Peer review plays an important role in maintaining the quality of science. Selection of peers is at the heart of the process by which science advances. Editors and others responsible for selecting a group of peers often rely on their position in a network by which experts in a field are linked to one another by bonds of common interest and recognized expertise. In this paper, we report one aspect of a study aimed at characterizing the structure of this network: the asymmetry of the fraction of experts receiving varying numbers of nominations as experts by peers. The distribution of such nominations is very skew, and we have found that a law of cumulative advantage provides the best theoretical approximation for the distribution of nominations, expecially when the overall pool of data is broken down into well-defined specialties.