# Identifying Influencers in Social Networks

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Works

## 3. Modeling and Methods

#### 3.1. Network Modeling

#### 3.2. Methods

- Susceptible (S) state, where a node is vulnerable to infection.
- Infectious (I) state, where a node tries to infect its susceptible neighbors.
- Recovered (R) state, where a node has recovered (or isolated) and can no longer infect others.

#### 3.3. Complexity Analysis

## 4. Experiments and Discussion

#### 4.1. Experimental Datasets

#### 4.2. Performance Comparison

#### 4.3. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Aldecoa, R.; Marín, I. Surprise maximization reveals the community structure of complex networks. Sci. Rep.
**2013**, 3, 1–9. [Google Scholar] [CrossRef] [Green Version] - Pei, S.; Muchnik, L.; Andrade, J.S., Jr.; Zheng, Z.; Makse, H.A. Searching for superspreaders of information in real-world social media. Sci. Rep.
**2014**, 4, 5547. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Xu, Z.; Rui, X.; He, J.; Wang, Z.; Hadzibeganovic, T. Superspreaders and superblockers based community evolution tracking in dynamic social networks. Knowl.-Based Syst.
**2019**, 105377. [Google Scholar] [CrossRef] - Malliaros, F.D.; Rossi, M.E.G.; Vazirgiannis, M. Locating influential nodes in complex networks. Sci. Rep.
**2016**, 6, 19307. [Google Scholar] [CrossRef] [PubMed] - Salavati, C.; Abdollahpouri, A.; Manbari, Z. BridgeRank: A novel fast centrality measure based on local structure of the network. Phys. A Stat. Mech. Appl.
**2018**, 496, 635–653. [Google Scholar] [CrossRef] - Huang, X.; Chen, D.; Ren, T. Social network coalescence based on multilayer network model. J. Nonlinear Convex Anal.
**2019**, 20, 1465–1474. [Google Scholar] - Wang, D.; Wang, H.; Zou, X. Identifying key nodes in multilayer networks based on tensor decomposition. Chaos Interdiscip. J. Nonlinear Sci.
**2017**, 27, 063108. [Google Scholar] [CrossRef] [Green Version] - Kivelä, M.; Arenas, A.; Barthelemy, M.; Gleeson, J.P.; Moreno, Y.; Porter, M.A. Multilayer networks. J. Complex Netw.
**2014**, 2, 203–271. [Google Scholar] [CrossRef] [Green Version] - Boccaletti, S.; Bianconi, G.; Criado, R.; Del Genio, C.I.; Gómez-Gardenes, J.; Romance, M.; Sendina-Nadal, I.; Wang, Z.; Zanin, M. The structure and dynamics of multilayer networks. Phys. Rep.
**2014**, 544, 1–122. [Google Scholar] [CrossRef] [Green Version] - Rodrigues, F.A. Network centrality: an introduction. In A Mathematical Modeling Approach from Nonlinear Dynamics to Complex Systems; Springer: Berlin, Germany, 2019; pp. 177–196. [Google Scholar]
- Peng, S.; Zhou, Y.; Cao, L.; Yu, S.; Niu, J.; Jia, W. Influence analysis in social networks: A survey. J. Netw. Comput. Appl.
**2018**, 106, 17–32. [Google Scholar] [CrossRef] - Grover, A.; Leskovec, J. node2vec: Scalable feature learning for networks. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, San Francisco, CA, USA, 13–17 August 2016; pp. 855–864. [Google Scholar]
- Jeong, H.; Mason, S.P.; Barabási, A.L.; Oltvai, Z.N. Lethality and centrality in protein networks. Nature
**2001**, 411, 41. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kitsak, M.; Gallos, L.K.; Havlin, S.; Liljeros, F.; Muchnik, L.; Stanley, H.E.; Makse, H.A. Identification of influential spreaders in complex networks. Nat. Phys.
**2010**, 6, 888. [Google Scholar] [CrossRef] [Green Version] - Albert, R.; Albert, I.; Nakarado, G.L. Structural vulnerability of the North American power grid. Phys. Rev. E
**2004**, 69, 025103. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Liu, Y.Y.; Slotine, J.J.; Barabási, A.L. Controllability of complex networks. Nature
**2011**, 473, 167. [Google Scholar] [CrossRef] - Yan, G.; Zhou, T.; Hu, B.; Fu, Z.Q.; Wang, B.H. Efficient routing on complex networks. Phys. Rev. E
**2006**, 73, 046108. [Google Scholar] [CrossRef] [Green Version] - Xie, W.; Yu, W.; Zou, X. Diversity-maintained differential evolution embedded with gradient-based local search. Soft Comput.
**2013**, 17, 1511–1535. [Google Scholar] [CrossRef] - Bonacich, P. Factoring and weighting approaches to status scores and clique identification. J. Math. Sociol.
