# A Feasible Temporal Links Prediction Framework Combining with Improved Gravity Model

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Works

Indicator | Topology | Definition ^{1} | Complexity ^{2} |
---|---|---|---|

CN [24] | Local | ${S}_{xy}=|\Gamma \left(x\right)\cap \Gamma \left(y\right)|$ | $O\left({n}^{2}\right)$ |

Salton [25] | Local | ${S}_{xy}=\frac{|\Gamma (x)\cap \Gamma (y\left)\right|}{\sqrt{{k}_{x}{k}_{y}}}$ | $O\left({n}^{2}\right)$ |

Jaccard [26] | Local | ${S}_{xy}=\frac{|\Gamma (x)\cap \Gamma (y\left)\right|}{|\Gamma (x)\cup \Gamma (y\left)\right|}$ | $O\left(2{n}^{2}\right)$ |

Sorenson [27] | Local | ${S}_{xy}=\frac{2\times |\Gamma (x)\cap \Gamma (y\left)\right|}{{k}_{x}+{k}_{y}}$ | $O\left({n}^{2}\right)$ |

HPI [28] | Local | ${S}_{xy}=\frac{|\Gamma (x)\cap \Gamma (y\left)\right|}{min({k}_{x},{k}_{y})}$ | $O\left({n}^{2}\right)$ |

HDI [29] | Local | ${S}_{xy}=\frac{|\Gamma (x)\cap \Gamma (y\left)\right|}{max({k}_{x},{k}_{y})}$ | $O\left({n}^{2}\right)$ |

LHN-I [29] | Local | ${S}_{xy}=\frac{|\Gamma (x)\cap \Gamma (y\left)\right|}{{k}_{x},{k}_{y}}$ | $O\left({n}^{2}\right)$ |

AA [30] | Local | ${S}_{xy}={\sum}_{z\in \Gamma \left(x\right)\cap \Gamma \left(y\right)}\frac{1}{log{k}_{z}}$ | $O\left(2{n}^{2}\right)$ |

RA [31] | Local | ${S}_{xy}={\sum}_{z\in \Gamma \left(x\right)\cap \Gamma \left(y\right)}\frac{1}{{k}_{z}}$ | $O\left(2{n}^{2}\right)$ |

PA [32] | Local | ${S}_{xy}={k}_{x}{k}_{y}$ | $O\left(2n\right)$ |

LP [33] | Semi-Local | $S={A}^{2}+\alpha \xb7{A}^{3}$ | $O\left({n}^{3}\right)$ |

Katz [34] | Global | $S={(I-\alpha \xb7A)}^{-1}-I$ | $O\left({n}^{3}\right)$ |

LHN-II [35] | Global | $S=2m{\lambda}_{1}{D}^{-1}{(I-\frac{\varphi}{{\lambda}_{1}})}^{-1}{D}^{-1}$ | $O\left({n}^{3}\right)$ |

LRW [36] | Semi-Local | ${S}_{xy}^{LRW}\left(t\right)={q}_{x}\xb7{\pi}_{xy}\left(t\right)+{q}_{y}\xb7{\pi}_{yx}\left(t\right)$ | $O\left(n{k}^{t}\right)$ |

SRW [36] | Semi-Local | ${S}_{xy}^{SRW}\left(t\right)={\sum}_{l=1}^{t}{S}_{xy}^{LRW}\left(l\right)={q}_{x}{\sum}_{l=1}^{t}{\pi}_{xy}\left(l\right)+{q}_{y}{\sum}_{l=1}^{t}{\pi}_{yx}\left(l\right)$ | $O\left(n{k}^{t}\right)$ |

RWR [37] | Semi-Local | ${S}_{xy}^{LRW}\left(t\right)={q}_{xy}+{q}_{yx}$ | $O\left({n}^{3}\right)$ |

ACT [38] | Semi-Local | ${S}_{xy}^{ACT}=\frac{1}{{l}_{xx}^{+}\xb7{l}_{yy}^{+}-2{l}_{xy}^{+}}$ | $O\left({n}^{3}\right)$ |

SimR [39] | Global | ${S}_{xy}^{SimR}=C\frac{{\sum}_{{v}_{z}\in \Gamma \left(x\right)}{\sum}_{{v}_{{z}^{\prime}}\in \Gamma \left(y\right)}{S}_{z{z}^{\prime}}^{SimR}}{{k}_{x}{k}_{y}}$ | $O\left({n}^{3}\right)$ |

Cos+ [40] | Semi-Local | ${S}_{xy}^{cos+}=cos{(x,y)}^{+}=\frac{{l}_{xy}^{+}}{\sqrt{{l}_{xx}^{+}\xb7{l}_{yy}^{+}}}$ | $O\left({n}^{3}\right)$ |

