# Computational Properties of General Indices on Random Networks

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## Abstract

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## 1. Introduction

## 2. Erdös–Rényi Random Networks

#### 2.1. Computational Properties of General Indices on Erdös–Rényi Random Networks

**(i)**- The curves of $\u2329{M}_{1}^{\alpha}\left({G}_{\mathrm{ER}}\right)\u232a$, $\u2329{M}_{2}^{\alpha}\left({G}_{\mathrm{ER}}\right)\u232a$, $\u2329{\chi}_{\alpha}\left({G}_{\mathrm{ER}}\right)\u232a$ and $\u2329IS{I}_{\alpha}\left({G}_{\mathrm{ER}}\right)\u232a$ show three different behaviors as a function of p depending on the value of $\alpha $: for $\alpha <{\alpha}_{0}$, they grow for small p, approach a maximum value and then decrease when p is further increased. For $\alpha >{\alpha}_{0}$, they are monotonically increasing functions of p. For $\alpha ={\alpha}_{0}$ the curves saturate above a given value of p. See Figure 1a, Figure 2a, Figure 3a and Figure 4a.
**(ii)**- ${\alpha}_{0}=0$ for $\u2329{M}_{1}^{\alpha}\left({G}_{\mathrm{ER}}\right)\u232a$, where $\u2329{M}_{1}^{{\alpha}_{0}}\left({G}_{\mathrm{ER}}\right)\u232a$ is the average number of non-isolated vertices $\u2329{V}_{\times}\left({G}_{\mathrm{ER}}\right)\u232a$; ${\alpha}_{0}=-1/2$ for $\u2329{M}_{2}^{\alpha}\left({G}_{\mathrm{ER}}\right)\u232a$, where $\u2329{M}_{2}^{{\alpha}_{0}}\left({G}_{\mathrm{ER}}\right)\u232a$ is the average Randic index $\u2329R\left({G}_{\mathrm{ER}}\right)\u232a$; ${\alpha}_{0}=-1$ for $\u2329{\chi}_{\alpha}\left({G}_{\mathrm{ER}}\right)\u232a$, where $2\u2329{\chi}_{-1}\left({G}_{\mathrm{ER}}\right)\u232a$ is the average Harmonic index $\u2329H\left({G}_{\mathrm{ER}}\right)\u232a$; and ${\alpha}_{0}=-1$ for $\u2329IS{I}_{\alpha}\left({G}_{\mathrm{ER}}\right)\u232a$, so $\u2329IS{I}_{-1}\left({G}_{\mathrm{ER}}\right)\u232a=\u2329{V}_{\times}\left({G}_{\mathrm{ER}}\right)\u232a$.
**(iii)**- All curves of $\u2329G{A}_{\alpha}\left({G}_{\mathrm{ER}}\right)\u232a$ grow linearly with p for all $\alpha $ and ${n}_{\mathrm{ER}}$; see the magenta dashed line in Figure 5a, plotted to guide the eye.
**(iv)**- For $\alpha =0$, see the red curves in Figure 2a, Figure 3a, Figure 4a and Figure 5a, $\u2329{M}_{2}^{0}\left({G}_{\mathrm{ER}}\right)\u232a$, $\u2329{\chi}_{0}\left({G}_{\mathrm{ER}}\right)\u232a$, $\u2329IS{I}_{0}\left({G}_{\mathrm{ER}}\right)\u232a$ and $\u2329G{A}_{0}\left({G}_{\mathrm{ER}}\right)\u232a$ give the average number of edges of the ER random network. That is$$\u2329{M}_{2}^{0}\left({G}_{\mathrm{ER}}\right)\u232a=\u2329{\chi}_{0}\left({G}_{\mathrm{ER}}\right)\u232a=\u2329IS{I}_{0}\left({G}_{\mathrm{ER}}\right)\u232a=\u2329G{A}_{0}\left({G}_{\mathrm{ER}}\right)\u232a=\frac{{n}_{\mathrm{ER}}({n}_{\mathrm{ER}}-1)}{2}p\phantom{\rule{4pt}{0ex}}.$$
**(v)**- When ${n}_{\mathrm{ER}}p\gg 1$, we can write ${k}_{u}\approx {k}_{v}\approx \u2329{k}_{\mathrm{ER}}\u232a$ in Equations (1)–(5), with$$\u2329{k}_{\mathrm{ER}}\u232a=({n}_{\mathrm{ER}}-1)p.$$Therefore, for ${n}_{\mathrm{ER}}p\gg 1$, the average values of the indices we are computing here are well approximated by:$$\u2329{M}_{1}^{\alpha}\left({G}_{\mathrm{ER}}\right)\u232a={n}_{\mathrm{ER}}{\left[({n}_{\mathrm{ER}}-1)p\right]}^{\alpha},$$$$\u2329{M}_{2}^{\alpha}\left({G}_{\mathrm{ER}}\right)\u232a=\frac{{n}_{\mathrm{ER}}}{2}{\left[({n}_{\mathrm{ER}}-1)p\right]}^{1+2\alpha},$$$$\u2329{\chi}_{\alpha}\left({G}_{\mathrm{ER}}\right)\u232a=\frac{{n}_{\mathrm{ER}}}{{2}^{1-\alpha}}{\left[({n}_{\mathrm{ER}}-1)p\right]}^{1+\alpha},$$$$\u2329IS{I}_{\alpha}\left({G}_{\mathrm{ER}}\right)\u232a={n}_{\mathrm{ER}}{\left[\frac{({n}_{ER}-1)p}{2}\right]}^{1+\alpha},$$$$\u2329G{A}_{\alpha}\left({G}_{\mathrm{ER}}\right)\u232a=\frac{{n}_{\mathrm{ER}}({n}_{\mathrm{ER}}-1)p}{{2}^{1+2\alpha}}.$$

