CST-Net: community-guided structural-temporal convolutional networks for popularity prediction

Feature-driven approaches

Great efforts have been made to investigate an extensive set of hand-crafted features for predicting the popularity of online contents, including tweets (Hong, Dan & Davison, 2011; Zhao et al., 2015), microblogs (Bao et al., 2013; Bian, Yang & Chua, 2014), videos (Pinto, Almeida & Goncalves, 2013; Chang et al., 2014; Ding et al., 2015), academic articles (Shen et al., 2014; Wang, Song & Barabási, 2012), to name a few. Most of the studies devote to identifying temporal properties (Szabo & Huberman, 2010; Pinto, Almeida & Goncalves, 2013), structural features of the cascade at the early stage (Bao et al., 2013; Cheng et al., 2014), and the content features (Hong, Dan & Davison, 2011; Ma, Sun & Cong, 2013) as the most predictive factors. A classic approach is to making predictions by applying regression models (Szabo & Huberman, 2010; Pinto, Almeida & Goncalves, 2013; Martin et al., 2016) or classification models (Shulman, Sharma & Cosley, 2016; Cheng et al., 2014). Szabo & Huberman (2010) found that the final popularity is reflected by the popularity in early period by investigating Digg and YouTube. A direct extrapolation method is then employed to predict the long-term popularity. Later, Cheng et al. (2014) discovered that temporal features are most predictive of a cascade’s final popularity. Besides, user features, particularly user’s influence in past are shown to be informative predictors (Bakshy et al., 2011; Feng et al., 2018). Recently, Martin et al. (2016) explored the limits of predictability in complex social systems, suggesting that even with unlimited data predictive performance would be bounded well below deterministic accuracy. Li et al. (2020) explored group-level features to predict the popularity of new Meetup groups. In general, these feature-driven approaches devote to verifying the effectiveness of a bag of hand-crafting features in order to learn a better prediction model. However, there are still numerous factors to be investigated and it is not flexible to apply them to a broad spectrum of data domains. Moreover, feature-driven approaches heavily rely on the quality of artificially designed features, which are hand-crafted based on human prior domain knowledge.

Generative models

The other line of enquiry, in contrast, treats the popularity dynamics as time series, making predictions by modeling the process through which individual items gain their attentions (Shen et al., 2014; Gao, Ma & Chen, 2015; Crane & Sornette, 2008; Lerman & Hogg, 2010). Crane & Sornette (2008) studied the relaxation response of a social system after endogenous and exogenous bursts of activity using the time series of daily views on YouTube, finding that most activity can be described accurately as a Poisson process. Subsequently, Shen et al. (2014) employed reinforced Poisson process to model the arriving process of article citations. Recently, Hawkes self-exciting process was employed to captures the triggering effect of each attention (Zhao et al., 2015; Bao et al., 2015; Mishra, Rizoiu & Xie, 2016; Bao & Zhang, 2017). Despite their initial success in leveraging fine-grained timing information in the event series, however they make stronger assumptions about the diffusion process. Mishra, Rizoiu & Xie (2016) proposed a marked Hawkes self-exciting point process, which intuitively aligns with the social factors responsible for diffusion of cascades: social influence of users, social memory and inherent content quality. Furthermore, Rizoiu et al. (2017, 2018) investigated the correlation between the endogenous response and the exogenous stimuli of a social system, and explored the connection between Hawkes point processes and SIR epidemic models. Xu et al. (2023) enabled integration of structural and temporal information in a diffusion process. Generative approaches learn the inherent characteristics of online content and propagation mechanism on different communication platforms, which achieve good interpretability. However, most generative approaches generally depend on specific assumptions that are unknown in reality so that their representation ability are relatively limited.