TS [41] | Global | ${S}^{Tr}={(I-\epsilon S)}^{-1}S$ | $O\left({n}^{3}\right)$ |

LowRank [11] | Global | $S={min}_{{X}^{*},E}\left|\right|{X}^{*}{\left|\right|}_{*}+\lambda {\left|\right|E\left|\right|}_{1}$ | $O\left(n{k}^{3}\right)$ |

MFI [42] | Global | $S={(I+\alpha \xb7L)}^{-1},\alpha >0$ | $O\left({n}^{3}\right)$ |

^{1}$\Gamma \left(x\right)$ and $\Gamma \left(y\right)$ denote the neighbors of node x and y, respectively; ${k}_{x}$ and ${k}_{y}$ are the degrees of node x and y, respectively; m is the number of edges; ${l}_{xy}^{+}$ denotes the element at row x, column y of the pseudo-inverse matrix ${L}^{+}$; ${q}_{xy}$ represents the probability of random walk from node x to node y; $\epsilon $ represents an adjustable parameter; ${\pi}_{xy}\left(l\right)$ is the random walk probability from node x to node y at time l;

^{2}n denotes the number of nodes; k is the average degree of nodes; and t is the step of random walk steps.

## 3. Modeling and Methods

#### 3.1. Model

#### 3.2. Definitions

**Step 1:**Obtain all the node pairs at $T-1$ layer as the target-predicting links.

**Step 2:**Collect all the existing links from the start time to time $T-1$, marked as training set.

**Step 3:**Calculate the likelihood of node pairs in the training set in terms of Equation (7).

**Step 4:**Sort the possible links in descending order and obtain the top-k results as the predicted result.

Algorithm 1: Temporal links prediction framework. |

#### 3.3. Complexity Analysis

## 4. Experiments and Discussion

#### 4.1. Experimental Datasets

#### 4.2. Performance Comparison

#### 4.3. Parameters Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AA | Adamic-Adar |

ACT | Average commute time |

ARIMA | Autoregressive Integrated Moving Average |

AUC | Area Under the receiver operating characteristic Curve |

CN | Common Neighbors |

DS | Dynamic Similarity |

EPS | Exponential Smoothing |

GR | Gravity |

JC | Jaccard |

LR | Linear Regression |

LRW | Local Random Walk |

MFI | Matrix-forest index |

PCA | Principal Component Analysis |

RA | Resource Allocation |

RWR | Random Walk with Restart |

SimR | SimRank |

SRW | Superposed Random Walk |

TMLP | Time-aware Multi-relational Link Prediction |

TS | Transferring Similarity |

## References

- Wasserman, S.; Faust, K. Social Network Analysis: Methods and Applications; Cambridge University Press: Cambridge, UK, 1994; Volume 8. [Google Scholar]
- Antonacci, G.; Fronzetti Colladon, A.; Stefanini, A.; Gloor, P. It is rotating leaders who build the swarm: Social network determinants of growth for healthcare virtual communities of practice. J. Knowl. Manag.
**2017**, 21, 1218–1239. [Google Scholar] [CrossRef] [Green Version] - Sett, N.; Basu, S.; Nandi, S.; Singh, S.R. Temporal link prediction in multi-relational network. World Wide Web
**2018**, 21, 395–419. [Google Scholar] [CrossRef] - Getoor, L.; Diehl, C.P. Link mining: A survey. ACM SIGKDD Explor. Newslett.
**2005**, 7, 3–12. [Google Scholar] [CrossRef] - Srinivas, V.; Mitra, P. Link Prediction Using Thresholding Nodes Based on Their Degree. In Link Prediction in Social Networks; Springer: Berlin/Heidelberg, Germany, 2016; pp. 15–25. [Google Scholar]
- Oyama, S.; Hayashi, K.; Kashima, H. Cross-temporal link prediction. In Proceedings of the 2011 IEEE 11th International Conference on Data Mining, Vancouver, BC, Canada, 11–14 December 2011; pp. 1188–1193. [Google Scholar]
- Slokom, M.; Ayachi, R. A New Social Recommender System Based on Link Prediction Across Heterogeneous Networks. In Proceedings of the International Conference on Intelligent Decision Technologies, Sorrento, Italy, 17–19 June 2017; pp. 330–340. [Google Scholar]
- Kim, W.; Kwon, K.; Kwon, S.; Lee, S. The identification power of smoothness assumptions in models with counterfactual outcomes. Quantit. Econ.