**1972**, 2, 113–120. [Google Scholar] [CrossRef] - McPherson, M.; Smith-Lovin, L.; Cook, J.M. Birds of a feather: Homophily in social networks. Annu. Rev. Sociol.
**2001**, 27, 415–444. [Google Scholar] [CrossRef] [Green Version] - Borgatti, S.P. Centrality and network flow. Soc. Netw.
**2005**, 27, 55–71. [Google Scholar] [CrossRef] - Freeman, L.C. A set of measures of centrality based on betweenness. Sociometry
**1977**, 40, 35–41. [Google Scholar] [CrossRef] - Freeman, L.C. Centrality in social networks conceptual clarification. Soc. Netw.
**1978**, 1, 215–239. [Google Scholar] [CrossRef] [Green Version] - Bonacich, P. Power and centrality: A family of measures. Am. J. Sociol.
**1987**, 92, 1170–1182. [Google Scholar] [CrossRef] - Brin, S.; Page, L. The anatomy of a large-scale hypertextual web search engine. Comput. Netw. ISDN Syst.
**1998**, 30, 107–117. [Google Scholar] [CrossRef] - Watts, D.J.; Strogatz, S.H. Collective dynamics of ‘small-world’networks. Nature
**1998**, 393, 440. [Google Scholar] [CrossRef] [PubMed] - 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] - Chen, D.B.; Gao, H.; Lü, L.; Zhou, T. Identifying influential nodes in large-scale directed networks: The role of clustering. PLoS ONE
**2013**, 8, e77455. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ma, L.l.; Ma, C.; Zhang, H.F.; Wang, B.H. Identifying influential spreaders in complex networks based on gravity formula. Phys. A Stat. Mech. Appl.
**2016**, 451, 205–212. [Google Scholar] [CrossRef] [Green Version] - Gleiser, P.M.; Danon, L. Community structure in jazz. Adv. Complex Syst.
**2003**, 6, 565–573. [Google Scholar] [CrossRef] [Green Version] - Newman, M.E. Modularity and community structure in networks. Proc. Natl. Acad. Sci. USA
**2006**, 103, 8577–8582. [Google Scholar] [CrossRef] [Green Version] - Colizza, V.; Pastor-Satorras, R.; Vespignani, A. Reaction-diffusion processes and metapopulation models in heterogeneous networks. Nat. Phys.
**2007**, 3, 276–282. [Google Scholar] [CrossRef] [Green Version] - Li, Z.; Ren, T.; Ma, X.; Liu, S.; Zhang, Y.; Zhou, T. Identifying influential spreaders by gravity model. Sci. Rep.
**2019**, 9, 8387. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hirsch, J.E. An index to quantify an individual’s scientific research output. Proc. Natl. Acad. Sci. USA
**2005**, 102, 16569–16572. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lü, L.; Zhang, Y.C.; Yeung, C.H.; Zhou, T. Leaders in social networks, the delicious case. PLoS ONE
**2011**, 6, e21202. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zhuang, Y.; Yağan, O. Information propagation in clustered multilayer networks. IEEE Trans. Netw. Sci. Eng.
**2016**, 3, 211–224. [Google Scholar] [CrossRef] - Basaras, P.; Iosifidis, G.; Katsaros, D.; Tassiulas, L. Identifying influential spreaders in complex multilayer networks: A centrality perspective. IEEE Trans. Netw. Sci. Eng.
**2017**, 6, 31–45. [Google Scholar] [CrossRef] - Cozzo, E.; Kivelä, M.; De Domenico, M.; Solé, A.; Arenas, A.; Gómez, S.; Porter, M.A.; Moreno, Y. Clustering coefficients in multiplex networks. arXiv
**2013**, arXiv:1307.6780. [Google Scholar] - Rahmede, C.; Iacovacci, J.; Arenas, A.; Bianconi, G. Centralities of nodes and influences of layers in large multiplex networks. J. Complex Netw.
**2018**, 6, 733–752. [Google Scholar] [CrossRef] [Green Version] - Paidar, M.; Chaharborj, S.S.; Harounabadi, A. Identifying Top-k Most Influential Nodes by using the Topological Diffusion Models in the Complex Networks. Network
**2017**, 4, 5. [Google Scholar] [CrossRef] [Green Version] - Ohara, K.; Saito, K.; Kimura, M.; Motoda, H. Resampling-based predictive simulation framework of stochastic diffusion model for identifying top-K influential nodes. Int. J. Data Sci. Anal.
**2020**, 9, 175–195. [Google Scholar] [CrossRef] - Tang, J.; Zhang, R.; Yao, Y.; Yang, F.; Zhao, Z.; Hu, R.; Yuan, Y. Identification of top-k influential nodes based on enhanced discrete particle swarm optimization for influence maximization. Phys. A Stat. Mech. Appl.
**2019**, 513, 477–496. [Google Scholar] [CrossRef] - Silber, M.D. The Al Qaeda Factor: Plots against the West; University of Pennsylvania Press: Philadelphia, PA, USA, 2011. [Google Scholar]
- Hethcote, H.W. The mathematics of infectious diseases. SIAM Rev.
