Abstract
In the present paper, we study a specific partition of a given network and establish the moderate deviation principle of modularity for the partition when the size of the network gets large.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alert R, Barabási AL (2002) Statistical mechanics of complex networks. Rev Modern Phys 74(1):47–97
Bahadur R (1960) On the asymptotic efficiency of tests and estimators. Sankhyā 22:229–252
Bitseki Penda SV, Djellout H, Proïa F (2014) Moderate deviations for the Durbin–Watson statistic related to the first-order autoregressive process. ESAIM Probab Stat 18:308–331
Chernoff H (1952) A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann Math Statist 23:493–507
Dembo A, Zeitouni O (1998) Large deviations: techniques and applications, 2nd edn. Springer, New York
Djellout H (2002) Moderate deviations for martingale differences and applications to \(\phi \)-mixing sequences. Stoch Stoch Rep 73(1–2):37–63
Fortunato S (2010) Community detection in graphs. Phys Rep 486(3–5):75–174
Giné E, Latał R, Zinn J (2000) Exponential and moment inequalities for \(U\)-statistics. High Dimensional Probability, II (Seattle, WA, 1999). Birkhäuser Boston, Boston, pp 13–38
Hoeffding W (1963) Probability inequalities for sums of bounded random variables. J Am Stat Assoc 58:13–30
Jackson MO (2008) Social and economic networks. Princeton University Press, Princeton
Lancichinetti A, Radicchi F, Ramasco JJ (2010) Statistical significance of communities in networks. Phys Rev E 046110
Li Y, Qi YC (2020) Asymptotic distribution of modularity in networks. Metrika 83(4):467–484
Ma R, Barnett I (2021) The asymptotic distribution of modularity in weighted signed networks. Biometrika 108(1):1–16
Newman MEJ (2003) The structure and function of complex networks. SIAM Rev 45(2):167–256
Newman MEJ (2006) Modularity and community structure in networks. Proc Natl Acad Sci USA 103(23):8577–8582
Newman MEJ (2010) Networks: an introduction. Oxford University Press, Oxford
Newman MEJ, Girvan M (2004) Finding and evaluating community structure in networks. Phys Rev E 69:026113
Puhalskii A (1994) Large deviations of semimartingales via convergence of the predictable characteristics. Stochastics Stochastics Rep 49(1–2):27–85
Puhalskii A (1997) Large deviations of semimartingales: a maxingale problem approach. I. Limits as solutions to a maxingale problem. Stochastics Stochastics Rep 61(3–4):141–243
Puhalskii A, Spokoiny V (1998) On large-deviation efficiency in statistical inference. Bernoulli 4:203–272
Rosvall M, Bergstrom CT (2010) Mapping change in large networks. PLoS ONE 5(1):e8694
Zhang JF, Chen YJ (2017) A hypothesis testing framework for modularity based network community detection. Stat Sin 27(1):437–456
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work is supported by NSFC (11971154) and the Foundation of Young Scholar of the Educational Department of Henan province Grant 2019GGJS012.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Yin, Q., Miao, Y., Wang, Z. et al. Moderate deviation principle of modularity in network. Metrika 87, 281–298 (2024). https://doi.org/10.1007/s00184-023-00914-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00184-023-00914-4