# Effect of Weight Thresholding on the Robustness of Real-World Complex Networks to Central Node Attacks

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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 a simple local centrality measure defined as the number of links connected to it. The degree ${k}_{i}$ of node i is given by

- Strength (Str): A node’s strength is the sum of the weights of links connected to that node. It is a weighted version of the degree centrality [6].

- Betweenness (Bet): Betweenness of a node is the number of shortest paths (between all the pairs of nodes) passing through it [3,4,5]. This binary metric defines the shortest path between two nodes as the minimum number of links needed to travel from one node to another. Mathematically, betweenness ${b}_{i}$of node i is:

- Weighted betweenness (WBet): Weighted betweenness of a node is defined as the number of weighted shortest paths passing through that node [7].

#### 2.3. Network Robustness Indicator

#### 2.4. Weight Thresholding

Algorithm 1: Methodology of WT analysis. | |

Procedure Weight Thresholding (G, N, L) | |

1: | WT = {0.0, 0.05, 0.1, …………., 0.85, 0.9} |

2: | for each WT |

3: | for i = 1 to m |

4: | link_set = {links in the increasing order of their weight} |

5: | weak_linkset = {WT fraction of weak links from link_set} |

6: | G′ = G − weak_linkset |

7: | Initial attack (G′, N, L′) |

8: | Recalculated attack (G′, N, L′) |

Procedure Initial attack (G′, N, L′) | |

1: | Find Initial LCC |

2: | for i = 1 to n |

3: | node_set = { nodes of G′ in the decreasing order of centrality measure } |

4: | while (LCC ! = 1) |

5: | Remove a node x from the G′ (in the order of node_set) |

6: | Find LCC of new network |

7: | node_set = node_set − x |

Procedure Recalculated attack (G′, N, L′) | |

1: | Find Initial LCC |

2: | for i = 1 to n |

3: | while (LCC ! = 1) |

4: | Calculate centrality meaures |

5: | node_set = { nodes of G′ in the decreasing order of centrality measure } |

6: | Remove a node x from the G′ (in the order of node_set) |

7: | Find LCC of new network |

8: | node_set = node_set − x |

## 3. Results and Discussion

#### Analyzing Node Centrality Ranking under WT by Kendall’s Tau Coefficient

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Albert, R.; Jeong, H.; Barabasi, A.-L. Error and attack tolerance of complex networks. Nature
**2000**, 406, 378–382. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Albert, R.; Barabasi, A.-L. Statistical mechanics of complex networks. Rev. Mod. Phys.
**2002**, 74, 47–97. [Google Scholar] [CrossRef] [Green Version] - Holme, P.; Kim, B.J.; Yoon, C.N.; Han, S.K. Attack vulnerability of complex networks. Phys. Rev. E
**2002**, 65, 056109. [Google Scholar] [CrossRef] [Green Version] - Bellingeri, M.; Cassi, D.; Vincenzi, S. Efficiency of attack strategies on complex model and real-world networks. Phys. A Stat. Mech. Its Appl.
**2014**, 414, 174–180. [Google Scholar] [CrossRef] [Green Version] - Iyer, S.; Killingback, T.; Sundaram, B.; Wang, Z. Attack robustness and centrality of complex networks. PLoS ONE
**2013**, 8, e59613. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bellingeri, M.; Cassi, D. Robustness of weighted networks. Phys. A Stat. Mech. Its Appl.
**2018**, 489, 47–55. [Google Scholar] [CrossRef] - Nguyen, Q.; Nguyen, N.K.K.; Cassi, D.; Bellingeri, M. New betweenness centrality node attack strategies for real-world complex weighted networks. Complexity
**2021**, 2021, 1677445. [Google Scholar] [CrossRef] - Allesina, S.; Pascual, M. Googling food webs: Can an eigenvector measure species’ importance for coextinctions? PLoS Comput. Biol.
**2009**, 5, e1000494. [Google Scholar] [CrossRef] [Green Version] - Lekha, D.S.; Balakrishnan, K. Central attacks in complex networks: A revisit with new fallback strategy. Phys. A Stat. Mech. Its Appl.
**2020**, 549, 124347. [Google Scholar] [CrossRef] - Divya, P.B.; Lekha, D.S.; Johnson, T.; Balakrishnan, K. Vulnerability of link-weighted complex networks in central attacks and fallback strategy. Phys. A Stat. Mech. Its Appl.
