Small worlds and clustering in spatial networks

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. arXiv:1909.00226v1 [physics.soc-ph]

Geometric renormalization unravels self-similarity of the multiscale human connectome

“…5A, Euclidean distances alone do not contain enough information to explain the connectivity properties of the MH connectome (45). In fact, networks generated by a purely geometric model based on Euclidean distances would have the small-world property if and only if pij ∼ x −β ij , with β ∈ (d , 2d ) and d the dimension of the underlying space (46). However, these networks would be homogeneous in their node-degree distribution, in contrast to the observed heterogeneity in human connectomes.…”

Geometric detection of hierarchical backbones in real networks

“…The S 1 model is able to reproduce many of the features widely observed in real complex networks, such as scale-freeness, high levels of clustering and the small-world property, among others [19]. Interestingly, the S 1 model is the only model able to produce maximum entropy ensembles with power-law degree distributions and clustering and without nonstructural degree correlations [23]. Moreover, the model also allows us to construct geometric maps of real networks through a process called network embedding.…”

“…We characterize the local contribution of a link or a node to the hierarchical structure of a network by measuring its hierarchy load, which depends on status, similarity, and the reference provided by a null model to discount the effects of random fluctuations. We use the S 1 as a null model since it is the maximum entropy model for geometric networks with heterogeneous degrees [23]. It provides expectations for the distribution of hierarchy loads in a pure random assignment of angular positions of nodes given a degree distribution and a level of clustering, so that anomalous fluctuations can be detected.…”

“…We can take advantage of this heterogeneity to filter out the connections that dominate the hierarchical organization of the network in terms of popularity and similarity, what we name the hierarchical similarity backbone (HSB). A hierarchical similarity backbone contains the links that represent statistically significant contributions with respect to the null hypothesis given by the S 1 model, that is the only model able to produce maximum entropy ensembles constrained by power-law degree distributions and clustering and without nonstructural degree correlations [23].…”

“…Moreover, we exploit the great heterogeneity found in the hierarchy load of links to propose a filtering method, the similarity filter, that offers a practical procedure to extract a hierarchical similarity backbone of a network. The obtained similarity backbones contain the links that represent statistically relevant interactions with respect to the maximally random geometric network organization, constrained by node degrees and clustering [23]. The similarity filter preserves network features at all scales while pruning a large number of links, in the spirit of the disparity filter for weighted networks [24].…”