# Graph Metrics for Network Robustness—A Survey

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## Abstract

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## 1. Introduction

## 2. Methods and Notation

#### 2.1. Network Robustness

#### 2.2. Layer Models and Graph Models of the Internet

#### 2.3. Challenges and Aspects of Robustness

#### 2.4. Metric Selection and Article Structure

#### 2.5. Features of Metrics

#### 2.6. List of Symbols

$G(V,E)$ | Graph G with a set of vertices V and set of edges E. |

$v=\left|V\right|$ | Number of nodes in the graph. |

$e=\left|E\right|$ | Number of edges in the graph. |

${V}_{i}$ | set of nodes that are neighbors of node i in a simple graph |

$\overline{x}$ | Arithmetical mean of ${x}_{y}$ for all nodes/edges/etc. y. |

$\mathbf{A}=[{a}_{ij}]$ | Adjacency matrix; ${a}_{ij}=1$ if an edge from node i to j exists, ${a}_{ij}=0$ else. |

${\mathbf{A}}^{w}=[{w}_{ij}]$ | Weight matrix; ${w}_{ij}$ is the edge weight between nodes i and j. |

${k}_{i}^{(in/out)}$ | (In-/out-) degree of node i (Section 3.1). |

${s}_{i}^{(in/out)}$ | (In-/out-) strength of node i (Section 3.2). |

${d}_{(m,n)}$ | Minimum hop distance from node m to node n (Section 6). |

${d}_{(m,n)}^{w}$ | Minimum weighted distance from node m to node n (Section 6). |

## 3. Adjacency

#### 3.1. Node Degree (Degree Centrality)

L, R | simple, directed | local impact & liability/global | static | $\left(\right)$ | ✓ |

#### 3.2. Strength

L, R | weighted (un)directed | local impact/global | static | $\left(\right)$ | ✓ |

#### 3.3. Entropy

L, R | simple (all) | global | static | $[0,log(v-1)]$ | ✓ |

#### 3.4. Skewness

L, R | simple (all) | global | static | $\left(\right)$ | ✓ |

#### 3.5. Vulnerability Function

L, R | simple graph | global | static | $\left(\right)$ | ✓ |

#### 3.6. Assortative Coefficient

L, R | simple graph | global | static | $\left(\right)$ | ✓ |

#### 3.7. Average Neighbor Connectivity

L, R | (un)weighted (directed) | global | static | $\left(\right)$ | ✓ |

#### 3.8. Rich-Club Connectivity (Rich-Club Coefficient)

L, R | – all – | global | static | $[0,\infty ]$ | ✓ |

#### 3.9. Recapitulation

## 4. Clustering

#### 4.1. Clustering Coefficient (Transitivity)

L, R | – all – | local impact/global | static | $\left(\right)$ | ✓ |

#### 4.2. Edge Clustering Coefficient

L, R | simple (all) | local impact | static | $\left(\right)$ | ✓ |

#### 4.3. Modularity

L, To, R | simple (weighted) | global | static | $\left(\right)$ | ✓ |

#### 4.3.1. Modularity Matrix

#### 4.3.2. Edge Betweenness Partitioning Algorithm

#### 4.4. Z-Score of the Within Module-Degree

L, R | simple (all) | local impact | static | $\left(\right)$ | ✓ |

#### 4.5. Participation Coefficient

L, To, R | simple (weighted) | local impact and liability | static | $\left(\right)$ | ✓ |

#### 4.6. Recapitulation

## 5. Connectivity

#### 5.1. Vertex-, Edge- and Conditional Connectivity (Cohesion, Adhesion and P-Connectivity)

To, R | simple graph | global | worst-case | $\left(\right)$ | NP-complete |

#### 5.2. Sparsity

To, R | simple graph | global | worst-case | $\left(\right)$ | ✓ |

#### 5.3. Cheeger Constant (Isoperimetric Number)

To, R | simple graph | global | worst-case | $\left(\right)$ | NP-complete |

#### 5.4. Minimum m-Degree

To, R | simple graph | global | worst-case | $\left(\right)$ | NP-C. and $\mathcal{O}\left(\right)open="("\; close=")">(\begin{array}{c}v\\ m\end{array})e$ |

#### 5.4.1. Network Partitioning Algorithm

#### 5.5. Ratio of Disruption

To, R | simple graph | global | worst-case | $\left(\right)$ | NP-complete |

#### 5.6. Local Delay Resilience (Resilience)

To, R, Tr | simple, directed | local liability/global | worst-case | $\left(\right)$ | NP-hard ($\mathcal{O}(e)$) |

#### 5.7. Toughness, Integrity, Scattering Number

To, R | simple graph | global | worst-case | N/A | NP-complete |

#### 5.8. Tenacity

To, R | simple graph | global | worst-case | N/A | NP-complete |

#### 5.9. Percolation Threshold

To, R | simple graph | global | failures | $\left(\right)$ | ✓ |

#### 5.10. Reliability Polynomial

To, R | simple graph | global | failures | $\left(\right)$ | ✓ |

#### 5.11. Partition Resilience Factor (Resilience Factor)

To, R | simple graph | global | failures | $\left(\right)$ | ✓ |

#### 5.12. Analysis of Disconnected Components and Reachability (Flow Robustness)

To, R | simple graph | global | dynamic | different codomains | ✓ |

#### 5.13. Recapitulation

## 6. Distance

#### 6.1. Average Shortest Path Length (Closeness Centrality/ Hop Count/ Average Diameter/ Network Diameter/ Geodesic Distance)

Tr | simple (all) | local impact/global | static | $\left(\right)$/$\left(\right)$ | ✓($\mathcal{O}(v+e)$/$\mathcal{O}(ve)$) |

#### 6.2. Global Network Efficiency (Harmonic Centrality/Closeness Centrality)

Tr | simple (all) | local impact/global | static | $[0,1]$ | ✓($\mathcal{O}(v+e)$/$\mathcal{O}(ve)$) |

#### 6.3. Local Network Efficiency (Cyclic Coefficent)

Tr | simple (all) | local impact and liability/global | static | $\left(\right)$ | ✓ |

#### 6.4. Characteristic Path Length

Tr | labeled (all) | local impact | static | $\left(\right)open="["\; close="]">1,\infty $ | ✓ |

#### 6.5. Expansion

R, Tr | (un)directed (weighted) | local impact/global | static | $\left(\right)$ | ✓ |

