^{1}Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma, Parma, Italy^{2}Istituto Nazionale di Fisica Nucleare, Gruppo Collegato di Parma, Parma, Italy^{3}Dipartimento di Fisica, Politecnico di Milano, Milano, Italy^{4}Center for Nano Science and Technology@PoliMi, Istituto Italiano di Tecnologia, Milan, Italy^{5}Faculty of Fundamental Sciences, Van Lang University, Ho Chi Minh City, Vietnam^{6}Department of Physics, International University, Ho Chi Minh City, Vietnam^{7}Vietnam National University, Ho Chi Minh City, Vietnam

In this study, we simulate the degree and betweenness node attack over a large set of 200 real-world networks from different areas of science. We perform an initial node attack approach, where the node centrality rank is computed at the beginning of the simulation, and it is not updated along the node removal process. We quantify the network damage by tracing the largest connected component (

## 1 Introduction

Networks can model many real-world complex systems, where nodes (vertices) represent the constituent components and links (edges) describe the relationships among the node components [1, 2]. A paramount issue in complex network science is to determine the robustness of the overall system to the failure or attack of its nodes [3–10]. On the other hand, the robustness in complex networks is a problem closely related to understanding which kind of node removal (attack) strategy is the most effective in damaging the network [3, 11–14]. The node attack may model different real-world problems of high interest, such as the nodes/species extinction in ecological networks [15–17], the aging of nodes/chromophores in the photosynthetic network [18], the vaccination of nodes/individuals in social networks [19–22], or the malfunctioning of nodes/routers in computer networks [23, 24].

Network robustness to node attack may change in real-world networks with different structures [11]. Iyer et al. [3] studied network robustness as a function of the node clustering coefficient (or node transitivity). This study demonstrates that networks with higher clustering coefficients are more robust, with the most critical effect for the node degree and node betweenness attack. Nguyen and Trang [25] studied the Facebook social network. They found that those networks with higher modularity, i.e., networks presenting communities of nodes that are highly connected among them, have lower robustness to node removal. Zhou et al. [26] observed that increasing the assortativity of a network makes the network more robust against node removal and the network less stable. Nguyen et al. [27] showed that machine learning approaches unveil the degree assortativity, global closeness, and average node degree as the most critical factors in predicting the robustness (

Network science research shows contrasting outcomes about the role of the network structure in affecting its robustness to node attacks. On one hand, these studies are often based on small datasets of real-world networks, and they need more (robust) statistical analyses. On the other hand, research outcomes generally restrict the investigation, focusing on a few structural features of the networks, thus lacking a wide comparison of network structural indicators (NSIs) to forecast network robustness. For these reasons, understanding which structural features of real-world networks affect their robustness to node removal is still an urgent problem in network science.

In this research, we implement two well-known node attack strategies, i.e., the degree and betweenness node removal over a large set of 200 real-world networks from different areas of science.

We quantify the network functioning damage along the node attack sequence using the largest connected component (*quasi*-zero [29].

Then, to understand how the network structure affects the network robustness (and the node attack efficacy), we correlate

We find that the Estrada heterogeneity (

## 2 Methods

### 2.1 The node attack strategies

We simulated two classic node attack (removal) strategies. The first is the removal of nodes according to their degree (*DEG*), i.e., the number of links to the node [3, 4, 31]. The *DEG* strategy removes nodes in decreasing order of connectivity, i.e., the most connected nodes (hubs) are removed first. The second node attack strategy removes nodes in decreasing order of betweenness centrality (*BET*) [3, 7, 32]. The betweenness centrality is a node centrality based on the shortest paths between node pairs (also called geodesic paths). The shortest path between two nodes is the minimum number of links required to travel from one node to another [33]. The betweenness centrality of a node returns the number of shortest paths from every node pair of the network passing along that node. The betweenness

We perform an “initial node attack approach,” i.e., the node centrality rank is computed at the beginning of the simulation, and it is not updated along the node removal process [11]. The “initial node attack approach” differs from the recalculated (also named adaptive) node attack, in which node centralities are updated after node removals [11, 28]. The initial node attack describes the case where it is not possible to collect information about node features during the node removal process, such as vaccinating nodes/individuals in a social contact network with limited resources (limited time or vaccines) [34] or attacking nodes/routers in a computer network with a simultaneous node attack [28].

