# 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

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**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