# Social Influence Prediction with Train and Test Time Augmentation for Graph
Neural Networks

Hongbo Bo  Department of Computer Science
University of Bristol
Bristol, UK
<EMAIL_ADDRESS>Ryan McConville  Jun Hong  Department of Engineering
Mathematics
University of Bristol
Bristol, UK
<EMAIL_ADDRESS>Department of Computer Science
and Creative Technologies
University of the West of England
Bristol, UK
<EMAIL_ADDRESS>Weiru Liu  Department of Engineering Mathematics
University of Bristol
Bristol, UK
<EMAIL_ADDRESS>
###### Abstract

Data augmentation has been widely used in machine learning for natural
language processing and computer vision tasks to improve model performance.
However, little research has studied data augmentation on graph neural
networks, particularly using augmentation at both train- and test-time.
Inspired by the success of augmentation in other domains, we have designed a
method for social influence prediction using graph neural networks with train-
and test-time augmentation, which can effectively generate multiple augmented
graphs for social networks by utilising a variational graph autoencoder in
both scenarios. We have evaluated the performance of our method on predicting
user influence on multiple social network datasets. Our experimental results
show that our end-to-end approach, which jointly trains a graph autoencoder
and social influence behaviour classification network, can outperform state-
of-the-art approaches, demonstrating the effectiveness of train- and test-time
augmentation on graph neural networks for social influence prediction. We
observe that this is particularly effective on smaller graphs.

###### Index Terms:

graph neural networks, social network analysis, social influence analysis,
augmentation

##  I Introduction

Graph neural networks (GNNs) [1] have been shown to be effective in various
graph machine learning tasks, such as link prediction and node classification.
The rapid growth of online social networks has led to the development of
numerous methods for studying social behaviour online. However, many learning
tasks on social networks have relied heavily on manual feature extraction.
GNNs have provided an alternative to this with their ability to automatically
learn representations end-to-end. One such task of interest, which has been
shown to be enhanced using GNNs, is social influence prediction [2].

Data augmentation [3], which increases the amount of data available by
creating informative variations of existing data, can improve the performance
of machine learning models and has been widely used in many machine learning
tasks [4, 5, 6]. In the fields of computer vision (CV) [7, 3] and natural
language processing (NLP) [8], the combination of data augmentation methods
and deep neural networks have been shown to be effective. By performing data
augmentation, model performance can be improved as it facilitates the neural
network to learn generalizable features related to the task. While GNNs have
become a popular research field, little research has focused on using data
augmentation for GNNs, especially in terms of using augmentation at both
train- and test-time. Motivated by the success of data augmentation in CV and
NLP, we study whether data augmentation at not only train-time, but also test-
time, can improve the performance of GNNs, particularly on the task of social
influence prediction.

Extending the work of DeepInf [2], we have developed a method, AugInf, for
social influence prediction with both train- and test-time augmentation for
GNNs. In this method, the augmented graphs are first generated by utilising a
variational graph autoencoder (VGAE) [9] and then social influence is
predicted by joint training of both a Graph Auto-Encoder (GAE) [9] and a GNN
prediction module based on either a Graph Convolutional Network (GCN) [10] or
a Graph Attention Network (GAT) [11]. We have compared the performance of
AugInf with several state-of-the-art GNN approaches by experimenting on
numerous social networks. Our experimental results show that AugInf can
improve prediction performance on several of these social networks.

In summary, our contributions are as follows. First, we propose a joint
training approach consisting of Graph Auto-Encoder (GAE) [9] and GNN
prediction module. The GNN prediction module is implemented as either a Graph
Attention Network (GAT) [11] or a Graph Convolutional Network (GCN) [10]. The
joint training approach optimizes for two tasks simultaneously. The first is
to obtain more effective latent representations of input graphs, while the
second is to utilize the resulting representations to improve social influence
predictive performance. Additionally, we use the augmentation approach at both
train- and test-time to further improve the joint training model performance,
which we demonstrate with an ablation study. To the best of our knowledge, we
are the first to explore test-time augmentation on GNNs, and therefore the
first to combine both train-time and test-time augmentation on GNNs. Finally,
we conduct an experimental evaluation on multiple different social networks
comparing our proposed method with two state-of-the-art methods.

