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On the Rényi index of random graphs

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  • Mingao Yuan

    (North Dakota State University)

Abstract

Networks (graphs) permeate scientific fields such as biology, social science, economics, etc. Empirical studies have shown that real-world networks are often heterogeneous, that is, the degrees of nodes do not concentrate on a number. Recently, the Rényi index was tentatively used to measure network heterogeneity. However, the validity of the Rényi index in network settings is not theoretically justified. In this paper, we study this problem. We derive the limit of the Rényi index of a heterogeneous Erdös–Rényi random graph and a power-law random graph, as well as the convergence rates. Our results show that the Erdös–Rényi random graph has asymptotic Rényi index zero and the power-law random graph (highly heterogeneous) has asymptotic Rényi index one. In addition, the limit of the Rényi index increases as the graph gets more heterogeneous. These results theoretically justify the Rényi index is a reasonable statistical measure of network heterogeneity. We also evaluate the finite-sample performance of the Rényi index by simulation.

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  • Mingao Yuan, 2024. "On the Rényi index of random graphs," Statistical Papers, Springer, vol. 65(3), pages 1773-1803, May.
    • Handle: RePEc:spr:stpapr:v:65:y:2024:i:3:d:10.1007_s00362-023-01463-8
    DOI: 10.1007/s00362-023-01463-8
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    References listed on IDEAS

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    1. Peter J. Bickel & Purnamrita Sarkar, 2016. "Hypothesis testing for automated community detection in networks," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(1), pages 253-273, January.
    2. Chun-Xiao Nie, 2021. "Studying the correlation structure based on market geometry," Journal of Economic Interaction and Coordination, Springer;Society for Economic Science with Heterogeneous Interacting Agents, vol. 16(2), pages 411-441, April.
    3. Nie, Chun-Xiao & Song, Fu-Tie & Li, Sai-Ping, 2016. "Rényi indices of financial minimum spanning trees," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 444(C), pages 883-889.
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    5. Nie, Chun-Xiao & Song, Fu-Tie, 2019. "Global Rényi index of the distance matrix," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 514(C), pages 902-915.
    6. Eliazar, Iddo, 2011. "Randomness, evenness, and Rényi’s index," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(11), pages 1982-1990.
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    Cited by:

    1. Wenli Deng & Jinglong Wang & Xianyi Wu & Huan Xi, 2025. "A novel goodness of fit test for the truncated and non-truncated Yule distributions," Statistical Papers, Springer, vol. 66(4), pages 1-18, June.

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