Deep learning is widely used in areas such as image recognition, natural language processing, and recommendation systems with its powerful learning and representation capabilities [17,18]. However, its susceptibility to learn spurious relations affected by confounding variables has been plaguing researchers in the field of deep learning [19]. Since causal inference [20] can control the impact of confounders and evaluate the causal effect of feature variables on predicted targets by do-calculus, i.e., P(Y|do(X)). Recently, there is an increasing number of studies introducing causal inference into deep learning models to implement causal deep learning [11–16]. For example, Yang et al. [11] improved the attention mechanism based on the front-door criteria of causal inference to help visual language models control the influence of confounding association. Liu et al. [13] developed the CMCIR framework, which leverages causal reasoning to learn cross-modal information, enhancing the framework’s ability to understand the causal, logical, and spatiotemporal dynamics between video and language content. This approach effectively addresses event-level visual question answering tasks that demand complex reasoning. Some studies [14,15] have incorporated causal inference to assess the impact of item popularity on recommendations within recommender systems. These frameworks help to reduce popularity bias, ensuring that recommendations rely solely on user and product characteristics.

Predicting reposts plays a key role in opinion analysis and recommendation systems [4,21]. The majority of repost prediction methods concentrate on using machine learning models to identify and learn relevant features. For instance, Jiang et al. [22] extended the probabilistic matrix factorization method by introducing two additional matrices: a social influence matrix based on social network structure and interaction history, and a message similarity matrix based on document semantics. By optimizing the latent feature space of users and messages, the model separately learns social influence and message semantic features, thereby enhancing its predictive performance on user reposting behavior. Safari et al. [23] designed a V-DBNC model based on a dynamic Bayesian network, which can predict users’ reposting behavior by learning user behavior patterns, reposting path structure, and social influence between users. In recent years, the development of deep learning technology has highlighted the significant advantages of neural networks over traditional machine learning methods in feature representation. Consequently, neural network-based models for predicting user reposting behavior have gained increasing popularity [5,7]. For example, Wang et al. [24] proposed a dual autoencoder model to capture the features of user identity information, social relationships, and group reposting factors. The model also incorporates an attention mechanism to extract topic representations from users’ historical reposting activities, ultimately predicting their behavior by learning these multi-dimensional features. Literature [25] introduced the GODEN model, which combines ordinary differential equations with graph neural networks to capture both dynamic user interactions and static user relationships. The model represents the reposting propagation pattern through user and time context, employing a multi-head attention module to focus on various contextual information and predict the next user likely to forward the content.

Existing repost prediction models often neglect confounding variables, which can result in capturing spurious associations between feature variables and user behavior. This limitation hampers the generalization ability of these models. To address this issue and achieve Debiased Repost Prediction (DRP), it is essential to combine repost prediction models with causal inference techniques. However, causal inference typically relies on prior knowledge of the causal relationships, which is challenging to obtain in the context of real-world applications like repost prediction. Correspondingly, due to the lack of prior knowledge of causality, we cannot predefine a confounding variable dictionary [11,13,26] to help the DRP model learn causal associations. Therefore, in this paper, we aim to empower the DRP model to discover variable causality and to learn causal associations without predefining a confounder dictionary.

Recent causal discovery algorithms [27–29] are capable of identifying the structure of a causal Bayesian network from datasets containing n observations and p variables. In this context, a causal Bayesian network is represented as a Directed Acyclic Graph (DAG) with p nodes, where each node corresponds to a variable, and directed edges indicate causal relationships between variables. These algorithms aim to determine the causal Bayesian network that best fits the data, while adhering to the structural constraints of a DAG. We can formulate it as: (1)minℒ=ℒrec+αℒDAG+βℒs,where the term ℒrec measures the discrepancy between the observed distribution of variables and the distribution of variables generated by the causal Bayesian network. This term reflects the causal Bayesian network’s ability to explain observable data. The term ℒDAG quantifies the difference between the causal Bayesian network structure and the DAG structure, ensuring that no causal loop such as X→Y→Z→X occurs within the causal Bayesian network. The DAG structure avoids paradoxical situations where X causes itself since X cannot be its own cause. Finally, the L1 regularization term of the causal Bayesian network parameters, denoted as ℒs, is used to enforce sparsity in the graph, making it easier to interpret the causal graph.

