# A Novel Centrality for Finding Key Persons in a Social Network by the Bi-Directional Influence Map

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Introduction of Centralities

_{i}, can be represented as follows:

_{j}to v

_{k}and ${\sigma}_{{v}_{j}{v}_{k}}\left({v}_{i}\right)$ is the number of shortest paths from v

_{j}to v

_{k}that pass through vi. Although the previous three centralities are easily calculated, they only reflect the influence of vertices with respect to others in the topology of a social network without considering the influence of their neighbors/friends and cannot be used as a comprehensive centrality for measuring key persons. Hence, the eigenvector centrality is proposed to reflect the importance of neighbors.

## 3. Eigenvector Centrality

_{j,i}is the element at the jth row and ith column of the adjacency matrix which indicates the relationship from one vertex (row) to another (column) and λ is a fixed constant. For simplicity, we can represent Equation (4) as a matrix form:

**v**), which is an eigenvector of

**A**

^{T}and λ is the corresponding eigenvalue. Note that the initial EC(

**v**) can be set as 1, i.e., the all-one vector. Usually, we select the maximum eigenvalue, λ

_{max}, to ensure EC(

**v**) is large than the zero vector. According to the Perron–Frobenius theorem [23], for any a

_{ij}> 0, EC(

**v**) of

**A**with eigenvalue λ

_{max}such that ∀EC(

**v**) > 0.

_{j}**A**

^{T}be a row stochastic matrix, i.e., normalized

**A**

^{T}such that all sums of each row exactly equal to one. We can rewrite Equation (5) as follows:

_{max}= 1. In addition, we can also derive the eigenvector by calculating the limiting power of

**A**

^{T}according to Markov chain theory.

## 4. Katz Centrality

**1**denotes the one vector.

## 5. PageRank Algorithm

_{j,i}over the out-degree is to normalize the adjacency matrix into the stochastic matrix. In addition, we can re-write Equation (10) as the matrix form as follows:

**D**= diag(${d}_{1}^{out},{d}_{2}^{out},\dots ,{d}_{n}^{out})$ denotes a fixed out-degree matrix and

**A**

^{T}

**D**

^{−}

^{1}is a column stochastic matrix. Note that the since

**A**

^{T}

**D**

^{−}

^{1}is a column stochastic matrix, the damping factor α should be less than one to ensure that (

**I**− α

**A**

^{T}

**D**

^{−}

^{1}) is invertible. Although many variants of the PageRank have been proposed successively, the cores of the algorithms are similar.

## 6. HITS Algorithm

_{j}→ v

_{1}indicates that vertex v

_{j}points to v

_{i}. We can calculate the score vectors of the authority and hub of vertices, respectively, as follows:

**A**

^{T}

**A**and

**AA**

^{T}are called authority and hub matrices, respectively, and c(t) and c′(t) are constants which normalize the authority and hub score vectors. From Equations (14) and (15), it can be seen that the HITS algorithm is used to calculate the eigenvectors of

**A**

^{T}

**A**and

**AA**

^{T}. The HITS algorithm highlights that the centrality of a vertex should consider two different forces, namely, the authority and hub. However, it only proposes indices to measure the centralities of the authority and hub separately without a synthesized centrality.

## 7. Fuzzy Cognitive Map

_{ij}∈ [−1, 1] be the degree of influence from the ith concept, C

_{i}, to the jth concept, C

_{j}, where the sign indicates the positive or negative influence and −1 denotes a full negative impact and 1 expresses a full positive impact. Then, the influence of concept, x, can be calculated by the following equation:

## 8. Bi-Directional Influence Maps (BIM)

_{ri}denote the flows from vertex r to vertex i. We consider the centrality of a vertex in terms of two factors, namely, the amounts of inflow and outflow. In addition, we also define the reference of vertex i, R, as the vertex which link to vertex i (e.g., vertices r and s). For example, in the path from r to i, denoted by r → i, vertex r is the reference of vertex i.

_{i}indicates the input degree (number of inflows) of vertex i. Then, the outflow vertex from vertex i to vertex j at time t can be calculated as follows here:

_{i}indicates the output degree (number of outflows) of vertex i.

_{i}(t + 1)) in Equation (22) satisfies M(v

_{i}(t + 1)) ∈ (0, 1) and ${{\displaystyle \sum}}_{i=1}^{n}M\left({v}_{i}\left(t+1\right)\right)=1$. Hence, in this paper, we introduce two sigmoid functions, namely, the smoothstep and inverted smoothstep functions, to reflect the s-shape situation of updated centralities and restrict the input range between [0, 1], as shown in Figure 4.

