# Evolution of Robustness in Growing Random Networks

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## Abstract

**:**

## 1. Introduction

## 2. Kirchhoff Index

#### 2.1. Definitions

#### 2.2. Lower Bound on $K{f}_{1}$

## 3. Robustness of Growing Networks

#### 3.1. One New Node with a Single Connection ($m=1$)

#### Discussion

#### 3.2. One New Node with Two Connections ($m=2$)

#### 3.2.1. Discussion

#### 3.2.2. Remark

## 4. Conclusions

#### Outlook

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Evolution of the network from iteration t to $t+1$, where a new node (in red) connecting to a single existing node k (in black) has been added. The label of the new node is ${N}_{t+1}={N}_{t}+1$. No new path is created within the existing nodes.

**Figure 2.**Evolution of the Kirchhoff index divided by the number of nodes ${N}_{t}$ when a new node is connected to a single existing one at each new iteration. The initial network has ten nodes, and is obtained from a Watts–Strogatz rewiring procedure using nearest neighbors coupling [29]. The green curves correspond to twenty realizations starting from this initial network and recursively adding nodes while selecting the existing nodes to which they connect uniformly and at random. For large ${N}_{t}$, the green curves follow the scaling of Equation (14). The red and blue curves are obtained by selecting the least and most central existing nodes, respectively, in each iteration. When ${N}_{t}$ is large, the curves follow the scalings in Equations (16) and (18). The dotted, dashed, and dash-dotted black lines indicate ${N}_{t}^{2}$, ${N}_{t}\phantom{\rule{0.166667em}{0ex}}log{N}_{t}$, and ${N}_{t}$, respectively.

**Figure 3.**Evolution of the network from iteration t to $t+1$, where a new node (in red) is added that connects to two existing ones k and l (in black). The label of the new node is ${N}_{t+1}={N}_{t}+1$. In this case, a new path between k and l is created.

**Figure 4.**Evolution of the Kirchhoff index divided by the number of nodes ${N}_{t}$ when a new node is connected to two existing ones (k and l) in each new iteration. Two nodes are selected by minimizing/maximizing ${\mu}_{kl}\left(t\right)$, ${\rho}_{kl}\left(t\right)$, and ${\rho}_{kl}\left(t\right)-{\mu}_{kl}\left(t\right)$ for each new iteration. The meaning of each curve is shown in the legend. The initial network has ten nodes, and is obtained from a Watts–Strogatz rewiring procedure with nearest-neighbors coupling [29]. The black dash-dotted and dashed lines show the scalings ${N}_{t}$ and ${N}_{t}^{2}$, respectively. Note that in our simulations we ensured that $k\ne l$; however, we found similar scalings when relaxing this condition.

**Figure 5.**Evolution of the Kirchhoff index divided by the number of nodes ${N}_{t}$ when a new node is connected to two existing ones at each iteration. The two nodes are selected uniformly at random among the existing ones in each new iteration. Each grey line (twenty in total) is one realization of the process. The initial network has ten nodes and is obtained from a Watts–Strogatz rewiring procedure with nearest-neighbors coupling [29]. The black dashed line shows the linear scaling with ${N}_{t}$.

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**MDPI and ACS Style**

Tyloo, M.
Evolution of Robustness in Growing Random Networks. *Entropy* **2023**, *25*, 1340.
https://doi.org/10.3390/e25091340

**AMA Style**

Tyloo M.
Evolution of Robustness in Growing Random Networks. *Entropy*. 2023; 25(9):1340.
https://doi.org/10.3390/e25091340

**Chicago/Turabian Style**

Tyloo, Melvyn.
2023. "Evolution of Robustness in Growing Random Networks" *Entropy* 25, no. 9: 1340.
https://doi.org/10.3390/e25091340