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The spatial preferential attachment (SPA) is a model for complex networks. Inthe SPA model, nodes are embedded in a metric space, and each node has a sphereof influence whose size increases if the node gains an in-link, and otherwisedecreases with time. In this paper, we study the behaviour of the SPA modelwhen the distribution of the nodes is non-uniform. Specifically, the space isdivided into dense and sparse regions, where it is assumed that the denseregions correspond to coherent communities. We prove precise theoreticalresults regarding the degree of a node, the number of common neighbours, andthe average out-degree in a region. Moreover, we show how these theoreticallyderived results about the graph properties of the model can be used toformulate a reliable estimator for the distance between certain pairs of nodes,and to estimate the density of the region containing a given node.

Non-Uniform Distribution of Nodes in the Spatial Preferential Attachment Model