#### 2.2. Scaling Properties of General Indices on Erdös–Rényi Random Networks

## 3. Random Geometric Graphs

#### 3.1. The Average Degree of Random Geometric Graphs

#### 3.2. Computational Properties of General Indices on Random Geometric Graphs

**(i–v)**made in the previous section for ER random networks are also valid for RG graphs. Even though most statements are applicable to RG graphs by just replacing ${G}_{\mathrm{ER}}\to {G}_{\mathrm{RG}}$ and $p\to f\left(r\right)$, given the fact that this is the first study (to our knowledge) of average topological indices on RG graphs, we want to explicitly write three statements we consider relevant:

**(iii’)**- The curves $\u2329G{A}_{\alpha}\left({G}_{\mathrm{RG}}\right)\u232a$ grow with r, for all $\alpha $ and ${n}_{\mathrm{RG}}$, as $\u2329G{A}_{\alpha}\left({G}_{\mathrm{RG}}\right)\u232a\propto f\left(r\right)$; see the magenta dashed line in Figure 11a, plotted to guide the eye.
**(iv’)**- For $\alpha =0$, see the red curves in Figure 8a, Figure 9a, Figure 10a and Figure 11a, $\u2329{M}_{2}^{0}\left({G}_{\mathrm{RG}}\right)\u232a$, $\u2329{\chi}_{0}\left({G}_{\mathrm{RG}}\right)\u232a$, $\u2329IS{I}_{0}\left({G}_{\mathrm{RG}}\right)\u232a$ and $\u2329G{A}_{0}\left({G}_{\mathrm{RG}}\right)\u232a$ give the average number of edges of the RG graph. That is$$\u2329{M}_{2}^{0}\left({G}_{\mathrm{RG}}\right)\u232a=\u2329{\chi}_{0}\left({G}_{\mathrm{RG}}\right)\u232a=\u2329IS{I}_{0}\left({G}_{\mathrm{RG}}\right)\u232a=\u2329G{A}_{0}\left({G}_{\mathrm{RG}}\right)\u232a=\frac{{n}_{\mathrm{RG}}({n}_{\mathrm{RG}}-1)}{2}f\left(r\right)\phantom{\rule{4pt}{0ex}}.$$
**(v’)**- When ${n}_{\mathrm{RG}}r\gg 1$, we can write ${k}_{u}\approx {k}_{v}\approx \u2329{k}_{\mathrm{RG}}\u232a=({n}_{\mathrm{RG}}-1)f\left(r\right)$ in Equations (1)–(5). Therefore, for ${n}_{\mathrm{RG}}r\gg 1$, the average values of the indices we are computing here are well approximated by:$$\u2329{M}_{1}^{\alpha}\left({G}_{\mathrm{RG}}\right)\u232a={n}_{\mathrm{RG}}{\left[({n}_{\mathrm{RG}}-1)f\left(r\right)\right]}^{\alpha},$$$$\u2329{M}_{2}^{\alpha}\left({G}_{\mathrm{RG}}\right)\u232a=\frac{{n}_{\mathrm{RG}}}{2}{\left[({n}_{\mathrm{RG}}-1)f\left(r\right)\right]}^{1+2\alpha},$$$$\u2329{\chi}_{\alpha}\left({G}_{\mathrm{RG}}\right)\u232a=\frac{{n}_{\mathrm{RG}}}{{2}^{1-\alpha}}{\left[({n}_{\mathrm{RG}}-1)f\left(r\right)\right]}^{1+\alpha},$$$$\u2329IS{I}_{\alpha}\left({G}_{\mathrm{RG}}\right)\u232a={n}_{\mathrm{RG}}{\left[\frac{({n}_{RG}-1)f\left(r\right)}{2}\right]}^{1+\alpha},$$$$\u2329G{A}_{\alpha}\left({G}_{\mathrm{RG}}\right)\u232a=\frac{{n}_{\mathrm{RG}}({n}_{\mathrm{RG}}-1)f\left(r\right)}{{2}^{1+2\alpha}}.$$