##  II Related Work

##### Graph Neural Networks

Graph Neural Networks (GNNs) [12, 13] have rapidly grown to become a popular
research area, providing a highly competitive approach for tasks involving
graph data. One line of research focuses on unsupervised models, e.g. VGAE [9]
and Graphite [14]. These unsupervised variational models typically aim to use
generative modelling of graphs for graph reconstruction, link prediction and
clustering. Additionally, supervised models have attracted significant
attention, such as SCNN [15], ChebyNet [16], GAT [11] and GCN [10], which are
widely used in tasks where labelled data is available.

##### Data Augmentation

Data augmentation has been shown to be an effective approach in machine
learning which expands a dataset by producing transformed copies of data,
thereby making the model invariant to these transformations. Data augmentation
has been widely used to improve generalizability of machine learning models in
natural language processing (NLP) and computer vision (CV). Most of the work
on data augmentation has focused on improving augmentation at the training
phase, e.g., batch augmentation [5] and UDA [6]. There are also studies that
focus on augmentation during the testing phase [17]. However, data
augmentation for graph neural networks has only been recently studied, such as
SUBG-CON [18] and NodeAug [19]. Particularly, there is no research on test-
time augmentation for GNNs.

##### Social Influence

The Independent Cascade Model [20] and Linear Threshold Model [21] are classic
social network influence propagation models. Measuring social influence can be
broadly divided into two categories based on the methods used. The first
category typically utilizes ranking algorithms such as TwitterRank [22],
Truetop [23] and EIRank [24], to quantify each user’s influence. This category
of methods provide a coarse influence value for each user at a specific time-
point and do not model changes to influence, nor the direct effect of this
influence on others on the network. The second category of methods are based
on predictive models to estimate social influence change. For example, on the
global-level patterns of social influence, the DeepCas model [25] can predict
the information cascade by using recurrent neural networks. DeepInf [2] and
NNMLInf [26] consider a social influence prediction task as a label
classification task and directly predict the user behaviour with respect to
influence.

##  III Preliminaries

Let $G=(V,E)$ be an input graph which consists of a set of nodes $V$ and a set
of edges $E$, where $E\in V\times V$. In the most general sense, a social
network can be represented by a graph $G$, where $V$ represents users and $E$
is the set of directed edges representing how the users are connected. The
adjacency matrix $A$ is a (0,1) matrix with 0s on its diagonal (i.e. no self-
connection) which can represent graph $G$, where $A_{ij}=0$ indicates nodes
$i$ and $j$ are not connected. If $G$ is undirected, $A_{ij}=1$ indicates
nodes $i$ and $j$ are connected and $A$ is symmetric. If $G$ is directed,
$A_{ij}=1$ indicates there is a link from node $i$ to node $j$. A graph neural
network model is defined as $f(X,A)$, where $X$ is the node feature matrix of
$A$.

###  III-A Graph Convolutional Network

A Graph Convolutional Network (GCN) [10] is a semi-supervised learning
algorithm for graph data, typically used for node and graph classification, as
well as link prediction. A GCN model is typically formed by stacking multiple
GCN layers, and for each GCN layer, the inputs are the adjacency matrix $A$
and the features matrix, $H\in\mathbb{R}^{n\times F}$, where $n$ is the number
of vertices, and $F$ is the number of features. For each GCN layer:

$H^{(l+1)}=\sigma(\widetilde{D}^{-1/2}\widetilde{A}\widetilde{D}^{1/2}H^{(l)}W^{(l)}),$
(1)

where $\widetilde{A}$ is the adjacency matrix $A$ with added self-connections
(the diagonal elements of the matrix are 1), $\widetilde{D}$ is the diagonal
degree matrix where
${\widetilde{D}}_{ii}={\textstyle\sum_{j}}{\widetilde{A}}_{ij}$, and
$\sigma(\cdot)$ denotes an activation function. The input layer $H^{(0)}=X$.
We will experiment with the use of a GCN model for social influence
prediction.

