Evolution of Networks

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New results:

• An extensive review about of critical phenomena in complex networks and in interacting systems on them
Note that the cond-mat version of the review is essentially more detailed and comprehensive than the Rev. Mod. Phys. version.
Critical phenomena in complex networks,
A.V. Goltsev, S.N. Dorogovtsev, and J.F.F. Mendes, Rev. Mod. Phys. , (2008); arXiv:0705.0010 [cond-mat].

• Transition from infinite- to finite-dimensional system
We have analytically resolved this non-trivial problem for a generalization of random recursive trees where a probability of attachment to an existing vertex depends on its age.
Transition from small to large world in growing networks,
S. N. Dorogovtsev, P. L. Krapivsky, and J. F. F. Mendes, EPL 81, 30004 1-5 (2008); arXiv:0709.3094 [cond-mat].

• Laplacian spectra of complex networks and random walks on them
We resolved this problem for sparse uncorrelated networks. The results --- the Laplacian spectrum near the lower boundary and the asymptotics of the autocorrelator --- are determined by the minimum degrees in a network. Unfortunately, these asymptotics can be observed only in astronomically large networks.
Laplacian spectra of complex networks and random walks on them: Are scale-free architectures really important?,
A. N. Samukhin, S. N. Dorogovtsev, and J. F. F. Mendes, Phys. Rev. E , (2008); arXiv:0706.1176 [cond-mat].

Sergey Dorogovtsev