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Abstract
A large variety of interacting complex systems are characterized by interactions occurring between more than two nodes. These systems are described by simplicial complexes. Simplicial complexes are formed by simplices (nodes, links, triangles, tetrahedra etc.) that have a natural geometric interpretation. As such simplicial complexes are widely used in quantum gravity approaches that involve a discretization of spacetime. Here, by extending our knowledge of growing complex networks to growing simplicial complexes we investigate the nature of the emergent geometry of complex networks and explore whether this geometry is hyperbolic. Specifically we show that an hyperbolic network geometry emerges spontaneously from models of growing simplicial complexes that are purely combinatorial. The statistical and geometrical properties of the growing simplicial complexes strongly depend on their dimensionality and display the major universal properties of real complex networks (scale-free degree distribution, small-world and communities) at the same time. Interestingly, when the network dynamics includes an heterogeneous fitness of the faces, the growing simplicial complex can undergo phase transitions that are reflected by relevant changes in the network geometry.
Highlights
Simplicial complexes are the many-body generalization of networks [1,2,3,4,5,6] and they can encode interactions occurring between two or more nodes [7,8,9,10,11,12,13,14]
The emergent hidden geometry of growing simplicial complexes is hyperbolic, i.e. the hyperbolic geometry emerges spontaneously from the evolution of the simplicial complexes. In this way we provide evidence that hyperbolic network geometry emerges from growing simplicial complexes whose temporal evolution is purely combinatorial, i.e. it does not take into account the hidden geometry
We show a visualization of the model above and below the phase transition for dimension d = 2, 3 showing that the geometry of the boundary of simplicial complexes in d = 3 is strongly affected by the geometrical phase transition occurring in the model
Summary
Simplicial complexes are the many-body generalization of networks [1,2,3,4,5,6] and they can encode interactions occurring between two or more nodes [7,8,9,10,11,12,13,14]. One of the fundamental quests of quantum gravity is to describe the emergence of a continuous, finite dimensional space, using pre-geometric models, where space is an emergent property of a network or of a simplicial complex [21, 22, 24, 25]. This fundamental mathematical problem has its relevance in the field of network theory [16] where one of the major aim of network geometry is to characterize the continuous hidden metric behind the inherently discrete structure of complex networks. While the mathematical definition of the curvature of networks is a hot mathematical subject for which different definitions have been given [16, 33,34,35,36,37], most of the results obtained so far are related to the embeddings of complex networks in hyperbolic spaces [27,28,29,30, 32]
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