###  III-B Graph Attention Network

A Graph Attention Network (GAT) [11] is an attention-based version of GCN,
which incorporates self-attention mechanisms. The GAT layer performs the self-
attention mechanism for each node by introducing attention coefficients. The
self-attention mechanism is an attention function $attn$:

$\mathbb{R}^{F^{\prime}}\times\mathbb{R}^{F^{\prime}}\rightarrow{\mathbb{R}},$
(2)

where $F^{\prime}$ represents the number of output features of each node in a
GAT layer and the attention coefficient between each pair of nodes is
calculated as:

$e_{ij}=attn(Wh_{i},Wh_{j}),$ (3)

where $h_{i}$, $h_{j}\in\mathbb{R}^{F}$ and
$W\in\mathbb{R}^{F^{\prime}}\times\mathbb{R}^{F}$, and the attention function
$attn$ is instantiated with a dot product and a LeakyReLU [27] non-linearity.
The attention coefficient $e_{ij}$ is considered as the importance of node $j$
to node $i$. A softmax function is adopted to the normalized attention
coefficient to make it easier to calculate coefficients and compare them among
nodes:

$\alpha_{ij}=softmax_{j}(e_{ij})={\textstyle\frac{Exp(e_{ij})}{\sum_{k\in
N_{i}}Exp(e_{ik})}},$ (4)

where $N_{i}$ is the set of neighbour nodes of node $i$ and $\alpha_{ij}$ is
the coefficient for aggregating the calculation of the output feature
${h_{i}}^{\prime}$ which has incorporated neighborhood information:

${h_{i}}^{\prime}=\sigma({\sum_{j\in N_{i}}}\alpha_{ij}Wh_{j}),$ (5)

where $\sigma$ is a non-linear function. In order to make the self-attention
learning process more stable, it is known to be effective to use multi-head
attention to expand the attention mechanism. By using $K$ independent
attention mechanisms to perform transformations with Equation 5, and then
concatenating their features together, the following output can be obtained:

$h_{i}^{\prime}=\parallel_{k=1}^{K}\sigma(\sum_{j\in
N_{i}}\alpha_{ij}^{k}Wh_{j}^{k}),$ (6)

where $\parallel$ denotes the vector concatenation operation and the dimension
of $h_{i}^{\prime}$ is $KF^{\prime}$. We will experiment with the use of a GAT
model for social influence prediction.

###  III-C Graph Auto-Encoder

A Graph Auto-Encoder (GAE) [9] can utilize the GCN layers to obtain the latent
representations of the nodes in the graph through an encoder-decoder structure
to learn representations for downstream tasks, such as link prediction and
node classification. The encoder process can be expressed as:

$Z=GCN(X,A),$ (7)

where $Z\in\mathbb{R}^{n\times F}$ is the latent representation, which is also
referred to as an embedding. In this paper, we use a two layer GCN autoencoder
which can be defined as:

$GCN(X,A)=\widetilde{A}\sigma(\widetilde{A}XW_{0})W_{1},$ (8)

where $W_{0}$ and $W_{1}$ are the parameters to be learned and $\sigma$ is the
ReLU activation function. GAE uses the inner-product as a decoder to
reconstruct the original graph. In the training process of GAEs, cross entropy
is used as the loss function:

$\mathcal{L}=\frac{1}{N}\sum_{i=1}^{N}-(y_{i}\log{\widehat{y}_{i}}+(1-y_{i})\log(1-{\widehat{y}_{i}}))$
(9)

In the above equation, $y_{i}$ represents the value of an element in the
adjacency matrix $A$ (0 or 1), and $N$ is the size of $A$, $\widehat{y}_{i}$
represents the value of the corresponding element in the reconstructed
adjacency matrix $\widehat{A}$ (between 0 and 1). We use GAE in the joint
training model to obtain the latent representation.

###  III-D Variational Graph Auto-Encoder

A Variational Graph Auto-Encoder (VGAE) [9] uses latent variables for the
model to learn the distributions, and then samples from these distributions to
get latent representations. In VGAE, $Z$ is no longer obtained by a certain
function (such as eq. (7)), but by sampling from a Gaussian distribution. This
mechanism makes VGAE suitable for graph generation tasks. VGAE uses a 2-layer
GCN model (same as Equation. (8)) to calculate the mean $\mu$ and variance
$\sigma$ respectively to obtain a Gaussian distribution:

$\mu=GCN_{\mu}(X,A),$ (10) $\log_{\sigma}=GCN_{\sigma}(X,A),$ (11)

$W_{0}$ is the same in both $GCN_{\mu}$ and $GCN_{\sigma}$, but $W_{1}$ is
different. $Z$ can be calculated by reparameterization [28] and the decoder of
VGAE is also an inner-product. The loss function of VGAE is:

$\mathcal{L}={\mathcal{L}}_{CE}+{\mathcal{L}}_{KLD},$ (12)

Where ${\mathcal{L}}_{CE}$ is same as Equation 9 and ${\mathcal{L}}_{KLD}$ is
the KL divergence which is
$\frac{1}{2}\sum\nolimits_{i=1}^{d}(\sigma_{i}^{2}+\mu_{i}^{2}-\log(\sigma_{i}^{2})-1)$,
where $d$ is the dimension of $Z$. We use VGAE as part of the graph
augmentation process.

