Abstract
In recent years, how to prevent the widespread transmission of infectious diseases in communities has been a research hot spot. Tracing close contact with infected individuals is one of the most severe problems. In this work, we present a model called Follower Prediction Graph Network (FPGN) to identify highrisk visitors, which is known as follower prediction. The model is designed to identify visitors who may be infected with a disease by tracking their activities at the exact location of infected visitors. FPGN is inspired by the stateoftheart temporal graph edge prediction algorithm TGN and draws on the shortcomings of existing algorithms. It utilizes graph structure information based on (\(\alpha \), \(\beta \))core, time interval statistics by using the statistics of timestamp information, and a GATbased prediction module to achieve high accuracy in follower prediction. Extensive experiments are conducted on two real datasets, demonstrating the progress of FPGN. The experimental results show that FPGN can achieve the highest results compared with other SOTA baselines. Its AP scores are higher than 0.46, and its AUC scores are higher than 0.62.
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1 Introduction
In recent years, the outbreak of infectious diseases has caused great harm to human health and social stability. To prevent and control the spread of infectious diseases, it is necessary to predict and control the highrisk population accurately. In this regard, graph analysis has become an essential tool for epidemiologists [14, 32]. Temporal bipartite graphs represent the relationships between two sets of vertices while tracking their evolution over time. As a result, this structure is particularly useful in analyzing infectious disease transmission patterns. By analyzing these temporal bipartite graphs, we can identify gaps and clusters that may indicate potential transmission.
This paper focuses on identifying highrisk visitors who may be infected with a disease by tracking their activities at the same location as infected visitors. The identification allows public health authorities to take proactive measures to prevent the spread of infectious diseases. The objective is to predict the likelihood that two visitors (i.e., a follower and a leader) will visit the same location within a specific time window. This challenge is called the follower prediction problem.
Example
An example of the following behavior is shown in Figure 1. We assume that the time window to identify the following behavior in this figure is half an hour, i.e., if two visitors visit the same venue within half an hour, the second visitor is identified as the follower of the first visitor. We also assume that Bob (\(u_1\)) is an infected visitor. We can see that Bob went to the cafe (\(v_1\)) at 8 o’clock in the morning, and another visitor Sam (\(u_2\)) arrived within ten minutes after Bob’s arrival. Therefore, we consider Sam to be one of Bob’s followers. Although Lisa (\(u_3\)) and Bob also arrived at Gym(\(v_2\)) within ten minutes of each other, Lisa arrived at the venue before Bob did. Jack(\(u_4\)) arrived at the restaurant(\(v_4\)) three hours after Bob did. Therefore, neither Lisa nor Jack is a follower of Bob.
Challenge
The difficulty in solving this problem lies in two aspects: on the one hand, due to the special nature of the bipartite graph structure, there will be no connections between vertices of the same type in the dataset, while the goal of follower prediction is precisely to predict connections between vertices of the same type. On the other hand, the conditions for constituting the following behavior are strict. They require simultaneous consideration of both cases where two vertices are connected to the same vertex and their time difference for connecting.
To the best of our knowledge, there is no existing method proposed to solve the follower prediction problem. One possible solution is to use a highperformence GNN model [4, 15, 19, 42, 49] trained on datasets for this problem. However, these generic models do not effectively utilize timestamp information and have limited accuracy. A related problem of the follower prediction problem is edge prediction in temporal graphs, which has been addressed by various algorithms [3, 20, 33, 40, 47]. These methods accurately predict edges on temporal graphs but either rely solely on timestamps as sampling criteria or only learn individual activity information for each vertex. Consequently, models designed for other purposes may struggle with this specific follower prediction problem and even perform worse than general GNN models due to being misled by their original goals.
In order to better solve the problem of follower prediction, inspired by the stateoftheart temporal graph edge prediction algorithm TGN [33] and drawing on the shortcomings of existing algorithms, we propose a model called Follower Prediction Graph Network (FPGN). TGN significantly contributes to preserving temporal information by learning the memory of vertices. Therefore, the vertex memory obtained by TGN can help us solve the problem of predicting connections between sametype vertices on bipartite graphs. However, TGN cannot be directly applied to this problem. When TGN updates the memory of a vertex, it refers to its previous memories, allowing for sampling train sets based on chronological order. In other words, TGN updates vertex memories while sampling based on downstream task requirements. On the other hand, to predict followers accurately, it is necessary to anticipate the connections between sametype vertices in the bipartite graph that have been updated after all timestamps in the training set. This makes it impossible for TGN to be directly used for solving follower problems. Motivated by TGN and learned from his shortcomings, we design our model by optimizing the structure of TGN to let it update memories of all vertices beforehand and then train according to downstream needs.