#### 6.6. Effective Eccentricity and Effective Diameter

R, Tr | simple (all) | local impact/global | static | $\left(\right)$ | ✓ |

#### 6.7. Recapitulation

## 7. Throughput

#### 7.1. Betweenness Centrality and AS Hegemony

Tr | (un)weighted (directed) | local impact | static | $\left(\right)open="["\; close="]">0,\frac{v-1}{2}(v-2)$ ✓ ($\mathcal{O}(ve+{v}^{2}logv)$) |

#### 7.2. Central Point Dominance

Tr | simple (all) | global | static | $\left(\right)$ | ✓ |

#### 7.3. Effective Load

Tr | simple (all) | local impact and liability | dynamic | $\left(\right)$ | ✓ |

#### 7.4. Performance

Tr | node-weighted (all) | global | static | $\left(\right)$ | ✓ |

#### 7.5. Elasticity

Tr | simple (all) | global | worst-case | $\left(\right)$ | ✓ |

#### 7.6. Vulnerability Impact Factors

Tr | labeled weighted (directed) | local impact and liability/global | dynamic | $\left(\right)open="["\; close="]">0,\infty $ | ✓ |

#### 7.7. Survivability Function, Failures

A | weighted directed | global | failures | $[0,1]$ | ✓ |

#### 7.8. Recapitulation

## 8. Spectral Methods

#### 8.1. Eigenvector Centrality (Significance)

To, R | simple (all) | local impact and liability | static | $\left(\right)$ | ✓ |

#### 8.2. Symmetry Ratio

To, R | simple graph | global | static | $\left(\right)$ | ✓ |

#### 8.3. Spectral Cluster Identification

L, To, R | simple, directed | both | static | - | ✓ |

#### 8.4. Algebraic Connectivity (Laplacian Spectral Gap)

To, R | simple graph | global | static | $[0,\frac{v}{v-1}min\left\{{k}_{i}\right|i\in V\}]$ | ✓ |

#### 8.5. Good Expansion

To, R | simple, weighted | global | static | $\left(\right)$ | NP-hard |

#### 8.6. Number of Spanning Trees

To, R | simple graph | global | static | $[1,{\prod}_{i=1}^{v-1}{k}_{i}]$ | ✓ |

#### 8.7. Natural Connectivity (Natural Eigenvalue)

To, R | simple graph | global | static | $\left(\right)$ | ✓ |

#### 8.8. Random Walk ASPL

Tr | (un)weighted (directed) | local impact/global | static | $[0,\infty ]$ | ✓ |

#### 8.9. Current-Flow Closeness (Information Centrality)

Tr | (un)weighted (directed) | local impact/global | static | N/A | ✓ |

#### 8.10. Random Walk Betweenness

Tr | – all – | local impact | static | $[{k}_{min}v/a(G),{k}_{max}v(v-1)/a(G)]$ | ✓ |

#### 8.11. Current-Flow Betweenness

Tr | (un)weighted (directed) | local impact | static | N/A | ✓ |

#### 8.12. Network Criticality (Effective Resistance)

Tr | – all – | global | static | $[v/a(G),v(v-1)/a(G)]$ | ✓ |

#### 8.13. Recapitulation

## 9. Geographical Metrics

#### 9.1. Distance Strength and Outreach

P | geographical weighted (directed) | local impact | static | $\left(\right)$ | ✓ |

#### 9.2. Survivability Function, Geographical

P | geographical | global | dynamic | $[0,1]$ | ✓ |

#### 9.3. Pointwise Vulnerability (Information Centrality)

P | geographical (all) | local impact | dynamic | $\left(\right)$ | ✓ |

#### 9.4. Effective Geographical Path Diversity

P | geographical (directed) | local liability/global | static | $\left(\right)$ | ✓ |

#### 9.5. Recapitulation

## 10. Comparison and Discussion

#### 10.1. Backbone Identification and Evaluation

#### 10.2. Disconnection Robustness

#### 10.3. Transmission Speed Robustness

#### 10.4. Traffic Robustness

#### 10.5. Ambiguity and Correlations of Metrics

## 11. Transfer to Other Research Fields

## 12. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Summarizing Table

Section and Metric | Graphs | Scope | Static | Codo. | Robustness | Comp. | |||||||||

Section 3.1 | Node Degree/Degree-Freq. Distr. | s | [d] | l & i | / | g | s | v/∘ | [d] | [t] | [b] | ✓ | |||

Section 3.2 | Strength/Strength Distribution | w | [d] | i | / | g | s | ∞/ ∘ | [t] | [o] | [b] | ✓ | |||

Section 3.3 | Entropy | s | [w] | [d] | g | s | v | [d] | [t] | ✓ | |||||

Section 3.4 | Skewness | s | [w] | [d] | g | s | ∘ | [d] | [t] | ✓ | |||||

Section 3.5 | Vulnerability Function | s | g | s | ∘ | [d] | [t] | ✓ | |||||||

Section 3.6 | Assortative Coefficient | s | g | s | ∘ | [d] | ✓ | ||||||||

Section 3.7 | Average Neighbor Connectivity | s | [w] | [d] | g | s | v | [d] | [b] | ✓ | |||||

Section 3.8 | Rich-Club Connectivity | – all – | g | s | ∞ | [t] | [o] | [b] | ✓ | ||||||

Section 4.1 | Clustering Coefficient | – all – | i | / | g | s | ∘ | [d] | [t] | [o] | ✓ | ||||

Section 4.2 | Edge Clustering Coefficient | s | [w] | [d] | i | s | ${k}_{i}$ | [d] | [t] | ✓ | |||||

Section 4.3 | Modularity | s | [w] | g | s | ∘ | ✓ | ||||||||

Section 4.3.1 | Modularity Matrix | s | – both – | s | - | d | ✓ | ||||||||

Section 4.3.2 | Edge Betweenness Part. Algorithm | s | – both – | s | - | d | $<\mathcal{O}({v}^{3})$ | ||||||||

Section 4.4 | Z-Score of Within Module-Degree | s | [w] | [d] | i | s | ∞ | [d] | [t] | ✓ | |||||

Section 4.5 | Participation Coefficient | s | [w] | l and i | s | ∘ | [d] | [t] | ✓ | ||||||

Section 5.1 | Vertex-, Edge- and Cond. Connectivity | s | g | w | v | NP-c. | |||||||||

Section 5.2 | Sparsity | s | g | w | v | d | ✓ | ||||||||

Section 5.3 | Cheeger Constant | s | g | w | v | d | NP-c. | ||||||||

Section 5.4 | Minimum m-Degree | s | g | w | v | d | $\mathcal{O}((\begin{array}{c}v\\ m\end{array})e)$ | ||||||||

