Networks are natural geometric objects . Yet the discrete metric structure of shortest path lengths in a network , known as chemical distances, is definitely not the only reservoir of geometric distances that characterize many networks. The other forms of network-related geometries are the geometry of continuous latent spaces underlying many networks, and the effective geometry induced by dynamical processes in networks. A solid and rapidly growing amount of evidence shows that the three approaches are intimately related. Network geometry is immensely efficient in discovering hidden symmetries, such as scale-invariance, and other fundamental physical and mathematical properties of networks, as well as in a variety of practical applications, ranging from the understanding how the brain works, to routing in the Internet. Here, we review the most important theoretical and practical developments in network geometry in the last two decades, and offer perspectives on future research directions and challenges in this novel frontier in the study of complexity.
Real networks are finite metric spaces . Yet the geometry induced by shortest path distances in a network is definitely not its only geometry. Other forms of network geometry are the geometry of latent spaces underlying many networks, and the effective geometry induced by dynamical processes in networks. These three approaches to network geometry are all intimately related, and all three of them have been found to be exceptionally efficient in discovering fractality, scale-invariance, self-similarity, and other forms of fundamental symmetries in networks. Network geometry is also of great utility in a variety of practical applications, ranging from the understanding how the brain works, to routing in the Internet. Here, we review the most important theoretical and practical developments dealing with these approaches to network geometry in the last two decades, and offer perspectives on future research directions and challenges in this novel frontier in the study of complexity.
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discrete metric structure of shortest path lengths
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, known as chemical distances,
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the only reservoir of geometric distances that characterize many networks. The other forms of network-related geometries
its only geometry. Other forms of network geometry
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continuous
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A solid and rapidly growing amount of evidence shows that the three approaches are intimately related. Network geometry is immensely
These three approaches to network geometry are all intimately related, and all three of them have been found to be exceptionally
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hidden symmetries, such as
fractality,
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and other fundamental physical and mathematical properties of networks, as well as
self-similarity, and other forms of fundamental symmetries in networks. Network geometry is also of great utility
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in
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