It has recently been demonstrated that many biological networks exhibit a "scale-free" topology, for which the probability of observing a node with a certain number of edges (k) follows a power law: i.e., p(k) approximately k(-gamma). This observation has been reproduced by evolutionary models. Here we consider the network of protein-protein interactions (PPIs) and demonstrate that two published independent measurements of these interactions produce graphs that are only weakly correlated with one another despite their strikingly similar topology. We then propose a physical model based on the fundamental principle that (de)solvation is a major physical factor in PPIs. This model reproduces not only the scale-free nature of such graphs but also a number of higher-order correlations in these networks. A key support of the model is provided by the discovery of a significant correlation between the number of interactions made by a protein and the fraction of hydrophobic residues on its surface. The model presented in this paper represents a physical model for experimentally determined PPIs that comprehensively reproduces the topological features of interaction networks. These results have profound implications for understanding not only PPIs but also other types of scale-free networks.