Section 5.4.1 | Network Partitioning Algorithm | s | – both – | w | - | d | $<\mathcal{O}({e}^{2})$ | ||||||||

Section 5.5 | Ratio of Disruption | s | g | w | v | d | NP-c. | ||||||||

Section 5.6 | Local Decay Resilience | s | [d] | l | / | g | w | $v(h)$ | d | t | ✓ | ||||

Section 5.7 | Toughness/Integrity/Scatt. No. | s | g | w | N/A | d | NP-c. | ||||||||

Section 5.8 | Tenacity/Edge-T./Mixed-T. | s | g | w | N/A | d | NP-c. | ||||||||

Section 5.9 | Percolation Threshold | s | g | f | ∘ | d | ✓ | ||||||||

Section 5.10 | Reliability Polynomial | s | g | f | ∘ | d | b | ✓ | |||||||

Section 5.11 | Partition Resilience Factor | s | g | f | ∘ | d | b | ✓ | |||||||

Section 5.12 | Total Number of Isolated Components | s | g | d | v | d | ✓ | ||||||||

Frac. of Nodes in Largest Component | s | g | d | ∘ | d | ✓ | |||||||||

Average Size of Isolated Components | s | g | d | v | ✓ | ||||||||||

Distr. of Component Class Frequency | s | g | d | v | d | ✓ | |||||||||

Distr. of Rel. No. of Nodes per Class | s | g | d | ∘ | d | ✓ | |||||||||

Reachability | s | g | d | ∘ | d | ✓ | |||||||||

Section 6.1 | ASPL | s | [w] | [d] | i | / | g | s | v | t | ✓ | ||||

Diameter-Inverse-K | s | [w] | [d] | g | s | v | t | ✓ | |||||||

Diameter | s | [w] | [d] | g | s | ∞/v | t | ✓ | |||||||

Section 6.2 | Global Network Efficiency | s | [w] | [d] | i | / | g | s | ∘ | t | b | ✓ | |||

Harm. Mean of Geodesic Distances | s | g | s | ∞ | t | ✓ | |||||||||

Section 6.3 | Local Network Efficiency | s | [w] | [d] | l and i | / | g | s | ∘ | d | t | [o] | ✓ | ||

Cyclic Coefficient | s | [w] | [d] | l and i | / | g | s | ∘ | d | t | [o] | ✓ | |||

Section 6.4 | Characteristic Path Length | l | [w] | [d] | i | s | ∞/v | t | [b] | ✓ | |||||

Section 6.5 | Expansion | s | [w] | [d] | i | / | g | s | ∘ | [d] | t | b | ✓ | ||

Section 6.6 | Effective Eccentricity/Eff. Diameter | s | [w] | [d] | i | / | g | s | v | [d] | t | b | ✓ | ||

Section 7.1 | Betweenness Centrality, node/link | s | [w] | [d] | i | s | v | [d] | [t] | o | ✓ | ||||

AS Hegemony | s | i | s | N/A | [d] | [t] | o | ✓ | |||||||

Edge Degree | s | [w] | [d] | i | s | ${k}_{max}$ | ✓ | ||||||||

Section 7.2 | Central Point Dominance | s | [w] | [d] | g | s | ∘ | [d] | [t] | ✓ | |||||

Section 7.3 | Effective Load | s | [w] | [d] | l and i | d | v | o | ✓ | ||||||

Section 7.4 | Performance | $\tilde{w}$ | [w] | [d] | g | s | ∞ | o | ✓ | ||||||

Section 7.5 | Elasticity | s | [w] | [d] | g | d | ∘ | o | ✓ | ||||||

Section 7.6 | Vulnerability Impact Factors | l | w | [d] | l and i | / | g | d | ∞/∘ | o | ✓ | ||||

Section 7.7 | Survivability Function, Failures | w | d | g | f | ∘ | o | b | ✓ | ||||||

Section 8.1 | Eigenvector Centrality | s | [w] | [d] | l and i | s | ∘ | ✓ | |||||||

Section 8.2 | Symmetry Ratio | s | g | s | v | [o] | ✓ | ||||||||

Section 8.3 | Spectral Cluster Identification | s | [d] | – both – | s | - | d | [t] | [o] | ✓ | |||||

Section 8.4 | Algebraic Connectivity | s | g | s | $v,{k}_{min}$ | d | ✓ | ||||||||

Section 8.5 | Good Expansion | s | [w] | g | s | ∘ | d | [t] | [o] | NP-h. | |||||

Section 8.6 | Number of Spanning Trees | s | g | s | ${k}_{i}$ | d | ✓ | ||||||||

Section 8.7 | Natural Connectivity | s | g | s | v | d | ✓ | ||||||||

Section 8.8 | Random Walk ASPL | s | [w] | [d] | i | / | g | s | ∞ | t | ✓ | ||||

Section 8.9 | Current-Flow Closeness | s | [w] | [d] | i | / | g | s | N/A | t | ✓ | ||||

Section 8.10 | Random Walk Betweenness | – all – | i | s | $v,P(k)$ | [d] | [t] | o | ✓ | ||||||

Section 8.11 | Current-Flow Betweenness | s | g | s | v | [d] | [t] | o | ✓ | ||||||

Section 8.12 | Network Criticality | – all – | g | s | $v,a(G)$ | [d] | t | o | ✓ | ||||||

Section 9.1 | Distance Strength | g | [d] | i | s | ∞ | d | t | ✓ | ||||||

Section 9.2 | Survivability Function, Geographical | g | g | d | ∘ | d | ✓ | ||||||||

Outreach | g | w | [d] | i | s | ∞ | d | t | o | ✓ | |||||

Section 9.3 | Pointwise Vulnerability | g | [w] | [d] | i | d | ∘ | t | [o] | ✓ | |||||

Global Vulnerability | g | [w] | [d] | g | d | ∘ | t | ✓ | |||||||

Rel. Variance of Pointwise Vuln. | g | [w] | [d] | g | d | ∞ | t | ✓ | |||||||

Section 9.4 | Eff./Total Geograph. Path Diversity | g | [d] | l | / | g | s | ∘ | d | ✓ |

