Pattern formation in spatially distributed networks via spatially correlated preferential attachment

In this paper we propose a model of the spatially distributed network based on the spatially correlated preferential attachments. Nodes in the spatially distributed networks of the real word, such as various urban or biological networks, aren't establishing randomly: The probability of emergence of new nodes is higher in the area of already existing ones. In this work we unite two principles of the real network modeling: The correlated percolation model and preferential attachment. To regulate spatial limitations of the network, we use density gradient, which determines the decrease of the probability of the connection emergence between two nodes with increase of the distance between them. We also consider the consistency of our results in the context of the real-world system modeling.