Epidemic thresholds on scale-free graphs: the interplay between exponent   and preferential choice
  We show for a model of scale-free graphs with biased partner choice that knowing the exponent for the degree distribution is in general not sufficient to decide epidemic threshold properties for exponents less than three.We show that the connectivity between the high degree vertices and therefore the diameter is the relevant geometric quantity for epidemic threshold estimations.Absence of epidemic threshold happens precisely when a positive fraction of the nodes form a cluster of bounded diameter. 
Ph. Blanchard, Ch.-H. Chang, T. Krueger