FairnessAware Predictive Graph Learning in Social Networks
Abstract
:1. Introduction
 Biases Formulation: We formally define two biases, i.e., Preference and Favoritism that widely exist in current predictive learning models. Based on the formulation, we utilize modularity maximization to distinguish weak and strong links.
 Fairnessaware Predictive Graph Learning: We propose ACE, a novel predictive learning framework that seamlessly integrates link strength to differentiate the learning process and a dual propagation process.
 Realworld Social Networks Evaluation: We empirically verify the efficacy by experiments on link prediction. Experimental results demonstrate that ACE achieves great improvement and smaller extents of the two biases than nine baseline methods.
2. Related Work
3. Preliminaries
3.1. Graph
3.2. Biases
 Preference: If one method shows Preference to one side, it prefers to perform link prediction on that side so that it performs link prediction better on one side than on the other side.
 Favoritism: If one method shows Favoritism to one side, it favors one side and neglects the other side so that it gives higher scores to one side when performing link prediction.
4. The Design of ACE
4.1. Link Strength Learning
4.2. Dual Propagation
4.3. Supervised Learning
Algorithm 1 Training Process of ACE. 

5. Experiments
5.1. Datasets
 Node2vec learns embedding vectors through vertices sequences sampled by a random walk. LINE learns embedding vectors by preserving both firstorder and secondorder proximities. MNMF captures community structure through modularity and preserves secondorder proximities to learn embedding vectors.
 GCN [25], GAT [26], and DGI [27] are three GNNbased methods. GCN defines a layerwise propagation rule by spectral graph convolutions. GAT uses selfattention to assign a weight to each neighbor and employs multihead attention to keep stability. DGI learns vector representations by maximizing mutual information between patch representations and corresponding highlevel summaries of graphs.
 ACE_S: It only uses one part $Propagatio{n}_{S}$ of the dual propagation.
 ACE_W: It only uses one part $Propagatio{n}_{W}$ of the dual propagation.
 ${P}^{s}$: the example set of existent strong links.
 ${P}^{w}$: the example set of existent weak links.
 ${N}^{s}$: the example set of nonexistent strong links.
 ${N}^{w}$: the example set of nonexistent weak links.
5.2. Fairness Analysis
 ${P}^{s}$ vs. ${N}^{s}$: The experiment on it tells us the capacity with respect to predicting positive links on strong links.
 ${P}^{w}$ vs. ${N}^{w}$: The experiment on it tells us the capacity with respect to predicting positive links on weak links.
 ${P}^{s}$ vs. ${N}^{w}$: The experiment on it tells us the capacity with respect to predicting positive strong links that are mingled with negative weak links.
 ${P}^{w}$ vs. ${N}^{s}$: The experiment on it tells us the capacity with respect to predicting positive weak links that are mingled with negative strong links.
5.3. Parameter Sensitivity
5.4. Network Reconstruction
5.5. Gain Rate
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Notations  Definitions 

$\mathcal{G}=(\mathcal{V},\mathcal{E})$  The given graph G, node set V, and edge set E 
${\mathcal{E}}^{w}$  The set of weak link edges 
${\mathcal{E}}^{s}$  The set of strong link edges 
${\mathcal{E}}^{\prime}$  The set of edges in graph 
${\mathcal{N}}_{i}$  The set of neighbors linked to node i 
${\mathcal{N}}_{i}^{s}$  The set of strong neighbors linked to node i 
${\mathcal{N}}_{i}^{w}$  The set of weak neighbors linked to node i 
$\mathbf{M}$  The attribute matrix of graph 
$\mathbf{A}$  The adjacency matrix of graph 
$\mathbf{D}$  The degree matrix of graph 
$\mathbf{W},\mathbf{B},{\mathbf{R}}_{r}$  The training parameters 
$\Delta $  The batch size in training model 
$\delta ,\alpha ,\beta $  The attribute information matrix 
K  The layer number of autoencoder 
T  The number of the dual propagation 
Dataset  Link Strength  ACE_S  ACE_W  LINE  Node2Vec 

DBLP  Strong  83.5%  258.2%  9.2%  2.8% 
Weak  124.6%  152.9%  192.2%  116.5%  
LiveJournal  Strong  12.1%  40.2 %  88.7%  36.4% 
Weak  58.3%  −1.9%  1289.6%  218.8%  
Youtube  Strong  16.6%  21.5%  231.4%  46.7% 
Weak  17.4%  15.7%  1622.2%  124.6%  
Friendster  Strong  136.1%  152.0%  62.1%  31.8% 
Weak  inf  220.0%  inf  100%  
Dataset  link strength  MNMF  GCN  GAT  DGI 
DBLP  Strong  107.3%  2070.6%  1950.0%  301.1% 
Weak  301.3%  inf  9933.3%  4200.0%  
LiveJournal  Strong  55.5%  6.7%  5.4%  75.9% 
Weak  644.8%  8.6  8.1%  1211.3%  
Youtube  Strong  4540%  231.4%  136.7%  2220.0% 
Weak  inf  154.1  89.0%  1191.2%  
Friendster  Strong  25.6%  109.6%  269.3%  216.1% 
Weak  166.7%  inf  inf  300.0% 
Dataset  ACE  ACE_S  ACE_W  CN  AA  JI 

DBLP  0.751  0.748  0.554  0.659  0.659  0.659 
LiveJournal  0.977  0.957  0.729  0.942  0.942  0.892 
Youtube  0.935  0.874  0.839  0.692  0.694  0.667 
Friendster  0.781  0.570  0.525  0.612  0.615  0.607 
Dataset  LINE  Node2Vec  MNMF  GCN  GAT  DGI 
DBLP  0.712  0.691  0.727  0.686  0.700  0.663 
LiveJournal  0.702  0.883  0.872  0.937  0.943  0.726 
Youtube  0.783  0.738  0.643  0.845  0.855  0.624 
Friendster  0.603  0.630  0.700  0.617  0.585  0.531 
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Wang, L.; Yu, S.; Febrinanto, F.G.; Alqahtani, F.; ElTobely, T.E. FairnessAware Predictive Graph Learning in Social Networks. Mathematics 2022, 10, 2696. https://doi.org/10.3390/math10152696
Wang L, Yu S, Febrinanto FG, Alqahtani F, ElTobely TE. FairnessAware Predictive Graph Learning in Social Networks. Mathematics. 2022; 10(15):2696. https://doi.org/10.3390/math10152696
Chicago/Turabian StyleWang, Lei, Shuo Yu, Falih Gozi Febrinanto, Fayez Alqahtani, and Tarek E. ElTobely. 2022. "FairnessAware Predictive Graph Learning in Social Networks" Mathematics 10, no. 15: 2696. https://doi.org/10.3390/math10152696