Most of real-world networks are extremely compact, infinite-dimensional objects. Consequently, any cooperative model on any of these network substrates is surely in situation above the upper critical dimension. This is why critical phenomena in these models should be precisely described in the framework of a mean field approach. Nonetheless, due to specific architectures of complex networks, these mean field theories are surprisingly non-standard.
We discuss the unusual critical phenomena in complex networks by using representative examples: the Ising and Potts models, the percolation and its generalizations, etc. Remarkably, the critical behaviours are very different in equilibrium and growing networks. We explain that in a class of growing networks, the percolation and Ising models may even demonstrate a critical singularity of the Berezinskii-Kosterlitz-Thouless kind. We also touch upon the bootstrap (k-core) percolation problem and the k-core organization of complex networks.
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S N Dorogovtsev, A V Goltsev, J F F Mendes, Ising model on networks with an arbitrary distribution of connections, Phys. Rev. E 66, 016104 (2002).
S N Dorogovtsev, J F F Mendes, A N Samukhin, Anomalous percolation properties of growing networks, Phys. Rev. E 64, 066110 (2001).
M Bauer, S Coulomb, S N Dorogovtsev, Phase transition with the Berezinskii-Kosterlitz-Thouless singularity in the Ising model on a growing network, Phys. Rev. Lett. 94, 200602 (2005).
S N Dorogovtsev, A V Goltsev, J F F Mendes, k-core organization of complex networks, Phys. Rev. Lett. 96, 17 February (2006).