Influence d'une contamination initiale sur une dynamique spatiale non itérative

Hardouin, Cécile

Journal de la Société française de statistique & Revue de statistique appliquée, Tome 149 (2008) no. 4, pp. 107-129.

$n$ consommateurs répartis sur un réseau spatial $S$ choisissent tour à tour entre deux standards $A$ et $B$ suivant des règles locales. Un unique balayage du réseau est effectué, c’est-à-dire que la dynamique est non itérative. Dans ce cas, et contrairement aux dynamiques itératives ergodiques, les caractéristiques de la configuration spatiale finale du réseau dépendent de la configuration initiale et ne peuvent pas être évaluées mathématiquement. Nous en faisons l’étude empirique par simulation, pour un certain nombre de règles d’adoption bien spécifiées. L’objectif central de ce travail est de voir quel est l’effet d’une contamination initiale, ou effet de dumping, sur le standard $A$ au taux $\tau$ sur la répartition spatiale finale. On évaluera en particulier de manière empirique la fréquence finale du standard $A$, la corrélation spatiale, ainsi que des mesures d’aggrégation et de connexité. Pour chacun de ces indicateurs, on constate que l’effet du dumping est d’autant plus important que le taux de contamination initial est faible.

This paper explores the diffusion of technological innovations, under a simple and real framework: $n$ agents spread on a spatial network $S$, choosing between two competitive technologies $A$ and $B$. The choice is unique, made individually, one by one, according to a sequential assignment rule. We consider various rules, but depending on the local context. In the case of iterative dynamics, it is possible to characterize the probability distribution of limits configurations. We consider here a non iterative dynamics with a unique scan of $S.$ Therefore, we don’t know the final configuration, since there is no asymptotics in space nor in time. We propose to study empirically this situation, in the case of an initial occurence of standards $A$ at a given rate $\tau$. In particular, we will study the behaviour of the final frequency of standards $A,$ spatial correlation, and we give some connectedness and clustering indexes. For each index, we see that if the initial rate of contamination $\tau$ is lower, the dumping effect becomes more important.

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