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The evolution of molecular and phenotypic traits is commonly modelled using Markov processes along a phylogeny. This phylogeny can be a tree, or a network if it includes reticulations, representing events such as hybridization or admixture. Computing the likelihood of data observed at the leaves is costly as the size and complexity of the phylogeny grows. Efficient algorithms exist for trees, but cannot be applied to networks. We show that a vast array of models for trait evolution along phylogenetic networks can be reformulated as graphical models, for which efficient belief propagation algorithms exist. We provide a brief review of belief propagation on general graphical models, then focus on linear Gaussian models for continuous traits. We show how belief propagation techniques can be applied for exact or approximate (but more scalable) likelihood and gradient calculations, and prove novel results for efficient parameter inference of some models. We highlight the possible fruitful interactions between graphical models and phylogenetic methods. For example, approximate likelihood approaches have the potential to greatly reduce computational costs for phylogenies with reticulations. This article is part of the theme issue ‘“A mathematical theory of evolution”: phylogenetic models dating back 100 years’. 

Within-species trait variation may be the result of genetic variation, environmental variation, or measurement error, for example. In phylogenetic comparative studies, failing to account for within-species variation has many adverse effects, such as increased error in testing hypotheses about evolutionary correlations, biased estimates of evolutionary rates, and inaccurate inference of the mode of evolution. These adverse effects were demonstrated in studies that considered a tree-like underlying phylogeny. Comparative methods on phylogenetic networks are still in their infancy. The impact of within-species variation on network-based methods has not been studied. Here, we introduce a phylogenetic linear model in which the phylogeny can be a network to account for within-species variation in the continuous response trait assuming equal within-species variances across species. We show how inference based on the individual values can be reduced to a problem using species-level summaries, even when the within-species variance is estimated. Our method performs well under various simulation settings and is robust when within-species variances are unequal across species. When phenotypic (within-species) correlations differ from evolutionary (between-species) correlations, estimates of evolutionary coefficients are pulled towards the phenotypic coefficients for all methods we tested. Also, evolutionary rates are either underestimated or overestimated, depending on the mismatch between phenotypic and evolutionary relationships. We applied our method to morphological and geographical data from Polemonium. We find a strong negative correlation of leaflet size with elevation, despite a positive correlation within species. Our method can explore the role of gene flow in trait evolution by comparing the fit of a network to that of a tree. We find marginal evidence for leaflet size being affected by gene flow and support for previous observations on the challenges of using individual continuous traits to infer inheritance weights at reticulations. Our method is freely available in the Julia package PhyloNetworks.