For both the node attack strategies, in the case of ties, i.e., nodes with equal ranking, we randomly sort their sequence. We perform 10^{3} simulations for each node attack strategy. We implemented the node attack simulations using the igraph package of the R program. The simulations are carried out on the high-performance computing (HPC) cluster of the “Università degli Studi di Parma.”

### 2.2 Real-world networks

We analyzed a large dataset of real-world network systems composed of 200 networks from different fields of science. The real-world networks analyzed here come from social, biological, Internet, road, transportation, neuronal, and ecological networks. The networks analyzed here are undirected (i.e., do not account for link directionality) and unweighted (do not account for link weight). The number of network nodes ranges from

### 2.3 Network structure indexes

We considered 20 different NSIs from the network science literature, graph theory, and chemical graph theory to predict

### 2.4 The network robustness

To evaluate the networks’ response to node attack, we trace *quasi*-zero [29]. This work defines

**FIGURE 1**. *)*. The percolation threshold *q*-value at which

### 2.5 The linear regression models

We perform regression model analyses to understand the relationship between NSI and the *x* is expressed by the following linear equation:

where *a* is the intercept and ^{2}), also named the coefficient of determination, measures how close the data points are to the fitted line. Higher R^{2} denotes better regression fitting models [48].

Then, we perform MLR models. MLR is an extension of SLR for multi-dimension variables

where *t*-value. The *t*-value used in MLR is the *t* di-student statistic value from a two-sided *t*-test. The larger the absolute value of the *t*-test statistic, the less likely the results occurred by chance [48]. For this, larger absolute *t*-values are associated with better predictors (NSIs).

We use the *lm* function of the R program to perform the SLR and MLR models. The fitting process is computed using the OLS method, which estimates the coefficients by minimizing an appropriate loss function [49].

Last, we perform the Pearson correlation coefficient (*t*-value. Last, we furnish the *p*-value to show the statistical significance of each model.

## 3 Results

Figure 2 shows the scatterplots of *q*_{c} vs. NSIs for the *DEG* node attack strategy. Figure 3 shows the scatterplots of *q*_{c} vs. NSIs for the *BET* node attack strategy.

**FIGURE 2**. Scatterplots of the percolation threshold (*q*_{c}*)* vs. the network structural indicators (NSIs) for the *DEG* node attack strategy, removing nodes with higher degrees first.

**FIGURE 3**. Scatterplots of the percolation threshold (*q*_{c}*)* vs. the network structural indicators (NSIs) for the *BET* node attack strategy, removing nodes with higher betweenness first.

Table 2 shows the outcomes of the SLR model. The best NSI to fit an SLR model with *DEG* (*p*-value <10^{–4}, R^{2} = 0.567) and *BET* (*p*-value <10^{–4}, R^{2} = 0.671) strategies. SLR *p*-values and the highest R^{2} for both node attack strategies (Table 2). The *q*_{c} decreases as a function of

**TABLE 2**. Single linear regression model outcomes. The best significant predictor with the highest R^{2} value is in bold.

Table 3 shows the outcomes of the MLR model. The best NSI to predict *DEG* (*t*-value = −11.9, *p*-value <10^{–23}) and *BET* (*t*-value = −11.8, *p*-value <10^{–23}) strategies. MLR estimates a negative correlation between

**TABLE 3**. Multiple linear regression model outcomes. The best significant predictor with the highest absolute *t*-value is in bold.

Table 4 summarizes the *DEG* (*t*-value = −11.9, *p*-value <10^{–23}) and *BET* (*t*-value = −11.8, *p*-value <10^{–23}) strategies. The *DEG* and −20.035 for *BET*, Table 4).

**TABLE 4**. Pearson correlation coefficient test outcomes. The best significant predictor with the highest absolute *t*-value is in bold.

