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This paper is a direct continuation of an earlier work, where we studiedErd\"os-R\'enyi random graphs perturbed by an interaction Hamiltonian favouringthe formation of short cycles. Here, we generalize these results. We keep thesame interaction Hamiltonian but let it act on general graphs with uncorrelatednodes and an arbitrary given degree distribution. It is shown that the resultsobtained for Erd\"os-R\'enyi graphs are generic, at the qualitative level.However, scale-free graphs are an exception to this general rule and exhibit asingular behaviour, studied thoroughly in this paper, both analytically andnumerically.Comment: 7 pages, 7 eps figures, 2-column revtex format, references adde

Perturbing general uncorrelated networks