This paper presents a geometric framework for studying complex networks, proposing that hyperbolic geometry underlies these networks. The authors show that heterogeneous degree distributions and strong clustering in complex networks emerge naturally from the properties of hyperbolic geometry. They also demonstrate that if a network has a metric structure and a heterogeneous degree distribution, it has an effective hyperbolic geometry. The framework maps network edges to non-interacting fermions, with energies corresponding to hyperbolic distances between nodes. The network ensemble includes standard configuration models and random graphs as limiting cases. The authors show that targeted transport processes in networks with strong heterogeneity and clustering are maximally efficient, and this efficiency is robust to network damage. The paper also discusses the relationship between hyperbolic geometry and complex network topology, showing that hyperbolic geometry naturally leads to heterogeneous degree distributions and strong clustering. The authors further demonstrate that heterogeneous degree distributions imply hyperbolic geometry, and that the two are equivalent. The paper concludes that hyperbolic geometry naturally emerges from network heterogeneity, and that complex networks have optimal structures for efficient routing.This paper presents a geometric framework for studying complex networks, proposing that hyperbolic geometry underlies these networks. The authors show that heterogeneous degree distributions and strong clustering in complex networks emerge naturally from the properties of hyperbolic geometry. They also demonstrate that if a network has a metric structure and a heterogeneous degree distribution, it has an effective hyperbolic geometry. The framework maps network edges to non-interacting fermions, with energies corresponding to hyperbolic distances between nodes. The network ensemble includes standard configuration models and random graphs as limiting cases. The authors show that targeted transport processes in networks with strong heterogeneity and clustering are maximally efficient, and this efficiency is robust to network damage. The paper also discusses the relationship between hyperbolic geometry and complex network topology, showing that hyperbolic geometry naturally leads to heterogeneous degree distributions and strong clustering. The authors further demonstrate that heterogeneous degree distributions imply hyperbolic geometry, and that the two are equivalent. The paper concludes that hyperbolic geometry naturally emerges from network heterogeneity, and that complex networks have optimal structures for efficient routing.