Besides, FPGN can also consider the correlation information between vertex activities to enhance the model’s ability to solve follower prediction problems. Specifically, in addition to learning which vertices will be followers of their leaders, FPGN will further analyze the time window in which these following behaviors occur, allowing the model to grasp the degree of association or closeness between these followers and their leaders. This strategy can help FPGN achieve the ability to predict the following behaviors whose conditions are strict.
Moreover, when some followers often follow several other fixed leaders, the density of the graph composed of them will be relatively high. Therefore, information about the structure of the graph, especially the density, is essential in the follower prediction problem. A typical way to find dense subgraphs on a bipartite graph is biclique [1, 58]. More specifically, biclique aims to find subgraphs on a bipartite graph where any vertex connects to all of the vertices belonging to another class. Thus, it can be assumed that any vertex in the biclique is closely related to all other vertices in this subgraph. However, on the one hand, biclique is an NPhard problem that is difficult to obtain quickly by conventional ways [26, 27, 31]. On the other hand, biclique is too strict to provide richer structural information for the model. Therefore, we choose to use the more flexible (\(\alpha \), \(\beta \))core [23] to extract the structural information of the graph. The advantage of using (\(\alpha \), \(\beta \))core is that it allows the size of \(\alpha \) and \(\beta \) to be controlled according to actual needs so that the model can flexibly obtain the required structural information while reducing computational costs. With the help of (\(\alpha \), \(\beta \))core and time interval statistics, combined with the mapping method based on UniMP [37], FPGN can effectively solve the problem of difficulttopredict relationships between vertices of the same type on temporal bipartite graphs, obtain highly accurate follower prediction results.
Contributions
The contributions of this paper are summarized as follows:

As far as we know, FPGN is the first model proposed to solve the problem of follower prediction.

By relying on improvements to TGN and carefully designed modules that make good use of temporal statistical information and graph structure information, FPGN has achieved extremely high accuracy in follower prediction.

Extensive experiments are carried out on two real datasets, and the results proved the effectiveness of FPGN.
2 Background
2.1 Preliminaries
The temporal bipartite graph is denoted as \(G = (U, V, T, A)\), where \(U = \{u_1, u_2, \dots \}\) is a visitor set; \(V = \{v_1, v_2, \dots \}\) is a location set; \(T = \{t_{(u_1, v_1)}, t_{(u_2, v_1)}, \dots \}\) is a set of times when a visitor arrives at a location, where \(t_{(u_1, v_1)} = \{t_{(u_1, v_1)}^{(1)}, t_{(u_1, v_1)}^{(2)}, \dots \}\) is a set of timestamps to record each timestamp of a visitor arriving at the exact location; \(A = \{{\textbf {A}}^{t_1}, {\textbf {A}}^{t_2}, \dots \}\) is the set of adjacency matrix, where \({\textbf {A}}^{t_1}\) denotes all visitor records for at time \(t_1\). If u and v connect with each other at time t, then \({\textbf {A}}^{t}_{u,v} = 1\), otherwise \({\textbf {A}}^{t}_{u,v} = 0\).
Definition 1
(Minimum Following Time Interval) Given a leader u, a follower \(u'\), and a location v that the follower follows the leader to reach, the definition of the minimum following time interval:
Definition 2
((\(\alpha \), \(\beta \))core) Given a bipartite graph \(G = (U, V)\) and two integers \(\alpha \) and \(\beta \), the (\(\alpha \), \(\beta \))core of G is made up of two subsets of vertices: \(U' \in U\) and \(V' \in V\). These subsets induce a bipartite subgraph g that is the largest subgraph of G where all vertices in \(U'\) have a degree of at least \(\alpha \) and all vertices in \(V'\) have a degree of at least \(\beta \).
2.2 Problem statement
Given a temporal bipartite graph G, the task of follower prediction is to predict whether \(u'\) will go to the location v where u has visited in a certain time window \(\tau _t > 0\). That is, for visitor u and \(u'\), if \(\{ 0 \le \Delta (u, u', v) \le \tau _t  v \in V \} \ne \phi \), then \(u'\) is a follower of leader u.
Generally speaking, graph neural networks can be summarized by the following formula:
where \({\textbf {h}}^{(k)}_u\) represents the hidden representation of vertex u at layer k, this vector captures important information about the vertex and its relationships with other vertices in the graph; \({\textbf {h}}^{(k1)}_u\) refers to the hidden representation of vertex u at the previous layer (\(k1\)). In other words, it is the input to the current layer k; \(\mathcal {AGG}^{(k)}\{{\textbf {h}}^{(k)}_{u'}; u' \in N(u)\}\) is the aggregation function, which combines the hidden representations of neighboring vertices \(u' \in N(u)\) to produce a summary of their information. The specific function used can vary (e.g., mean, max, sum, etc.); \(\mathcal {COM}^{(k)}(\cdot )\) is the update function, which takes the aggregated information and the previously hidden representation of the vertex and produces a new hidden representation for the vertex at the current layer k. This function allows the vertex to incorporate information from its neighbors into its own representation. The core of the graph neural network model consists of these elements, which allow vertices to acquire representations that encompass both the structure and features of the graph.