#### 3.3. Scaling Properties of General Indices on Random Geometric Graphs

## 4. Discussion and Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**,

**b**) Average first variable Zagreb index $\u2329{M}_{1}^{\alpha}\left({G}_{\mathrm{ER}}\right)\u232a$ as a function of the probability p of ER random networks of size ${n}_{\mathrm{ER}}=1000$. Here we show curves for $\alpha \in [-2,2]$ in steps of $0.2$ (from bottom to top). The red (blue) curve in (

**a**) corresponds to $\alpha =0$ [$\alpha =1$]. The red dashed lines in (

**b**) are Equation (8). The blue dashed line in (

**b**) marks $\u2329{k}_{\mathrm{ER}}\u232a=10$. (

**c**) $\u2329{M}_{1}^{\alpha}\left({G}_{\mathrm{ER}}\right)\u232a$ as a function of the probability p of ER random networks of four different sizes ${n}_{\mathrm{ER}}$. (

**d**) $\u2329{M}_{1}^{\alpha}\left({G}_{\mathrm{ER}}\right)\u232a/{n}_{\mathrm{ER}}$ as a function of the average degree $\u2329{k}_{\mathrm{ER}}\u232a$. Same curves as in panel (

**c**). The inset in (

**d**) is the enlargement of the cyan rectangle.

**Figure 2.**(

**a**,

**b**) Average second variable Zagreb index $\u2329{M}_{2}^{\alpha}\left({G}_{\mathrm{ER}}\right)\u232a$ as a function of the probability p of ER random networks of size ${n}_{\mathrm{ER}}=1000$. Here we show curves for $\alpha \in [-2,2]$ in steps of $0.2$ (from bottom to top). The red (blue) curve in (

**a**) corresponds to $\alpha =0$ [$\alpha =1$]. The red dashed lines in (

**b**) are Equation (9). The blue dashed line in (

**b**) marks $\u2329{k}_{\mathrm{ER}}\u232a=10$. (

**c**) $\u2329{M}_{2}^{\alpha}\left({G}_{\mathrm{ER}}\right)\u232a$ as a function of the probability p of ER random networks of four different sizes ${n}_{\mathrm{ER}}$. (

**d**) $\u2329{M}_{2}^{\alpha}\left({G}_{\mathrm{ER}}\right)\u232a/{n}_{\mathrm{ER}}$ as a function of the average degree $\u2329{k}_{\mathrm{ER}}\u232a$. Same curves as in panel (

