# Robustness of Real-World Networks after Weight Thresholding with Strong Link Removal

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Real-World Networks

#### 2.2. Attack Strategies

- Degree (Deg): The degree of a node is the number of links connected to it [10,29,30,31,32]. The degree ${k}_{i}$ of node i is given by the following:$${k}_{i}={\sum}_{j=1}^{N}{a}_{ij},$$

- Strength (Str): A node’s strength is the total weight of the links connected to that node [33], also called a weighted degree.

- Weighted Betweenness (WBet): Weighted betweenness of a node is defined as the number of weighted shortest paths passing through that node [34].

#### 2.3. Weight Thresholding

#### 2.4. Network Robustness Indicator

- One way was to normalize the LCC after node removal using the initial LCC value (before node attack) of the network after WT. In this case, we considered each thresholded network an independent network, and we did not account for the LCC decrease directly caused by the WT procedure.
- A second way was to normalize the LCC after node removal using the initial LCC at WT = 0, i.e., we normalized using the LCC of the original network. In this second case, we also considered the LCC decrease triggered by the link removal of the WT procedure. This normalization was intended to analyze the joint effect of the weight thresholding and node attack to decrease the LCC (total LCC decrease).

_{tot}) as the measure computed with the second LCC normalization. Table 2 lists the abbreviations used in this manuscript.

## 3. Results and Discussion

#### 3.1. Robustness against WT

#### 3.2. Robustness to WT and Node Attack

_{tot}). The R

_{tot}for the node attack strategies, Ran, Deg, Str, Bet, and WBet, is represented in Figure 1 and Figure 2.

_{tot}(solid lines) follows a similar pattern of robustness decrease for all the attack strategies except Ran (see green dotted and solid lines). In networks such as C. Elegans, Human12a, and E. Coli, the joint effect of thresholding and node attacks (R

_{tot}) returns roughly the same robustness computed with the first normalization procedure (R). In all other networks, we can observe only a small difference in the values of these two types of robustness when focusing on targeted attacks. Differently, the robustness of the networks against random removal is always lower when considering the joint effect of WT and random node attacks. The solid green lines describing the R

_{tot}decrease with increasing WT in Figure 1 and Figure 2 are significantly lower than the dotted green lines (R).

**(**Figure 2) and for the Cyp network for random node removal (Ran) (Figure 2) at the end of the WT procedure. The Buda network is a complex brain network where nodes are brain regions and links indicate electrical activity between them [28]. The Cyp network is a food web ecological network in which nodes are species and links depict trophic interactions among them [16]. Global node clustering <CC> is a simple measure evaluating the presence of communities of nodes in networks [28], and it is a measure that counts node triplets in the network. A triplet is three nodes connected by either two (open triplet) or three (closed triplet) links. <CC> is the ratio between the number of closed triplets and the total number of triplets (both open and closed) in the network [28]. The higher the <CC>, the higher the node’s tendency to cluster in communities.

#### 3.3. The Efficacy of the Node Attack Strategies

#### 3.4. Comparing Strong and Weak WT Procedures

_{tot}against the initial node attack when weak and strong WT procedures simplify networks. We do not find a clear trend; in some cases, weak WT triggers a faster robustness decrease, and in others, it is to the contrary. For example, the weak WT induces a higher decrease in robustness concerning the strong WT for the Eleg, Cyp (except under WBet initial attack), Air, and Cargo networks for both the initial (Figure 7, red lines) and recalculated attacks (Figure 9, green lines). These results agree with the study by Onnela et al. [13] on mobile communication networks [13]. Onnela et al. [13] show the counterintuitive consequence that real-world social networks are robust to removing the strong links but fall apart quickly if the weak links are removed. Onnela et al. [13] analyzed the network’s robustness to removing links only. Our study, on the other hand, analyzes the combined effect of removing links and then attacking the network by removing nodes. Despite the differences between Onnela et al. [13] and our approaches, similar systems’ responses are observed: for certain types of real-world networks, removing weak links can induce greater fragility. Our results show that this may happen not only in social networks [13] but also in transportation and biological networks.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The LCC after each weight thresholding (WT) value (left column), the robustness (R) of the network under the initial (middle column), and the recalculated attack strategies (right column) as a function of the weight thresholding (WT) value for the networks C. Elegans (Eleg), Caribbean (Carib), Human12a (Hum), Cypdry (Cyp), and E. Coli (Coli).

**Figure 2.**The LCC after each weight thresholding (WT) value (left column), the robustness (R) of the network under the initial (middle column), and the recalculated attack (right column) strategies as a function of the weight thresholding (WT) value for the networks Budapest (Buda), Cargoship (Cargo), US airports (Air), and Netscience (Net).

**Figure 3.**The LCC as a function of the fraction of nodes that removed q for Ran, Deg, Str, Bet, and WBet (both initial and recalculated) attacks in the Budapest network for WT values 0.75, 0.8, 0.85, and 0.9.

**Figure 4.**The LCC as a function of the fraction of nodes removed q for Ran, Deg, Str, Bet, and WBet (both initial and recalculated) attacks in the Netscience network for WT values 0.25, 0.45, 0.55, and 0.65.

**Figure 5.**Real-world network features as a function of WT for each network. $<k>$: average node degree; $<s>$: average node strength; $<w>$: average link weight; <CC>: global clustering coefficient. For the ease of analysis, the network features are normalized by their maximum value.