## 9. Empirical Studies

#### 9.1. Marvel Universe Dataset

#### 9.2. Facebook Dataset

## 10. Discussion

## 11. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Ahmed, H.M.S. A Proposal Model for Measuring the Impact of Viral Marketing through Social Networks on Purchasing Decision: An Empirical Study. Int. J. Cust. Relatsh. Mark. Manag. (IJCRMM)
**2018**, 9, 13–33. [Google Scholar] [CrossRef] [Green Version] - Al-Garadi, M.A.; Varathan, K.D.; Ravana, S.D.; Ahmed, E.; Mujtaba, G.; Khan, M.U.S.; Khan, S.U. Analysis of online social network connections for identification of influential users: Survey and open research issues. ACM Comput. Surv. (CSUR)
**2018**, 51, 1–37. [Google Scholar] [CrossRef] - Alkemade, F.; Castaldi, C. Strategies for the diffusion of innovations on social networks. Comput. Econ.
**2005**, 25, 3–23. [Google Scholar] [CrossRef] - Axelord, R. Structure of Decision: The Cognitive Maps of Political Elites; Princeton University Press: Princeton, NJ, USA, 1976. [Google Scholar]
- Bar-Yossef, Z.; Mashiach, L.T. Local Approximation of Pagerank and Reverse Pagerank. In Proceedings of the 17th ACM Conference on Information and Knowledge Management, Napa Valley, CA, USA, 26–30 October 2008. [Google Scholar]
- Beauchamp, M.A. An improved index of centrality. Behav. Sci.
**1965**, 10, 161–163. [Google Scholar] [CrossRef] - Bringmann, L.F.; Elmer, T.; Epskamp, S.; Krause, R.W.; Schoch, D.; Wichers, M.; Wigman, J.T.; Snippe, E. What do centrality measures measure in psychological networks? J. Abnorm. Psychol.
**2019**, 128, 892. [Google Scholar] [CrossRef] [Green Version] - Catanese, S.; De Meo, P.; Ferrara, E.; Fiumara, G.; Provetti, A. Extraction and analysis of facebook friendship relations. In Computational Social Networks; Springer: London, UK, 2012; pp. 291–324. [Google Scholar]
- Cha, M.; Benevenuto, F.; Haddadi, H.; Gummadi, K. The world of connections and information flow in twitter. IEEE Trans. Syst. Man Cybern. Part A Syst. Hum.
**2012**, 42, 991–998. [Google Scholar] - Cha, M.; Haddadi, H.; Benevenuto, F.; Gummadi, K.P. Measuring user influence in twitter: The million follower fallacy. In Proceedings of the Fourth International AAAI Conference on Weblogs and Social Media, Washington, DC, USA, 23–26 May 2010. [Google Scholar]
- Ding, C.; Chen, Y.; Fu, X. Crowd crawling: Towards collaborative data collection for large-scale online social networks. In Proceedings of the First ACM Conference on Online Social Networks, Boston, MA, USA, 7–8 October 2013; pp. 183–188. [Google Scholar]
- Easley, D.; Kleinberg, J. Networks, Crowds, and Markets; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Fogaras, D. Where to start browsing the web? In International Workshop on Innovative Internet Community Systems; Springer: Berlin/Heidelberg, Germany, 2003; pp. 65–79. [Google Scholar]
- Gyongyi, Z.; Garcia-Molina, H.; Pedersen, J. Combating web spam with trustrank. In Proceedings of the 30th International Conference on Very Large Data Bases (VLDB), Toronto, ON, Canada, 31 August–3 September 2004. [Google Scholar]
- Jabeur, L.B.; Tamine, L.; Boughanem, M. Active microbloggers: Identifying influencers, leaders and discussers in microblogging networks. In International Symposium on String Processing and Information Retrieval; Springer: Berlin/Heidelberg, Germany, 2012; pp. 111–117. [Google Scholar]
- Jiang, J.; Wilson, C.; Wang, X.; Sha, W.; Huang, P.; Dai, Y.; Zhao, B.Y. Understanding latent interactions in online social networks. ACM Trans. Web (TWEB)
