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Abstract
Networks with underlying metric spaces attract increasing research attention in network science, statistical physics, applied mathematics, computer science, sociology, and other fields. This attention is further amplified by the current surge of activity in graph embedding. In the vast realm of spatial network models, only a few reproduce even the most basic properties of real-world networks. Here, we focus on three such properties---sparsity, small worldness, and clustering---and identify the general subclass of spatial homogeneous and heterogeneous network models that are sparse small worlds and that have nonzero clustering in the thermodynamic limit. We rely on the maximum entropy approach where network links correspond to noninteracting fermions whose energy dependence on spatial distances determines network small worldness and clustering.
Highlights
In spatial networks, nodes are positioned in a geometric space, and the distances between them in the space affect their linking probability in the network [1]
We focus on three such properties—sparsity, small worldness, and clustering—and identify the general subclass of spatial homogeneous and heterogeneous network models that are sparse small worlds and that have nonzero clustering in the thermodynamic limit
We rely on the maximum entropy approach in which network links correspond to noninteracting fermions whose energy depends on spatial distances between nodes
Summary
Nodes are positioned in a geometric space, and the distances between them in the space affect their linking probability in the network [1]. Since random geometric graphs provide the simplest explanation for the emergence of clustering in complex networks, our models should be considered as the appropriate null models of clustering and degree distributions observed in many real-world networks. To obtain these results, we take a statistical physics stance in which we interpret spatial network models as probabilistic mixtures of grand canonical ensembles that maximize ensemble entropy under certain constraints, and are statistically unbiased. We call these mixtures hyper-grand-canonical ensembles, as some of their parameters are random
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