3 Model
3.1 Overview
The overall framework of the model is shown in Figure 2. FPGN is mainly composed of two parts, the information recording module, and the prediction module. In more detail, the role of the information recording module is to obtain the memory of each vertex and record the most recent neighbors with each vertex based on the information in a temporal bipartite graph. And relying on graph neural networks and multilayer perceptrons (MLP), and using the memory of vertices and the most recent neighbors of these vertices, the prediction module can predict followers. The model is described in detail in this section.
3.2 Preprocessing stage
FPGN aims to solve the follower prediction problem on featureless temporal bipartite graphs. With embedding, the ID of each vertex can be mapped to a vector \({\textbf {r}}_o \in {\textbf {R}}_o\). We take this vector as the initial feature of the vertex. However, the features generated in this way can only help the model distinguish vertices but cannot provide more information to help the model perform better. In order to enable the model to mine better vertex representations, we also need to obtain more information in the graph.
3.2.1 Graph structure information
To be able to help the model extract information about the graph structure, we compute (\(\alpha \), \(\beta \))core for each vertex. More specifically, we compute whether each vertex exists with a certain set of subgraphs satisfying (\(\alpha \), \(\beta \))core and treat that graph structure information as a part of the vertex features. Thanks to the graph structure information, FPGN can produce more precise results. Given an upper threshold and a lower threshold for \(\alpha \) and \(\beta \), respectively: \(\alpha ^{}_{\tau } \le \alpha \le \alpha ^{+}_{\tau }\), and \(\beta ^{}_{\tau } \le \beta \le \beta ^{+}_{\tau }\), and then determine whether each vertex is within these (\(\alpha \), \(\beta \))core subgraphs to obtain the graph structure characteristic matrix \({\textbf {R}}_s \in \mathbb {R}^{(U + V) \times d_s)}\), where \(d_s = (\alpha ^{+}  \alpha ^{} + 1) \times (\beta ^{+}  \beta ^{} + 1)\) equals the number of (\(\alpha \), \(\beta \))core queried. Each vertex’s graph structure feature \({\textbf {r}}_s \in \{0, 1\}^{d_s}\) is a part of \({\textbf {R}}_s\). If the value of a dimension of the feature is 0, it means that the vertex does not exist in the (\(\alpha \), \(\beta \))core corresponding to that dimension, and vice versa.
In the end, the characteristic matrix of the vertices used by FPGN is:
where  is concatenation operation. Generating the vertex features by using the above way can provide rich graph structure information for FPGN while avoiding increasing its time complexity. For convenience in the description, we represent \({\textbf {R}}(u)\) as a characterization of u.
3.2.2 Time interval statistics
It is crucial for the follower prediction problem to get information on the time interval between the follower’s journey to a certain location and that of the leader. This information can help the model discover the following behavior and solve the follower prediction problem. Firstly, the number of times the same follower and leader travel to the same location sequentially can reveal the correlation between them. This information can be obtained statistically using the following formula:
where \([\cdot ]\) is Iverson bracket. It is a useful tool for mapping any statement to a function of the free variables in that statement. This bracket serves as a kind of truthvalue indicator to assign a value of 1 to any true statement and 0 to any false statement:
And \({\textbf {S}}_{n, (u, u')}\) denotes the number of locations where the follower \(u'\) arrives to follow the leader u. \({\textbf {S}}_n\) can record the frequency of followup between two visitors very well, which can play a significant role in the following prediction.
Then we count the closeness of the following actions for each follower, i.e., the time interval between two visitors arriving at the same location. The smaller the time interval, the closer that follower is to that leader. However, the follower can follow the leader in multiple locations. To better inform the model, we calculated the average and variance of their time intervals. Given a leader u, a follower \(u'\), We can get the average statistics and variance statistics of \(\Delta \):
Using this statistical information, it is possible to help the model gain a better understanding of the relationship between visitors. This information can be used to build features that highlight the relationship between leaders and followers:
where Norm is the normalization operation. The features \({\textbf {S}}\) obtained will help the model optimize the parameters during the training process. The details of the training process will be explained in Section 3.5.
3.3 Information update stage
Motivated by TGN, FPGN extracts memory and saves the last neighbors for each vertex so that FPGN can build the subgraph of target vertices and achieve follower prediction. However, TGN can only train for the samples in each batch separately. In follower prediction, there is no guarantee that each round of batch can provide the prediction module with features for the vertices that need to be predicted. To solve the problem, we design the module of FPGN based on TGN. More specifically, FPGN updates only the memory and neighbors of each vertex in each batch. The prediction module and backpropagation will only be done after these updates are completed.