Deep learning models typically generate predictions by identifying associations between feature X and label Y, i.e., likelihood P(Y|X). However, when X and Y have a common ancestor Z (confounding variable, as shown in Fig. 2a), learning P(Y|X) will lead the model to learn spurious relationships [20]. Because

(2)P(Y|X)=∑zP(Y|X,z)P(z|X),the confounder Z introduces the bias through P(z|X). For instance, suppose that in the repost prediction scenario, P (z = public media | X = anime) is greater than P (z = friends | X = anime). In that case, most of the likelihood in Eq. (2) will arise from P (Y = repost | X = anime, z = public media) instead of P (Y = repost | X = anime, z = friends). When predicting a user’s repost behavior for posts about anime, the model tends to prioritize posts shared by the public media over the content of the post itself.

Therefore, recent deep learning models [11,19] implement causal learning by leveraging the do-calculus to block the causal relationship between Z and X, as shown in Fig. 2b. These models make predictions based on the causal association P(Y|do(X)) between X and Y, where (3)P(Y|do(X))=∑zP(Y|X,z)P(z).

Compared to Eq. (2), z is no longer affected by X, and thus the do-calculus deliberately forces X to incorporate every z fairly, subject to its prior P(z), into the prediction of Y.

In the context of the repost prediction, addressing the impact of confounding variables is equally crucial. Employing causal deep learning techniques enables these models to discern causal associations and enhances their generalization capabilities. However, two substantial challenges are encountered:

To perform causal deep learning (as illustrated in Fig. 2), a model needs to initially identify the confounding variable Z. However, in the context of repost prediction involving multiple variables such as social relationships (S), user interactions (D), user topic interests (I), and query posts (T), obtaining complete prior knowledge of the causal relationships between these variables poses a significant challenge. Depending solely on expert knowledge makes it difficult to ascertain the causal relationships between variables. For instance, in tasks such as Visual Question Answering (VQA), diverse experts may hold varying interpretations of the causal links between variables [11,13].

The repost prediction task can be seen as a classification task [5]. Given a query post T and a user u, the repost prediction task is to predict whether u will repost based on T and the microblogging data. These data include (1) past posts reposted by users I={t1,…,tl}; (2) the following network F={UF,EF}, where UF is the set of users and EF is the set of following relationships of users; (3) the temporal repost interaction network D={D1,…,Dd}, Dm=⟨Um,Em,dm⟩, where Um and Em are the sets of users and interaction relations in time slice dm, 1≤m≤d, respectively. To improve the generalization ability of the repost prediction model, we combine it with causal inference to control the impact of confounding variables to achieve Debiased Repost Prediction (DRP). However, as it is difficult for us to have complete prior knowledge of the causal relationships of variables, we introduce Causal Discovery methods into DRP model, i.e., CD-DRP model (see Fig. 3). The proposed CD-DRP model can achieve repost prediction by observing microblogging data without making assumptions about the causal relationships between variables.

The CD-DRP model diverges fundamentally from existing causal deep learning models, such as those presented in [11,19]. Firstly, the CD-DRP model operates independently of expert knowledge on causal relationships, autonomously discovering variable causality through the observation of microblogging data. This autonomy allows the model to adeptly handle repost prediction scenarios involving multiple variables and mitigate bias stemming from expert knowledge. Secondly, in the process of learning causal associations, the CD-DRP model obviates the need for manually constructing confounder dictionaries. This characteristic provides effective support for debiased repost prediction.

