Sylvie Huet, Margaret Edwards and Guillaume Deffuant (2007)
Taking into Account the Variations of Neighbourhood Sizes in the Mean-Field Approximation of the Threshold Model on a Random Network
Journal of Artificial Societies and Social Simulation
vol. 10, no. 1
<http://jasss.soc.surrey.ac.uk/10/1/10.html>
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Received: 30-Jun-2006 Accepted: 27-Sep-2006 Published: 31-Jan-2007
(1) |
(2) |
where V(e,A) and V(e,B) are the proportion of neighbours of individual e respectively of behaviour A or B and the gA, gB are parameters of the model.
(3) |
(4) |
(5) |
Note that the mean number of links for one node is z = (N-1)p. Therefore, we can write:
(6) |
where the last approximate equality expresses that a binomial law tends to a Poisson distribution when N tends to infinity. Since our simulations involve 10.000 individuals, we shall consider that it is justified to follow this approximation.
(7) |
(8) |
E | (9) |
where P(A,k_{A}/i) and P(B,k_{A}/i) are the respective probability of: choosing A (respectively B) given the number k_{A} of A neighbours over the neighbourhood size i. For i = 0, the probabilities of behaviour A and B are both equal to 0.5.
(10) |
(11) |
(12) |
(13) |
(14) |
(15) |
(16) |
Figure 1. Absolute value of difference between attraction basin limit values predicted by the two aggregated models for part of the tested values. We show here the most significant results |
Table 1: Experimental design for gA, β and average neighbourhood size — values of experimental design for initial proportion of A in the population are obtained as explained in the paragraph above | ||
Mean size of neighbourhood | gA (behaviour A payoff) | β (level of randomness in decision function) |
3 | 0,51 | 7 - 8 - 9 - 10 - 11 - 12 - 13 - 17 - 30 - 25 - 45 - 50 - 60 - 70 - 80 - 90 - 100 - 200 - 300 - 400 - 700 |
0,57 | 12 - 16 - 21 - 23 - 35 - 240 - 940 | |
5 | 0,55 | 7,6 - 8 - 10 - 12 - 14 - 16 - 20 - 25 - 50 - 60 - 80 - 250 - 500 - 800 - 1200 |
0,59 | 8,98 - 10 - 11 - 13 - 15 - 20 - 25 - 60 - 250 - 470 - 620 - 800 - 1200 | |
0,6 | 9,46 - 9,6 - 10 - 11 - 12 - 13 - 14 - 15 - 17 - 20 - 25 - 26 - 30 - 32 - 38 - 40 - 50 - 60 - 80 - 100 - 120 - 140 - 160 - 200 - 260 - 300 - 800 - 1200 | |
0,61 | 12 - 14 - 16 - 18 - 20 - 23 - 27 - 30 - 40 - 50 - 60 - 70 - 80 - 90 - 100 - 150 - 200 - 300 - 400 - 500 - 600 - 700 - 800 - 1200 | |
0,65 | 13,1 - 15 - 16 - 25 - 30 - 40 - 60 - 100 - 800 - 1200 | |
0,67 | 14,6 - 18 - 19 - 20 - 22 - 25 - 30 - 35 - 80 - 100 - 230 - 320 - 470 - 650 - 800 - 1200 | |
0,7 | 18,1 - 22 - 25 - 27 - 28 - 29 - 30 - 35 - 60 - 110 - 250 - 470 - 800 - 1200 | |
8 | 0,66 | 11 - 12 - 13 - 14 - 15 - 16 - 18 - 25 - 40 - 130 - 220 - 320 - 620 |
0,67 | 11 - 12 - 13 - 14 - 15 - 16 - 18 - 25 - 40 - 130 - 220 - 320 - 620 | |
0,74 | 20 - 21 - 22 - 23 - 24 - 25 - 40 - 130 - 220 - 320 - 620 | |
0,75 | 21 - 22 - 23 - 24 - 25 - 27 - 30 - 33 - 36 - 40 - 130 - 220 - 320 - 620 | |
0,76 | 21 - 22 - 23 - 24 - 25 - 27 - 30 - 33 - 36 - 40 - 130 - 220 - 320 - 620 | |
25 | 0,84 | 31 - 32 - 33 - 35 - 40 - 45 - 50 - 55 - 60 - 65 - 70 - 90 - 110 - 130 - 150 - 170 - 220 - 270 - 320 - 420 - 520 - 620 - 720 |
(17) |
Figure 2. Global improvement of our approximation |
Figure 3. Three models bifurcations conditions for 5 neighbours on average |
Figure 4. Three models bifurcations conditions for 3 neighbours mean |
Figure 5. Three models bifurcations conditions for 25 neighbours mean |
Figure 6. Evolution of % A behaviours for the variable neighbourhood aggregated model for β = 29, gA=0.7, 5 neighbours and 4% initial A behaviours |
Figure 7a. Evolution of % A behaviours for the variable neighbourhood aggregated model for β = 29, gA=0.7, 5 neighbours and 4% initial A behaviours |
Figure 7b. Details on the evolution of % A behaviours on time for the individual-based model for 200 runs for β = 29, gA=0.7, 5 neighbours and 4% initial A behaviours |
Figure 8. Aggregated model potential function for β = 29, gA=0.7, 5 neighbours |
The potential function shows that the aggregated model cannot increase to 100 % of A behaviour when it begins with less than 5% of A behaviour (see the zoom on the right hand side), because it is caught in the potential minimum which appears in the zoom. The stochastic events taking place in the individual based model allow it to climb out and to join 100% of A behaviour.
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