**c**). The inset in (

**d**) is the enlargement of the cyan rectangle.

**Figure 3.**(

**a**,

**b**) Average general sum-connectivity index $\u2329{\chi}_{\alpha}\left({G}_{\mathrm{ER}}\right)\u232a$ as a function of the probability p of ER random networks of size ${n}_{\mathrm{ER}}=1000$. Here we show curves for $\alpha \in [-2,2]$ in steps of $0.2$ (from bottom to top). The red (blue) curve in (

**a**) corresponds to $\alpha =0$ [$\alpha =1$]. The red dashed lines in (

**b**) are Equation (10). The blue dashed line in (

**b**) marks $\u2329{k}_{\mathrm{ER}}\u232a=10$. (

**c**) $\u2329{\chi}_{\alpha}\left({G}_{\mathrm{ER}}\right)\u232a$ as a function of the probability p of ER random networks of four different sizes ${n}_{\mathrm{ER}}$. (

**d**) $\u2329{\chi}_{\alpha}\left({G}_{\mathrm{ER}}\right)\u232a/{n}_{\mathrm{ER}}$ as a function of the average degree $\u2329{k}_{\mathrm{ER}}\u232a$. Same curves as in panel (

**c**). The inset in (

**d**) is the enlargement of the cyan rectangle.

**Figure 4.**(

**a**,

**b**) Average general inverse sum indeg index $\u2329IS{I}_{\alpha}\left({G}_{\mathrm{ER}}\right)\u232a$ as a function of the probability p of ER random networks of size ${n}_{\mathrm{ER}}=1000$. Here we show curves for $\alpha \in [-2,2]$ in steps of $0.2$ (from bottom to top). The red (blue) curve in (

**a**) corresponds to $\alpha =0$ [$\alpha =1$]. The red dashed lines in (

**b**) are Equation (11). The blue dashed line in (

**b**) marks $\u2329{k}_{\mathrm{ER}}\u232a=10$. (

**c**) $\u2329IS{I}_{\alpha}\left({G}_{\mathrm{ER}}\right)\u232a$ as a function of the probability p of ER random networks of four different sizes ${n}_{\mathrm{ER}}$. (

**d**) $\u2329IS{I}_{\alpha}\left({G}_{\mathrm{ER}}\right)\u232a/{n}_{\mathrm{ER}}$ as a function of the average degree $\u2329{k}_{\mathrm{ER}}\u232a$. Same curves as in panel (

**c**). The inset in (

**d**) is the enlargement of the cyan rectangle.

**Figure 5.**(

**a**,

**b**) Average general first geometric-arithmetic index $\u2329G{A}_{\alpha}\left({G}_{\mathrm{ER}}\right)\u232a$ as a function of the probability p of ER random networks of size ${n}_{\mathrm{ER}}=1000$. Here we show curves for $\alpha \in [-2,2]$ in steps of $0.4$ (from bottom to top). The red (blue) curve in (

**a**) corresponds to $\alpha =0$ [$\alpha =1$]. The red dashed lines in (

**b**) are Equation (12). The blue dashed line in (

**b**) marks $\u2329{k}_{\mathrm{ER}}\u232a=10$. The magenta dashed line in (

**a**) proportional to p is shown to guide the eye. (

**c**) $\u2329G{A}_{\alpha}\left({G}_{\mathrm{ER}}\right)\u232a$ as a function of the probability p of ER random networks of four different sizes ${n}_{\mathrm{ER}}$. (

**d**) $\u2329G{A}_{\alpha}\left({G}_{\mathrm{ER}}\right)\u232a/{n}_{\mathrm{ER}}$ as a function of the average degree $\u2329{k}_{\mathrm{ER}}\u232a$. Same curves as in panel (

**c**).

**Figure 6.**(

**a**) Average index $\u2329{M}_{1}^{1}\left({G}_{\mathrm{RG}}\right)\u232a/{n}_{\mathrm{RG}}$ as a function of the connection radius r of RG graphs of four different sizes ${n}_{\mathrm{RG}}$ (symbols). Full lines correspond to Equation (19). (

