Ecological networks such as plant-pollinator and host-parasite networks have structured interactions that define who interacts with whom. The structure of interactions also shapes ecological and evolutionary dynamics. Yet, there is significant ongoing debate as to whether certain structures, e.g., nestedness, contribute positively, negatively or not at all to biodiversity. We contend that examining variation in life history traits is key to disentangling the potential relationship between network structure and biodiversity. Here, we do so by analyzing a dynamic model of virus-bacteria interactions across a spectrum of network structures. Consistent with prior studies, we find plausible parameter domains exhibiting strong, positive relationships between nestedness and biodiversity. Yet, the same model can exhibit negative relationships between nestedness and biodiversity when examined in a distinct, plausible region of parameter space. We discuss steps towards identifying when network structure could, on its own, drive the resilience, sustainability, and even conservation of ecological communities.