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The function of a protein is defined by its interaction partners. Thus, topology-driven network alignment of the protein-protein interaction (PPI) networks of two species should uncover similar interaction patterns and allow identification of functionally similar proteins. Howver, few of the fifty or more algorithms for PPI network alignment have demonstrated a significant link between network topology and functional similarity, and none have recovered orthologs using network topology alone. We find that the major contributing factors to this failure are: (i) edge densities in current PPI networks are too low to expect topological network alignment to succeed; (ii) when edge densities are high enough, some measures of topological similarity easily uncover functionally similar proteins while others do not; and (iii) most network alignment algorithms fail to optimize their own topological objective functions, hampering their ability to use topology effectively. We demonstrate that SANA-the Simulated Annealing Network Aligner-significantly outperforms existing aligners at optimizing their own objective functions, even achieving near-optimal solutions when optimal solution is known. We offer the first demonstration of global network alignments based on topology alone that align functionally similar proteins with p-values in some cases below 1e-300. We predict that topological network alignment has a bright future as edge densities increase towards the value where good alignments become possible. We demonstrate that when enough common topology is present at high enough edge densities-for example in the recent, partly synthetic networks of the Integrated Interaction Database-topological network alignment easily recovers most orthologs, paving the way towards high-throughput functional prediction based on topology-driven network alignment.