**2018**, 9, 617–642. [Google Scholar] [CrossRef] - Liben-Nowell, D.; Kleinberg, J. The link-prediction problem for social networks. J. Am. Soc. Inf. Sci. Technol.
**2007**, 58, 1019–1031. [Google Scholar] [CrossRef] [Green Version] - Lü, L.; Zhou, T. Link prediction in complex networks: A survey. Phys. A Stat. Mech. Appl.
**2011**, 390, 1150–1170. [Google Scholar] [CrossRef] [Green Version] - Pech, R.; Hao, D.; Pan, L.; Cheng, H.; Zhou, T. Link prediction via matrix completion. EPL (Europhys. Lett.)
**2017**, 117, 38002. [Google Scholar] [CrossRef] [Green Version] - Munasinghe, L.; Ichise, R. Time aware index for link prediction in social networks. In Proceedings of the International Conference on Data Warehousing and Knowledge Discovery, Toulouse, France, 29 August–2 September 2011; pp. 342–353. [Google Scholar]
- Yasami, Y.; Safaei, F. A novel multilayer model for missing link prediction and future link forecasting in dynamic complex networks. Phys. A Stat. Mech. Appl.
**2018**, 492, 2166–2197. [Google Scholar] [CrossRef] - Kostakos, V. Temporal graphs. Phys. A Stat. Mech. Appl.
**2009**, 388, 1007–1023. [Google Scholar] [CrossRef] [Green Version] - Alhajj, R.; Rokne, J. Encyclopedia of Social Network Analysis and Mining; Springer: Berlin/Heidelberg, Germany, 2014. [Google Scholar]
- Casteigts, A.; Flocchini, P.; Quattrociocchi, W.; Santoro, N. Time-varying graphs and dynamic networks. Int. J. Parallel Emerg. Distrib. Syst.
**2012**, 27, 387–408. [Google Scholar] [CrossRef] - Hua, T.D.; Nguyen-Thi, A.T.; Nguyen, T.A.H. Link prediction in weighted network based on reliable routes by machine learning approach. In Proceedings of the 2017 4th NAFOSTED Conference on Information and Computer Science, Hanoi, Vietnam, 24–25 November 2017; pp. 236–241. [Google Scholar]
- Zhou, J.; Huang, D.; Wang, H. A dynamic logistic regression for network link prediction. Sci. China Math.
**2017**, 60, 165–176. [Google Scholar] [CrossRef] - Tabourier, L.; Bernardes, D.F.; Libert, A.S.; Lambiotte, R. RankMerging: A supervised learning-to-rank framework to predict links in large social networks. Mach. Learn.
**2019**, 108, 1729–1756. [Google Scholar] [CrossRef] [Green Version] - Hyndman, R.J.; Athanasopoulos, G. Forecasting: Principles and Practice; OTexts: Melbourne, Australia, 2018. [Google Scholar]
- Hyndman, R.; Koehler, A.B.; Ord, J.K.; Snyder, R.D. Forecasting with Exponential Smoothing: The State Space Approach; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2008. [Google Scholar]
- Divakaran, A.; Mohan, A. Temporal Link Prediction: A Survey. New Gener. Comput.
**2019**. [Google Scholar] [CrossRef] - Özcan, A.; Öğüdücü, Ş.G. Multivariate temporal link prediction in evolving social networks. In Proceedings of the 2015 IEEE/ACIS 14th International Conference on Computer and Information Science (ICIS), Las Vegas, NV, USA, 28 June–1 July 2015; pp. 185–190. [Google Scholar]
- Lorrain, F.; White, H.C. Structural equivalence of individuals in social networks. J. Math. Soc.
**1971**, 1, 49–80. [Google Scholar] [CrossRef] - Worth, D. Introduction to modern information retrieval. Aust. Acad. Res. Libr.
**2010**, 41, 305–306. [Google Scholar] [CrossRef] [Green Version] - Jaccard, P. Étude comparative de la distribution florale dans une portion des Alpes et des Jura. Bull. Soc. Vaudoise Sci. Nat.
**1901**, 37, 547–579. [Google Scholar] - Sorensen, T.A. A method of establishing groups of equal amplitude in plant sociology based on similarity of species content and its application to analyses of the vegetation on Danish commons. Biol. Skar.
**1948**, 5, 1–34. [Google Scholar] - Ravasz, E.; Somera, A.L.; Mongru, D.A.; Oltvai, Z.N.; Barabási, A.L. Hierarchical organization of modularity in metabolic networks. Science
**2002**, 297, 1551–1555. [Google Scholar] [CrossRef] [Green Version] - Molloy, M.; Reed, B. A critical point for random graphs with a given degree sequence. Random Struct. Algorithms
**1995**, 6, 161–180. [Google Scholar] [CrossRef] - Adamic, L.A.; Adar, E. Friends and neighbors on the web. Soc. Netw.
