Genomic duplication-divergence processes are the primary source of new protein functions and thereby contribute to the evolutionary expansion of functional molecular networks. Yet, it is still unclear to what extent such duplication-divergence processes also restrict by construction the emerging properties of molecular networks, regardless of any specific cellular functions. We address this question, here, focusing on the evolution of protein-protein interaction (PPI) networks. We solve a general duplication-divergence model, based on the statistically necessary deletions of protein-protein interactions arising from stochastic duplications at various genomic scales, from single-gene to whole-genome duplications. Major evolutionary scenarios are shown to depend on two global parameters only: (i) a protein conservation index (M), which controls the evolutionary history of PPI networks, and (ii) a distinct topology index (M ) controlling their resulting structure. We then demonstrate that conserved, nondense networks, which are of prime biological relevance, are also necessarily scale-free by construction, irrespective of any evolutionary variations or fluctuations of the model parameters. It is shown to result from a fundamental linkage between individual protein conservation and network topology under general duplication-divergence evolution. By contrast, we find that conservation of network motifs with two or more proteins cannot be indefinitely preserved under general duplication-divergence evolution (independently from any network rewiring dynamics), in broad agreement with empirical evidence between phylogenetically distant species. All in all, these evolutionary constraints, inherent to duplication-divergence processes, appear to have largely controlled the overall topology and scale-dependent conservation of PPI networks, regardless of any specific biological function.T he primary source of new protein functions is generally considered to originate from duplication of existing genes followed by functional divergence of their duplicate copies (1-3). In fact, duplication-divergence events have occurred and continue to occur at a wide range of genomic scales, from many independent duplications of individual genes † [10 Ϫ3 fixed events per gene per million years (MY) (4)] to rare but evolutionary dramatic duplications of entire genomes [one fixed event per 100-200 MY (5)]. For instance, there have been between two and four consecutive whole-genome duplications in all major eukaryote kingdoms in the past 300-500 MY (5). This actually amounts to a more-or-less similar contribution of new genes from whole-genome duplication as from individual gene duplications [i.e., one fixed event per 100-200 MY Ӎ 10 Ϫ3 fixed events per gene per MY, assuming a 10% fixation rate after a wholegenome duplication with Ϸ10,000 genes (5)].This succession of whole-genome duplications, together with the accumulation of individual gene duplications, must have greatly contributed to shaping the global structure of large biological network...