To facilitate the proposed CD-DRP model to learn microblogging data, we first map T, I, F, and D to a multidimensional feature space. Specifically, T and I composed of text use the pre-trained language model BERT [30] as the embedding method. If a query post is tq, its feature representation xt=BERT(tq), xt∈Rdt. dt is the dimension of the BERT feature. If the repost record of user u is Iu={t1u,…,tlu}, the feature of Iu is represented as Xu={x1u,…,xlu}, xju=BERT(tju), 1≤j≤l, xju∈Rdt. We perform an average pooling operation on Xu to obtain the feature representation xi of the user interests, xi=AvgPooling(Xu), xi∈Rdt. For the graph structure data F and D, we use node2vec [31] as the embedding method. The feature of F is represented as XF=node2vec(F), XF∈Rn×dg, n is the number of users, and dg is the node2vec feature dimension. Each row of XF represents the features xu of a user in the following network. Each time slice of repost interaction network D is also embedded with node2vec, XD={XD1,…,XDd}, XDm=node2vec(Dm), XD∈Rd×nm×dg, and nm is the number of users of time slice m. Take the features of user u in different time slices of the repost interaction network and do the averaging pooling operation to get the representation xd of the user interaction preference, xd=AvgPooling(xDu), xd∈Rdg,xDu={xD1u,…,xDdu}.

The aim of the causal discovery (CD) module is to identify the causal graph with the most explanatory power for the microblogging data and assist the debiased repost prediction module in identifying confounding variables. As variables like T, I, F, and D are multidimensional, discovering the causal graph through a single linear model or MLP, as in recent work [27,28], is not feasible. Therefore, based on the paradigm described in Section 3.1, we develop a deep generative network that can uncover the causal graph for multidimensional feature variables. This network can gradually minimize the difference between the observed and conditional expectation E(X|Parent(X)) of the multidimensional variable X by modifying its parameters. Specifically, the causal discovery module comprises a learnable parameter matrix (causal graph) 𝒜 and multiple parallel MLPs for generation (as illustrated in Fig. 3), where (4)𝒜=[0wtiwtfwtdwtywit0wifwidwiywftwfi0wfdwfywdtwdiwdf0wdywytwyiwyfwyd0],the parameters in the columns of matrix 𝒜 represent the weight of the incoming edges or the influence of other variables on a given variable. For instance, the first column of 𝒜, denoted as 𝒜:𝒯=[0,wit,wft,wdt,wyt], represents the edges pointing to node T from nodes T, I, F, D, and Y. The main diagonal elements of 𝒜 are fixed at 0 to indicate that each variable cannot be self-explained. The parental information of any variable X expresses as Parent(X)=𝒜:𝒳𝒯⋅ℳ, where ℳ=[T,I,F,D,Y]. Given the parental information, the conditional expectation of X is (5)X~=E(X|Parent(X)) =MLP(A:XT⋅ℳ)=σ(W(h)σ(…σ(W(1)A:XT⋅ℳ))),where σ is the activation function, e.g., ReLU, and W(k) is the parameter of the k-th perceptron layer, 1≤k≤h. According to Eq. (5), we can obtain the conditional expectations of different variables from a number of MLPs. The generated microblogging data can be represented as ℳ~=[T~,I~,F~,D~,Y~].

During training, the causal discovery module continually optimizes the parameters of both 𝒜 and MLPs to minimize the reconstruction loss of the microblogging data, (6)ℒrec=RMSE(ℳ,ℳ~),ultimately achieves the goal of uncovering the causal relationships between variables and understanding the mechanisms that drive the generation of data. Here, RMSE is the root mean square error, which is used to measure the discrepancy between ℳ and ℳ~. In addition, in order to ensure 𝒜 is a sparse DAG, the causal discovery module also imposes the following constraints: (7)ℒDAG=tr(e𝒜⋅𝒜)−d, (8)ℒs=|𝒜|1,where ℒDAG is proposed by [27], tr(⋅) denotes the trace of the matrix, e𝒜⋅𝒜 is the matrix exponential of Hadamard product of 𝒜, d is the number of elements of the main diagonal of 𝒜, and |⋅|1 is the L1 parametrization. Therefore, the objective of the causal discovery module is (9)ℒCD=ℒrec+αℒDAG+βℒs =RMSE(ℳ,ℳ~)+α(tr(e𝒜⋅𝒜)−d)+β|𝒜|1.

