Abstract
We ascertain the modularity-like objective function whose optimization is equivalent to the maximum likelihood in annotated networks. We demonstrate that the modularity-like objective function is a linear combination of modularity and conditional entropy. In contrast with statistical inference methods, in our method, the influence of the metadata is adjustable; when its influence is strong enough, the metadata can be recovered. Conversely, when it is weak, the detection may correspond to another partition. Between the two, there is a transition. This paper provides a concept for expanding the scope of modularity methods.
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This research uses data from Add Health, a program project directed by K. M. Harris and designed by J. R. Udry, P. S. Bearman, and K. M. Harris at the University of North Carolina at Chapel Hill, and funded by grant P01-HD31921 from the Eunice Kennedy Shriver National Institute of Child Health and Human Development, with cooperative funding from 23 other federal agencies and foundations. Special acknowledgment is due R. R. Rindfuss and B. Entwisle for assistance in the original design. Information on how to obtain the Add Health data files is available on the Add Health website (http://www.cpc.unc.edu/addhealth). No direct support was received from grant P01-HD31921 for this analysis.
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Acknowledgements
This work was funded by the National Natural Science Foundation of China (Grant Nos. 11275186, 91024026, and FOM2014OF001).
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Xie, JR., Wang, BH. Modularity-like objective function in annotated networks. Front. Phys. 12, 128903 (2017). https://doi.org/10.1007/s11467-017-0657-y
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DOI: https://doi.org/10.1007/s11467-017-0657-y