**2013**, 7, 1–39. [Google Scholar] [CrossRef] - Katz, L. A new status index derived from sociometric analysis. Psychometrika
**1953**, 18, 39–43. [Google Scholar] [CrossRef] - Keener, J.P. The Perron–Frobenius theorem and the ranking of football teams. SIAM Rev.
**1993**, 35, 80–93. [Google Scholar] [CrossRef] - Kempe, D.; Kleinberg, J.; Tardos, É. Maximizing the spread of influence through a social network. In Proceedings of the Ninth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, Washington, DC, USA, 24–27 August 2003; pp. 137–146. [Google Scholar]
- Kim, E.S.; Han, S.S. An analytical way to find influencers on social networks and validate their effects in disseminating social games. In Proceedings of the 2009 International Conference on Advances in Social Network Analysis and Mining, Athens, Greece, 20–22 July 2009; IEEE: Piscataway, NJ, USA, 2009; pp. 41–46. [Google Scholar]
- Kleinberg, J.M. Authoritative sources in a hyperlinked environment. In Proceedings of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, San Francisco, CA, USA, 25–27 January 1998; pp. 668–677. [Google Scholar]
- Chakrabarti, S.; Dom, B.; Raghavan, P.; Rajagopalan, S.; Gibson, D.; Kleinberg, J. Automatic resource compilation by analyzing hyperlink structure and associated text. Comput. Netw. ISDN Syst.
**1998**, 30, 65–74. [Google Scholar] [CrossRef] [Green Version] - Kosko, B. Fuzzy cognitive maps. Int. J. Man Mach. Studies
**1986**, 24, 65–75. [Google Scholar] [CrossRef] - Kwok, N.; Hanig, S.; Brown, D.J.; Shen, W. How leader role identity influences the process of leader emergence: A social network analysis. Leadersh. Q.
**2018**, 29, 648–662. [Google Scholar] [CrossRef] [Green Version] - Mislove, A.; Marcon, M.; Gummadi, K.P.; Druschel, P.; Bhattacharjee, B. Measurement and analysis of online social networks. In Proceedings of the 7th ACM SIGCOMM Conference on Internet Measurement, San Diego, CA, USA, 24–26 October 2007; pp. 29–42. [Google Scholar]
- Nieminen, J. On the centrality in a graph. Scand. J. Psychol.
**1974**, 15, 332–336. [Google Scholar] [CrossRef] [PubMed] - Page, L.; Brin, S.; Motwani, R.; Winograd, T. The Pagerank Citation Ranking: Bringing Order to the Web; Stanford InfoLab: Stanford, CA, USA, 1999. [Google Scholar]
- 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] - Saito, K.; Kimura, M.; Ohara, K.; Motoda, H. Super mediator–A new centrality measure of node importance for information diffusion over social network. Inf. Sci.
**2016**, 329, 985–1000. [Google Scholar] [CrossRef] [Green Version] - Shelton, R.C.; Lee, M.; Brotzman, L.E.; Crookes, D.M.; Jandorf, L.; Erwin, D.; Gage-Bouchard, E.A. Use of social network analysis in the development, dissemination, implementation, and sustainability of health behavior interventions for adults: A systematic review. Soc. Sci. Med.
**2019**, 220, 81–101. [Google Scholar] [CrossRef] [PubMed] - Silva, A.; Guimarães, S.; Meira, W., Jr.; Zaki, M. ProfileRank: Finding relevant content and influential users based on information diffusion. In Proceedings of the 7th Workshop on Social Network Mining and Analysis, Chicago, IL, USA, 11 August 2013; pp. 1–9. [Google Scholar]
- Stylios, C.D.; Groumpos, P.P. Mathematical formulation of fuzzy cognitive maps. In Proceedings of the 7th Mediterranean Conference on Control and Automation, Akko, Israel, 1–4 July 2019; pp. 2251–2261. [Google Scholar]
- Tunkelang, D. TunkRank: A Twitter Analog to PageRank. 2009. Available online: http.thenoisychannel.com/2009/01/13/atwitter-analog-to-pagerank (accessed on 20 September 2020).
- Weng, J.; Lim, E.P.; Jiang, J.; He, Q. Twitterrank: Finding topic-sensitive influential twitterers. In Proceedings of the Third ACM International Conference on Web Search and Data Mining, New York, NY, USA, 3–6 February 2010; pp. 261–270. [Google Scholar]
- Rodríguez-Velázquez, J.A.; Balaban, A.T. Two new topological indices based on graph adjacency matrix eigenvalues and eigenvectors. J. Math. Chem.
**2019**, 57, 1053–1074. [Google Scholar] [CrossRef] [Green Version]

