The Anna Karenina principle (AKP), applied to microbial communities, is the statement (see Holmes, et al., PMID:22319561) that disturbed states have many more configurations than undisturbed. Inspiration for the principle is found in the first sentence of Tolstoy's book Anna Karenina, "Happy families are all alike; every unhappy family is unhappy in its own way."
In studies of the human microbiome, there seem to be a small number of qualitatively similar microbial population structures, or "enterotypes", among comparatively healthy adults in industrialized nations. Holmes, et al. use the AKP to suggest that the greater microbiome variance they observe for obese twins, compared to lean twins, indicates a disturbed (unhappy) enterotype in obesity.
Holmes, et al. credit popularization of the term Anna Karenina principle to Jared Diamond's book, Guns, Germs, and Steel, but the meaning is different when applied to failure modes for endeavors such as domesticating large animals: many ways to fail, few ways to succeed.