##  IV Method

We propose a method for social influence prediction which consists of a joint
training of the GAE and the GNN prediction module with both train- and test-
time augmentation. For clarity, as different strategies are used for train-
and test-time augmentation, we demonstrate the entire method in Figure 1 with
shared boxes showing training and testing stages, respectively.

Figure 1: The AugInf method. AugInf will first obtain the predicted edge
probability matrix $M$ by using the VGAE [9]. A threshold hyperparameter will
control the amount of edges added during the augmentation. We extend the work
of [4] to perform multiple augmentations, randomly sampling a subset of edges
to add to each original subgraph. (a) For train-time augmentation, the method
will then integrate all augmentations together for the joint training process.
The joint training model has two steps: the GAE step will learn the latent
representations of the input data for the prediction stage; the prediction
module (GAT or GCN) will produce the social influence predictions. The losses
of these two stages (decoder loss and prediction loss) will be combined and
propagated backwards to jointly update the model. (b) For test-time
augmentation, AugInf will generate several augmentations of each test
(validation) example, learning a representation for each before producing the
influence prediction. AugInf will calculate the average of the predictions to
produce the final social influence prediction.

###  IV-A Data Augmentation

Before the joint training, we first obtain augmentations of our data by
adapting a similar idea to [4]. The approach for augmentation consists of two
steps: (1) using the variational graph auto-encoder (VGAE) [9] to obtain edge
probabilities for all possible and existing edges in graph $G$, (2) using the
predicted edge probabilities, with a threshold set to stochastically add new
edges, creating a modified graph $G_{m}$, which is used as input to the joint
training process. The key idea of this augmentation approach is to use
information inherent in the graph to predict which non-existent edges are
likely to be added to the augmented graph to improve generalization. As a VGAE
is a generative model, we utilize VGAE as part of our graph augmentation
process.

The VGAE consists of a two-layer GCN encoder and an inner-product decoder:

$M=\sigma(ZZ^{T}),$ (13)

where $Z$ is the latent representation, $\sigma$ is an element-wise sigmoid
function and $M$ is the predicted (symmetric) edge probability matrix produced
by the inner-product decoder. We can add edges in graph $G$ to obtain the
augmentation graph $G_{m}$ according to the probabilities in $M$.
Corresponding to $G=(V,E)$, $G_{m}$ can be expressed as $G_{m}=(V,E_{m})$, and
for each node $v\in V$, there are different edge sets $e$ and $e_{m}$ in $G$
and $G_{m}$ respectively.

For efficiency, the input graph we will use is not an entire graph $G$, but a
set of subgraphs sampled by random walks on $G$, resulting in $N$ subgraphs,
following the approach of DeepInf [2]. We will perform augmentation on each of
the subgraphs separately, generating in total $Q$ augmentations for each of
the $N$ subgraphs. For the probability matrix $M_{i}$ of each subgraph, we
will set a threshold, and augment the subgraph by adding number of edges that
have probabilities higher than this threshold. This threshold, as a
hyperparameter of AugInf, will be discussed in the experimental section of
this paper.

###  IV-B Representation Learning

AugInf will primarily use a GAE to learn a representation of the graph via a
transformation of the graph structure into a low-dimensional latent space.
Additionally, consistent with DeepInf, AugInf will also use DeepWalk [29] to
generate features of both the original graph $G$ and $Q$ augmented graphs as
extra features for the social influence prediction.

###  IV-C Neural Network Model

We have implemented two variants of AugInf based on GCN and GAT, denoted by
AugInf-GCN and AugInf-GAT. In our joint training model, AugInf-GCN uses GCN
layers to obtain the output, while AugInf-GAT uses multi-head GAT layers.

