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16090029

Abstract: Complex networks describe interacting systems ranging from the brain to the Internet. While so far the geometrical nature of complex networks has been mostly neglected, the novel field of network geometry is crucial for gaining a deeper theoretical understanding of the architecture of complexity. At the same time, network geometry is at the heart of quantum gravity, since many approaches to quantum gravity assume that space-time is discrete and network-like at the quantum level. In network geometry a crucial problem is the identifications of mechanisms to describe emergent network geometry. In this talk, after an introduction to complex network theory, l will present a class of nonâ€“equilibrium models [1-4] in which geometrical properties of the networks emerge spontaneously from their dynamics. Specifically we will discuss the model called network geometry with flavor (NGF).The NGF can generate discrete geometries of different nature, ranging from chains and higher dimensional manifolds to scale-free complex networks. 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.

[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, arxiv:1511.04539 (2015)

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