**b**) $\u2329{M}_{1}^{1}\left({G}_{\mathrm{RG}}\right)\u232a/{n}_{\mathrm{RG}}$ as a function of the average degree $\u2329{k}_{\mathrm{RG}}\u232a$. Same data as in panel (

**a**).

**Figure 7.**(

**a**,

**b**) Average first variable Zagreb index $\u2329{M}_{1}^{\alpha}\left({G}_{\mathrm{RG}}\right)\u232a$ as a function of the connection radius r of RG graphs of size ${n}_{\mathrm{RG}}=1000$. Here we show curves for $\alpha \in [-2,2]$ in steps of $0.2$ (from bottom to top). The red (blue) curve in (

**a**) corresponds to $\alpha =0$ [$\alpha =1$]. The red dashed lines in (

**b**) are Equation (21). The blue dashed line in (

**b**) marks $\u2329{k}_{\mathrm{RG}}\u232a\approx 10$. (

**c**) $\u2329{M}_{1}^{\alpha}\left({G}_{\mathrm{RG}}\right)\u232a$ as a function of the connection radius r of RG graphs of four different sizes ${n}_{\mathrm{RG}}$. (

**d**) $\u2329{M}_{1}^{\alpha}\left({G}_{\mathrm{RG}}\right)\u232a/{n}_{\mathrm{RG}}$ as a function of the average degree $\u2329{k}_{\mathrm{RG}}\u232a$. Same curves as in panel (

**c**). The inset in (

**d**) is the enlargement of the cyan rectangle.

**Figure 8.**(

**a**,

**b**) Average second variable Zagreb index $\u2329{M}_{2}^{\alpha}\left({G}_{\mathrm{RG}}\right)\u232a$ as a function of the connection radius r of RG graphs of size ${n}_{\mathrm{RG}}=1000$. Here we show curves for $\alpha \in [-2,2]$ in steps of $0.2$ (from bottom to top). The red (blue) curve in (

**a**) corresponds to $\alpha =0$ [$\alpha =1$]. The red dashed lines in (

**b**) are Equation (22). The blue dashed line in (

**b**) marks $\u2329{k}_{\mathrm{RG}}\u232a\approx 10$. (

**c**) $\u2329{M}_{2}^{\alpha}\left({G}_{\mathrm{RG}}\right)\u232a$ as a function of the connection radius r of RG graphs of four different sizes ${n}_{\mathrm{RG}}$. (

**d**) $\u2329{M}_{2}^{\alpha}\left({G}_{\mathrm{RG}}\right)\u232a/{n}_{\mathrm{RG}}$ as a function of the average degree $\u2329{k}_{\mathrm{RG}}\u232a$. Same curves as in panel (

**c**). The inset in (

**d**) is the enlargement of the cyan rectangle.

**Figure 9.**(

**a**,

**b**) Average general sum-connectivity index $\u2329{\chi}_{\alpha}\left({G}_{\mathrm{RG}}\right)\u232a$ as a function of the connection radius r of RG graphs of size ${n}_{\mathrm{RG}}=1000$. Here we show curves for $\alpha \in [-2,2]$ in steps of $0.2$ (from bottom to top). The red (blue) curve in (

**a**) corresponds to $\alpha =0$ [$\alpha =1$]. The red dashed lines in (

**b**) are Equation (23). The blue dashed line in (

**b**) marks $\u2329{k}_{\mathrm{RG}}\u232a\approx 10$. (

**c**) $\u2329{\chi}_{\alpha}\left({G}_{\mathrm{RG}}\right)\u232a$ as a function of the connection radius r of RG graphs of four different sizes ${n}_{\mathrm{RG}}$. (

**d**) $\u2329{\chi}_{\alpha}\left({G}_{\mathrm{RG}}\right)\u232a/{n}_{\mathrm{RG}}$ as a function of the average degree $\u2329{k}_{\mathrm{RG}}\u232a$. Same curves as in panel (

**c**). The inset in (

**d**) is the enlargement of the cyan rectangle.

**Figure 10.**(

**a**,

**b**) Average general inverse sum indeg index $\u2329IS{I}_{\alpha}\left({G}_{\mathrm{RG}}\right)\u232a$ as a function of the connection radius r of RG graphs of size ${n}_{\mathrm{RG}}=1000$. Here we show curves for $\alpha \in [-2,2]$ in steps of $0.2$ (from bottom to top). The red (blue) curve in (