Influence Matrix | C1 | C2 | C3 | C4 | C5 |
---|---|---|---|---|---|

C1 | ±0.3 | ±0.5 | 0 | ±0.4 | ±0.1 |

C2 | 0 | ±0.2 | ±0.6 | 0 | ±0.5 |

C3 | 0 | ±0.3 | 0 | 0 | ±0.2 |

C4 | 0 | ±0.4 | ±0.7 | 0 | ±0.7 |

C5 | 0 | 0 | 0 | ±0.3 | 0 |

Equilibrium Influence | C1 | C2 | C3 | C4 | C5 |
---|---|---|---|---|---|

All Positive Influence | 0.7177 | 0.8804 | 0.8766 | 0.794 | 0.8945 |

Rank | 5 | 2 | 3 | 4 | 1 |

All Negative Influence | 0.6042 | 0.4207 | 0.4558 | 0.544 | 0.42 |

Rank | 1 | 4 | 3 | 2 | 5 |

Transition Function | Mathematical Equation |
---|---|

Linear | $f(x)=x$ |

Softmax | $f({x}_{i})=\frac{{e}^{{x}_{i}}}{{\displaystyle \sum _{j=1}^{n}{e}^{{x}_{j}}}},\forall i=1,\dots ,n$ |

Restricted logistic | $f(x)=\frac{1}{1+{e}^{-10\ast (x-0.5)}}$ |

Smoothstep | $f(x)=(3-2x){x}^{2}$ |

Inverted smoothstep | $f(x)=x(2{x}^{2}-3x+2)$ |

**Table 4.**Centrality comparisons between different algorithms in the toy example. BIM: Bi-directional influence map.

Centrality | A | B | C | D | E | F |
---|---|---|---|---|---|---|

PageRank | 0.1304 | 0.1161 | 0.1649 | 0.1321 | 0.2142 | 0.2423 |

Rank | 5 | 6 | 3 | 4 | 2 | 1 |

BIM (linear, γ = 1) | 0.0858 | 0.0804 | 0.1547 | 0.0984 | 0.2605 | 0.3202 |

Rank | 5 | 6 | 3 | 4 | 2 | 1 |

BIM (linear, γ = 0) | 0.2369 | 0.1758 | 0.1105 | 0.1576 | 0.2064 | 0.1127 |

Rank | 1 | 3 | 6 | 4 | 2 | 5 |

BIM (linear, γ = 0.5) | 0.1778 | 0.1127 | 0.1371 | 0.1381 | 0.2193 | 0.2150 |

Rank | 3 | 6 | 5 | 4 | 1 | 2 |

BIM (softmax, γ = 0.5) | 0.1709 | 0.1560 | 0.1601 | 0.1611 | 0.1776 | 0.1742 |

Rank | 3 | 6 | 5 | 4 | 1 | 2 |

BIM (restricted, γ = 0.5) | 0.1692 | 0.1606 | 0.1630 | 0.1636 | 0.1728 | 0.1708 |

Rank | 3 | 6 | 5 | 4 | 1 | 2 |

BIM (smoothstep, γ = 0.5) | 0.1708 | 0.1565 | 0.1605 | 0.1615 | 0.1769 | 0.1738 |

Rank | 3 | 6 | 5 | 4 | 1 | 2 |

BIM (inverted, γ = 0.5) | 0.1795 | 0.1285 | 0.1483 | 0.1516 | 0.1981 | 0.1940 |

Rank | 3 | 6 | 5 | 4 | 1 | 2 |

Statistics of the Network | Value |
---|---|

Average number of neighbors | 37.333 |

Network diameter | 8 |

Characteristic path length | 2.937 |

Clustering coefficient | 0.400 |

Network density | 0.003 |

Multi-edge node pairs | 64,216 |

Number of self-loops | 2232 |

Top 5 Key Persons | 1st Place | 2nd Place | 3rd Place | 4th Place | 5th Place |
---|---|---|---|---|---|

PageRank | Spider Man | Captain America | Iron Man | Wolverine | Thor |

BIM (linear, γ = 0.5) | Spider Man | Captain America | Iron Man | Wolverine | Thing |

BIM (softmax, γ = 0.5) | Spider Man | Captain America | Iron Man | Wolverine | Thing |

BIM (restricted, γ = 0.5) | Spider Man | Captain America | Iron Man | Wolverine | Thing |

BIM (smoothstep, γ = 0.5) | Spider Man | Captain America | Iron Man | Wolverine | Thing |