**2000**, 42, 599–653. [Google Scholar] [CrossRef] [Green Version] - Guan-Rong, C.; Xiao-Fan, W.; Xiang, L. Introduction to Complex Networks: Models, Structures and Dynamics; Higher Education Press: Beijing, China, 2012. [Google Scholar]
- Krackhardt, D. Assessing the political landscape: Structure, cognition, and power in organizations. Adm. Sci. Q.
**1990**, 35, 342–369. [Google Scholar] [CrossRef] - Zachary, W.W. An information flow model for conflict and fission in small groups. J. Anthropol. Res.
**1977**, 33, 452–473. [Google Scholar] [CrossRef] [Green Version] - Lusseau, D.; Schneider, K.; Boisseau, O.J.; Haase, P.; Slooten, E.; Dawson, S.M. The bottlenose dolphin community of Doubtful Sound features a large proportion of long-lasting associations. Behav. Ecol. Sociobiol.
**2003**, 54, 396–405. [Google Scholar] [CrossRef] - Tsvetovat, M.; Kouznetsov, A. Social Network Analysis for Startups: Finding Connections on the Social Web; OŔeilly Media, Inc.: Sebastopol, CA, USA, 2011. [Google Scholar]
- Knuth, D.E. The Stanford GraphBase: A Platform for Combinatorial Algorithms; SODA: Austin, TX, USA, 1993; Volume 93, pp. 41–43. [Google Scholar]
- Shen-Orr, S.S.; Milo, R.; Mangan, S.; Alon, U. Network motifs in the transcriptional regulation network of Escherichia coli. Nat. Genet.
**2002**, 31, 64–68. [Google Scholar] [CrossRef] [PubMed] - Rossi, R.; Ahmed, N. The network data repository with interactive graph analytics and visualization. In Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence, Austin, TX, USA, 25–30 January 2015. [Google Scholar]
- Duch, J.; Arenas, A. Community detection in complex networks using extremal optimization. Phys. Rev. E
**2005**, 72, 027104. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Cho, A.; Shin, J.; Hwang, S.; Kim, C.; Shim, H.; Kim, H.; Kim, H.; Lee, I. WormNet v3: A network-assisted hypothesis-generating server for Caenorhabditis elegans. Nucleic Acids Res.
**2014**, 42, W76–W82. [Google Scholar] [CrossRef] [Green Version] - Action, R. The Rise of the Medici. Am. J. Sociol.
**1993**, 98, 1259–1319. [Google Scholar] - Krackhardt, D. Cognitive social structures. Soc. Netw.
**1987**, 9, 109–134. [Google Scholar] [CrossRef] - Vickers, M.; Chan, S. Representing Classroom Social Structure; Victoria Institute of Secondary Education: Melbourne, Australia, 1981. [Google Scholar]
- Kapferer, B. Strategy and Transaction in an African Factory: African Workers and Indian Management in a Zambian Town; Manchester University Press: Manchester, UK, 1972. [Google Scholar]
- Lazega, E. The Collegial Phenomenon: The Social Mechanisms of Cooperation Among Peers in a Corporate Law Partnership; Oxford University Press on Demand: Oxford, UK, 2001. [Google Scholar]
- Snijders, T.A.; Pattison, P.E.; Robins, G.L.; Handcock, M.S. New specifications for exponential random graph models. Sociol. Methodol.
**2006**, 36, 99–153. [Google Scholar] [CrossRef] - Stark, C.; Breitkreutz, B.J.; Reguly, T.; Boucher, L.; Breitkreutz, A.; Tyers, M. BioGRID: a general repository for interaction datasets. Nucleic Acids Res.
**2006**, 34, D535–D539. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Magnani, M.; Micenkova, B.; Rossi, L. Combinatorial analysis of multiple networks. arXiv
**2013**, arXiv:1303.4986. [Google Scholar] - De Domenico, M.; Solé-Ribalta, A.; Gómez, S.; Arenas, A. Navigability of interconnected networks under random failures. Proc. Natl. Acad. Sci. USA
**2014**, 111, 8351–8356. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Cardillo, A.; Gómez-Gardenes, J.; Zanin, M.; Romance, M.; Papo, D.; Del Pozo, F.; Boccaletti, S. Emergence of network features from multiplexity. Sci. Rep.
**2013**, 3, 1344. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kendall, M.G. A new measure of rank correlation. Biometrika
**1938**, 30, 81–93. [Google Scholar] [CrossRef]