**2003**, 25, 211–230. [Google Scholar] [CrossRef] [Green Version] - Zhou, T.; Lü, L.; Zhang, Y.C. Predicting missing links via local information. Eur. Phys. J. B
**2009**, 71, 623–630. [Google Scholar] [CrossRef] [Green Version] - Barabási, A.L.; Albert, R. Emergence of scaling in random networks. Science
**1999**, 286, 509–512. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lü, L.; Jin, C.H.; Zhou, T. Similarity index based on local paths for link prediction of complex networks. Phys. Rev. E
**2009**, 80, 046122. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Katz, L. A new status index derived from sociometric analysis. Psychometrika
**1953**, 18, 39–43. [Google Scholar] [CrossRef] - Leicht, E.A.; Holme, P.; Newman, M.E. Vertex similarity in networks. Phys. Rev. E
**2006**, 73, 026120. [Google Scholar] [CrossRef] [Green Version] - Liu, W.; Lü, L. Link prediction based on local random walk. EPL (Europhys. Lett.)
**2010**, 89, 58007. [Google Scholar] [CrossRef] [Green Version] - Vragović, I.; Louis, E. Network community structure and loop coefficient method. Phys. Rev. E
**2006**, 74, 016105. [Google Scholar] [CrossRef] - Klein, D.J.; Randić, M. Resistance distance. J. Math. Chem.
**1993**, 12, 81–95. [Google Scholar] [CrossRef] - Jeh, G.; Widom, J. SimRank: A measure of structural-context similarity. In Proceedings of the eighth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, Edmonton, AB, Canada, 23–26 July 2002; pp. 538–543. [Google Scholar]
- Fouss, F.; Pirotte, A.; Renders, J.M.; Saerens, M. Random-walk computation of similarities between nodes of a graph with application to collaborative recommendation. IEEE Trans. Knowl. Data Eng.
**2007**, 19, 355–369. [Google Scholar] [CrossRef] - Sun, D.; Zhou, T.; Liu, J.G.; Liu, R.R.; Jia, C.X.; Wang, B.H. Information filtering based on transferring similarity. Phys. Rev. E
**2009**, 80, 017101. [Google Scholar] [CrossRef] [Green Version] - Chebotarev, P.Y.; Shamis, E. A matrix-forest theorem and measuring relations in small social group. Avtomatika i Telemekhanika
**1997**, 58, 125–137. [Google Scholar] - 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] - Paranjape, A.; Benson, A.R.; Leskovec, J. Motifs in temporal networks. In Proceedings of the Tenth ACM International Conference on Web Search and Data Mining, Cambridge, UK, 6–10 February 2017; pp. 601–610. [Google Scholar]
- Fawcett, T. An introduction to ROC analysis. Pattern Recognit. Lett.
**2006**, 27, 861–874. [Google Scholar] [CrossRef] - 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] - Chen, D.; Kong, L.; Wang, D.; Huang, X.; Fang, B. TNLCD: A Feasible Algorithm for Local Community Discovery in Temporal Networks. In FSDM; IOS Press: Amsterdam, The Netherlands, 2018; pp. 459–464. [Google Scholar]
- Wang, P.; Xu, B.; Wu, Y.; Zhou, X. Link prediction in social networks: The state-of-the-art. Sci. China Inf. Sci.
**2015**, 58, 1–38. [Google Scholar] [CrossRef] [Green Version] - 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; O’Reilly Media, Inc.: Sebastopol, CA, USA, 2011. [Google Scholar]
- Newman, M.E. 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. Community structure in social and biological networks. Proc. Natl. Acad. Sci. USA
**2002**, 99, 7821–7826. [Google Scholar] [CrossRef] [Green Version] - Watts, D.J.; Strogatz, S.H. Collective dynamics of ‘small-world’ networks. Nature
**1998**, 393, 440. [Google Scholar] [CrossRef] - Hu, H.B.; Wang, X.F. Unified index to quantifying heterogeneity of complex networks. Phys. A Stat. Mech. Appl.
**2008**, 387, 3769–3780. [Google Scholar] [CrossRef]