Based on the discovered causal graph, we can identify confounders that affect the generalizability of the repost prediction model, i.e., variables that affect both predictor X and label Y such as Z in Fig. 2a. To control this influence, we devise the Debiased Repost Prediction (DRP) module, which introduces causal inference to the repost prediction model as mentioned in Section 3.2. For the predictor X affected by confounders, given its feature x and user repost behavior Y=y, we can write the causal association between X and Y as ∑zP(y|x,z)P(z). Here, the conditional probability P(y|x,z) can be parameterized by a Softmax layer, i.e., P(y|x,z)=Softmax(f(x,z)), where f(⋅) is a function that calculates the probability of various user behaviors. ∑z⋅P(z) is the expectation based on the prior distribution of z. Therefore, we implement DRP as (10)P(Y|do(X))=Ez[Softmax(f(x,z))].

However, Ez requires expensive sampling, which can significantly affect the training efficiency. Therefore, we adopt the scheme of [11,26] and use the Normalized Weighted Geometric Mean (NWGM) [32] to approximate this expectation, i.e., (11)Ez[Softmax(f(x,z))]≈Softmax(Ez[f(x,z)]).

By setting f(⋅) to a linear model, f(x,z)=Wxx+Wzz, Eq. (10) can be written as: (12)P(Y|do(X))=Softmax(Wxx+Wz⋅Ez(z)),where Wx and Wz are linear neural network parameters. As soon as Ez(z) is obtained, we can evaluate the causal association between X and Y according to Eq. (12). For a predictor Xn that is not affected by confounding variables, given Xn=xn, the causal association between Xn and Y is P(y|xn), implemented as: (13)P(Y|Xn)=Softmax(Wnxn),with Wn is the parameter of the linear neural network. Based on the causal association of different predictors with user behavior Y, the DRP module makes the prediction (as shown in Fig. 3): (14)y~=Softmax(Wxx+Wz⋅Ez(z))+Softmax(Wnxn).

During training, the DRP module takes the cross entropy ℒp between the predicted value y~ and the label y as the objective function, (15)ℒp=−∑yylog⁡(y~),to optimize the parameters and improve its prediction performance.

It should be noted that CD-DRP lacks prior knowledge of the causal relationships and therefore cannot identify Z before training to construct a confounder dictionary that represents P(z), as demonstrated in recent studies [11,13,26]. Instead, CD-DRP needs to estimate P(z) through training, utilizing information on confounders that are scattered across different data batches. It presents a challenge in computing Ez(z). To overcome this issue, we devise a novel confounder memory network. This network is parameterized as a matrix 𝒞, 𝒞∈Rs×dc. While training, the DRP module retrieves 𝒞’s memory z~ for different data batches of confounder z, and continuously reduces the reconstruction loss of z~ and z to optimize 𝒞’s parameters. In this way, 𝒞 is able to perceive P(z). Specifically, when retrieving 𝒞’s memory of z, the DRP module first calculates the association of z to each piece of information stored by 𝒞: (16)αz=Softmax(zT𝒞/dc),the information about z is z′=αz𝒞. Subsequently, DRP employs multiple self-attention [33] layers to represent the memory z~. (17)z~=Self−attention(N)(…Self-attention(1)(αz𝒞)).

The reconstruction loss of z is the root mean square error: (18)ℒzrec=RMSE(z,z~).

After multiple rounds of training, 𝒞 progressively saves information on confounding variables in different data batches. Finally, DRP performs an average pooling operation on 𝒞 to calculate Ez(z): (19)Ez(z)=AvgPooling(𝒞).

To summarize, the DRP module is designed to achieve two objectives: to perceive the distribution of confounding variables and to predict user behavior. The objective for the module is denoted as ℒDRP. (20)ℒDRP=ℒp+ℒzrec    =−∑yylog⁡(y~)+RMSE(z,z~).