3.3.1 Memory update
To better learn the temporal graph information, FPGN mines the vertex memory with temporal information. Given a visitor record triplet (u, v, t), which means that the visitor u arrives at the location v at timestamp t with message \({\textbf {M}}_{(u,v)} = {\textbf {R}}(u)  {\textbf {R}}(v)\). The memory of time t for u is defined as \({\textbf {F}}_u(t)\), and \({\textbf {F}}_u(0) = \{0\}^{1 \times d}\) where d is the dimension of the memory. The memory of u and v is calculated separately. For the sake of description, we only describe the process of updating u’s memory.
To better explore the historical activity message of vertices, all activity timestamps \(T_u(t)\) of u earlier than time t will be extracted and synthesized into vectors \({\textbf {T}}_u(t) \in \mathbb {R}^{1 \times T_u(t)}\). And memories \({\textbf {M}}_u(t)\) of u at those times are also involved in the vertex memory update task. FPGN first encodes the activity of u:
where \(\cos (\cdot )\) is the cosine function; \({\textbf {W}}_t\) is a matrix of trainable parameters and \({\textbf {b}}_t\) is a trainable parameter; \(t_u\) is the timestamp of the last activity of u; \({\textbf {I}}_{(t,u)} \in \mathbb {R}^{1 \times T_u(t)}\) is a vector whose elements are all 1 with dimensions \(1 \times T_u(t)\). It should be noted that \(t_u\) is initialized with a value of 0.
In order to extract information about the past activities of u, we need to extract all neighbors of u before time t from the adjacency matrix set A. This can be done by defining the set of neighbors of u at time t as follows:
where \({\textbf {A}}^{t}\) is the adjacency matrix at time t, and \({\textbf {A}}_{u, v}^{t} = 1\) if u and v connected with each other at time t. After obtaining these neighboring vertices, we can then get the message between u and them, these neighbors’ memories at that time, and the past memories of u at all timestamps before t:
then, concatenate \({\textbf {X}}_u\), \({\textbf {X}}_{\mathcal {N}_u^t}\) and \({\textbf {M}}_{\mathcal {N}_u^t}\), as well as \({\textbf {X}}_{t, u}\) to obtain the initial memory of u:
where \({\textbf {X}}_{t, u}^{init}\) is the initial memory of u at timestamp t.
Because early activity records should not have too much influence on the present, and each vertex memory at every moment contains activity information before that moment, to make FPGN pay more attention to recent activity information and reduce its computational cost, we only keep the latest \(\mathcal {K}\) sets of initial memory \(\bar{{\textbf {X}}}_u^{init}(t)\) in \({\textbf {X}}_u^{init}(t)\).
LSTM is a widely used recurrent neural network architecture that is particularly suited to processing sequential data, such as speech and text [35, 38]. There are many works that handle temporal graphs using LSTM and have achieved good performance [6, 50] nowadays. However, recent research has found that the newer Gated Recurrent Unit (GRU) [10] architecture offers comparable performance to LSTM while being more computationally efficient [5, 5, 51], making it an attractive alternative for processing temporal graph datasets. Therefore, in this work, we chose to use GRU to obtain vertex memories.
To be more precise, during each time step, a GRU unit utilizes the initial memories \(\bar{{\textbf {X}}}_u^{init}(t)\) and the most recent memories of u, denoted as \({\textbf {F}}_{u}(t')\), to compute the reset gate \({\textbf {p}}_{t}\) and update gate \({\textbf {q}}_{t}\) in the following manner:
where \(\sigma \) is nonlinear activation functions, \({\textbf {W}}_{px}\), \({\textbf {W}}_{pf}\) and \({\textbf {W}}_{qx}\), \({\textbf {W}}_{qf}\) are weight matrices, and \({\textbf {b}}_{p}\), \({\textbf {b}}_{q}\) are bias vectors. The reset gate controls how much of the previous memories should be ignored, while the update gate controls how many new memories should be added to the initial memories. The candidate memory \(\widetilde{{\textbf {F}}}_{u}(t)\) is then computed as follows:
where \(\odot \) denotes the elementwise product, \({\textbf {W}}_{x}\) and \({\textbf {W}}_{f}\) are weight matrices and \({\textbf {b}}_f\) is a bias vector. Finally, the new memory of u \({\textbf {F}}_{u}(t)\) is calculated as a linear interpolation between the previous memory and the candidate memory, controlled by the update gate:
GRU’s mechanism allows it to process temporal graphs by selectively retaining or discarding information as time passes. This feature enables GRU to store and analyze data over an extended period, making it an effective tool for handling dynamic and evolving datasets. After completing the training of all data through the above method, we can obtain the final vertex memories:
where \(\Gamma _{n}\) is the last timestamp of vertex n.
3.3.2 Neighbors update
Each round of memory update for each vertex requires tracing the vertices that were previously connected to it. Therefore, after each round of vertex memory update for a triplet (u, v, t), the edges involved in that round of update need to be saved into the adjacency matrix \({\textbf {A}}^t\), that is to say, \({\textbf {A}}^t_{u,v} = 1\).