Deep learning methods

In order to deal with such complex popularity dynamics and automatically leverage predictive information from raw data, in recent year, there has been heightened research interest regarding the popularity prediction methods based on deep learning (Bourigault et al., 2014; Du et al., 2016; Li et al., 2017; Cao et al., 2017; Xiao et al., 2017; Zhao, Zhang & Feng, 2022; Bao, Yan & Yang, 2024). Du et al. (2016) viewed the intensity function of a temporal point process as a nonlinear function of the history, and employed a recurrent neural network (RNN) to automatically learn a representation of influences from the event history. Later, Xiao et al. (2017) modeled both background and history effect by two RNN, respectively. The recent successes of deep learning in a wide range of areas also inspires some end-to-end deep learning framework for popularity prediction. Li et al. (2017) presented DeepCas model that learn the representation of cascade graphs in an end-to-end manner, which significantly improve the performance of popularity prediction over strong baselines. Subsequently, Cao et al. (2017) proposed DeepHawkes model, which inherits the high interpretability of Hawkes process and possesses the high predictive power of deep learning methods, bridging the gap between prediction and understanding of information cascades. This line of works can automatically learn the valuable information from raw data. More recently, Zhao, Zhang & Feng (2022) utilized the temporal and structural information of cascade networks as input to predict the future growth of information cascades. Bao, Yan & Yang (2024) proposed a learning framework for popularity prediction via modeling both the temporal evolution in a separate snapshot and the inherent temporal dependencies among different snapshots based on the dynamic evolution process. However, existing works mainly focus on sampling sequences from cascade graphs and then feeding the sequences into recurrent neural networks. In this article, we take advantage of both the effective information demonstrated by traditional methods and the power of end-to-end deep learning framework for popularity prediction. Existing methods mostly utilize the order relationship between forwarding users to model the sequential information, while ignoring the impact of temporal features that mainly dominates the future popularity.

Problem definition

Let $\mathcal{M}$ denote a set of items, e.g., microblogs or academic articles. Suppose the set $\mathcal{M}$ contains M items, it is denoted by $\mathcal{M}=\left\{m^i\right\}(1\le i\le M)$.

Definition 1 (cascade) For each item $m^i$, a cascade is characterized by ${\mathcal{C}}^i=\left\{(u_k^i,v_k^i,t_k^i)\right\}$, where the tuple $(u_k^i,v_k^i,t_k^i)$ corresponds to the $k$-th forwarding, meaning that user $v_k^i$ forwards the item $m^i$ from user $u_k^i$ at time $t_k^i$.

Definition 2 (popularity prediction) Given the cascades in the observation time window $[0,T)$, we aim to predict the incremental popularity $\mathrm{\Delta }{N}_T^i$ between observed popularity $N_T^i$ and final popularity $N_{\mathrm{\infty }}^i$ of each cascade ${\mathcal{C}}^i$, where $\mathrm{\Delta }{N}_T^i$ is denoted as $\mathrm{\Delta }{N}_T^i=N_{\mathrm{\infty }}^i-N_T^i$.

In the definition, T indicates the earliness of the prediction and refers to the length of training period. It is worth noting that we predict the incremental popularity instead of the final popularity to avoid the intrinsic correlation between the observed popularity and the final popularity, which forms a more challenging scenario for popularity prediction.

Convolutional neural networks

Convolutional neural networks (CNNs) have proven to be effective models for tackling a variety of visual tasks. For each convolutional layer, a set of filters are learned to express local spatial connectivity patterns along input channels. By stacking a series of convolutional layers interleaved with non-linearities and downsampling, CNNs are capable of capturing hierarchical patterns with global receptive fields as powerful image descriptions. Recently, residual learning allows such networks to have a deeper structure (He et al., 2016a), which gains state-of-the-art results on multiple challenging recognition tasks.