## References

- Ho, S.C.; Kauffman, R.J.; Liang, T.P. A growth theory perspective on B2C e-commerce growth in Europe: An exploratory study. Electron. Commer. Res. Appl.
**2007**, 6, 237–259. [Google Scholar] [CrossRef] - Doerr, C.; Kuipers, F.A. All quiet on the Internet front? IEEE Commun. Mag.
**2014**, 52, 46–51. [Google Scholar] [CrossRef] - Albert, R.; Jeong, H.; Barabási, A.L. Error and attack tolerance of complex networks. Nature
**2000**, 406, 378–382. [Google Scholar] [CrossRef] [PubMed][Green Version] - Magnien, C.; Latapy, M.; Guillaume, J.L. Impact of random failures and attacks on poisson and power-law random networks. ACM Comput. Surv.
**2011**, 43, 13:1–13:31. [Google Scholar] [CrossRef] - Baumann, A.; Fabian, B. How robust is the Internet?—Insights from graph analysis. In International Conference on Risks and Security of Internet and Systems (CRiSIS 2014): Risks and Security of Internet and Systems; Springer: Berlin/Heidelberg, Germany, 2014; Volume 8924, pp. 247–254. [Google Scholar]
- Sterbenz, J.P.; Hutchison, D.; Çetinkaya, E.K.; Jabbar, A.; Rohrer, J.P.; Schöller, M.; Smith, P. Resilience and survivability in communication networks: Strategies, principles, and survey of disciplines. Comput. Netw.
**2010**, 54, 1245–1265. [Google Scholar] [CrossRef] - Fabian, B.; Tilch, G.; Ermakova, T. A Multilayer Graph Model of the Internet Topology; Technical Report; Zenodo (CERN): Geneva, Switzerland, 2017. [Google Scholar] [CrossRef]
- Fabian, B.; Baumann, A.; Lackner, J. Topological Analysis of Cloud Service Connectivity. Comput. Ind. Eng.
**2015**, 88, 151–165. [Google Scholar] [CrossRef] - Albert, R.; Barabási, A.L. Statistical mechanics of complex networks. Rev. Mod. Phys.
**2002**, 74, 47. [Google Scholar] [CrossRef][Green Version] - Pastor-Satorras, R.; Vespignani, A. Evolution and Structure of the Internet: A Statistical Physics Approach; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Barabási, A.L. Network Science; Cambridge University Press: Cambridge, UK, 2016. [Google Scholar]
- Costa, L.d.F.; Rodrigues, F.A.; Travieso, G.; Villas Boas, P.R. Characterization of complex networks: A survey of measurements. Adv. Phys.
**2007**, 56, 167–242. [Google Scholar] [CrossRef][Green Version] - Tangmunarunkit, H.; Govindan, R.; Shenker, S.; Estrin, D. The impact of routing policy on Internet paths. In Proceedings of the Twentieth Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM 2001), Anchorage, AK, USA, 22–26 April 2001; pp. 736–742. [Google Scholar]
- Tangmunarunkit, H.; Govindan, R.; Jamin, S.; Shenker, S.; Willinger, W. Network topology generators: Degree-based vs. structural. ACM SIGCOMM Comput. Commun. Rev.
**2002**, 32, 147. [Google Scholar] [CrossRef] - Doyle, J.C.; Alderson, D.L.; Li, L.; Low, S.; Roughan, M.; Shalunov, S.; Tanaka, R.; Willinger, W. The “robust yet fragile” nature of the Internet. Proc. Natl. Acad. Sci. USA
**2005**, 102, 14497–14502. [Google Scholar] [CrossRef] [PubMed][Green Version] - Crucitti, P.; Latora, V.; Porta, S. Centrality measures in spatial networks of urban streets. Phys. Rev. E Stat. Nonlinear Soft Matter Phys.
**2006**, 73, 1–5. [Google Scholar] [CrossRef][Green Version] - Faloutsos, M.; Faloutsos, P.; Faloutsos, C. On power-law relationships of the Internet topology. In ACM SIGCOMM Computer Communication Review; ACM: New York, NY, USA, 1999; Volume 29, pp. 251–262. [Google Scholar]
- Watts, D.J.; Strogatz, S.H. Collective dynamics of ’small-world’ networks. Nature
**1998**, 393, 440–442. [Google Scholar] [CrossRef] [PubMed] - Erdös, P.; Rényi, A. On random graphs, I. Publ. Math. (Debrecen)
**1959**, 6, 290–297. [Google Scholar] - Barabási, A.L.; Albert, R. Emergence of scaling in random networks. Science
**1999**, 286, 509–512. [Google Scholar] [CrossRef][Green Version] - Klemm, K.; Eguíluz, V.M. Growing scale-free networks with small-world behavior. Phys. Rev. E Stat. Nonlinear Soft Matter Phys.
**2002**, 65, 057102. [Google Scholar] [CrossRef][Green Version] - Li, L.; Alderson, D.; Willinger, W.; Doyle, J. A first-principles approach to understanding the Internet’s router-level topology. ACM SIGCOMM Comput. Commun. Rev.
**2004**, 34, 3–14. [Google Scholar] [CrossRef] - Aggarwal, C.; Subbian, K. Evolutionary network analysis: A survey. ACM Comput. Surv.
**2014**, 47, 10:1–10:36. [Google Scholar] [CrossRef] - Çetinkaya, E.K.; Alenazi, M.J.; Peck, A.M.; Rohrer, J.P.; Sterbenz, J.P. Multilevel resilience analysis of transportation and communication networks. Telecommun. Syst.
**2015**, 60, 515–537. [Google Scholar] [CrossRef][Green Version] - Muro, M.A.D.; Valdez, L.D.; Rego, H.H.A.; Buldyrev, S.V.; Stanley, H.E.; Braunstein, L.A. Cascading Failures in Interdependent Networks with Multiple Supply-Demand Links and Functionality Thresholds. Sci. Rep.
**2017**, 7, 15059. [Google Scholar] [CrossRef] [PubMed][Green Version] - Kamisinski, A.; Cholda, P.; Jajszczyk, A. Assessing the structural complexity of computer and communication networks. ACM Comput. Surv.
**2015**, 47, 66:1–66:36. [Google Scholar] [CrossRef] - Baumann, A.; Fabian, B. Vulnerability against Internet disruptions—A graph-based perspective. In International Conference on Critical Information Infrastructures Security; Springer: Berlin, Germany, 2015; Volume 9578, pp. 120–131. [Google Scholar]
- Hakimi, S.L. On realizability of a set of integers as degrees of the vertices of a linear graph. I. J. Soc. Ind. Appl. Math.