**2021**, 590, 126667. [Google Scholar] - Nie, T.; Guo, Z.; Zhao, K.; Lu, Z.-M. New attack strategies for complex networks. Phys. A Stat. Mech. Its Appl.
**2015**, 424, 248–253. [Google Scholar] [CrossRef] - Nie, T.; Guo, Z.; Zhao, K.; Lu, Z.-M. The dynamic correlation between degree and betweenness of complex network under attack. Phys. A Stat. Mech. Its Appl.
**2016**, 457, 129–137. [Google Scholar] [CrossRef] - Zhang, J.; Xu, X.-K.; Li, P.; Zhang, K.; Small, M. Node importance for dynamical process on networks: A multiscale characterization. Chaos
**2011**, 21, 016107. [Google Scholar] [CrossRef] [PubMed] - Li, C.; Wang, L.; Sun, S.; Xia, C. Identification of influential spreaders based on classified neighbors in realworld complex networks. Appl. Math. Comput.
**2018**, 320, 512–523. [Google Scholar] [CrossRef] - Bellingeri, M.; Bevacqua, D.; Scotognella, F.; Cassi, D. The heterogeneity in link weights may decrease the robustness of real-world complex weighted networks. Sci. Rep.
**2019**, 9, 10692. [Google Scholar] [CrossRef] [Green Version] - Crucitti, P.; Latora, V.; Marchiori, M.; Rapisarda, A. Efficiency of scale-free networks: Error and attack tolerance. Phys. A Stat. Mech. Its Appl.
**2003**, 320, 622–642. [Google Scholar] [CrossRef] [Green Version] - Bellingeri, M.; Bevacqua, D.; Scotognella, F.A.; Alfieri, R.; Cassi, D. A comparative analysis of link removal strategies in real complex weighted networks. Sci. Rep.
**2020**, 10, 3911. [Google Scholar] [CrossRef] [PubMed] [Green Version] - He, S.; Li, S.; Ma, H. Effect of edge removal on topological and functional robustness of complex networks. Phys. A Stat. Mech. Its Appl.
**2009**, 388, 2243–2253. [Google Scholar] [CrossRef] [Green Version] - Granovetter, M.S. The strength of weak ties. Am. J. Sociol.
**1973**, 78, 1360–1380. [Google Scholar] [CrossRef] [Green Version] - Pajevic, S.; Plenz, D. The organization of strong links in complex networks. Nat. Phys.
**2012**, 8, 429–436. [Google Scholar] [CrossRef] [Green Version] - Garas, A.; Argyrakis, P.; Havlin, S. The structural role of weak and strong links in a financial market network. Eur. Phys. J. B
**2008**, 63, 265–271. [Google Scholar] [CrossRef] [Green Version] - Yan, X.; Jeub, L.G.S.; Flammini, A.; Radicchi, F.; Fortunato, S. Weight thresholding on complex networks. Phys. Rev. E
**2018**, 98, 042304. [Google Scholar] [CrossRef] [Green Version] - Csermely, P. Weak Links: The Universal Key to the Stability of Networks and Complex Systems; Springer: Berlin/Heidelberg, Germany, 2009. [Google Scholar]
- Radicchi, F.; Ramasco, J.J.; Fortunato, S. Information filtering in complex weighted networks. Phys. Rev. E
**2011**, 83, 046101. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bartoldson, B.; Morcos, A.; Barbu, A.; Erlebacher, G. The generalization-stability tradeoff in neural network pruning. Adv. Neural Inf. Process. Syst.
**2020**, 33, 20852–20864. [Google Scholar] - Freund, A.J.; Giabbanelli, P.J. An experimental study on the scalability of recent node centrality metrics in sparse complex networks. Front. Big Data
**2022**, 5, 797584. [Google Scholar] [CrossRef] - Srinivasan, S.; Das, S.; Bhowmick, S. Application of Graph Sparsification in Developing Parallel Algorithms for Updating Connected Components. In Proceedings of the IEEE International Parallel and Distributed Processing Symposium Workshops, Chicago, IL, USA, 23–27 May 2016. [Google Scholar]
- Namaki, A.; Shirazi, A.H.; Raei, R.; Jafari, G. Network analysis of a financial market based on genuine correlation and threshold method. Phys. A Stat. Mech. Its Appl.