Formally, a residual unit with an identity mapping is defined as:

(1) $${\mathbf{X}}_{l+1}={\mathbf{X}}_l+\mathcal{F}({\mathbf{X}}_l,{\mathcal{W}}_l),$$where in the Eq. (1) ${\mathbf{X}}_l$ and ${\mathbf{X}}_{l+1}$ are the input and output of the $l$-th residual unit respectively; $\mathcal{F}$ is a residual function, e.g., a stack of two $3\ast 3$ convolution layers; ${\mathcal{W}}_l$ is a set of weights (and biases) associated with the $l$-th residual unit (He et al., 2016b).

To improve the ability of representation of a network by explicitly modeling the interdependencies between the channels of its convolutional features, Hu, Shen & Sun (2018) introduce a new architectural unit, which is termed as the “Squeeze-and-Excitation” (SE) block.

User embedding

Feature-based approaches have demonstrated that user features, such as, number of fans and past success, are effective for popularity prediction (Shulman, Sharma & Cosley, 2016; Cheng et al., 2014). However, it is still unclear which features is the best important for us to represent and distinguish users or users’ influence. Recently, with the development of node embedding technology, learning users’ low-dimensional representations directly from history data has been widely adopted. Therefore, in this article, we propose to learn users’ representations by using their historic interactions, which are regarded as the input feature vectors for the following community detection algorithm.

Generally, the static social graph among users are treated as input to learn their representations. However, it is usually intractable to obtain the entire social graph structure. Moreover, previous works have proved that historic interactions among users are more effective for measuring the influence and susceptibility among them (Aral & Walker, 2012). Hence, instead of social graph, we construct the interaction graph $G=\left(V,E\right)$ directly from historic interactions, where V is the set of users and E is the set of edges. Note that each edge in E represents a retweet in microblogging network or a citation in academic network respectively.

Specifically, we employ Node2Vec (Grover & Leskovec, 2016) to characterize each user, which is a framework with outstanding performance for the continuous feature representation of nodes in large scale networks. The learned user embeddings are regared as input vectors for constructing community interaction matrix.

Community interaction matrix construction

In this article, we aim to model information diffusion from the mesoscopic perspective of communities. We suppose that information propagation between users corresponds to the diffusion among different communities. As shown in Fig. 2, the observed episode of each cascade is transformed to the form of community interaction matrix. Specifically, information diffusion is regarded as the inflow and outflow among different communities. Therefore, it is significantly important for us to firstly detect the communities among users. In previous section, we obtain the learned user embeddings from user interaction graph. Here, we exploit the mini-batch k-means++ proposed by Arthur & Vassilvitskii (2007) to cluster users into $s$ communities. The community list is represented as L and $s$ is the number of communities. The approach can handle large-scale networks with low computational complexity and has superior performance in community detection.

Given a cascade, it can be represented as the form of community interaction matrix $\mathbf{W}\in {\mathbb{R}}^{s\ast s}$, where the ( $j,k$)-th value denotes the number of forwardings from community $j$ to community $k$ within the observation time window. Therefore, the problem of predicting the incremental popularity can be transformed into forecasting the information flow among different communities.

Now, we have obtained the $s$ detected communities among users based on their historic interactions. It is important and challenging for the downstream task to organize the community interaction matrix $\mathbf{W}$. When communities are arranged randomly in the matrix, the similarity between adjacent communities is uncertain and neglected obviously in the community interaction matrix $\mathbf{W}$. To extract deep features from user interactions, it is necessary to acquire the local proximity of the input community interaction matrix $\mathbf{W}$. Therefore, it is significantly important to assemble the community interaction matrix appropriately, which makes these communities with more interactions to get closer in the matrix $\mathbf{W}$. In this article, we propose an algorithm to assemble communities, preserving the proximity among them. Details are shown in Algorithm 1. Firstly, based on the obtained detected communities, the largest community is chosen and denoted by $L_m$, where $m=\lceil \frac{s}{2}\rceil$. According to some similarity measurement for communities, we obtain two other communities which are most similar with $L_m$, denoted by $L_{m-1}$ and $L_{m+1}$ respectively. Then, we continue to choose the most similar community in the rest and add it to the list denoted by $L_{m-2}$. Similarly, we add another one to the list denoted by $L_{m+2}$. The process continues until all communities are added to the list L. Finally, we reassemble the community interaction matrix $\mathbf{W}$ according to the order in the list L.

