The following discussion is an edited summary of the public debate started during the conference "Growing Networks and Graphs in Statistical Physics, Finance, Biology and Social Systems" held in Rome in September 2003. Drafts documents were circulated electronically among experts in the field and additions and follow-up to the original discussion have been included. Among the scientists participating to the discussion L. A. N. Amaral, A. Barrat, A. L. Barabasi, G. Caldarelli, P. De Los Rios, A. Erzan, B. Kahng, R. Mantegna, J. F. F. Mendes, R. Pastor-Satorras, A. Vespignani are acknowledged for their contributions and editing.