**Figure 1.**Network of 9/11 terrorists. (

**a**) The monolayer network representation, the size of a node represents its degree; (

**b**) The 9/11 terrorists’ interactions represented by a multilayer network model, where ${L}_{1}$ presents confirmed close contact, ${L}_{2}$ layer shows various recorded interactions, ${L}_{3}$ contains potential or planed or unconfirmed interactions; (

**c**) The super-adjacency matrix representation.

**Figure 2.**The illustration of parameters in the susceptible–infected–recovered (SIR) model and corresponding differential equations.

**Figure 3.**The process of SIR model on Krackhardt’s Kite network. In panel (

**a**), all the nodes are in Susceptible state; while we select one node to be infected, and the neighbors will be infected soon, as shown in panel (

**b**); Finally, the network will reach a stable state, i.e., the number of recovered nodes will reach a maximum, as shown in panel (

**c**).

**Figure 4.**The varying susceptible, infectious and recovered nodes with the increasing of iterations.

**Figure 6.**The comparison of recovered nodes in 9/11 terrorists network by initially setting every node infected separately. The recovered nodes and the corresponding INF values are normalized, which are marked in cyan and red, respectively. The comparison of the nodes in three layers are plotted in facet ${L}_{1}$,${L}_{2}$ and ${L}_{3}$.

**Figure 7.**The varying subgraphs of 12 monolayer networks with the removal of the most influential nodes repeatedly with each centrality indicator.

**Figure 8.**The algorithms’ performance comparison for varying k (ranging from 0 to 20 percent of total nodes), measured by the recovered nodes. The betweenness centrality and the proposed INF measure are very competitive than the others.

**Figure 9.**The computational time comparison of different indicators. The accumulation of running time on the 12 real-world datasets has exhibited that the proposed INF measure is much more efficient than the competitors.