For simplicity, in this article, the similarity between two communities $i$ and $j$ is defined as:

(2) $$s(i,j)=U_{ij}/D_i^{out}+U_{ji}/D_j^{out}+U_{ij}/D_j^{in}+U_{ji}/D_i^{in},$$where in the Eq. (2) $U_{ij}$ represents the historic interaction from community $i$ to community $j$, such as, the number of retweets or citations from community $i$ to community $j$. $D_i^{out}$ and $D_i^{in}$ represent the outdegree and indegree of community $i$, respectively. Note that other similarity measurements can also be adopted. Here we treat investigating the effect of different similarity measurements as a future work.

Convolutional architecture

Once we have obtained the community interaction matrix $\mathbf{W}$, an effective convolution neural network can be applied to extract the deep features of a specific cascade within the observation time window. In the proposed CST-Net, we utilize the convolutional neural network framework ResNet (He et al., 2016a) to predict the future incremental popularity. The “Squeeze-and-Excitation” block proposed by Hu, Shen & Sun (2018) is adopted to improve the performance of our model as a new architectural unit.

To capture the temporal features of a cascade, we divide the observation time window into three fragments, denoting recent time, near history and distant history. Then, they are fed into three channels of the convolution neural network. Moreover, we build multiple labels for each cascade which denotes the incremental popularity during the observation time window. Afterwards, we can obtain the learned representation for each cascade.

In addition, we extract some predictive structural and temporal features for each cascade. More specifically, we extract the structural features of the observed cascade as a measure of centrality and density, such as the number of leaf nodes, the number of first layer nodes, average and max length of retweet or citation path. To capture the temporal dynamics of each cascade, we extract the mean time interval between each retweet or citation, the time latency of the first retweet or citation, the cumulative popularity, incremental popularity every 10 min for Sina Weibo and every 1 year for APS.

Finally, we feed the learned representation of each cascade graph and the aforementioned extracted features, denoted by ${\mathbf{h}}^{\mathbf{i}}$, into a fully-connected layer (Eq. (3)) to predict the incremental popularity:

(3) $$\mathrm{\Delta }{N}_T^i=FC\left({\mathbf{h}}^{\mathbf{i}}\right).$$

Datasets

In order to demonstrate the effectiveness and generalizability of the proposed CST-Net, we evaluate it on two scenarios of popularity prediction. One of the scenario is forecasting the incremental popularity of retweet cascades in a social network and the other is predicting the incremental size of article citation cascades. In this article, the data were collected as previously described in DeepHawkes (Cao et al., 2017), where retweets and citations were considered as cascades, respectively.

Sina Weibo dataset: This data is collected from the most popular micro-blogging service in China, namely Sina Weibo, which has more than 500 million registered users and generates about 100 million messages per day. It consists of all the original microblogs that were submitted on June 1, 2016, and for each microblog, we collect its retweets within the next 24 h. In Sina Weibo, a cascade is generated by an original tweet and all of its retweets, and its popularity is the number of retweets. Figure 3A shows that the retweet cascade popularity exhibits a power-law distribution.

Generally, the life cycle of a microblog is relatively short. In Fig. 4A, we can see that the retweet number within 24 h $N_{24hours}$ after publication have a good approximation to the final popularity $N_{\mathrm{\infty }}$ for each microblog. In our experiment, we set T, the observation time window of a cascade since the original microblog was released, from a range of 1, 2 and 3 h. For each T, we only consider the cascades with no less than 10 retweets and no more than 1,000 retweets in the observation window. Figure 5 shows that due to the diurnal rhythm of users, microblogs released in different period may have different popularity dynamic patterns. Therefore, we only consider microblogs published between 8:00 and 18:00. We split the remaining cascades into training set, validation set and test set by the first 70%, the middle 15% and the last 15% according to the publication time. Detailed statistics of the preprocessed dataset are shown in Table 1.

