Several species of yeast, including the baker's yeast Saccharomyces cerevisiae, underwent a genome duplication roughly 100 million years ago. We analyze genetic networks whose members were involved in this duplication. Many networks show detectable redundancy and strong asymmetry in their interactions. For networks of co-expressed genes, we find evidence for network partitioning whereby the paralogs appear to have formed two relatively independent subnetworks from the ancestral network. We simulate the degeneration of networks after duplication and find that a model wherein the rate of interaction loss depends on the "neighborliness" of the interacting genes produces networks with parameters similar to those seen in the real partitioned networks. We propose that the rationalization of network structure through the loss of pair-wise gene interactions after genome duplication provides a mechanism for the creation of semi-independent daughter networks through the division of ancestral functions between these daughter networks.