Metric | Topology | Complexity | Advantages | Disadvantages |
---|---|---|---|---|

DC [19] | Local | $O\left(n\right)$ | simple | incapable of dealing with “bridge” nodes |

BC [22] | Global | $O(nm+{n}^{2}logn)$ | finding “bridge-like” nodes | cannot differentiate most marginal nodes |

CC [23] | Global | $O(nm+{n}^{2}logn)$ | finding “nearest” nodes | incapable in disconnected graphs |

EC [24] | Global | $O(n+m)$ | consider both of the quality and quantity of neighbors | may be non-convergent |

PR [25] | Global | $O(n+m)$ | efficient, widely applied in search engine | may be non-convergent |

H-index [34] | Semi-local | $O(n+m)$ | famous for academic evaluation | lack of global information |

k-shell [14] | Global | $O(n+m)$ | suitable for large-scale networks | indistinguishable |

LR [35] | Global | $O(n+m)$ | no parameters; robustness | may be non-convergent |

GR, GR+ [29] | Global | $O\left({n}^{3}\right)$ | accuracy | high complexity in k-shell |

LGR [33] | Semi-local | $O\left({n}^{2}\right)$ | simple and capable in most cases | additional parameters R determination |

**Table 2.**Classic node centralities comparison of kite network. The maximum centralities are marked in bold.

ID | Name | Degree | Betweenness | Closeness | Katz | Eigenvector | INF |
---|---|---|---|---|---|---|---|

1 | Andre | 0.4444 | 0.0231 | 0.5294 | 0.3307 | 0.3522 | 0.7211 |

2 | Beverley | 0.4444 | 0.0231 | 0.5294 | 0.3307 | 0.3522 | 0.7211 |

3 | Carol | 0.3333 | 0.0000 | 0.5000 | 0.3006 | 0.2858 | 0.6495 |

4 | Diane | 0.6667 | 0.1019 | 0.6000 | 0.3907 | 0.4810 | 0.8273 |

5 | Ed | 0.3333 | 0.0000 | 0.5000 | 0.3006 | 0.2858 | 0.6495 |

6 | Fernando | 0.5556 | 0.2315 | 0.6429 | 0.3595 | 0.3977 | 0.7830 |

7 | Garth | 0.5556 | 0.2315 | 0.6429 | 0.3595 | 0.3977 | 0.7830 |

8 | Heather | 0.3333 | 0.3889 | 0.6000 | 0.2887 | 0.1959 | 0.7109 |

9 | Ike | 0.2222 | 0.2222 | 0.4286 | 0.2431 | 0.0481 | 0.7914 |

10 | Jane | 0.1111 | 0.0000 | 0.3103 | 0.2168 | 0.0112 | 0.6225 |

**Table 3.**Averaging recovered nodes and iterations times of each node as initially infected spreaders under 10,000 times SIR stimulations with parameters setting $\beta =0.35$, $\gamma $ = 1.

ID | Name | Recovered Nodes | Iterations |
---|---|---|---|

1 | Andre | 4.8015 | 3.3399 |

2 | Beverley | 4.7902 | 3.3392 |

3 | Carol | 4.3485 | 3.1529 |

4 | Diane | 5.3182 | 3.3684 |

5 | Ed | 4.3336 | 3.1455 |

6 | Fernando | 5.0856 | 3.3251 |

7 | Garth | 5.0338 | 3.2994 |

8 | Heather | 4.0765 | 2.9625 |

9 | Ike | 2.6060 | 2.1806 |

10 | Jane | 1.8086 | 1.7019 |

Dataset Name | $\left|\mathit{V}\right|$ | |E| | <k> | <d> | |C| | r | |H| | ${\mathit{\beta}}_{\mathit{c}}$ |
---|---|---|---|---|---|---|---|---|

Club [47] | 34 | 78 | 4.5882 | 2.4082 | 0.5706 | −0.4756 | 1.6933 | 0.1477 |

Dolphins [48] | 62 | 159 | 5.1290 | 3.3570 | 0.2590 | −0.0436 | 1.3268 | 0.1723 |