Additionally, after acquiring \({\textbf {F}}\), utilizing all the connection information for improved follower predictions is imperative. Thus, amalgamating all the adjacency matrices from each time step into a single matrix becomes necessary:
3.4 Prediction module
Given a visitor u, by utilizing \({\textbf {A}}_a\), we can extract all the locations u have visited, represented as \(\mathcal {N}_a(u)\). It should be noted that to ensure FPGN can retain the memory of each vertex during the learning process, vertex u will be included in \(\mathcal {N}_a(u)\). Then, FPGN can use the memories of these vertices (e.g., the memory of u is \({\textbf {F}}_u\)) as their features to further learn about the characteristics of these two visitors through GNNs.
In FPGN, we use Graph Attention Network (GAT) [42] for vertex representation learning. GAT is a multihead attention mechanismbased GNN model that enables efficient and effective vertex and edge relationship modeling in graphstructured data. One of the key advantages of the GAT model is that it can learn to dynamically assign different weights to different vertices and edges in the graph while considering the neighborhood and global information. This is achieved using a selfattention mechanism that allows the model to selectively attend to different subsets of vertices and edges based on their relevance to the target vertex or subgraph. With the attention mechanism of GAT, FPGN can better determine which locations are more important for determining the relationship between two visitors. Moreover, using multiple attention heads allows the model to learn diverse representations of the graph, thereby improving the robustness and generalization capabilities of the model.
GAT in FPGN can be mathematically formulated as follows: Given a visitor vertex u with memory \({\textbf {F}}_u\), and the locations \(\mathcal {N}_a(u)\) visited by u, the output features of the lth layer can be computed as:
where K is the number of attention heads, \(\sigma \) is a nonlinear activation function (e.g., ReLU), \({\textbf {W}}^{(l)}_{k}\) is the weight matrix for the kth attention head at the lth layer, and \(a_{k, uv}^{(l)}\) is the attention weight assigned to the edge from vertex u to vertex v by the kth attention head at lth layer, \({\textbf {h}}_{u}^{(l)}\) is lth layer representation of u, and \({\textbf {h}}_u^{(0)} = {\textbf {F}}_u\), \({\textbf {h}}_v^{(0)} = {\textbf {F}}_v\).
Follow prediction problem involves bipartite graph datasets. However, the conventional GAT model is inefficient in distinguishing between different types of vertices on the bipartite graph and cannot handle edge messages. To overcome these limitations, we drew inspiration from UniMP [37] and mapped different types of vertices differently while utilizing edge information. More specifically, for the cth head in layer l, the representations of u, v, and their edge, as well as the attention weight between them, can be calculated using the following method:
where \(a^{(l)}_{c,uv}\) is the new attention weight, \(<{\textbf {h}}_{1}, {\textbf {h}}_{2}> = \exp (\frac{{\textbf {h}}_{1}^T {\textbf {h}}_{2}}{\sqrt{d_h}})\) is exponential scale dotproduct function and \(d_h\) is the hidden size of each head, \({\textbf {W}}_{c,u}^{(l)}\), \({\textbf {W}}_{c,v}^{(l)}\), and \({\textbf {W}}_{c,m}^{(l)}\) represent weight matrices, while \({\textbf {b}}_{c,u}\), \({\textbf {b}}_{c,v}\), and \({\textbf {b}}_{c,m}\) are bias vectors at the lth layer. Additionally, the variables \({\textbf {h}}_{c,u}^{(l)}\) and \({\textbf {h}}_{c,v}^{(l)}\) represent the lth layer representations of u, v; \({\textbf {h}}_{m,uv}\) and \({\textbf {h}}_{c,uv}\) represent the message and hidden representation of their edge. It is important to note that for the initial layer, we have \({\textbf {h}}_{c,u}^{(0)}={\textbf {F}}_u \), \({\textbf {h}}_{c,v}^{(0)}={\textbf {F}}_v\), and each layer of \({\textbf {h}}_{m,uv}\) can be generated as follows:
where timestamp t indicates the occurrence of the current connection, allowing the model to leverage user activity information.
After the above adjustments, corresponding adjustments also need to be made to the mapping function for different types of vertices. For example, the process of mapping u is as follows:
where \({\textbf {W}}^{(l)}_{\eta , u}\), \({\textbf {W}}^{(l)}_{\eta , v}\), \({\textbf {W}}^{(l)}_{c}\), \({\textbf {W}}^{(l)}_{g}\) are weight matrices and \({\textbf {b}}_{\eta ,u}^{(l)}\), \({\textbf {b}}_{\eta ,v}^{(l)}\) are bias vectors, \(\textrm{LayerNorm}(\cdot )\) is layer normalization function.