##### Jointly trained model with train- and test-time augmentation

We utilize joint training of both the GAE and the GNN prediction module to
achieve two objectives: (1) obtain a more effective latent feature
representation of the input graphs, (2) using these representations with the
GNN prediction module to more accurately predict social influence. We
experiment with the GNN prediction module implemented as either a GAT or a GCN
for the social influence prediction.

The train-time augmentation process is shown in Figure 1 (a). The inputs are
the subgraphs sampled from graph $G$, along with their augmentations. We
utilize the learned representation $Z$ as one of the GNN prediction module
features, along with influence features and pretrained DeepWalk embeddings.
The influence features are provided by the authors of DeepInf [2], which
record the user behaviour status (i.e. have they taken an action) and whether
the user is the ego [2] user. The output of the GNN prediction module is a
2-dimension representation for each user (each node), corresponding to the
negative log-likelihood. The test-time augmentation process is shown in Figure
1 (b). For each test graph, we generate a number of augmented versions of the
graph, learning representations and producing influence predictions for each,
with the average of these predictions forming the final social influence
prediction.

##### Loss

The loss function of the joint model consists of two parts:

$L=L_{D}+L_{GAE},$ (14)

where $L_{D}$ is the loss of the final prediction and the ground truth, which
is calculated by a negative log-likelihood, and $L_{GAE}$ is the loss of GAE
which is calculated by the cross entropy of the decoded reconstructed graphs
and the input graphs.

##  V Experiments

###  V-A Datasets

We evaluate using four datasets across different social network domains,
namely OAG111OAG dataset details: www.openacademic.ai/ (Open Academic Graph),
Digg222Digg dataset details: www.isi.edu/ lerman/downloads/digg2009.html/,
Twitter333Twitter dataset details: snap.stanford.edu/data/higgs-twitter.html
and Weibo444Weibo dataset details: www.aminer.cn/influencelocality.

  * •

The OAG graph consists of academic graphs representing the co-author network
in which the citation behaviours, which we are predicting, are defined as the
influence action behaviour.

  * •

The Digg dataset contains the timestamped voting behaviours of users on
stories on a social news aggregation website. The edges of Digg graph are
defined as following relationships and the influence actions, which we are
predicting, are voting behaviours.

  * •

The Twitter dataset has been built by collecting Twitter data corresponding to
tweets collected before, during and after the announcement of the discovery of
the Higgs boson in 2012. The graph is defined as a friendship network, and the
social action, which we are predicting, is defined as whether a user retweets
Higgs boson tweets.

  * •

The Weibo graph was built from 100 randomly selected users and their followers
and followees. The social network is defined as a friendship network, and the
social action, which we are predicting, is defined as retweeting behaviors in
the Weibo social network.

These datasets were used previously by Qiu et al. [2]. Qiu et al. [2] sampled
the entire social network into sub-networks with 50 nodes in each sub-network
by using a random walk with restart, extracted features for each node and
provided a ground-truth for the dataset. The statistics of the three datasets
are shown in Table I.

TABLE I: The statistics of the datasets. $|V|$ and $|E|$ are the total numbers
of nodes and edges of the original dataset respectively and $N$ is the number
of subgraphs after preprocessing.

| OAG | Digg | Twitter | Weibo
---|---|---|---|---
$|V|$ | 953,675 | 279,630 | 456,626 | 1,776,950
$|E|$ | 4,151,463 | 1,548,126 | 12,508,413 | 308,489,739
$N$ | 499,848 | 24,428 | 362,888 | 779,164

###  V-B Evaluation Metrics

We analyze several hyperparameters in our model and study how different
hyperparameters may affect prediction performance. The performance is
evaluated in terms of Area Under Curve (AUC) and $F_{1}$ score ($F_{1}$). We
compare our model with the state-of-the-art DeepInf [2] and PSCN [30]
algorithms.

###  V-C Experiment Setup

In our experiments we apply three augmentations to each graph with the
augmentation hyperparameter threshold value set to 0.8 and train for 500
epochs. We will discuss the performance of varying these parameters in a later
section. For the GAE component, each of the two hidden layers contain 64
hidden units for Digg and Twitter, and 32 for OAG and Weibo. They are trained
with the Adagrad optimizer, using a 0.2 learning rate for OAG and Weibo, 0.05
for Digg and 0.1 for Twitter. Weight decay is set to 0.0005 all datasets
except Digg, where it is 0.001. Additionally, we use dropout rate of 0.2. For
the GNN prediction module, the first and second layers each contain 128 hidden
units and the third layer, as the output layer, has two hidden units. There
are eight attention heads in each GAT layer, which means each head needs to
process 16 hidden units for Digg and Twitter, with four attention heads for
OAG and Weibo, which means each head needs to process 32 hidden units. The
nonlinear activation function we use for both augmentation and prediction
($\sigma$ in Eq. 1 and 5) is the exponential linear unit (ELU) [31].

