# Computational Properties of General Indices on Random Networks

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Facultad de Ciencias Químicas, Benemérita Universidad Autónoma de Puebla, Puebla 72570, Mexico

^{2}

Departamento de Ingeniería, Universidad Popular Autónoma del Estado de Puebla, Puebla 72410, Mexico

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Departamento de Matemática Aplicada e Estatística, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo—Campus de São Carlos, Caixa Postal 668, São Carlos 13560-970, Brazil

^{4}

Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apartado Postal J-48, Puebla 72570, Mexico

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Facultad de Matemáticas, Universidad Autónoma de Guerrero, Carlos E. Adame No.54 Col. Garita, Acapulco 39650, Mexico

^{*}

Author to whom correspondence should be addressed.

^{†}

The authors contributed equally to this work.

Received: 30 June 2020 / Revised: 31 July 2020 / Accepted: 7 August 2020 / Published: 11 August 2020

(This article belongs to the Special Issue Analytical and Computational Properties of Topological Indices)

We perform a detailed (computational) scaling study of well-known general indices (the first and second variable Zagreb indices, ${M}_{1}^{\alpha}\left(G\right)$ and ${M}_{2}^{\alpha}\left(G\right)$ , and the general sum-connectivity index, ${\chi}_{\alpha}\left(G\right)$ ) as well as of general versions of indices of interest: the general inverse sum indeg index $IS{I}_{\alpha}\left(G\right)$ and the general first geometric-arithmetic index $G{A}_{\alpha}\left(G\right)$ (with $\alpha \in \mathbb{R}$ ). We apply these indices on two models of random networks: Erdös–Rényi (ER) random networks ${G}_{\mathrm{ER}}({n}_{\mathrm{ER}},p)$ and random geometric (RG) graphs ${G}_{\mathrm{RG}}({n}_{\mathrm{RG}},r)$ . The ER random networks are formed by ${n}_{\mathrm{ER}}$ vertices connected independently with probability $p\in [0,1]$ ; while the RG graphs consist of ${n}_{\mathrm{RG}}$ vertices uniformly and independently distributed on the unit square, where two vertices are connected by an edge if their Euclidean distance is less or equal than the connection radius $r\in [0,\sqrt{2}]$ . Within a statistical random matrix theory approach, we show that the average values of the indices normalized to the network size scale with the average degree $\u27e8k\u27e9$ of the corresponding random network models, where $\u27e8{k}_{\mathrm{ER}}\u27e9=({n}_{\mathrm{ER}}-1)p$ and $\u27e8{k}_{\mathrm{RG}}\u27e9=({n}_{\mathrm{RG}}-1)(\pi {r}^{2}-8{r}^{3}/3+{r}^{4}/2)$ . That is, $\u27e8X\left({G}_{\mathrm{ER}}\right)\u27e9/{n}_{\mathrm{ER}}\approx \u27e8X\left({G}_{\mathrm{RG}}\right)\u27e9/{n}_{\mathrm{RG}}$ if $\u27e8{k}_{\mathrm{ER}}\u27e9=\u27e8{k}_{\mathrm{RG}}\u27e9$ , with X representing any of the general indices listed above. With this work, we give a step forward in the scaling of topological indices since we have found a scaling law that covers different network models. Moreover, taking into account the symmetries of the topological indices we study here, we propose to establish their statistical analysis as a generic tool for studying average properties of random networks. In addition, we discuss the application of specific topological indices as complexity measures for random networks.
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*Keywords:*computational analysis of networks; general topological indices; Erdös–Rényi networks; random geometric graphs

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

**MDPI and ACS Style**

Aguilar-Sánchez, R.; Herrera-González, I.F.; Méndez-Bermúdez, J.A.; Sigarreta, J.M. Computational Properties of General Indices on Random Networks. *Symmetry* **2020**, *12*, 1341.

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