Given two visitor vertices u and \(u'\), their memories are processed through GAT, and the final output of the last layer \({\textbf {h}}_{u}^{(L)}\) and \({\textbf {h}}_{u'}^{(L)}\) can be used to predict their relationship with the help of the following module:
where \({\textbf {W}}_{p_1}\), \({\textbf {W}}_{p_2}\), \({\textbf {W}}_{y}\) are weight matrices and \({\textbf {b}}_{p_1}\), \({\textbf {b}}_{p_2}\), \({\textbf {b}}_{y}\) are bias vectors, \(\hat{y}_{u, u'} \in [0, 1]\) is the probability that \(u'\) follows u predicted by FPGN.
3.5 Training objective
The follower prediction problem can be approached as a binary classification task. As such, the model can be trained using binary crossentropy as the loss function. The binary crossentropy loss function is commonly used in machine learning to train models for binary classification tasks. It measures the difference between the predicted and ground truth by using the logarithmic loss. The formula for it can be written as follows:
where \(D_{train}\) is the training set, \(D_{train}\) is the size of training set, \(y_{u, u'}\) represents the ground truth indicating whether \(u'\) is a follower of u. Specifically, if the result is positive, then \(y_{u, u'} = 1\); otherwise, it equals 0. \(\hat{y}_{u,u'}\) refers to the predicted probability of a positive outcome. \(y_{u,u'}\log \hat{y}_{u,u'}\) heavily penalizes misclassification of positive instances by the model. Similarly, \((1y_{u,u2})\log (1\hat{y}_{u11_u})\) penalizes misclassification of negative instances. Moreover, it is evident that the loss function necessitates the predicted values to be within the range of 0 and 1, denoted as \(0 \le \hat{y}_{u, u'} \le 1\). As a result, before feeding into this loss function, the predicted outcomes will undergo a Sigmoid transformation.
Furthermore, as explained in Section 3.2.2, time interval statistics are also important information that helps the model solve the follower prediction problem. Therefore, FPGN also uses the following functions during training:
where \({\textbf {W}}_{s_1}\), \({\textbf {W}}_{s_2}\), \({\textbf {W}}_{s_3}\) are weight matrices and \({\textbf {b}}_{s_1}\), \({\textbf {b}}_{s_2}\), \({\textbf {b}}_{s_3}\) are bias vectors, \(\cdot ^2_2\) representing L2 norm.
The final loss function can be represented as:
where \(\lambda \) is a hyperparameter used to adjust the weights of two sets of loss functions. Since \(\mathcal {L}_m\) is the main loss function of FPGN, while \(\mathcal {L}_c\) mainly plays a supporting role, therefore in actual training, \(\lambda \) will be adjusted to a relatively small value.
4 Experiment
This section first introduces the experimental setup, then explains the method of generating the dataset, followed by an introduction to the baselines compared with FPGN, and finally presents a large number of experimental results. These results demonstrate the excellent performance of FPGN. In addition, this section also shows the results of ablation experiments, proving that each part of FPGN plays a positive role in improving model accuracy.
4.1 Experimental setup
In all precision experiments, the hyperparameters of FPGN remain the same. Among them, epoch is 340, learning rate is \(1 \times 10^{4}\), batch size is \(1 \times 10^{4}\), the number of selected latest initial memory sets in \({\textbf {X}}_u^{init}(t)\) is \(\mathcal {K}=10\), the number of heads in GAT is \(C=2\), vertex ID embedding dimension is 32, \(\alpha _{\tau }^{} = \alpha _{\tau }^{+} = 2\), \(\beta _{\tau }^{} = 1\), \(\beta _{\tau }^{+} = 20\), all module hidden layer dimensions are 64, loss function weight \(\lambda =0.05\) and Adam optimizer is chosen. The algorithm we choose to implement (\(\alpha \), \(\beta \))core is proposed by work [23]. In order to improve the efficiency of the model, we use the source code provided by the authors ^{Footnote 1}, and embed it into Python code using SWIG ^{Footnote 2}. The experiment is conducted on a server with two Intel(R) Xeon(R) Silver 4208 CPUs, one NVIDIA RTX 6000, and 64GB of memory.
In our experiment, we utilize two widely adopted evaluation metrics of edge prediction on temporal bipartite graphs, namely Average Precision (AP) and AUC, as the metrics of the follower prediction problem. The definitions of these metrics are as follows:
4.1.1 Dataset
To the best of our knowledge, there is currently no opensource dataset available for predicting followers. To address this gap, we utilized the widelyused temporal bipartite graph datasets named Wikipedia and Reddit [16, 21, 33, 47] to create our own dataset. The statistical details of these two datasets are presented in Table 1.
In the follower prediction problem, we consider all connections to only have the meaning of “visit,” so the dataset we use does not include edge features. We use \(80\%\) of the data in each dataset as the training set and the remaining \(20\%\) as the test set. Additionally, \(20\%\) of the training set is treated as a validation set. It should be noted that in order to ensure that the model does not test visitors who have not seen before, we remove visitors that only appear in the test set.