**a**) corresponds to $\alpha =0$ [$\alpha =1$]. The red dashed lines in (

**b**) are Equation (24). The blue dashed line in (

**b**) marks $\u2329{k}_{\mathrm{RG}}\u232a\approx 10$. (

**c**) $\u2329IS{I}_{\alpha}\left({G}_{\mathrm{RG}}\right)\u232a$ as a function of the connection radius r of RG graphs of four different sizes ${n}_{\mathrm{RG}}$. (

**d**) $\u2329IS{I}_{\alpha}\left({G}_{\mathrm{RG}}\right)\u232a/{n}_{\mathrm{RG}}$ as a function of the average degree $\u2329{k}_{\mathrm{RG}}\u232a$. Same curves as in panel (

**c**). The inset in (

**d**) is the enlargement of the cyan rectangle.

**Figure 11.**(

**a**,

**b**) Average general first geometric-arithmetic index $\u2329G{A}_{\alpha}\left({G}_{\mathrm{RG}}\right)\u232a$ as a function of the connection radius r of RG graphs of size ${n}_{\mathrm{RG}}=1000$. Here we show curves for $\alpha \in [-2,2]$ in steps of $0.4$ (from bottom to top). The red curve in (a) corresponds to $\alpha =0$. The red dashed lines in (

**b**) are Equation (25). The blue dashed line in (

**b**) marks $\u2329{k}_{\mathrm{RG}}\u232a\approx 10$. The magenta dashed line in (

**a**) proportional to $f\left(r\right)$ is shown to guide the eye. (

**c**) $\u2329G{A}_{\alpha}\left({G}_{\mathrm{RG}}\right)\u232a$ as a function of the connection radius r of RG graphs of four different sizes ${n}_{\mathrm{RG}}$. (

**d**) $\u2329G{A}_{\alpha}\left({G}_{\mathrm{RG}}\right)\u232a/{n}_{\mathrm{RG}}$ as a function of the average degree $\u2329{k}_{\mathrm{RG}}\u232a$. Same curves as in panel (

**c**).

**Figure 12.**(

**a**) $\u2329{M}_{1}^{\alpha}\left(G\right)\u232a/n$, (

**b**) $\u2329{M}_{2}^{\alpha}\left(G\right)\u232a/n$, (

**c**) $\u2329{\chi}_{\alpha}\left(G\right)\u232a/n$ and (

**d**) $\u2329IS{I}_{\alpha}\left(G\right)\u232a/n$ as a function of the average degree $\u2329k\u232a$ for ER random networks (red lines) and RG graphs (black lines). We choose $n=1000$ for ER random networks and $n=250$ for RG graphs.

**Figure 13.**(

**a**) $\u2329{M}_{1}^{0}\left(G\right)\u232a/n$, (

**b**) $\u2329{M}_{2}^{-1/2}\left(G\right)\u232a/n$, (

**c**) $\u2329{\chi}_{-1}\left(G\right)\u232a/n$ and (

**d**) $\u2329IS{I}_{-1}\left(G\right)\u232a/n$ as a function of the average degree $\u2329k\u232a$ for ER random networks (red lines) and RG graphs (black lines). We choose $n=1000$ for both ER random networks and RG graphs. Magenta dashed lines indicate $\u2329k\u232a=1/10$ and $\u2329k\u232a=10$.

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**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.
https://doi.org/10.3390/sym12081341

**AMA Style**

Aguilar-Sánchez R, Herrera-González IF, Méndez-Bermúdez JA, Sigarreta JM.
Computational Properties of General Indices on Random Networks. *Symmetry*. 2020; 12(8):1341.
https://doi.org/10.3390/sym12081341

**Chicago/Turabian Style**

Aguilar-Sánchez, R., I. F. Herrera-González, J. A. Méndez-Bermúdez, and José M. Sigarreta.
2020. "Computational Properties of General Indices on Random Networks" *Symmetry* 12, no. 8: 1341.
https://doi.org/10.3390/sym12081341