The descriptions of different modules of the proposed CD-DRP model in Sections 4.2–4.4 are summarized here as Algorithm 1. Specifically, the CD-DRP model first uses BERT [30] and node2vec [31] to map query posts T, user repost records I, following network F, and repost interaction network D to a multidimensional feature space (Section 4.2). Then, the CD-DRP model identifies the parents of different variables based on the causal graph 𝒜 and generates different variables according to Eq. (5). By optimizing Eq. (9), the CD-DRP model is able to update the parameters of the causal discovery module and find the causal graph 𝒜 with the best explanatory power for the microblogging data (Section 4.3). Finally, the CD-DRP model identifies the confounder Z with the help of 𝒜. The distribution of Z in the training set is perceived by Eqs. (16) and (17). By optimizing the loss calculated by Eq. (20), CD-DRP is able to make prediction with the help of Eq. (14) (Section 4.4).

Datasets The proposed CD-DRP model is assessed through experiments using the Twitter-Dynamic-Net (TDN)¹1

https://www.aminer.cn/data-sna#Twitter-Dynamic-Net (accessed on 13 November 2024).

Baselines We conduct a comparative analysis of the proposed method against various baseline models, encompassing random models, those with simple structures (such as LR, SUA-ACNN [34], GraphSAGE [35], and DynamicGCN [36]), and those with complex structures (including AMNL [37], DFMF [38], GFCI [39], and GCRec [15]):

Random: The prediction of reposting behavior, whether it occurs or not, is randomly determined in this model, which establishes the distribution of positive and negative instances and serves as the baseline. Any model capable of learning from data to predict reposting behavior is expected to outperform this Random model.

Metrics The experiments use the popular Accuracy (A) and F1-score (F) to quantify the repost prediction performance of the proposed model and baselines. The definition of A and F are as follows: (21)A=tp+tntp+fn+tn+fp,

(22)F=2×P×RP+RP=tptp+fpR=tptp+fn,where tp is the number of true positive, tn is the number of true negative, fp the number of false positive, and fn is the number of false negative.

We implement the proposed model and all baseline models (excluding LR) using PyTorch, while LR is implemented with scikit-learn’s built-in function. The pretrained BERT parameters are obtained from the links provided in the footnotes³3

Parameters from https://huggingface.co/roberta-base for TDN (accessed on 13 November 2024)

Generalizability is important for repost prediction models because the goal of these models is to be able to accurately predict users’ repost behavior for new, unseen posts. A model with poor generalizability may suffer from overfitting problems, i.e., it performs well on training data but fails to make accurate predictions on testing data. Therefore, we analyze the generalizability of the proposed CD-DRP model and baselines in terms of both prediction performance and overfitting degree.

Prediction Comparison Table 2 records the prediction performance of the proposed CD-DRP model and baselines on the testing set of six sample datasets. From Table 2, we can find that:

Each of the simple baseline methods demonstrates superior performance compared to the Random model. This observation indicates that the inclusion of the query post, user repost records, following networks, and repost interactions as features is advantageous for predicting user behavior. Notably, GraphSAGE and DynamicGCN exhibit comparable performance, with a marginal difference in average test Accuracy of less than 0.7%. Moreover, both GraphSAGE and DynamicGCN outperform LR and SUA-ACNN in the majority of cases.

Overfitting Comparison As described in Section 5.2, we utilize the F1-score as an early stopping metric during training and assess the model’s repost prediction ability on the training set. To evaluate the degree of overfitting, we examine the difference, denoted as ΔF, between the F1-score of the model on the training and testing sets. A smaller value of ΔF indicates a closer alignment between the model’s prediction performance on the training and testing sets, thereby suggesting a lower degree of overfitting. Since generalizability encompasses multiple aspects, we aim for a balance between predictive performance and overfitting. Ideally, a model with good generalizability should exhibit strong prediction performance while minimizing overfitting.

In this study, we compare the degree of overfitting of the proposed CD-DRP model and the 4 best baselines in terms of prediction performance (shown in Fig. 4). As can be seen from Fig. 4, these models have an F1-score distributed between 0.96 and 1.00 at the end of the training. They all fit the training data well. When combined with their F1-score on the testing set recorded in Table 2, we can see that CD-DRP model has the smallest ΔF on most of the dataset samples (except WNT Sample 3). Specifically, the overfitting of CD-DRP model was reduced by 7.44%, 9.18%, 11.96%, and 16.84% compared to GCRec, AMNL, DFMF, and GFCI, respectively. This result shows that the proposed CD-DRP model combined with causal discovery and inference can provide a more realistic understanding of microblogging data and alleviate the overfitting problem.

