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Routing packets on the growing and changing underlying structure of the Internet is challenging and currently based only on local connectivity. Here, a global Internet map is devised: with a greedy forwarding algorithm, it is robust with respect to network growth, and allows speeds close to the theoretical best.

A framework is developed in which new connections to a growing network optimize geometric trade-offs between popularity and similarity, instead of simply preferring popular nodes; this approach accurately describes the large-scale evolution of various networks.

Bipartite networks link two different kinds of nodes, but standard one-mode projections can distort their structure. Here, the authors introduce a geometric model and the B-Mercator algorithm that embed both node types in a shared hyperbolic space, yielding accurate maps that improve analysis, prediction, and synthetic network generation.

Embedding of complex networks in the latent geometry allows for a better understanding of their features. The authors propose a framework for mapping complex networks into high-dimensional hyperbolic space to capture their intrinsic dimensionality, navigability and community structure.

Reducing of dimension is often necessary to detect and analyze patterns in large datasets and complex networks. Here, the authors propose a method for detection of the intrinsic dimensionality of high-dimensional networks to reproduce their complex structure using a reduced tractable geometric representation.

Network geometry is an emerging framework used to describe several topological and organizational features of complex networks. Now this approach has been extended to directed networks, which contain both symmetric and asymmetric interactions.

Complex networks have been conjectured to be hidden in metric spaces, which offer geometric interpretation of networks’ topologies. Here the authors extend this concept to weighted networks, providing empirical evidence on the metric natures of weights, which are shown to be reproducible by a gravity model.

Complex networks are not obviously renormalizable, as different length scales coexist. Embedding networks in a geometrical space allows the definition of a renormalization group that can be used to construct smaller-scale replicas of large networks.

Multiplex networks are shown to harbour significant correlations between layers. A framework describing the correlations enables multilayer community and link detection, and reveals that they improve navigation — but only when they’re strong.

Many complex properties of real networks appear as consequences of a small set of their basic properties. Here, the authors show thatdk-random graphs that reproduce degree distributions, degree correlations, and clustering in real networks, reproduce a variety of their other properties as well.

In many real-world processes that can be mapped onto complex networks—from cell signalling to transporting people—communication between distant nodes is surprisingly efficient, considering that no node has a full view of the entire network. A framework sets out to explain why ‘navigability’ is so efficient in these networks.

Geometric renormalization reveals hidden network symmetries by scaling them down while retaining key features. Extended to weighted networks, in which link intensities matter, here the authors present empirical evidence and theory to justify selecting links with maximum weights across increasingly longer length scales to reduce resolution, enabling self-similar replicas and study of size-dependent phenomena.

Geometric insights into the structure and function of complex networks have led to exciting developments in network science. This Review Article summarizes progress in network geometry, its theory, and applications to biological, sociotechnical and other real-world networks.

The Berezinskii, Kosterlitz, and Thouless (BKT) model shows that there are finite temperature phase transitions driven not by symmetry breaking, but rather by topological defects such as vortices. The authors show that a transition occurring in a general class of sparse space random networks model is topological in nature with no broken symmetry