Abstract
The structure of complex networks is an important aspect in the study of the real network data. Quite often, it is desirable to know the division of the network into communities. A large number of community detection algorithms have been proposed to probe the community structure of complex networks. For a specific partition of a given network, we show that the distribution of modularity under a null hypothesis of free labeling is asymptotically normal when the size of the network gets large. The significance of the partition is defined based on this asymptotic distribution, which can help assess its goodness. Two different partitions can also be compared statistically. Simulation studies and real data analyses are performed for illustration.
Similar content being viewed by others
References
Albert R, Barabási A-L (2002) Statistical mechanics of complex networks. Rev Mod Phys 74:47–97
Cliff AD, Ord JK (1981) Spatial processes: models and applications. Pion Limited, London
Csárdi G, Nepusz T (2006) The igraph software package for complex network research. InterJ Complex Syst 1695:1–9
Donetti L, Muñoz MA (2004) Detecting network communities: a new systematic and efficient algorithm. J Stat Mech Theory Exp 2004:P10012
Faust K, Wasserman S (1992) Blockmodels: interpretation and evaluation. Soc Netw 14:5–61
Fortunato S (2010) Community detection in graphs. Phys Rep 486:75–174
Geary RC (1954) The contiguity ratio and statistical mapping. Inc Stat 5:115–146
Hall P, Heyde CC (1980) Martingale limit theory and its application. Academic Press, London
Jackson MO (2010) Social and economic networks. Princeton University Press, Princeton
Lancichinetti A, Radicchi F, Ramasco JJ (2010) Statistical significance of communities in networks. Phys Rev E 81:046110
Moran PA (1950) Notes on continuous stochastic phenomena. Biometrika 37:17–23
Newman ME (2003) The structure and function of complex networks. SIAM Rev 45:167–256
Newman ME (2006) Modularity and community structure in networks. Proc Nat Acad Sci 103:8577–8582
Newman ME (2010) Networks: an introduction. Oxford University Press, Oxford
Newman ME, Girvan M (2004) Finding and evaluating community structure in networks. Phys Rev E 69:026113
Riordan O, Selby A (2000) The maximum degree of a random graph. Comb Probab Comput 9:549–572
Rosvall M, Bergstrom CT (2010) Mapping change in large networks. PLoS ONE 5:e8694
Zachary WW (1977) An information flow model for conflict and fission in small groups. J Anthropol Res 33:452–473
Zhang J, Chen Y (2016) A hypothesis testing framework for modularity based network community detection. Stat Sin 27:437–456
Acknowledgements
The authors would like to thank the Editor-in-Chief, an associate editor, and two anonymous reviewers for their valuable suggestions and comments which help improve the presentation of this paper.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
On behalf of all authors, the corresponding authors states that there is no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Li, Y., Qi, Y. Asymptotic distribution of modularity in networks. Metrika 83, 467–484 (2020). https://doi.org/10.1007/s00184-019-00740-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00184-019-00740-7