This section presents the causal graphs identified by CD-DRP model in Fig. 5. It can be observed that query posts T, user interests I, following relations F, and interaction relations D are the parents of the repost behavior Y, implying that these variables contribute to predicting repost behavior. This finding is reinforced by the results presented in Table 3. For instance, the simple structural model DynamicGCN, which uses information from D, outperforms Random across various datasets. Meanwhile, the following relationships F serve as the parent node for both query posts T and user behaviors Y. This means that the posts a user sees and his repost behavior are influenced by the following relationship, which aligns with previous research findings [40,41]. However, we also observe a discrepancy in the causal graphs identified between the TDN and WNT datasets. Specifically, in the TDN dataset, the variable I serves as the parent node of D (I→D), while in the WNT dataset, I functions as the parent node of T (I→T). This phenomenon might be influenced by disparities between the X and Sina Weibo platforms. For example, Li et al. [42] found that users on platform X tend to engage in interactions with sports organizations, whereas Sina Weibo users prefer collecting information and expressing support for sports teams.

To validate the causal relationships that exhibit differences between the TDN and WNT datasets, this section further conducts a causal analysis of these variations. We employ the method adopted by [43,44] to validate the causal relationships I→D depicted in Fig. 5a and I→T illustrated in Fig. 5b. Specifically, we leverage Permutation Importance (PI) for causal validation. The directions of causal relationships I→D and I→T are permuted. We train the CD-DRP model to capture variable associations by adapting to the modified causal graphs. The performance of the updated model CD-DRP^m is then evaluated in terms of prediction Accuracy and F1-score. A decrease in model prediction performance following the modification suggests the correctness of the original I→D and I→T relationships. Conversely, a performance improvement implies that the initial I→D and I→T relationships were incorrect. Table 3 provides a clear comparison of performance metrics, indicating that the CD-DRP^m model consistently exhibits lower performance across all sampled datasets when compared to the CD-DRP model. This observation underscores the significance of the causal relationship where I serves as the parent node of D in the TDN dataset and as the parent node of T in the WNT dataset. Understanding these causal connections is crucial for enhancing the model’s comprehension of microblogging data, leading to improved predictions of user retweet behavior.

As detailed in Section 1, our primary objective is to safeguard the repost prediction model from being influenced by confounding factors, which could otherwise result in the learning of spurious relationships between features and user repost behaviors. To achieve this, we introduce causal discovery and inference methods into the repost prediction model, referred to as CD-DRP. The CD-DRP model comprises two key modules: a causal discovery module responsible for identifying causal relationships among variables, and a prediction module designed to mitigate the impact of confounding variables. Additionally, we incorporate a confounder memory network to enhance CD-DRP’s ability to perceive the distribution of confounders.

In this study, we investigate the effects of these modules on the performance of CD-DRP model through an ablation study. Firstly, we randomly disrupt the causal relationships identified by the causal discovery module and employ the disrupted relationships to guide the prediction module in generating debiased repost predictions. This variant is denoted as CD-DRP (w/o Causal Discovery). Secondly, we omit the randomization of the causal discovery module’s output, but refrain from making debiased predictions. Instead, we directly input different features into Multi-Layer Perceptrons (MLPs) for prediction. This model is labeled as CD-DRP (w/o Debiased Prediction). Lastly, we eliminate the confounder memory network and simply employ the mean of the pretrained features of confounders for debiased repost prediction. This variant is referred to as CD-DRP (w/o Memory Network).