###  V-D Experimental Results

We report the performance of our method (AugInf-GAT and AugInf-GCN) over ten
runs. The mean and standard deviations are shown in Table II. For clarity, the
input features of our models are influence features and pretrained DeepWalk
embeddings, consistent with the end-to-end method DeepInf [2], but without any
hand-crafted vertex features. As an experimental comparison, we used the
state-of-the-art GNN methods DeepInf and PSCN [30] as the baselines.

TABLE II: The performance of two AugInf models on different datasets, along
with the performance of the baselines without vertex features.

Dataset | Model | AUC | $F_{1}$
---|---|---|---
Digg | DeepInf-GAT |  0.8882($\pm$0.011) |  0.7052($\pm$0.010)
DeepInf-GCN |  0.8372($\pm$0.007) |  0.6404($\pm$0.006)
PSCN |  0.8499($\pm$0.013) |  0.6636($\pm$0.011)
AugInf-GCN |  0.8580($\pm$0.013) |  0.6711($\pm$0.024)
AugInf-GAT |  0.9067($\pm$0.011) |  0.7385($\pm$0.017)
Twitter | DeepInf-GAT |  0.7843($\pm$0.002) |  0.5484($\pm$0.002)
DeepInf-GCN |  0.7615($\pm$0.004) |  0.5271($\pm$0.004)
PSCN |  0.7664($\pm$0.004) |  0.5319($\pm$0.006)
AugInf-GCN |  0.7694($\pm$0.003) |  0.5360($\pm$0.006)
AugInf-GAT |  0.7861($\pm$0.004) |  0.5486($\pm$0.007)
OAG | DeepInf-GAT |  0.6814($\pm$0.006) |  0.4544($\pm$0.004)
DeepInf-GCN |  0.6281($\pm$0.002) |  0.4231($\pm$0.003)
PSCN |  0.6556($\pm$0.006) |  0.4372($\pm$0.029)
AugInf-GCN |  0.6334($\pm$0.003) |  0.4229($\pm$0.011)
AugInf-GAT |  0.6889($\pm$0.012) |  0.4602($\pm$0.004)
Weibo | DeepInf-GAT |  0.8212($\pm$0.003) |  0.5770($\pm$0.003)
DeepInf-GCN |  0.7706($\pm$0.002) |  0.5312($\pm$0.004)
PSCN |  0.8012($\pm$0.004) |  0.5625($\pm$0.002)
AugInf-GCN |  0.7531($\pm$0.006) |  0.5147($\pm$0.004)
AugInf-GAT |  0.8124($\pm$0.005) |  0.5785($\pm$0.007)

Our proposed method AugInf-GAT achieves better performance over most of the
baselines in terms of AUC. However, consistent with all models, the GCN-based
approaches do not perform well, as GCN uses the unweighted aggregations over
the neighbours’ representations when calculating a node’s representations, a
mechanism that appears not to be suitable for tasks like social influence
prediction, which benefits from considering the importance of neighbouring
nodes.

Referring to the statistics of the datasets in Table I, we can see that the
performance improvement of AugInf-GAT is particularly clear on the datasets
with much fewer edges (Digg) but limited on datasets with more edges (Twitter
and Weibo). We believe that this is because the Twitter and Weibo datasets
contain enough edges to learn a sufficiently comprehensive representation,
hence less benefit is gained from the augmentation. We will further
investigate the effect of removing edges from graphs as part of data
augmentation in future work. Nonetheless, particularly for smaller graphs, we
believe our proposed approach of train- and test-time augmentation can provide
additional performance.

####  V-D1 Hyperparameter Analysis

We conduct an hyperparameter analysis on the Digg dataset with the same
hyperparameters values mentioned previously, unless stated otherwise.

