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Abstract Diverse viruses and their hosts are interconnected through complex networks of infection, which are thought to influence ecological and evolutionary processes, but the principles underlying infection network structure are not well understood. Here we focus on network dimensionality and how it varies across 37 networks of viruses infecting eukaryotic phytoplankton and bacteria. We find that dimensionality is often strikingly low, with most networks being one‐ or two‐dimensional, although dimensionality increases with network richness, suggesting that the true dimensionality of natural systems is higher. Low‐dimensional networks generally exhibit a mixture of host partitioning among viruses and nestededness of host ranges. Networks of bacteria‐infecting and eukaryote‐infecting viruses possess comparable distributions of dimensionality and prevalence of nestedness, indicating that fundamentals of network structure are similar among domains of life and different viral lineages. The relative simplicity of many infection networks suggests that coevolutionary dynamics are often driven by a modest number of underlying mechanisms.