**1963**, 11, 496–506. [Google Scholar] [CrossRef] - Ghedini, C.G.; Ribeiro, C.H.C. Rethinking failure and attack tolerance assessment in complex networks. Phys. A Stat. Mech. Its Appl.
**2011**, 390, 4684–4691. [Google Scholar] [CrossRef] - Xia, Y.; Hill, D.J. Attack vulnerability of complex communication networks. IEEE Trans. Circuits Syst. II Express Briefs
**2008**, 55, 65–69. [Google Scholar] [CrossRef] - Dall’Asta, L.; Barrat, A.; Barthélemy, M.; Vespignani, A. Vulnerability of weighted networks. J. Stat. Mech. Theory Exp.
**2006**, 2006, P04006. [Google Scholar] [CrossRef][Green Version] - Barrat, A.; Barthelemy, M.; Pastor-Satorras, R.; Vespignani, A. The architecture of complex weighted networks. Proc. Natl. Acad. Sci. USA
**2004**, 101, 3747–3752. [Google Scholar] [CrossRef] [PubMed][Green Version] - Wang, B.; Tang, H.; Guo, C.; Xiu, Z. Entropy optimization of scale-free networks’ robustness to random failures. Phys. A Stat. Mech. Its Appl.
**2006**, 363, 591–596. [Google Scholar] [CrossRef][Green Version] - Park, S.T.; Pennock, D.M.; Giles, C.L. Comparing static and dynamic measurements and models of the Internet’s AS topology. In Proceedings of the Twenty-third Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM 2004), Hong Kong, China, 7–11 March 2004; Volume 3, pp. 1616–1627. [Google Scholar]
- Criado, R.; Flores, J.; Hernández-Bermejo, B.; Pello, J.; Romance, M. Effective measurement of network vulnerability under random and intentional attacks. J. Math. Model. Algorithms
**2005**, 4, 307–316. [Google Scholar] [CrossRef] - Yazdani, A.; Jeffrey, P. A note on measurement of network vulnerability under random and intentional attacks. arXiv
**2010**, arXiv:1006.2791. [Google Scholar] - Mahadevan, P.; Krioukov, D.; Fomenkov, M.; Dimitropoulos, X.; Claffy, K.; Vahdat, A. The Internet AS-level topology: Three data sources and one definitive metric. ACM SIGCOMM Comput. Commun. Rev.
**2006**, 36, 17–26. [Google Scholar] [CrossRef][Green Version] - Newman, M.E.J. Assortative mixing in networks. Phys. Rev. Lett.
**2002**, 89, 208701. [Google Scholar] [CrossRef][Green Version] - Liu, J.; Zhou, M.; Wang, S.; Liu, P. A comparative study of network robustness measures. Front. Comput. Sci.
**2017**, 11, 568–584. [Google Scholar] [CrossRef] - Mahadevan, P.; Hubble, C.; Krioukov, D.; Huffaker, B.; Vahdat, A. Orbis: Rescaling degree correlations to generate annotated Internet topologies. ACM SIGCOMM Comput. Commun. Rev.
**2007**, 37, 325–336. [Google Scholar] [CrossRef] - Colizza, V.; Flammini, A.; Serrano, M.A.; Vespignani, A. Detecting rich-club ordering in complex networks. Nat. Phys.
**2006**, 2, 110–115. [Google Scholar] [CrossRef] - Opsahl, T.; Colizza, V.; Panzarasa, P.; Ramasco, J.J. Prominence and control: The weighted rich-club effect. Phys. Rev. Lett.
**2008**, 101, 168702. [Google Scholar] [CrossRef] [PubMed][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] - Edwards, B.; Hofmeyr, S.; Stelle, G.; Forrest, S. Internet topology over time. arXiv
**2012**, arXiv:1202.3993. [Google Scholar] - Sato, Y.; Ata, S.; Oka, I. A strategic approach for re-organizing the Internet topology by applying social behavior dynamics. J. Netw. Syst. Manag.
**2009**, 17, 208–229. [Google Scholar] [CrossRef] - Oliveira, R.V.; Zhang, B.; Zhang, L. Observing the evolution of Internet as topology. ACM SIGCOMM Comput. Commun. Rev.
**2007**, 37, 313. [Google Scholar] [CrossRef][Green Version] - Soffer, S.N.; Vázquez, A. Network clustering coefficient without degree-correlation biases. Phys. Rev. E Stat. Nonlinear Soft Matter Phys.
**2005**, 71, 2–5. [Google Scholar] [CrossRef][Green Version] - Newman, M.E.J. Scientific collaboration networks. I. Network construction and fundamental results. Phys. Rev. E Stat. Nonlinear Soft Matter Phys.
**2001**, 64, 016131. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ravasz, E.; Barabási, A.L. Hierarchical organization in complex networks. Phys. Rev. E
**2003**, 67, 026112. [Google Scholar] [CrossRef] [PubMed][Green Version] - Jabbar, A. A Framework to Quantify Network Resilience and Survivability. Ph.D. Thesis, University of Kansas, Lawrence, KS, USA, 2010. [Google Scholar]
- Barthélemy, M.; Barrat, A.; Pastor-Satorras, R.; Vespignani, A. Characterization and modeling of weighted networks. Phys. A Stat. Mech. Its Appl.
**2005**, 346, 34–43. [Google Scholar] [CrossRef][Green Version] - Onnela, J.P.; Saramäki, J.; Kertész, J.; Kaski, K. Intensity and coherence of motifs in weighted complex networks. Phys. Rev. E Stat. Nonlinear Soft Matter Phys.
**2005**, 71, 065103. [Google Scholar] [CrossRef][Green Version] - Opsahl, T.; Panzarasa, P. Clustering in weighted networks. Soc. Netw.
**2009**, 31, 155–163. [Google Scholar] [CrossRef] - Wasserman, S.; Faust, K. Social Network Analysis: Methods and Applications; Cambridge University Press: Cambridge, UK, 1994; Volume 8. [Google Scholar]
- Radicchi, F.; Castellano, C.; Cecconi, F.; Loreto, V.; Parisi, D. Defining and identifying communities in networks. Proc. Natl. Acad. Sci. USA
