- Published 2013

In the recent development of network sciences, spatial constrained networks have become an object of extensive investigation. Spatial constrained networks are embedded in configuration space. Their structures and dynamics are influenced by spatial distance. This is proved by more and more empirical data on real systems showing exponential or power laws spatial distance distribution of links. In this dissertation, we focus on the structure of spatial network with power law spatial distribution. Several mechanisms of structure formation and diffusion dynamics on these networks are considered. First we propose an evolutionary network constructed in the configuration space with a competing mechanism between the degree and the spatial distance preferences. This mechanism is described by a ki ∑ j kj + (1− a) r −α ni ∑ j r −α nj , where ki is the degree of node i and rni is the spatial distance between nodes n and i. By adjusting parameter a, the network can be made to change continuously from the spatial driven network (a = 0) to the scale-free network (a = 1). The topological structure of our model is compared to the empirical data from email network with good agreement. On this basis, we focus on the diffusion dynamics on spatial driven network (a = 0). The first model we used is frequently employed in the study of epidemic spreading: the spatial susceptible-infected-susceptible (SIS) model. Here the spreading rate between two connected nodes is inversely proportional to their spatial distance. The result shows that the effective spreading time increases with increasing α. The existence of generic epidemic threshold is observed, whose value depends on parameter α. The maximum epidemic threshold and the minimum stationary ratio of infected nodes simultaneously locate in the interval 1.5 < α < 2. Since the spatial driven network has well defined spatial distance, this model offers an occasion to study the diffusion dynamics by using the usual techniques of statistical mechanics. First, considering the fact that the diffusion is anomalous in general due to the important long-range spreading, we introduce a composite diffusion coefficient which is the sum of the usual diffusion constant D of the Fick’s laws applied over different possible transfer distances on the network. As expected, this composite coefficient decreases with increasing α and is a good measure of the efficiency of the diffusion. Our second approach to this anomalous diffusion is to calculate the mean square displacement 〈l2〉 to identify a te l-0 08 12 60 4, v er si on 1 12 A pr 2 01 3 diffusion constant D and the degree of the anomalousness γ with the help of the power law 〈l2〉 = 4Dt. D behaviors in the same way as D, i.e., it decreases with increasing α. γ is smaller than unity (subdiffusion) and tends to one (normal diffusion) as α increases.

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@inproceedings{Hui2013LUNAMU,
title={LUNAM Universit{\'e} , Universit{\'e} du Maine Th{\`e}se de Doctorat Sp{\'e}cialit{\'e} : Physique},
author={Zi Hui and Mouad Lamrani},
year={2013}
}