Most proteins do not evolve in isolation, but as components of complex genetic networks. Therefore, a protein's position in a network may indicate how central it is to cellular function and, hence, how constrained it is evolutionarily. To look for an effect of position on evolutionary rate, we examined the protein-protein interaction networks in three eukaryotes: yeast, worm, and fly. We find that the three networks have remarkably similar structure, such that the number of interactors per protein and the centrality of proteins in the networks have similar distributions. Proteins that have a more central position in all three networks, regardless of the number of direct interactors, evolve more slowly and are more likely to be essential for survival. Our results are thus consistent with a classic proposal of Fisher's that pleiotropy constrains evolution.