**Figure 6.**Best attack strategy returning the lowest R value for each real-world network and each WT value. In each cell, we indicate the best attack strategy and its R

_{tot}value. The R

_{tot}value is computed by normalizing the LCC with the initial LCC for WT = 0. Colors indicate the different attack strategies.

**Figure 7.**Best attack strategy returning the lowest R value for each real-world network and each WT value. In each cell, we indicate the best attack strategy and its R value. The R value is computed by normalizing the LCC with the initial LCC at each WT value. Colors indicate the different attack strategies.

**Figure 8.**Kendall’s tau coefficient (τ) for centrality measures Deg, Str, Bet, and WBet. Correlation is measured between the initial network’s node rank and the network’s node rank after WT. We compute τ using the top 30% of nodes of the network. Solid lines indicate τ for WT with strong link removal; dashed lines indicate τ for WT with weak link removal as in [2].

**Figure 9.**Comparison between the total robustness (R

_{tot}) against weak and strong WT procedures. Network robustness under the initial attack (dotted lines) and recalculated attack (solid lines) strategies as a function of the weight thresholding (WT) value for the networks C. Elegans (Eleg), Caribbean (Carib), Human12a (Hum), Cypdry (Cyp), E. Coli (Coli), Budapest (Buda), Cargoship (Cargo), US airports (Air), and Netscience (Net).

**Table 1.**Statistics of real-world networks. N number of nodes; L number of links; <k> average node degree; <w> average link weight; <CC> global clustering coefficient; LCC size of the largest connected component.

Networks | Key | Ref. | Type | Node | Link | Weight | N | L | <k> | <w> | <CC> | LCC |
---|---|---|---|---|---|---|---|---|---|---|---|---|

C. Elegans | Eleg | [17,18] | Biological | Neurons | Neurons connection | Number of Connections | 297 | 2344 | 15.8 | 3.761 | 0.181 | 297 |

Cargoship | Cargo | [19] | Transport | Ports | Route | Shipping journeys | 834 | 4348 | 10.4 | 97.709 | 0.222 | 821 |

US airport | Air | [20] | Transport | Airports | Route | Passengers | 500 | 2979 | 11.9 | 152320.2 | 0.351 | 500 |

E. Coli | Coli | [19,21] | Biological | Metabolites | Common reaction | Number of Common reactions | 1100 | 3636 | 6.61 | 1.364 | 0.139 | 1100 |

Netscience | Net | [22] | Social | Authors | Coauthorship | Number of Common papers | 1461 | 2741 | 3.75 | 0.434 | 0.693 | 379 |

Human 12a | Hum | [23,24] | Biological | Brain regions | Connection between regions | Connection density | 501 | 6038 | 24.1 | 0.01 | 0.457 | 501 |

Caribbean | Carib | [25,26] | Ecological Food web | Species | Trophic relation | Amount of biomass | 249 | 3503 | 28.13 | 0.067 | 0.172 | 249 |

CypDry | Cyp | [16,27] | Ecological Food web | Species | Trophic relation | Amount of biomass | 66 | 503 | 15.24 | 0.358 | 0.421 | 65 |

Budapest | Buda | [28] | Biological | Brain regions | Neural connection | Amount of track flow | 480 | 1000 | 4.167 | 5.024 | 0.120 | 467 |

Abbreviation | Full Name |
---|---|

WT | Weight thresholding |

LCC | Size of largest connected component |

N | Number of nodes |

L | Number of links |

<w> | Average link weight |

<k> | Average node degree |

<CC> | Global clustering coefficient |

Ran | Random node attack |

Deg | Degree node attack |

Str | Strength node attack |

Bet | Betweenness node attack |

WBet | Weighted Betweenness node attack |

G | Weighted network |

G’ | Thresholded network |

L’ | Number of links in G’ |

q | Fraction of nodes removed |

R | Robustness |

R_{tot} | Total Robustness |

<s> | Average node strength |

Initial_Weak WT | WT by weak link removal with initial node attack strategy |

Initial_Strong WT | WT by strong link removal with initial node attack strategy |

Recalculated_Weak WT | WT by weak link removal with recalculated node attack strategy |

Recalculated_Strong WT | WT by strong link removal with recalculated node attack strategy |

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**MDPI and ACS Style**

John, J.M.; Bellingeri, M.; Lekha, D.S.; Cassi, D.; Alfieri, R.
Robustness of Real-World Networks after Weight Thresholding with Strong Link Removal. *Mathematics* **2024**, *12*, 1568.
https://doi.org/10.3390/math12101568

**AMA Style**

John JM, Bellingeri M, Lekha DS, Cassi D, Alfieri R.
Robustness of Real-World Networks after Weight Thresholding with Strong Link Removal. *Mathematics*. 2024; 12(10):1568.
https://doi.org/10.3390/math12101568

**Chicago/Turabian Style**

John, Jisha Mariyam, Michele Bellingeri, Divya Sindhu Lekha, Davide Cassi, and Roberto Alfieri.
2024. "Robustness of Real-World Networks after Weight Thresholding with Strong Link Removal" *Mathematics* 12, no. 10: 1568.
https://doi.org/10.3390/math12101568