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Ideas, behaviors, and opinions spread through social networks. If the probability of spreading to a new individual is a non-linear function of the fraction of the individuals' affected neighbors, such a spreading process becomes a "complex contagion". This non-linearity does not typically appear with physically spreading infections, but instead can emerge when the concept that is spreading is subject to game theoretical considerations (e.g. for choices of strategy or behavior) or psychological effects such as social reinforcement and other forms of peer influence (e.g. for ideas, preferences, or opinions). Here we study how the stochastic dynamics of such complex contagions are affected by the underlying network structure. Motivated by simulations of complex epidemics on real social networks, we present a general framework for analyzing the statistics of contagions with arbitrary non-linear adoption probabilities based on the mathematical tools of population genetics. Our framework provides a unified approach that illustrates intuitively several key properties of complex contagions: stronger community structure and network sparsity can significantly enhance the spread, while broad degree distributions dampen the effect of selection. Finally, we show that some structural features can exhibit critical values that demarcate regimes where global epidemics become possible for networks of arbitrary size. Our results draw parallels between the competition of genes in a population and memes in a world of minds and ideas. Our tools provide insight into the spread of information, behaviors, and ideas via social influence, and highlight the role of macroscopic network structure in determining their fate.