##### Number of Augmentations

For the data augmentation, we analyze the effect of the number of
augmentations on the performance of AugInf. We successively apply one to eight
augmentations while leaving the other parameters constant. The results of this
are shown in Figure 2. When we apply four augmentations, the highest
performance is achieved. After that, as the number of augmentations increase,
the AUC score stabilises. Interestingly, the $F_{1}$ score does not appear to
be affected by the number of augmented graphs and remains stable between 0.73
and 0.74.

Figure 2: The performance of AugInf-GAT on Digg as we vary the number of
augmentations. The value of threshold is set to 0.8.

##### The Threshold for Augmentation

Another parameter we analyze is the threshold that determines which edges may
be added. The results of this are shown in Figure 3. When the threshold is set
to 0.8 for Digg dataset, our method achieves the highest performance, while on
average the number of edges per dataset increases by $2.7\%$. As we increase
the number of added edges, the performance of our method decreases.

Figure 3: The performance of AugInf-GAT on Digg as we vary the parameter that
controls the minimum quality of edges that are added for augmentation. The
number of augmentations is set to three.

###  V-E Ablation Study

We have further evaluated AugInf-GAT and AugInf-GCN on the individual
components of our approach, to determine the contribution of each component to
the overall performance.

There are three main components in our approach: (1) train-time augmentation
(2) test-time augmentation and (3) the jointly trained model. We have
evaluated the following combinations on the Digg dataset:

  * •

Ablation $\\#1$: The method contains only a GNN prediction module. We use this
set of experiments as a baseline to compare with other combinations when other
components are added.

  * •

Ablation $\\#2$: The method contains the jointly trained model with no
augmentation. The purpose of this experiment is to measure the effect of joint
training without augmentation.

  * •

Ablation $\\#3$: The method uses a GNN prediction module with train-time
augmentation only. The purpose of this experiment is to measure the effect of
train-time augmentation.

  * •

Ablation $\\#4$: The method uses a GNN prediction module with test-time
augmentation only. The purpose of this experiment is to measure the effect of
test-time augmentation.

  * •

Ablation $\\#5$: The method uses a GNN prediction module with both train-time
and test-time augmentation. The purpose of this experiment is to measure the
effect of both forms of augmentation together.

  * •

Ablation $\\#6$: The method contains the joint training model with train-time
augmentation only. The purpose of this experiment is to evaluate the extent of
which using train-time augmentation improves the joint training model.

  * •

Ablation $\\#7$: The method contains the joint train-model with test-time
augmentation only. The purpose of this experiment is to evaluate the extent of
which using test-time augmentation improves the joint training model.

  * •

Complete $\\#8$: The method is our complete method, AugInf, consisting of all
components.

The results with the GAT prediction module are shown in Figure 4. Through the
comparison with $\\#1$ and $\\#2$, we can see the jointly trained model
contributes to the performance improvement. Through the comparison among the
ablations $\\#1$, $\\#3$, $\\#4$ and $\\#5$, we observe that the augmentations
can improve the GAT performance, but there is little difference between using
train-time augmentation and test-time augmentation separately or together,
without the jointly trained model. The result of version $\\#8$, the complete
AugInf method, shows using train- and test-time augmentation, along with the
jointly trained model, has the highest AUC score and $F_{1}$ score.

Figure 4: The performance of the ablation study using GAT where $\\#8$ is our
complete method consisting of all components, AugInf.

The results with the GCN prediction module are shown in Figure 5. The AUC
scores of this set of experiments are between 0.83 and 0.86. We can see that
all components have contributed to some extent to the performance of the GCN
prediction module, but overall, AugInf when using a GCN for social influence
prediction achieves lower performance than AugInf when using a GAT for social
influence prediction.

Figure 5: The performance of using different combinations of components with
GCN, where $\\#8$ is our complete model consisting of all components, AugInf.

##  VI Conclusions

In this paper we proposed a new social influence prediction method, AugInf,
which incorporates train- and test-time augmentation with a jointly trained
graph neural network approach. During training, this method takes into account
the losses of both the graph representation learning and downstream social
influence prediction task. We improve performance by applying numerous
augmentations to the graphs using variational graph auto-encoders at both
train- and test-time. Via an ablation study we show that the jointly trained
model obtains more effective latent feature representations by using the joint
loss along with both the train- and test-time augmentations. We compare our
proposed end-to-end method with the state-of-the-art on several social network
datasets. The experimental results show that our proposed method, AugInf-GAT,
can improve the performance of predicting social influence on a number of
social networks, and in particular, on the smallest of the social network
graphs.

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