**2011**, 390, 3835–3841. [Google Scholar] [CrossRef] - Lynall, M.-E.; Bassett, D.S.; Kerwin, R.; McKenna, P.J.; Kitzbichler, M.; Muller, U.; Bullmore, E. Functional connectivity and brain networks in schizophrenia. J. Neurosci.
**2010**, 30, 9477–9487. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Allesina, S.; Bodini, A.; Bondavalli, C. Secondary extinctions in ecological networks: Bottlenecks unveiled. Ecol. Model.
**2006**, 194, 150–161. [Google Scholar] [CrossRef] - Garrison, K.A.; Scheinost, D.; Finn, E.S.; Shen, X.; Constable, R.T. The (in) stability of functional brain network measures across thresholds. Neuroimage
**2015**, 118, 651–661. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zhang, J.; Eytan, M.; David, H. Enhancing network robustness via shielding. IEEE/ACM Trans. Netw.
**2017**, 25, 2209–2222. [Google Scholar] [CrossRef] - Liu, M.; Qi, X.; Pan, H. Optimizing communication network geodiversity for disaster resilience through shielding approach. Reliab. Eng. Syst. Saf.
**2022**, 228, 108800. [Google Scholar] [CrossRef] - Xiao, S.; Xiao, G. On imperfect node protection in complex communication networks. J. Phys. A Math. Theor.
**2011**, 44, 055101. [Google Scholar] [CrossRef] - Colizza, V.; Pastor-Satorras, R.; Vespignani, A. Reaction-diffusion processes and metapopulation models in heterogeneous. Nat. Phys.
**2007**, 3, 276–282. [Google Scholar] [CrossRef] [Green Version] - Newman, M.E.J. Finding community structure in networks using the eigenvectors of matrices. Phys. Rev.
**2006**, 74, 036104. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Watts, D.J.; Strogatz, S.H. Collective dynamics of ‘small-world’ networks. Nature
**1998**, 393, 440–442. [Google Scholar] [CrossRef] - Latora, V.; Nicosia, V.; Russo, G. Complex Networks: Principles, Methods and Applications; Cambridge University Press: Cambridge, UK, 2017. [Google Scholar]
- Allard, A.; Serrano, M.A.; Garcia-Perez, G.; Boguna, M. The geometric nature of weights in real complex networks. Nat. Commun.
**2017**, 8, 14103. [Google Scholar] [CrossRef] - Serrano, A.M.; Boguna, M.; Sagues, F. Uncovering the hidden geometry behind metabolic networks. Mol. BioSyst.
**2012**, 8, 843–850. [Google Scholar] [CrossRef] [Green Version] - Avena-Koenigsberger, A.; Goni, J.; Betzel, R.F.; Heuvel, M.P.V.D.; Griffa, A.; Hagmann, P.; Thiran, J.-P.; Sporns, O. Using Pareto optimality to explore the topology and dynamics of the human connectome. Philos. Trans. R. Soc. Lond. B Biol. Sci.
**2014**, 369, 20130530. [Google Scholar] [CrossRef] [Green Version] - Hagmann, P.; Cammoun, L.; Gigandet, X.; Meuli, R.; Honey, C.J.; Wedeen, V.J.; Sporns, O. Mapping the structural core of human cerebral cortex. PLoS Biol.
**2008**, 6, 1479–1493. [Google Scholar] [CrossRef] - Bellingeri, M.; Bodini, A. Food web’s backbones and energy delivery in ecosystems. Oikos
**2016**, 125, 586–594. [Google Scholar] [CrossRef] - Opitz, S. Trophic Interactions in Caribbean Coral Reefs; International Center for Living Aquatic Resources Management (ICLARM): Penang, Malaysia, 1996. [Google Scholar]
- Heymans, J.J.; Ulanowicz, R.E.; Bondavalli, C. Network analysis of the South Florida Everglades graminoid marshes and comparison with nearby cypress ecosystems. Ecol. Model.
**2002**, 149, 5–23. [Google Scholar] [CrossRef] - Bellingeri, M.; Vincenzi, S. Robustness of empirical food webs with varying consumer’s sensitivities to loss of resources. J. Theor. Biol.
**2013**, 333, 18–26. [Google Scholar] [CrossRef] [PubMed] - Szalkai, B.; Kerepesi, C.; Varga, B.; Grolmusz, V. The Budapest Reference Connectome Server v2.0. Neurosci. Lett.
**2015**, 595, 60–62. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Cohen, R.; Erez, K.; Ben-Avraham, D.; Havlin, S. Breakdown of the internet under intentional attack. Phys. Rev. Lett.
**2001**, 86, 3682–3685. [Google Scholar] [CrossRef] [Green Version] - Brandes, U. A faster algorithm for betweenness centrality. J. Math. Sociol.
**2004**, 25, 163–177. [Google Scholar] [CrossRef] - Jordán, F. Keystone Species and Food Webs. Philos. Trans. R. Soc. Lond. Ser. B Biol. Sci.
**2009**, 364, 1733–1741. [Google Scholar] [CrossRef] [Green Version] - Nguyen, Q.; Pham, H.-D.; Cassi, D.; Bellingeri, M. Conditional attack strategy for real-world complex networks. Phys. A Stat. Mech. Its Appl.
**2019**, 530, 121561. [Google Scholar] [CrossRef] - Kendall, M.G. The treatment of ties in ranking problems. Biometrika
**1945**, 33, 239–251. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Left chart: LCC size as a function of the fraction of nodes removed q for initial attacks in C. elegans network. Right chart: Robustness R of each attack strategy.

