Complex networks exhibit a characteristic bundle of properties: sparsity, small-world distances, heavy-tailed degree distributions, high clustering, and hierarchical organization. Hyperbolic random graph models reproduce all of these by embedding nodes in hyperbolic space and connecting nearby pairs — the geometry generates the structure.
The statistical-mechanics formulation reveals that these network models are fermionic systems. Each possible link is either present or absent — binary occupation, exactly like a fermion in a quantum state. The maximum-entropy distribution over network configurations, subject to the observed properties as constraints, takes the form of a Fermi-Dirac distribution over links. The connection probability between two nodes follows a logistic function of their hyperbolic distance, with temperature controlling the sharpness of the geometric cutoff.
This framing explains a known anomalous phase transition. At low temperature, the network is geometric — links strongly respect hyperbolic distance, clustering is high, communities are spatially coherent. At high temperature, geometry dissolves — connections randomize, clustering drops, the hyperbolic embedding becomes meaningless. The transition between these phases is temperature-dependent and sharp. In the fermionic language, it is a Fermi surface phenomenon: the occupation function (link probability as a function of distance) goes from a sharp step function to a smooth decay as temperature increases.
The structural claim: the Fermi-Dirac distribution is not an analogy. It is the maximum-entropy distribution for binary random variables (links) subject to geometric constraints. The exclusion principle (at most one link between any pair) is literally the fermionic condition. The physics of fermions and the statistics of networks share the same mathematical skeleton.
(arXiv:2603.18170)