911 [49] | 69 | 159 | 4.6087 | 2.4672 | 0.4698 | −0.0380 | 1.7304 | 0.1434 |

Lesmis [50] | 77 | 254 | 6.5974 | 2.6411 | 0.5731 | −0.1652 | 1.8273 | 0.0905 |

Escherichia [51] | 97 | 212 | 4.3711 | 5.5369 | 0.3675 | 0.4116 | 1.2367 | 0.2270 |

Eron [52] | 143 | 623 | 8.7133 | 2.9670 | 0.4339 | −0.0195 | 1.4829 | 0.0839 |

Jazz [30] | 198 | 2742 | 27.6970 | 2.2350 | 0.6175 | 0.0202 | 1.3951 | 0.0266 |

USAir [32] | 332 | 2126 | 12.8072 | 2.7381 | 0.6252 | −0.2079 | 3.4639 | 0.0231 |

NS [31] | 379 | 914 | 4.8232 | 6.0419 | 0.7412 | −0.0817 | 1.6630 | 0.1424 |

C.elegans [53] | 453 | 2032 | 9.0066 | 2.6638 | 0.6465 | −0.2197 | 4.4782 | 0.0254 |

DMLC [54] | 659 | 1570 | 4.7648 | 2.6370 | 0.3279 | −0.1914 | 14.8897 | 0.0143 |

Power [26] | 4941 | 6594 | 2.6691 | 18.9892 | 0.0801 | 0.0035 | 1.4504 | 0.3483 |

**Note:**$\left|V\right|$ and $\left|E\right|$ denotes the number of nodes and edges, respectively. <k> is the average degree; <d> is the average shortest path length; $\left|C\right|$ is the average clustering index; <r> is the assortativity coefficient; $\left|H\right|$ is the degree heterogeneity and ${\beta}_{c}$ represents the epidemic threshold of the SIR model.

**Club**contains the friendships between the 34 members of a karate club at a US university.

**Dolphins**dataset is a animals social network.

**911**represents a monolayer terrorist network of September 11 attacks.

**Lesmis**is the coappearance network of characters in the novel Les Miserables.

**Escherichia**represetns transcriptional regulation networks in cells orchestrate gene expression, where nodes are operons, and each edge is directed from an operon that encodes a transcription factor to an operon that it directly regulates (an operon is one or more genes transcribed on the same mRNA).

**Eron**is a email network collected from Eron company.

**Jazz**lists the collaboration patterns of jazz musicians.

**USAir**is an undirected weighted network as obtained by considering the 500 US airports with the largest amount of traffic from publicly available data. Nodes represent US airports and edges represent air travel connections among them.

**NS**represents coauthorships between 379 scientists whose research centers on the properties of networks of one kind or another.

**C.elegans**represents the edges of the metabolic network of C.elegans.

**DMLC**represents the inferred Links by small/medium-scale rotein-protein interactions (collected from protein-protein interaction data bases).

**Power**is a power grid of the western United States.

Dataset Name | $\left|\mathit{L}\right|$ | $\left|\mathit{V}\right|$ | |E| | |${\mathit{E}}_{\mathit{A}}$| | |${\mathit{E}}_{\mathit{C}}$| | <k> | <d> | |C| |
---|---|---|---|---|---|---|---|---|

Padgett [55] | 2 | 26 | 46 | 35 | 11 | 3.5385 | 2.6923 | 0.1441 |

Krackhardt [56] | 3 | 63 | 307 | 244 | 63 | 9.746 | 2.1731 | 0.3943 |

Vickers [57] | 3 | 87 | 605 | 518 | 87 | 13.908 | 2.1802 | 0.4823 |

Kapferer [58] | 4 | 150 | 769 | 552 | 217 | 10.2533 | 2.5889 | 0.3002 |

Lazega [59,60] | 3 | 211 | 2051 | 1842 | 209 | 19.4408 | 2.3958 | 0.3938 |

humanHIV1 [61] | 5 | 1195 | 1504 | 1269 | 235 | 2.5172 | 4.1385 | 0.0221 |

CS-Aarhus [62] | 5 | 224 | 948 | 620 | 328 | 8.4643 | 3.1847 | 0.3603 |

LondonTransport [63] | 3 | 399 | 472 | 441 | 31 | 2.3659 | 14.2989 | 0.0243 |

EUAirTransportation [64] | 37 | 2034 | 15199 | 3588 | 11611 | 14.9449 | 3.5087 | 0.5969 |

