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Abstract: Simplicial complexes are generalized network structures able to encode interactions occurring between more than two nodes. Simplicial complex describe a large variety of complex interacting systems ranging from brain networks, to social and collaboration networks. Additionally simplicial complexes have a geometrical interpretation and for this reasons they have been widely used in quantum gravity. Simplicial complexes are the ideal structures to characterize emergent network geometry [1-3] in which geometrical properties of the networks emerge spontaneously from their dynamics. Here we propose a general model for growing simplicial complexes called network geometry with flavor (NGF) [4]. This model deepens our understanding of growing complex networks and reveals the important effect that the dimensionality of growing simplicial complexes have on their evolution. The NGF can generate discrete geometries of different nature, ranging from chains and higher dimensional manifolds to scale-free networks with small-world properties, scale-free degree distribution and non-trivial community structure. We find that, for NGF with dimension greater than one, scale-free topologies emerge also without including an explicit preferential attachment because and efficient preferential attachment mechanism naturally emerges from the dynamical rules. Interestingly the NGF with fitness of the nodes reveals relevant relations with quantum statistics. In fact the faces of the NGF have generalized degrees that follow either the Fermi-Dirac, Boltzmann or Bose-Einstein statistics depending on their flavor and on their dimensionality. Specifically, NGFs with flavor s=-1, when constructed in dimension d=3 gluing tetrahedra along their triangular faces, have the generalized degrees of the triangular faces, of the links, and of the nodes following respectively the Fermi-Dirac, the Boltzmann or the Bose-Einstein distribution.
We will characterize the  emergent geometry of NGF as hyperbolic [5] and describe  its properties in any dimension.

[1] G. Bianconi, Interdisciplinary and physics challenges in network theory, EPL 111, 56001 (2015).
[2] Z. Wu, G. Menichetti C. Rahmede G. Bianconi, Emergent Complex Network Geometry, Scientific Reports 5, 10073 (2015).
[3] G. Bianconi and C. Rahmede, Complex Quantum Network Manifolds are Scale-Free in d>2, Scientific Reports 5, 13979 (2015)
[4] G. Bianconi and C. Rahmede, Network geometry with flavor: from complexity to quantum geometry,  Physical Review E, 93, 032315. (2016).
[5] G. Bianconi and C. Rahmede, Emergent hyperbolic network geometry Scientific Reports 7,  41974 (2017).