The outcomes of the ablation study are presented in Fig. 6. From the results, it becomes evident that CD-DRP (w/o Causal Discovery) exhibits the poorest prediction performance. In terms of F1-score, it performs 4.04% lower than CD-DRP and even 0.28% lower than CD-DRP (w/o Debiased Prediction), which does not account for the impact of confounding variables. These findings highlight the crucial role of the causal relationships identified by the causal discovery module in guiding the prediction module to identify confounding variables and improve the generalization ability of CD-DRP model. Furthermore, by comparing the performance of CD-DRP (w/o Debiased Prediction) and CD-DRP, it becomes evident that controlling the impact of confounding variables is essential for CD-DRP. Without such control, the F1-score of CD-DRP experiences an average decrease of 3.74% across the six datasets examined. Lastly, upon removing the confounder memory network, the Accuracy and F1-score of CD-DRP demonstrate varying degrees of reduction across different datasets, with the exception of TDN Sample 2. On average, CD-DRP (w/o Memory Network) exhibits 1.69% lower Accuracy and 1.90% lower F1-score compared to CD-DRP model. These findings emphasize the significance of the confounder memory network in assisting CD-DRP model in better perceiving the distribution of confounding variables.

The CD-DRP model contains some unlearnable hyperparameters. We choose the following aspects to analyze the influence of hyperparameter settings on its prediction performance: (1) the learning rate, (2) the number N of self-attention layers of the confounder memory network, and (3) the size of the confounder memory network. We change one parameter in turn and fix other parameters to test the prediction performance of CD-DRP model in TDN Sample 2 and WNT Sample 2, the experimental results are shown in Fig. 7. From Fig. 7, it can be seen that the learning rate has a large influence on CD-DRP model. It is suitable for CD-DRP model to be trained at a small learning rate. Meanwhile, combined with the results on TDN and WNT, we find that CD-DRP model is not sensitive to the number of self-attention layers of its memory network. This indicates that if the application requires higher time efficiency, we can appropriately reduce the number of self-attention layers to achieve the goal of improving the model efficiency without affecting the model performance. Finally, it can also be seen from the results that CD-DRP model with a memory network size setting of 768×768 performs slightly worse than other settings. Therefore, the memory network size of CD-DRP model should be selected as 512×768 or smaller.

The experiments also analyze the time efficiency of different models to provide a basis for comparison and selection for practical applications. Specifically, we test the time it takes for different models to complete a batch training on TDN Sample 2 (as shown in Table 4). Based on the records in Table 4, it can be seen that DFMF has the best time efficiency, followed by GraphSAGE, and CD-DRP₁ and AMNL are close to and at a good level. Therefore, combined with their prediction performance (see Table 2), we can give preference to DFMF in application scenarios that require high time efficiency. In the scenario that needs to balance time efficiency and prediction power, we can choose the CD-DRP with a simplified structure (reduce the number of self-attention layers), such as CD-DRP₁.

The above multi-dimensional quantitative analysis confirms that the CD-DRP model proposed in this paper demonstrates strong predictive performance and reduced overfitting. For instance, on the TDN sample dataset 2 (see Table 5), the CD-DRP model achieves a 1.87% improvement over the top-performing baseline, GCRec. Additionally, in terms of overfitting (measured by the F1-score difference between the end of training and testing), the CD-DRP model shows an 11.21% lower overfitting rate compared to the GCRec model. However, experimental results also reveal a gap between the CD-DRP model’s testing F1-score and its F1-score at the end of training, indicating some degree of overfitting. Several factors may contribute to this: (1) The CD-DRP model’s causal discovery paradigm (see Section 3.1) is data-driven, aiming to align the variable generation and observational distributions to uncover the causal structure most likely to represent the underlying variable generation mechanism. This may mean that the discovered causal diagram reflects only part of the true causal structure. (2) Test data distribution may differ from that of the training data. Since repost prediction requires a temporal component, training data is based on user activity from the past, while test data is drawn from later periods. Over time, online social network services can undergo changes influenced by various internal and external factors.

In prediction tasks using causal inference methods, confounding variable modeling typically relies on prior expert knowledge, with existing studies [11,15,26] identifying confounding variables by establishing a dictionary of these variables or pre-calculating their distributions before model training. However, in this study’s context, these solutions are challenging to implement due to the limited prior knowledge of relevant causal relationships. To address this, we design a confounder memory network to simulate pre-representation of confounder features. This network identifies confounding variables during causal inference and dynamically updates its memory of these variables across data batches. Given its flexibility, this network is also suitable for other data mining tasks, supporting prediction tasks where causal relationships are not pre-defined.