Climate change-driven stressors threaten the persistence of coral reefs worldwide. Symbiotic relationships between scleractinian corals and photosynthetic endosymbionts (genus Symbiodinium) are the foundation of reef ecosystems, and these associations are differentially impacted by stress. Here, we couple empirical data from the coral reefs of Moorea, French Polynesia, and a network theoretic modeling approach to evaluate how patterns in coral-Symbiodinium associations influence community stability under climate change. To introduce the effect of climate perturbations, we simulate local 'extinctions' that represent either the loss of coral species or the ability to engage in symbiotic interactions. Community stability is measured by determining the duration and number of species that persist through the simulated extinctions. Our results suggest that four factors greatly increase coral-Symbiodinium community stability in response to global changes: (i) the survival of generalist hosts and symbionts maximizes potential symbiotic unions; (ii) elevated symbiont diversity provides redundant or complementary symbiotic functions; (iii) compatible symbiotic assemblages create the potential for local recolonization; and (iv) the persistence of certain traits associate with symbiotic diversity and redundancy. Symbiodinium may facilitate coral persistence through novel environmental regimes, but this capacity is mediated by symbiotic specificity, association patterns, and the functional performance of the symbionts. Our model-based approach identifies general trends and testable hypotheses in coral-Symbiodinium community responses. Future studies should consider similar methods when community size and/or environmental complexity preclude experimental approaches.