2 u ≤ g < 3 u, defined by the set of retained features. The groups in grey have a negative attitude |
Table 3: communicated features for each group in the case 2 u ≤ g < 3 u or 2 U | |||||
Group | Media | {U} | {u*} | {UU} | {Uu*} |
Communicated features | U, u | none | u | U | u |
(2) |
with :
S_{0}: proportion of individuals with a void list of retained features,
S_{U}: proportion of individuals with a list of retained features containing only U,
S u*: proportion of individuals following all trajectories beginning with u,
S UU*: proportion of individuals following all trajectories beginning with UU,
S Uu*: proportion of individuals following all trajectories beginning with Uu.
f: frequency of feature diffusion by the media.
Figure 9. Comparison of trajectories of each groups of aggregated and IBM model for 2 u ≤ g < 3 u. One measure of the IBM's replicas is put all the ten measures of the aggregated model |
Figure 10. Comparison of probability of U emission due to interaction with the probability of U emission due to medium for the "decrease" interaction" effect case |
We see on figure 10 that the probability of U emission by interaction (from 0 to 0.0042) is always lower than the probability of U emission by the media (0.4).
Figure 11. Final percentage of negative individuals for various values of g and for three dynamics: isolated individuals (no interaction), interaction with any feature transmission, interaction with only congruent transmission (average on 100 replicas). The errors bars represent the minimum and maximum values on 100 replicas |
We observe on the left, for 0 ≤ g < 1.5 u, that both interaction cases lead to a higher part of the population exhibiting the primacy effect.
Figure 12. Comparison of the final part of negatives for various values of the frequency parameter f in case "without interaction" and case "with interaction" with the aggregated models: on the left, "increase" interaction effect case (for u ≤ g < 1.5 u); on the right, "decrease" interaction" effect case at bottom (for 2 u ≤ g < 3 u) |
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