**Figure 2.**Supra-adjacency matrix representation. The diagonal blocks with different colors represent the network structures at different snapshots.

**Figure 3.**Illustration of $\tau $ function with different parameters. $\alpha $ varies in [0, 1], which depicts the attenuation level. The larger is $\alpha $, the less effect there is on the current x.

**Figure 4.**Email-Eu-core-temporal-Dept3 network for link prediction in the dynamic network (daily, weekly, and monthly separated).

**Figure 6.**Analyzing parameter $\alpha $ in the temporal network with different separations. In general, the obtained AUC is larger with $\alpha $ increasing when the networks are daily and weekly separated.

**Figure 8.**Illustration of network features with varying numbers of slicing. T is the number of slices ranging from 17 to 249. <C>, <H>, and <k> are the average clustering coefficient [54], average heterogeneity [55], and average degree of the nodes in all slices, respectively. R is the ratio of intralayer edges comparing interlayer edges. In general, R declines when the temporal network is separated into more slices.

**Figure 9.**Relationship of performance with T (i.e., number of slices) and R (i.e., intralayer edges comparing interlayer edges). AUC declines with T increasing (or R declining) in general.

Dataset Name | $\left|\mathit{V}\right|$ | |E| | <k> | |C| | <c> | |D| | r |
---|---|---|---|---|---|---|---|

Zachary karate Club [49] | 34 | 78 | 4.59 | 0.57 | 2.41 | 2.22 | −0.48 |

Dolphins social network [50] | 62 | 159 | 5.13 | 0.26 | 3.06 | 3.36 | −0.04 |

Terriers of 9/11 [51] | 69 | 159 | 4.61 | 0.47 | 1.76 | 3.22 | −0.04 |

NEUSNCP dataset ^{1} | 89 | 365 | 4.10 | 0.54 | 3.15 | 1.92 | −0.40 |

Books about US politics [52] | 105 | 411 | 8.40 | 0.49 | 5.26 | 3.08 | −0.13 |

American college football network [53] | 115 | 613 | 10.66 | 0.40 | 10.23 | 2.51 | 0.16 |

Scientist collaboration network [52] | 1589 | 2742 | 4.60 | 0.64 | 0.08 | 5.99 | −0.09 |

**Note:**$\left|V\right|$ denotes the number of nodes; $\left|E\right|$ denotes the number of edges; <k> is the average degree; $\left|C\right|$ is the average clustering index; <c> is the average connectivity; $\left|D\right|$ is the average shortest path; and r represents assortativity coefficient.

^{1}We developed an experimental social platform and invited hundreds of users to register. Data availability: https://www.neusncp.com/api/about.

Dataset Name | $\left|\mathit{V}\right|$ | $\left|\mathit{E}\right|$ | Days |
---|---|---|---|

Email-Eu-core temporal network | 986 | 332,334 | 803 |

Email-Eu-core-temporal-Dept1 | 309 | 61,046 | 803 |

Email-Eu-core-temporal-Dept2 | 162 | 46,772 | 803 |

Email-Eu-core-temporal-Dept3 | 89 | 12,216 | 803 |

Email-Eu-core-temporal-Dept4 | 142 | 48,141 | 803 |

Dataset Name | GR | AA | RA | JC | PA | CN |
---|---|---|---|---|---|---|

Zachary karate Club | 0.8790 | 0.8784 | 0.8784 | 0.6281 | 0.8773 | 0.8433 |

Dolphins social network | 0.7442 | 0.7428 | 0.7425 | 0.7431 | 0.6621 | 0.7379 |

Terriers of 9/11 | 0.9374 | 0.9339 | 0.9371 | 0.9151 | 0.7144 | 0.9103 |

NEUSNCP dataset | 0.9110 | 0.9105 | 0.9096 | 0.8855 | 0.6725 | 0.9012 |

Books about US politics | 0.8310 | 0.8299 | 0.8299 | 0.8397 | 0.2573 | 0.8304 |

American college football network | 0.8775 | 0.8750 | 0.8769 | 0.7494 | 0.8381 | 0.8657 |

Scientist collaboration network | 0.9431 | 0.9431 | 0.9431 | 0.9430 | 0.6725 | 0.9429 |

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**MDPI and ACS Style**

Huang, X.; Chen, D.; Ren, T.
A Feasible Temporal Links Prediction Framework Combining with Improved Gravity Model. *Symmetry* **2020**, *12*, 100.
https://doi.org/10.3390/sym12010100

**AMA Style**

Huang X, Chen D, Ren T.
A Feasible Temporal Links Prediction Framework Combining with Improved Gravity Model. *Symmetry*. 2020; 12(1):100.
https://doi.org/10.3390/sym12010100

**Chicago/Turabian Style**

Huang, Xinyu, Dongming Chen, and Tao Ren.
2020. "A Feasible Temporal Links Prediction Framework Combining with Improved Gravity Model" *Symmetry* 12, no. 1: 100.
https://doi.org/10.3390/sym12010100