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Background: The notion of centrality is used to identify "important" nodes in social networks. Importance of nodes is not well-defined, and many different notions exist in the literature. The challenge of defining centrality in meaningful ways when network edges can be positively or negatively weighted has not been adequately addressed in the literature. Existing centrality algorithms also have a second shortcoming, i.e., the list of the most central nodes are often clustered in a specific region of the network and are not well represented across the network. Methods: We address both by proposing Ablatio Triadum (ATria), an iterative centrality algorithm that uses the concept of "payoffs" from economic theory. Results: We compare our algorithm with other known centrality algorithms and demonstrate how ATria overcomes several of their shortcomings. We demonstrate the applicability of our algorithm to synthetic networks as well as biological networks including bacterial co-occurrence networks, sometimes referred to as microbial social networks. Conclusions: We show evidence that ATria identifies three different kinds of "important" nodes in microbial social networks with different potential roles in the community. [ABSTRACT FROM AUTHOR]