Network complexity and topological phase transitions

A new type of collective excitations, due exclusively to the topology of a complex random network that can be characterized by a fractal dimension $D_F$, is investigated. We show analytically that these excitations generate phase transitions due to the non-periodic topology of the $D_F>1$ complex network. An Ising system, with long range interactions over such a network, is studied in detail to support the claim. The analytic treatment is possible because the evaluation of the partition function can be decomposed into closed factor loops, in spite of the architectural complexity. This way we compute the magnetization distribution, magnetization loops, and the two point correlation function; and relate them to the network topology. In summary, the removal of the infrared divergences leads to an unconventional phase transition, where spin correlations are robust against thermal fluctuations.