Abstract
Summary statistics play an important role in network data analysis. They can provide us with meaningful insight into the structure of a network. The Randić index is one of the most popular network statistics that has been widely used for quantifying information of biological networks, chemical networks, pharmacologic networks, etc. A topic of current interest is to find bounds or limits of the Randić index and its variants. A number of bounds of the indices are available in literature. Recently, there are several attempts to study the limits of the indices in the Erdős–Rényi random graph by simulation. In this paper, we shall derive the limits of the Randić index and its variants of an inhomogeneous Erdős–Rényi random graph. Our results charaterize how network heterogeneity affects the indices and provide new insights about the Randić index and its variants. Finally we apply the indices to several real-world networks.
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References
Abbe E (2018) Community detection and stochastic block models: recent developments. J Mach Learn Res 18:1–86
Bickel PJ, Sarkar P (2016) Hypothesis testing for automated community detection in networks. J R Stat Soc Ser B 78:253–273
Bollobás B, Erdos P (1998) Graphs of extremal weights. Ars Comb 50:225–233
Bollobás B, Erdős P, Sarkar A (1999) Extremal graphs for weights. Discrete Math 200:5–19
Bonchev D, Trinajstic N (1978) On topological characterization of molecular branching. Int J Quantum Chem Quantum Chem Symp 12:293–303
Cavers M, Fallat S, Kirkland S (2010) On the normalized Laplacian energy and general Randić index \(r_1\) of graphs. Linear Algebra Appl 433:172–190
Chakrabarty A, Hazra SR, Hollander FD, Sfragara M (2020) Large deviation principle for the maximal eigenvalue of inhomogeneous Erdős-Rényi random graphs. J Theor Probab. https://doi.org/10.1007/s10959-021-01138-w
Chakrabarty A, Chakrabarty S, Hazra RS (2020) Eigenvalues outside the bulk of of inhomogeneous Erdős-Rényi random graphs. J Stat Phys 181:1746–1780
Chakrabarty A, Hazra SR, Hollander FD, Sfragara M (2021) Spectra of adjacency and Laplacian matrices of inhomogeneous Erdős-Rényi random graphs. Random Matrices Theory Appl 10(1):215009
Chiasserini CF, Garetto M, Leonardi E (2016) Social network de-anonymization under scale-free user relations. IEEE/ACM Trans Netw 24(6):3756–3769
Das KC, Sun S, Gutman I (2017) Normalized Laplacian eigenvalues and Randić energy of graphs. MATCH Commun Math Comput Chem 77:45–59
Dattola S et al (2021) Testing graph robustness indexes for EEG analysis in Alzheimer’s disease diagnosis. Electronics 10:1440
De Meo P et al (2018) Estimating graph robustness through the Randic index. IEEE Trans Cybern 48(11):3232–3242
Doslic T et al (2020) On generalized Zagreb indices of random graphs. MATCH Commun Math Comput Chem 84:499–511
Estrada E (2010) Quantifying network heterogeneity. Phys Rev E 82:066102
Fajtlowicz S (1987) On conjectures of Graffiti-II. Congr Numer 60:187–197
Favaron O, Mahéo M, Saclé JF (1993) Some eigenvalue properties in graphs (conjectures of Graffiti-II). Discrete Math 111:197–220
Fourches D, Tropsha A (2013) Using graph indices for the analysis and comparison of chemical datasets. Mol Inform 32(9–10):827–842
Goldenberg A et al (2010) A survey of statistical network models. Found Trends Mach Learn 2(2):129–233
Kolaczyk E (2009) Statistical analysis of network data. Springer, Berlin
Li X, Shi Y (2008) A survey on the Randic index. MATCH Commun Math Comput Chem 59:127–156
Li S, Shi L, Gao W (2021) Two modified Zagreb indices for random structures. Main Group Met Chem 44:150–156
Ma YD et al (2018) From the connectivity index to various Randic-type descriptors. MATCH Commun Math Comput Chem 80:85–106
Martinez-Martinez CT, Mendez-Bermudez JA, Rodriguez J, Sigarreta J (2020) Computational and analytical studies of the Randić index in Erdős-Rényi models. Appl Math Comput 377:125137
Martinez-Martinez CT, Mendez-Bermudez JA, Rodriguez J, Sigarreta J (2021) Computational and analytical studies of the harmonic index on Erdős-Rényi models. MATCH Commun Math Comput Chem 85:395–426
Newman M (2009) Networks: an introduction. Oxford University Press, Oxford
Nikolic S et al (2003) The Zagreb indices 30 years after. Croat Chem Acta 76:113–124
Randić M (1975) Characterization of molecular branching. J Am Chem Soc 97(23):6609–6615
Randić M (2008) On history of the Randić index and emerging hostility toward chemical graph theory. MATCH Commun Math Comput Chem 59:5–124
Randic M, Novi CM, Plavsic D (2016) Solved and unsolved problems in structural chemistry. CRC Press, Boca Raton
RodrIguez JM, Sigarreta JM (2017) New results on the harmonic index and its generalizations. MATCH Commun Math Comput Chem 78:387–404
Yu L, Xu J, Lin X (2021) The power of D-hops in matching power-law graphs. Proc ACM Meas Anal Comput Syst 5(2):1–43
Zhong L (2012) The harmonic index for graphs. Appl Math Lett 25:561–566
Zhou B, Trinajstic N (2009) On a novel connectivity index. J Math Chem 46:1252–1270
Zhou B, Trinajstic N (2010) On general sum-connectivity index. J Math Chem 47:210–218
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Yuan, M. On the Randić index and its variants of network data. TEST 33, 155–179 (2024). https://doi.org/10.1007/s11749-023-00887-6
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DOI: https://doi.org/10.1007/s11749-023-00887-6