**2004**, 101, 2658–2663. [Google Scholar] [CrossRef][Green Version] - Newman, M.E.J. Detecting community structure in networks. Eur. Phys. J. B-Condens. Matter Complex. Syst.
**2004**, 38, 321–330. [Google Scholar] [CrossRef] - Newman, M.E.J.; Girvan, M. Finding and evaluating community structure in networks. Phys. Rev. E
**2004**, 69, 026113. [Google Scholar] [CrossRef] [PubMed][Green Version] - Newman, M.E.J. Modularity and community structure in networks. Proc. Natl. Acad. Sci. USA
**2006**, 103, 8577–8582. [Google Scholar] [CrossRef][Green Version] - Girvan, M.; Newman, M.E.J. Community structure in social and biological networks. Proc. Natl. Acad. Sci. USA
**2002**, 99, 7821–7826. [Google Scholar] [CrossRef] [PubMed][Green Version] - Tyler, J.R.; Wilkinson, D.M.; Huberman, B.A. Email as spectroscopy: Automated discovery of community structure within organizations. In First International Conference on Communities and Technologies (C&T 2003); Kluwer: Dordrecht, The Netherlands, 2003; pp. 81–96. [Google Scholar]
- Guimera, R.; Amaral, L.A.N. Functional cartography of complex metabolic networks. Nature
**2005**, 433, 895–900. [Google Scholar] [CrossRef] [PubMed][Green Version] - Mohar, B. Isoperimetric numbers of graphs. J. Comb. Theory Ser. B
**1989**, 47, 274–291. [Google Scholar] [CrossRef] - Chartrand, G.; Lesniak, L. Graphs and Digraphs; Wadsworth & Brooks/Cole: Monterey, CA, USA, 1996. [Google Scholar]
- Garey, M.R.; Johnson, D.S. Computers and Intractability: A Guide to the Theory of NP-Completeness; W. H. Freeman and Company: New York, NY, USA, 1979; Volume 58. [Google Scholar]
- Harary, F. Conditional connectivity. Networks
**1983**, 13, 347–357. [Google Scholar] [CrossRef] - Bui, T.N.; Jones, C. Finding good approximate vertex and edge partitions is NP-hard. Inf. Process. Lett.
**1992**, 42, 153–159. [Google Scholar] [CrossRef] - Sreenivasan, S.; Cohen, R.; López, E.; Toroczkai, Z.; Stanley, H.E. Structural bottlenecks for communication in networks. Phys. Rev. E Stat. Nonlinear Soft Matter Phys.
**2007**, 75, 1–4. [Google Scholar] [CrossRef] [PubMed][Green Version] - Merris, R. Laplacian matrices of graphs: A survey. Linear Algebra Appl.
**1994**, 197, 143–176. [Google Scholar] [CrossRef][Green Version] - Boesch, F.; Thomas, R. On graphs of invulnerable communication nets. IEEE Trans. Circuit Theory
**1970**, 17, 183–192. [Google Scholar] [CrossRef] - Fiduccia, C.M.; Mattheyses, R.M. A linear-time heuristic for improving network partitions. In Papers on Twenty-Five Years of Electronic Design Automation; ACM: New York, NY, USA, 1988; pp. 241–247. [Google Scholar]
- Wang, Y.; Xiao, S.; Xiao, G.; Fu, X.; Cheng, T.H. Robustness of complex communication networks under link attacks. In Proceedings of the International Conference on Advanced Infocomm Technology (ICAIT ’08), Hangzhou, China, 25–27 October 2008; pp. 1–7. [Google Scholar]
- Lipman, M.; Pippert, R. Toward a measure of vulnerability II. The ratio of disruption. In Graph Theory with Applications to Algorithms and Computer Science; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 1985; pp. 507–517. [Google Scholar]
- Karypis, G.; Kumar, V. A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput.
**1998**, 20, 359–392. [Google Scholar] [CrossRef] - Chvátal, V. Tough graphs and hamiltonian circuits. Discret. Math.
**1973**, 5, 215–228. [Google Scholar] [CrossRef][Green Version] - Barefoot, C.A.; Entringer, R.; Swart, H. Vulnerability in graphs—A comparative survey. J. Combin. Math. Combin. Comput
**1987**, 1, 13–22. [Google Scholar] - Jung, H.A. On a class of posets and the corresponding comparability graphs. J. Comb. Theory Ser. B
**1978**, 24, 125–133. [Google Scholar] [CrossRef][Green Version] - Zhang, S. Cycles in Weighted Graphs and Related Topics; Twente University Press: Enschede, The Netherlands, 2002. [Google Scholar]
- Moazzami, D.; Salehian, B. On the edge-tenacity of graphs. Int. Math. Forum
**2008**, 3, 929–936. [Google Scholar] - Piazza, B.L.; Robertst, F.S.; Stueckle, S.K. Edge-tenacious networks. Networks
**1995**, 25, 7–17. [Google Scholar] [CrossRef] - Cohen, R.; Erez, K.; Ben-Avraham, D.; Havlin, S. Resilience of the Internet to random breakdowns. Phys. Rev. Lett.
**2000**, 85, 4626. [Google Scholar] [CrossRef] [PubMed][Green Version] - Page, L.B.; Perry, J.E. Reliability polynomials and link importance in networks. IEEE Trans. Reliab.
**1994**, 43, 51–58. [Google Scholar] [CrossRef] - Page, L.B.; Perry, J.E. A practical implementation of the factoring theorem for network reliability. IEEE Trans. Reliab.
**1988**, 37, 259–267. [Google Scholar] [CrossRef] - Salles, R.M.; Marino, D.A., Jr. Strategies and metric for resilience in computer networks. Comput. J.
**2011**, 55, 728–739. [Google Scholar] [CrossRef] - Sun, S.; Liu, Z.; Chen, Z.; Yuan, Z. Error and attack tolerance of evolving networks with local preferential attachment. Phys. A Stat. Mech. Its Appl.
**2007**, 373, 851–860. [Google Scholar] [CrossRef] - Magoni, D. Tearing down the Internet. IEEE J. Sel. A. Commun.
**2006**, 21, 949–960. [Google Scholar] [CrossRef] - Xiao, S.; Xiao, G.; Cheng, T.H. Tolerance of local information-based intentional attacks in complex networks. J. Phys. A Math. Theor.
**2010**, 43, 335101. [Google Scholar] [CrossRef][Green Version] - Çetinkaya, E.K.; Peck, A.M.; Sterbenz, J.P. Flow robustness of multilevel networks. In Proceedings of the 9th International Conference on the Design of Reliable Communication Networks (DRCN), Budapest, Hungary, 4–7 March 2013; pp. 274–281. [Google Scholar]
- Brandes, U.; Fleischer, D. Centrality measures based on current flow. In Annual Symposium on Theoretical Aspects of Computer Science; Springer: Berlin, Germany, 2005; pp. 533–544. [Google Scholar]
