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The tree of blobs of a species network shows only the tree-like aspects of relationships of taxa on a network, omitting information on network substructures where hybridization or other types of lateral transfer of genetic information occur. By isolating such regions of a network, inference of the tree of blobs can serve as a starting point for a more detailed investigation, or indicate the limit of what may be inferrable without additional assumptions. Building on our theoretical work on the identifiability of the tree of blobs from gene quartet distributions under the Network Multispecies Coalescent model, we develop an algorithm, TINNiK, for statistically consistent tree of blobs inference. We provide examples of its application to both simulated and empirical datasets, utilizing an implementation in the MSCquartets 2.0 R package. 

Reticulations in a phylogenetic network represent processes such as gene flow, admixture, recombination and hybrid speciation. Extending definitions from the tree setting, an anomalous network is one in which some unrooted tree topology displayed in the network appears in gene trees with a lower frequency than a tree not displayed in the network. We investigate anomalous networks under the Network Multispecies Coalescent Model with possible correlated inheritance at reticulations. Focusing on subsets of 4 taxa, we describe a new algorithm to calculate quartet concordance factors on networks of any level, faster than previous algorithms because of its focus on 4 taxa. We then study topological properties required for a 4-taxon network to be anomalous, uncovering the key role of 32-cycles: cycles of 3 edges parent to a sister group of 2 taxa. Under the model of common inheritance, that is, when each gene tree coalesces within a species tree displayed in the network, we prove that 4-taxon networks are never anomalous. Under independent and various levels of correlated inheritance, we use simulations under realistic parameters to quantify the prevalence of anomalous 4-taxon networks, finding that truly anomalous networks are rare. At the same time, however, we find a significant fraction of networks close enough to the anomaly zone to appear anomalous, when considering the quartet concordance factors observed from a few hundred genes. These apparent anomalies may challenge network inference methods. 

As genomic-scale datasets motivate research on species tree inference, simulators of the multispecies coalescent (MSC) process have become essential for the testing and evaluation of new inference methods. However, the simulators themselves must be tested to ensure that they give valid samples. This work develops methods for checking whether a collection of gene trees is in accord with the MSC model on a given species tree. When applied to well-known simulators, we find that several give flawed samples. The tests presented are capable of validating both topological and metric properties of gene tree samples, and are implemented in a freely available R package MSCsimtester so that developers and users may easily apply them.