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Many physical and social systems are best described by networks. And the structural properties of these networks often critically determine the properties and function of the resulting mathematical models. An important method to infer the correlations between topology and function is the detection of community structure, which plays a key role in the analysis, design, and optimization of many complex systems. The nonnegative matrix factorization has been used prolifically to that effect in recent years, although it cannot guarantee balanced partitions, and it also does not allow a proactive computation of the number of communities in a network. This indicates that the nonnegative matrix factorization does not satisfy all the nonnegative low-rank approximation conditions. Here we show how to resolve this important open problem by optimizing the identifiability of community structure. We propose a new form of …