- Park, S.T.; Khrabrov, A.; Pennock, D.M.; Lawrence, S.; Giles, C.L.; Ungar, L.H. Static and dynamic analysis of the Internet’s susceptibility to faults and attacks. In Proceedings of the INFOCOM 2003—Twenty-Second Annual Joint Conference of the IEEE Computer and Communications, San Francisco, CA, USA, 30 March–3 April 2003; Volume 3, pp. 2144–2154. [Google Scholar]
- Ng, A.K.S.; Efstathiou, J. Structural robustness of complex networks. Phys. Rev.
**2006**, 3, 175–188. [Google Scholar] - Rochat, Y. Closeness centrality extended to unconnected graphs: The harmonic centrality index. In Proceedings of the 6th Applications of Social Network Analysis Conference, Zurich, Switzerland, 26–28 August 2009. [Google Scholar]
- Latora, V.; Marchiori, M. Efficient behavior of small-world networks. Phys. Rev. Lett.
**2001**, 87, 198701. [Google Scholar] [CrossRef][Green Version] - Gkantsidis, C.; Mihail, M.; Zegura, E. Spectral analysis of Internet topologies. In Proceedings of the IEEE INFOCOM 2003. Twenty-Second Annual Joint Conference of the IEEE Computer and Communications Societies (IEEE Cat. No.03CH37428), San Francisco, CA, USA, 30 March–3 April 2003. [Google Scholar]
- Opsahl, T.; Agneessens, F.; Skvoretz, J. Node centrality in weighted networks: Generalizing degree and shortest paths. Soc. Netw.
**2010**, 32, 245–251. [Google Scholar] [CrossRef] - Kim, H.J.; Kim, J.M. Cyclic topology in complex networks. Phys. Rev. E
**2005**, 72, 036109. [Google Scholar] [CrossRef] [PubMed][Green Version] - Trpevski, D.; Smilkov, D.; Mishkovski, I.; Kocarev, L. Vulnerability of labeled networks. Phys. A Stat. Mech. Its Appl.
**2010**, 389, 5538–5549. [Google Scholar] [CrossRef] - Palmer, C.R.; Siganos, G.; Faloutsos, M.; Faloutsos, C.; Gibbons, P.B. The connectivity and fault-tolerance of the Internet topology. In Proceedings of the Workshop on Network Related Data Management (NRDM 2001), Santa Barbara, CA, USA, 25 May 2001; pp. 1–9. [Google Scholar]
- Freeman, L.C. A set of measures of centrality based on betweenness. Sociometry
**1977**, 40, 35–41. [Google Scholar] [CrossRef] - Newman, M.E.J. A measure of betweenness centrality based on random walks. Soc. Netw.
**2005**, 27, 39–54. [Google Scholar] [CrossRef][Green Version] - Boccaletti, S.; Buldú, J.; Criado, R.; Flores, J.; Latora, V.; Pello, J.; Romance, M. Multiscale vulnerability of complex networks. Chaos Interdiscip. J. Nonlinear Sci.
**2007**, 17, 043110. [Google Scholar] [CrossRef][Green Version] - Holme, P.; Kim, B.J.; Yoon, C.N.; Han, S.K. Attack vulnerability of complex networks. Phys. Rev. E Stat. Nonlinear Soft Matter Phys.
**2002**, 65, 056109. [Google Scholar] [CrossRef][Green Version] - Fontugne, R.; Shah, A.; Aben, E. AS hegemony: A robust metric for AS centrality. In Proceedings of the SIGCOMM Posters and Demos 2017, Los Angeles, CA, USA, 22–24 August 2017; pp. 48–50. [Google Scholar]
- Holme, P.; Kim, B.J. Vertex overload breakdown in evolving networks. Phys. Rev. E Stat. Nonlinear Soft Matter Phys.
**2002**, 65, 066109. [Google Scholar] [CrossRef][Green Version] - Motter, A.E.; Lai, Y.C. Cascade-based attacks on complex networks. Phys. Rev. E Stat. Nonlinear Soft Matter Phys.
**2002**, 66, 065102. [Google Scholar] [CrossRef][Green Version] - Lee, E.J.; Goh, K.I.; Kahng, B.; Kim, D. Robustness of the avalanche dynamics in data-packet transport on scale-free networks. Phys. Rev. E Stat. Nonlinear Soft Matter Phys.
**2005**, 71, 1–5. [Google Scholar] [CrossRef][Green Version] - Zhang, Y.; Roughan, M.; Lund, C.; Donoho, D. An information-theoretic approach to traffic matrix estimation. In Proceedings of the Conference on Applications, Technologies, Architectures, and Protocols for Computer Communications (SIGCOMM ’03), Karlsruhe, Germany, 25–29 August 2003; pp. 301–312. [Google Scholar]
- Sydney, A.; Scoglio, C.; Schumm, P.; Kooij, R.E. ELASTICITY: Topological Characterization of Robustness in Complex Networks; Bionetics: Hyogo, Japan, 2008; Volume 16, p. 26. [Google Scholar]
- Sydney, A.; Scoglio, C.; Youssef, M.; Schumm, P. Characterising the robustness of complex networks. Int. J. Internet Technol. Secur. Trans.
**2010**, 2, 291–320. [Google Scholar] [CrossRef] - Dong, J.; Horvath, S. Understanding network concepts in modules. BMC Syst. Biol.
**2007**, 1, 24. [Google Scholar] [CrossRef] [PubMed][Green Version] - Hariri, S.; Qu, G.; Dharmagadda, T.; Ramkishore, M.; Raghavendra, C.S. Impact analysis of faults and attacks in large-scale networks. IEEE Secur. Priv.
**2003**, 99, 49–54. [Google Scholar] [CrossRef] - Molisz, W. Survivability function-a measure of disaster-based routing performance. IEEE J. Sel. Areas Commun.
**2004**, 22, 1876–1883. [Google Scholar] [CrossRef] - Medhi, D.; Ramasamy, K. Network Routing: Algorithms, Protocols, and Architectures, 2nd ed.; Morgan Kaufmann: Burlington, MA, USA, 2017. [Google Scholar]
- Yang, H.X.; Wang, W.X.; Xie, Y.B.; Lai, Y.C.; Wang, B.H. Transportation dynamics on networks of mobile agents. Phys. Rev. E
**2011**, 83, 016102. [Google Scholar] [CrossRef] [PubMed][Green Version] - Seary, A.J.; Richards, W.D. Spectral methods for analyzing and visualizing networks: An introduction. In Proceedings of the Workshop on Dynamic Social Network Modeling and Analysis, Washington, DC, USA, 7–9 November 2002; pp. 1–20. [Google Scholar]
- Golnari, G.; Zhang, Z.L.; Boley, D. Random walk fundamental tensor and its applications to network analysis. arxiv