To generate ground truth, we set a fixed ratio time window. Specifically, assuming the time span of the dataset is \(\Gamma \), two visitors u and \(u'\), and a location v, we consider u and \(u'\) are leader and follower when \(\Delta (u, u', v) \le \Gamma \times 10^{4}\). The negative samples are generated from samples other than the positive samples, and the number of negative samples is twice that of the positive samples. In the Wikipedia dataset, the links represent users’ edits to Wikipedia pages. The obtained Wikidata dataset can help FPGN predict those who follow other editors in editing Wikipedia pages. Reddit contains numerous user interaction information in numerous posts. The Reddit dataset obtained using the above method allows FPGN to predict followers who engage in reply conversations with specific users.
4.1.2 Compared methods
To the best of our knowledge, there is currently no model specifically optimized for the follower prediction problem. Therefore, we use six widely used GNN models: GCN [19], GAT [42], GE [53], SAGE [15], GNNFiLM [4] and GIN [49] as baselines, where GE is the general GNN layer adapted from [53]. All of these algorithms have the structure shown in (2), but differ in the way they aggregate and update the information.
In addition, since the edge prediction problem on temporal graphs is similar to the follower prediction problem, we also compare with stateoftheart algorithms JODIE [20] and NeurTW [16]. JODIE employs two recurrent neural networks to update a visitor’s and a location’s embedding at every interaction. In addition, this model has a novel projection operator that learns to estimate the embedding of the visitor at any time in the future. And to make the method scalable, a tBatch algorithm is developed that creates timeconsistent batches and leads to faster training. NeurTW conduct spatiotemporalbiased random walks to retrieve a set of representative motifs, enabling temporal vertices to be characterized effectively. With a component based on neural ordinary differential equations, the extracted motifs allow for irregularly sampled temporal vertices to be embedded explicitly over multiple different interaction time intervals. To enrich the supervision signals, a harder contrastive pretext task is designed for model optimization. It should be noted that TGN is also a stateoftheart algorithm for solving edge prediction problems on temporal graphs. However, as explained in Section 1, TGN cannot be directly applied to the follower prediction problem. Therefore, we do not compare TGN in accuracy evaluation experiments. Instead, we analyze the effectiveness of optimizing TGN for the follower prediction problem in ablation experiments and show that TGN does not outperform FPGN on this task.
4.2 Prediction accuracy evaluation
Table 2 shows the results of the accuracy experiments. From the table, we observe that FPGN outperforms all baseline models in terms of both AP and AUC on both datasets. Specifically, the AP of FPGN is 0.4620 on Wikipedia and 0.5457 on Reddit, which are the highest scores for both datasets. FIL achieves the secondhighest AP score of 0.4180 on Wikipedia, while GIN obtains the secondhighest AP score of 0.3717 on Reddit. In contrast, FPGN was able to improve by \(10.53\%\) on Wikipedia based on FIL, while on Reddit, it improved by \(46.81\%\) based on GIN. Similarly, on Wikipedia and Reddit datasets, FPGN outperforms other models with the highest AUC scores of 0.6250 and 0.7305, respectively. The secondbest AUC score on Wikipedia is achieved by FIL at 0.5646, while GE obtains the secondbest AUC score on Reddit at 0.5403. Compared to them, FPGN respectively increased by \(10.70\%\) and \(35.20\%\) based on their performance.
These results indicate that, on the one hand, because these six general GNN baseline models (i.e., GCN, GAT, GE, GraphSAGE, FIL and GIN) are designed for general graph structures, they do not perform as well as FPGN, which is specifically optimized for the follower prediction problem. This demonstrates the importance of problemspecific optimization in graphbased models. On the other hand, compared with two stateoftheart temporal graph edge prediction models (i.e., JODIE and NeurTW), FPGN still achieves better performance. Due to the heavy emphasis on temporal information processing in these two baselines, their efficiency is very low. In experiments conducted on the Reddit dataset, both JODIE and NeurTW were unable to obtain experimental results within 48 hours. In addition, because they are unable to fully utilize the time window information during follower and leader activity processes, they are more easily misled and achieve worse results than general GNN models. More specifically, the architecture of JODIE and NeurTW makes them more suitable for utilizing historical information to predict whether a particular event will occur. However, when it comes to predicting the time difference of future events, their method of handling timestamp information has a negative effect on prediction.
In summary, the proposed FPGN model is highly effective in predicting followers on both datasets. It outperforms six general GNN models and two stateoftheart temporal graph edge prediction models. These results demonstrate the exceptional performance of FPGN on this task.
4.3 Ablation experiment
In order to better achieve the follower prediction task, FPGN strengthened the utilization of time interval statistics information and graph structure information. To demonstrate the advanced nature of these optimizations, we conducted ablation experiments. Specifically, we compared FPGN with only improved TGN (referred to as FGN), FPGN without graph structure information (referred to as FPGN w/o S) and FPGN without time interval statistics information (referred to as FPGN w/o T). The experimental results on Wikipedia are shown in Figure 3.