BIM (inverted, γ = 0.5) | Spider Man | Captain America | Iron Man | Wolverine | Thing |

BIM (linear, γ = 0.1 | Spider Man | Iron Man | Wolverine | Thing | Scarlet Witch |

BIM (softmax, γ = 0.1 | Spider Man | Iron Man | Wolverine | Thing | Scarlet Witch |

BIM (restricted, γ = 0.1 | Spider Man | Iron Man | Wolverine | Thing | Scarlet Witch |

BIM (smoothstep, γ = 0.1 | Spider Man | Iron Man | Wolverine | Thing | Scarlet Witch |

BIM (inverted, γ = 0.1 | Spider Man | Iron Man | Wolverine | Thing | Scarlet Witch |

Statistics of the Network | Value |
---|---|

Average number of neighbors | 43.691 |

Network diameter | 17 |

Characteristic path length | 4.368 |

Clustering coefficient | 0.303 |

Network density | 0.005 |

Multi-edge node pairs | 0 |

Number of self-loops | 0 |

Centrality | 1st | 2nd | 3rd | 4th | 5th | 6th | 7th | 8th | 9th | 10th |
---|---|---|---|---|---|---|---|---|---|---|

Out-degree | 107 | 351 | 352 | 1821 | 0 | 348 | 2126 | 2995 | 366 | 2944 |

In-degree | 1373 | 1490 | 1285 | 3445 | 1312 | 1215 | 3443 | 1318 | 3439 | 3441 |

Betweenness | 351 | 352 | 1203 | 371 | 891 | 1142 | 572 | 1710 | 1821 | 119 |

Out-closeness | 1007 | 58 | 0 | 348 | 350 | 359 | 362 | 1539 | 366 | 1573 |

In-closeness | 2173 | 1503 | 1497 | 1501 | 1490 | 1495 | 1504 | 1496 | 2232 | 2168 |

Hubs | 352 | 3002 | 2995 | 2944 | 2993 | 2962 | 2964 | 3058 | 2976 | 3044 |

Authorities | 3441 | 3445 | 3431 | 3443 | 3438 | 3407 | 3456 | 3439 | 3457 | 3429 |

PageRank | 1396 | 2933 | 3478 | 1387 | 1373 | 1503 | 1392 | 3975 | 3477 | 1395 |

BIM (linear, γ = 1) | 1373 | 1490 | 1285 | 1312 | 3445 | 1318 | 1215 | 1253 | 1320 | 1289 |

BIM (softmax, γ = 1) | 1373 | 1490 | 1285 | 1312 | 3445 | 1215 | 1318 | 1253 | 1320 | 1289 |

BIM (inverted, γ = 1) | 1373 | 1490 | 1285 | 1312 | 3445 | 1215 | 1318 | 1253 | 1320 | 1289 |

BIM (linear, γ = 0.5) | 107 | 352 | 351 | 1821 | 1373 | 1490 | 348 | 1285 | 2126 | 3445 |

BIM (softmax, γ = 0.5) | 107 | 351 | 352 | 1821 | 348 | 1373 | 366 | 1490 | 1285 | 349 |

BIM (inverted, γ = 0.5) | 107 | 351 | 352 | 1821 | 348 | 1373 | 366 | 1490 | 1285 | 349 |

BIM (linear, γ = 0) | 107 | 352 | 351 | 1821 | 348 | 1063 | 2944 | 2126 | 2962 | 2964 |

BIM (softmax, γ = 0) | 107 | 351 | 352 | 1821 | 348 | 366 | 349 | 2126 | 2130 | 1 |

BIM (inverted, γ = 0) | 107 | 351 | 352 | 1821 | 348 | 366 | 349 | 2126 | 2130 | 1 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chen, C.-Y.; Huang, J.-J.
A Novel Centrality for Finding Key Persons in a Social Network by the Bi-Directional Influence Map. *Symmetry* **2020**, *12*, 1747.
https://doi.org/10.3390/sym12101747

**AMA Style**

Chen C-Y, Huang J-J.
A Novel Centrality for Finding Key Persons in a Social Network by the Bi-Directional Influence Map. *Symmetry*. 2020; 12(10):1747.
https://doi.org/10.3390/sym12101747

**Chicago/Turabian Style**

Chen, Chin-Yi, and Jih-Jeng Huang.
2020. "A Novel Centrality for Finding Key Persons in a Social Network by the Bi-Directional Influence Map" *Symmetry* 12, no. 10: 1747.
https://doi.org/10.3390/sym12101747