**2018**, arXiv:1801.08583. [Google Scholar] - Bonacich, P. Factoring and weighting approaches to status scores and clique identification. J. Math. Sociol.
**1972**, 2, 113–120. [Google Scholar] [CrossRef] - Tauro, S.L.; Palmer, C.; Siganos, G.; Faloutsos, M. A simple conceptual model for the Internet topology. In Proceedongs of the IEEE Global Telecommunications Conference (GLOBECOM’01). San Antonio, TX, USA, 25–29 November 2001; Volume 3, pp. 1667–1671. [Google Scholar]
- Dekker, A.H.; Colbert, B. The symmetry ratio of a network. In Proceedings of the 2005 Australasian Symposium on Theory of Computing (CATS ’05), Antonio, TX, USA, 25–29 November 2005; Volume 41. [Google Scholar]
- Manzano, M.; Calle, E.; Harle, D. Quantitative and qualitative network robustness analysis under different multiple failure scenarios. In Proceedings of the 3rd International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT), Budapest, Hungary, 5–7 October 2011; pp. 1–7. [Google Scholar]
- Mihail, M.; Papadimitriou, C. On the eigenvalue power law. In Randomization and Approximation Techniques in Computer Science; Springer: Berlin, Germany, 2002; pp. 254–262. [Google Scholar]
- Fiedler, M. Algebraic connectivity of graphs. Czechoslov. Math. J.
**1973**, 23, 298–305. [Google Scholar] [CrossRef] - Wu, J.; Tan, Y.J.; Deng, H.Z.; Li, Y.; Liu, B.; Lv, X. Spectral measure of robustness in complex networks. arXiv
**2008**, arXiv:0802.2564. [Google Scholar] [CrossRef] - Jamakovic-Kapic, A.; Uhlig, S. Influence of the network structure on robustness. In Proceedings of the 15th IEEE International Conference on Networks (ICON 2007), Adelaide, SA, Australia, 19–21 November 2007; pp. 278–283. [Google Scholar]
- Mohar, B. Some applications of Laplace eigenvalues of graphs. Graph Symmetry Algebr. Methods Appl.
**1991**, 497, 225–275. [Google Scholar] - Estrada, E. Spectral scaling and good expansion properties in complex networks. EPL (Europhys. Lett.)
**2006**, 73, 649. [Google Scholar] [CrossRef] - Szabó, G.J.; Alava, M.; Kertész, J. Geometry of minimum spanning trees on scale-free networks. Phys. A Stat. Mech. Its Appl.
**2003**, 330, 31–36. [Google Scholar] [CrossRef] - Zhang, Z.; Liu, H.; Wu, B.; Zhou, S. Enumeration of spanning trees in a pseudofractal scale-free web. EPL (Europhys. Lett.)
**2010**, 90, 68002. [Google Scholar] [CrossRef][Green Version] - Estrada, E. Characterization of 3D molecular structure. Chem. Phys. Lett.
**2000**, 319, 713–718. [Google Scholar] [CrossRef] - Wu, J.; Barahona, M.; Tan, Y.J.; Deng, H.Z. Spectral measure of structural robustness in complex networks. IEEE Trans. Syst. Man Cybern. Part A Syst. Hum.
**2011**, 41, 1244–1252. [Google Scholar] [CrossRef] - Tizghadam, A.; Leon-Garcia, A. Robust network planning in nonuniform traffic scenarios. Comput. Commun.
**2011**, 34, 1436–1449. [Google Scholar] [CrossRef] - Liew, S.C.; Lu, K.W. A framework for network survivability characterization. In Proceedings of the ICC’92, Conference Record, SUPERCOMM/ICC’92, Discovering a New World of Communications, Chicago, IL, USA, 14–18 June 1992; pp. 405–410. [Google Scholar]
- Gol’dshtein, V.; Koganov, G.A.; Surdutovich, G.I. Vulnerability and hierarchy of complex networks. arXiv
**2004**, arXiv:cond-mat/0409298. [Google Scholar] - Cheng, Y.; Li, J.; Sterbenz, J.P.G. Path Geo-Diversification: Design and Analysis; IEEE: Piscatway, NJ, USA, 2013; pp. 46–53. [Google Scholar]
- Fabian, B.; Baumann, A.; Ehlert, M.; Ververis, V.; Ermakova, T. CORIA - Analyzing Internet connectivity risks using network graphs. In Proceedings of the IEEE International Conference on Communications (IEEE ICC 2017), Paris, France, 21–25 May 2017. [Google Scholar]
- Strogatz, S.H. Exploring complex networks. Nature
**2001**, 410, 268–276. [Google Scholar] [CrossRef][Green Version] - Knoke, D.; Yang, S. Social Network Analysis, 3rd ed.; Quantitative Applications in the Social Sciences; SAGE Publishing: Newbury Park, CA, USA, 2019. [Google Scholar]
- Thadakamaila, H.P.; Raghavan, U.N.; Kumara, S.; Albert, R. Survivability of multiagent-based supply networks: A topological perspect. IEEE Intell. Syst.
**2004**, 19, 24–31. [Google Scholar] [CrossRef] - Kim, Y.; Choi, T.Y.; Yan, T.; Dooley, K. Structural investigation of supply networks: A social network analysis approach. J. Oper. Manag.
**2011**, 29, 194–211. [Google Scholar] [CrossRef] - Háznagy, A.; Fi, I.; London, A.; Nemeth, T. Complex network analysis of public transportation networks: A comprehensive study. In Proceedings of the 2015 International Conference on Models and Technologies for Intelligent Transportation Systems (MT-ITS), Budapest, Hungary, 3–5 June 2015; pp. 371–378. [Google Scholar] [CrossRef][Green Version]
- Lischke, M.; Fabian, B. Analyzing the Bitcoin Network: The First Four Years. Future Internet
**2016**, 8, 7. [Google Scholar] [CrossRef][Green Version]

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Oehlers, M.; Fabian, B.
Graph Metrics for Network Robustness—A Survey. *Mathematics* **2021**, *9*, 895.
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Graph Metrics for Network Robustness—A Survey. *Mathematics*. 2021; 9(8):895.
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2021. "Graph Metrics for Network Robustness—A Survey" *Mathematics* 9, no. 8: 895.
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