## 4 Discussion

The

The third and fourth ring roads of Beijing City, the capital of China, are the real-world networks of the lowest

The ^{2} of SLR is much higher for *t*-value for

The assortativity coefficient

Given a certain node degree heterogeneity, assortative networks should have, on average, lower *p*-value < 0.001) in our real-world network dataset and confirms this hypothesis, i.e., higher values of

**FIGURE 4**. Scatterplot of the assortativity coefficient (*A)* vs. the Estrada heterogeneity (*p*-value < 0.001).

Consequently, assortative networks should show higher robustness to node attack and higher *A* (Figures 2, 3), and all models SLR, MLR (Tables 2, 3 respectively), and the

To further investigate the relationship between node degree heterogeneity and network robustness, we perform an MLR model holding only *DEG* strategy and presents the lowest *t*-value, whereas *BET* strategy and presents a smaller *t*-value than

**TABLE 5**. Multiple linear regression model

## 5 Conclusion

Investigating node attack strategies provides valuable insights into enhancing network robustness by anticipating potential threats and identifying components that need protection. On the other side of the coin, node attack research plays a crucial role when the aim is to perform a fast network disruption, such as halting the spread of a disease or stopping the diffusion of a computer virus. Here, we investigate the relationship between the network structure and its robustness to node attack in a large dataset of real-world networks. Our results indicate that the degree heterogeneity of connected nodes negatively affects the network robustness. Specifically, the

This paper presents some limitations that may open new lines of research. First, we perform linear regression models only. The relationship between NSIs and the percolation threshold qc of the real-world networks may follow nonlinear models. Therefore, a natural extension of this research may consider nonlinear regression models, such as logistic, monomolecular, or exponential functions, to describe the relationship between the structure and the percolation threshold of real-world networks. Then, we adopt an initial node attack approach to study network robustness. Future research may analyze the robustness of real-world networks using recalculated node attacks, in which node ranking is updated after each node removal. Last, it would be interesting to investigate how NSIs correlate with other robustness indexes besides

## Data availability statement

The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author.

## Author contributions

MB, RA, and DC conceived the research. MB wrote the simulation codes. MB and RA performed the simulations. MB performed statistical analyses. All authors contributed to the article and approved the submitted version.

## Funding

This research is funded by a grant from the Italian Ministry of Foreign Affairs and International Cooperation, by the Ecosister project, funded under the National Recovery and Resilience Plan (NRRP), and Mission 4 Component 2 Investment 1.5—Call for tender No. 3277 of 30/12/2021 of Italian Ministry of University and Research funded by the European Union—NextGenerationEU Award Number: Project code ECS00000033, Concession Decree No. 1052 of 23/06/2022 adopted by the Italian Ministry. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. (816313)). This work is supported by the Vietnam’s Ministry of Science and Technology (MOST) under the Vietnam-Italy scientific and technological cooperation program for the period 2021–2023. This work is supported by the Vietnam National University Ho Chi Minh City (VNU-HCM), Ho Chi Minh City, Vietnam, under grant number B2018-42-01. This research is funded by a grant from the Italian Ministry of Foreign Affairs and International Cooperation.

## Acknowledgments

MB, MT, DC, and RA acknowledge the Italian Ministry of Foreign Affairs and International Cooperation. The authors are greatly thankful to Van Lang University, Vietnam, for providing the budget for this study. This research has benefited from the high-performance computing (HPC) cluster of the Università degli Studi di Parma. They thank Fabio Sartori for the revision of the first manuscript draft. They also thank Prof. Stefano Poletti for the intriguing discussions about this research.

## Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

## Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphy.2023.1245564/full#supplementary-material

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Keywords: complex network, network robustness and resilience, machine learning, node attack sequence, statistical physics

Citation: Bellingeri M, Turchetto M, Scotognella F, Alfieri R, Nguyen N-K-K, Nguyen Q and Cassi D (2023) Forecasting real-world complex networks’ robustness to node attack using network structure indexes. *Front. Phys.* 11:1245564. doi: 10.3389/fphy.2023.1245564

Received: 23 June 2023; Accepted: 22 September 2023;

Published: 11 October 2023.

Edited by:

Nuno Crokidakis, Fluminense Federal University, BrazilReviewed by:

Divya Sindhu Lekha, Indian Institute of Information Technology, IndiaJihui Han, Zhengzhou University of Light Industry, China

Copyright © 2023 Bellingeri, Turchetto, Scotognella, Alfieri, Nguyen, Nguyen and Cassi. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Michele Bellingeri, michele.bellingeri@unipr.it