After analyzing the experimental results, it can be found that both the AP and AUC metrics show that FGN is the worst, and FPGN is the best. This is because TGN can only make good use of the updated information of each vertex and fully consider the information at different times for the same vertex, but cannot effectively utilize time information between two different vertices. Therefore, although FGN performs better in follower prediction compared to JODIE and NeurTW, its performance is still worse than FPGN as well as FPGN w/o S and FPGN w/o T.
By comparing the effects of FPGN w/o S and FPGN w/o T, it can be found that the accuracy of FPGN w/o T is higher than that of FPGN w/o S. This is because (\(\alpha \), \(\beta \))core can extract the activity information of each vertex very well. There is no doubt that more active vertices are more likely to exhibit the following behavior. Therefore, the information extracted by (\(\alpha \), \(\beta \))core can better help the model predict future following behavior.
5 Related works
Graph Neural Networks
Graph Neural Networks (GNNs) have gained significant attention in recent years due to their ability to represent and learn from graphs and other information [4, 17, 22, 25, 48]. One of the earliest works in the development of GNNs is the Graph Convolutional Network (GCN) [19]. GCN is a type of neural network that operates directly on graphs, allowing it to learn representations that capture the underlying structure of the graph. Another notable work is Graph Attention Networks (GAT) [42]. GAT uses attention mechanisms to weigh the importance of each vertex’s neighbors, allowing the network to focus on the most relevant information for each vertex. Other related works in the field of GNNs include GraphSAGE [15], GIN [49], etc. These approaches have been used in various applications such as social network analysis, protein function prediction, and knowledge graph completion [43]. In addition, there are some GNN models that have been specifically optimized for bipartite graph problems [54, 56, 57]. However, these methods cannot utilize temporal information and are therefore not suitable for application on temporal bipartite graphs.
GNNs on Temporal Graphs
In recent years, graph neural network models specifically designed for temporal graphs have been continuously emerging. Early models for temporal graphs focused on DTDGs. There are two encoding methods for DTDGs: one is to aggregate graph snapshots and apply static methods [13, 34, 36, 59], and the other is to modify the behavior of subsequent snapshots through random walks [12, 29, 46]. Based on CTDGs, several methods have recently been proposed for temporal graphs, including random walk models [2, 30] and sequencebased methods [20, 28, 39]. What’s more, many temporal graph architectures focus on updating vertexwise memory when new interactions occur [3, 20, 33, 40, 47]. These algorithms can fully utilize the connectivity and temporal information and perform better. However, there is great confusion in predicting follower problems for these methods because it is difficult to distinguish the order of vertex activity and time difference. This makes these algorithms even misled by temporal information when making follower predictions, resulting in low prediction accuracy.
Cohesive Subgraph Mining
Many algorithms are now available for finding dense subgraphs, such as clique [8, 9], quasiclique [41], kcore [7, 18], and ktruss [11, 44], etc. At the same time, there are also many methods specifically designed to search for dense subgraphs on bipartite graphs, such as biclique [1, 58], kbitruss [45], bitriangle [52], (\(\alpha \), \(\beta \))core [23], \(\delta \)quasibiclique [24] and kbiplex [55], etc.
These algorithms cannot utilize temporal information or directly implement edge classification, so they cannot be directly used to solve the follower prediction problem. However, followers often exist in denser subgraphs, so these algorithms can provide good auxiliary support for solving this problem.
6 Conclusion
This paper introduces a novel approach to solving the problem of follower prediction in temporal bipartite graphs using a model called Follower Prediction Graph Network (FPGN). The model is designed to accurately predict highrisk visitors by tracking their activities in the same location as infected visitors. FPGN is inspired by the stateoftheart temporal graph edge prediction algorithm TGN and addresses its shortcomings. The high accuracy of the FPGN is attributed to its welldesigned approach to addressing this problem, including making full use of time windows and global structure information. The results of extensive experiments conducted on two real datasets demonstrate the effectiveness of FPGN. In conclusion, this work represents a significant advancement in the problem of follower prediction and provides a new and effective approach to solving this challenging problem.
Availability of data and materials
The code and data are available at https://github.com/yujianke100/FPGN.
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Jianke Yu, Xianhang Zhang, Hanchen Wang, and Xiaoyang Wang wrote the main manuscript text. Wenjie Zhang and Ying Zhang provided key ideas of the proposed model. All authors reviewed the manuscript.
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Yu, J., Zhang, X., Wang, H. et al. FPGN: follower prediction framework for infectious disease prevention. World Wide Web 26, 3795–3814 (2023). https://doi.org/10.1007/s11280023012058
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DOI: https://doi.org/10.1007/s11280023012058