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Introgression is a pervasive biological process, and many statistical methods have been developed to infer its presence from genomic data. However, many of the consequences and genomic signatures of introgression remain unexplored from a methodological standpoint. Here, we develop a model for the timing and direction of introgression based on the multispecies network coalescent, and from it suggest new approaches for testing introgression hypotheses. We suggest two new statistics, D  1 and D  2 , which can be used in conjunction with other information to test hypotheses relating to the timing and direction of introgression, respectively. D  1 may find use in evaluating cases of homoploid hybrid speciation (HHS), while D  2 provides a four-taxon test for polarizing introgression. Although analytical expectations for our statistics require a number of assumptions to be met, we show how simulations can be used to test hypotheses about introgression when these assumptions are violated. We apply the D  1 statistic to genomic data from the wild yeast Saccharomyces paradoxus -a proposed example of HHS-demonstrating its use as a test of this model. These methods provide new and powerful ways to address questions relating to the timing and direction of introgression.

The Timing and Direction of Introgression Under the Multispecies Network Coalescent