**Figure 2.**Steeper (black) and gentle (blue) degradation of a network functionality along the removal of fraction (q) of components (either nodes or links) of the network.

**Figure 3.**LCC after each weight thresholding (WT) value (

**left column**), robustness (R) of the network under initial (

**middle column**), and recalculated attack strategies (

**right column**) as a function of weight thresholding (WT) value for the networks C. elegans (Eleg), Caribbean (Carib), Human12a (Hum), Cypdry (Cyp), and E. coli (Coli).

**Figure 4.**LCC after each weight thresholding (WT) value (

**left column**), robustness (R) of the network under initial (

**middle column**), and recalculated attack (

**right column**) strategies as a function of weight thresholding (WT) value for the networks Budapest (Buda), Cargoship (Cargo), US Airports (Air), and Netscience (Net).

**Figure 5.**Kendall’s tau coefficient ($\tau )$ for centrality measures Deg, Str, Bet, and WBet. Correlation is measured between the top 30% of nodes of the initial network with each thresholded network.

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

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

C. elegans | Eleg | [37,38] | Biological | Neurons | Neurons connection | Number of Connections | 297 | 2344 | 15.8 | 3.761 | 297 |

Cargoship | Cargo | [39] | Transport | Ports | Route | Shipping journeys | 834 | 4348 | 10.4 | 97.709 | 821 |

US airport | Air | [35] | Transport | Airports | Route | Passengers | 500 | 2979 | 11.9 | 152,320.2 | 500 |

E. coli | Coli | [39,40] | Biological | Metabolites | Common reaction | Number of Common reactions | 1100 | 3636 | 6.61 | 1.364 | 1100 |

Netscience | Net | [36] | Social | authors | Coauthorship | Number of Common papers | 1461 | 2741 | 3.75 | 0.434 | 379 |

Human12a | Hum | [41,42] | Biological | Brain regions | Connection between regions | Connection density | 501 | 6038 | 24.1 | 0.01 | 501 |

Caribbean | Carib | [43,44] | Ecological Food web | Species | Trophic relation | Amount of biomass | 249 | 3503 | 28.13 | 0.067 | 249 |

CypDry | Cyp | [45,46] | Ecological Food web | Species | Trophic relation | Amount of biomass | 66 | 503 | 15.24 | 0.358 | 65 |

Budapest | Buda | [47] | Biological | Brain regions | Neural connection | Amount of track flow | 480 | 1000 | 4.167 | 5.024 | 467 |

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## Share and Cite

**MDPI and ACS Style**

John, J.M.; Bellingeri, M.; Lekha, D.S.; Cassi, D.; Alfieri, R.
Effect of Weight Thresholding on the Robustness of Real-World Complex Networks to Central Node Attacks. *Mathematics* **2023**, *11*, 3482.
https://doi.org/10.3390/math11163482

**AMA Style**

John JM, Bellingeri M, Lekha DS, Cassi D, Alfieri R.
Effect of Weight Thresholding on the Robustness of Real-World Complex Networks to Central Node Attacks. *Mathematics*. 2023; 11(16):3482.
https://doi.org/10.3390/math11163482

**Chicago/Turabian Style**

John, Jisha Mariyam, Michele Bellingeri, Divya Sindhu Lekha, Davide Cassi, and Roberto Alfieri.
2023. "Effect of Weight Thresholding on the Robustness of Real-World Complex Networks to Central Node Attacks" *Mathematics* 11, no. 16: 3482.
https://doi.org/10.3390/math11163482