**Note:**$\left|L\right|$ denotes the number of layers; $\left|V\right|$ and $\left|E\right|$ are the total number of nodes and edges, respectively; $|{E}_{A}|$ and $|{E}_{C}|$ denote the number of intralayer edges and interlayer edges, respectively. <k> is the average degree; <d> is the average shortest path length; $\left|C\right|$ is the average clustering index;

**Padgett**consists of 2 layers (marriage alliances and business relationships) describing florentine families in the Renaissance;

**Krackhardt**consists of 3 kinds of relationships (Advice, Friendship and “Reports to”) between managers of a high-tech company;

**Vickers**is collected by Vickers from 29 seventh grade students in a school in Victoria, Australia. Students are asked to nominate their classmates on a number of three kinds of relations;

**Kapferer**exhibits interactions in a tailor shop in Zambia (then Northern Rhodesia) over a period of ten months, where layers represent two different types of interaction, recorded at two different times (seven months apart) over a period of one month;

**Lazega**consists of three kinds of interactions (Co-work, Friendship and Advice) between partners and associates of a corporate law partnership;

**humanHIV1**represents the multiplex genetic and protein interactions network of the human HIV type 1;

**CS-Aarhus**consists of five kinds of online and offline relationships (Facebook, Leisure, Work, Co-authorship, Lunch) between the employees of Computer Science department at Aarhus;

**LondonTransport**is collected from the official website of Transport for London (https://www.tfl.gov.uk/). Nodes are train stations in London and edges encode existing routes between stations;

**EUAirTransportation**is composed by thirty-seven different layers each one corresponding to a different airline operating in Europe.

Dataset Name | DC | INF | LGR | BC | CC |
---|---|---|---|---|---|

Club | 0.2442 | 0.2513 | 0.1515 | 0.1016 | −0.0766 |

Dolphins | 0.0238 | 0.0344 | −0.0196 | −0.0323 | −0.0354 |

911 | 0.0878 | 0.1918 | 0.0426 | 0.0409 | −0.0895 |

Lesmis | 0.0909 | 0.1114 | 0.1032 | −0.1839 | −0.0519 |

Escherichia | 0.0726 | 0.0692 | 0.0009 | −0.0808 | 0.0971 |

Eron | 0.0454 | 0.0927 | −0.0056 | 0.0031 | 0.0107 |

Jazz | 0.069 | 0.0838 | 0.0325 | 0.0244 | 0.0312 |

USAir | −0.0358 | 0.0743 | −0.0402 | −0.0491 | −0.0311 |

NS | −0.0378 | 0.0302 | −0.0089 | −0.0143 | −0.0462 |

C.elegans | 0.0154 | 0.0697 | 0.0242 | 0.0185 | 0.0214 |

Power | −0.005 | 0.0272 | 0.0024 | −0.0015 | 0.0209 |

**Note:**Given a network, the parameters of SIR model are given with the transmission probability $\beta =0.35$ and recovering probability $\mu =1$ for simplicity. To obtain the standard ranking of nodes’ influences, we conducted 1000 independent simulations, in each process every node is selected once as the infect seed once. The best perfromed indicator for each network is emphasized by bold.

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Huang, X.; Chen, D.; Wang, D.; Ren, T.
Identifying Influencers in Social Networks. *Entropy* **2020**, *22*, 450.
https://doi.org/10.3390/e22040450

**AMA Style**

Huang X, Chen D, Wang D, Ren T.
Identifying Influencers in Social Networks. *Entropy*. 2020; 22(4):450.
https://doi.org/10.3390/e22040450

**Chicago/Turabian Style**

Huang, Xinyu, Dongming Chen, Dongqi Wang, and Tao Ren.
2020. "Identifying Influencers in Social Networks" *Entropy* 22, no. 4: 450.
https://doi.org/10.3390/e22040450