APS dataset: It comprises the articles published in all journals in American Physical Society from 1893 to 2009, consisting of 245,365 authors, 463,344 articles, and 4,692,026 citations. For each article, the dataset includes title, DOI, PACS code, date of publication, names and affiliations of authors, a list of the articles cited, and so on. Similar with the preprocessing in DeepHawkes (Cao et al., 2017), we consider all coauthors of a article as an author group, which is analogy to a distinct person in the scenario of Sina Weibo. In APS, the cascade can represent the citation relationship of a article. In Fig. 3B, the citation cascade popularity also follows a power-law distribution.

Similar to Sina Weibo, as shown in Fig. 4B, for each article, we utilize the citation count within $20$ years $N_{20years}$ to represent the final popularity $N_{\mathrm{\infty }}$ for each article. Accordingly, we only use articles published between 1893 and 1989. In APS dataset, we choose the observation time window from a range of 3, 6 and 9 years. For each observation time T, only those articles with more than 10 citations is retained. Finally, on the basis of the publication time, we take the first 70% as training set, the middle 15% as validation set and the last 15% as test set. Table 1 shows the statistics of the processed dataset.

Baselines

To evaluate the accuracy of our predictive model, we compare the proposed CST-Net model with state-of-the-art approaches. Specifically, the comparison methods in our experiments are listed as follows:

Features: Based on recent studies of popularity prediction, structural and temporal features provide strong evidence for the final popularity increment. Therefore, we extract all predictive features that could be generalized across datasets. These features include:

• Structural features. While it is often impossible to obtain the entire social network structure, we only extract the structural features of the observed cascade. We use the number of leaf nodes, the number of first layer nodes, average and max length of retweet or citation path as a measure of centrality and density.

• Temporal features: To capture the temporal dynamics of a cascade, We extract the mean time interval between each retweet or citation, the time latency of the first retweet or citation, the cumulative popularity, incremental popularity every 10 min for Sina Weibo and every 1 year for APS.

Once all the above predictive features are collected, we feed these feature vectors into a linear regression model with L2 regularization and a multi-layer perceptron (MLP), denoted as Feature-linear and Feature-deep, respectively.

DeepCas (Li et al., 2017) is state-of-the-art deep learning method for popularity prediction, which automatically learns the representation of individual cascade graph as a whole in an end-to-end manner and predicts the future incremental popularity.

DeepHawkes (Cao et al., 2017) leverages end-to-end deep learning to make an analogy to interpretable factors of Hawkes process, which inherits the high interpretability of Hawkes process and possesses the high predictive power of deep learning methods.

CasCN (Chen et al., 2019) conceptualizes the cascade graph as a discrete sequence of subgraphs, which introduces a graph convolutional network designed to capture local structural information within each snapshot, while recurrent neural networks are used to learn sequential information.

GTGCN (Yang et al., 2022) utilizes a temporal encoder to process timestamp information and applies an improved graph convolutional network to capture both structural and temporal information, combining temporal encoding with a gated recurrent unit to capture temporal dependencies across multiple snapshots.

Evaluation metric

In order to validate the prediction performance of the CST-Net model, we compare it with the existing prediction models, in terms of two metrics: mean square log-transformed error (MSLE) and mean relative log-transformed error (MRLE), following the practice in previous work of popularity prediction (Li et al., 2017; Cao et al., 2017). Let $\mathrm{\Delta }{\hat{N}}_T^i$ be the predicted incremental popularity for item $i$ up to time T, and $\mathrm{\Delta }{N}_T^i$ be the observed incremental popularity. The MSLE (Eq. (4)) measures the average deviation between the predicted and observed incremental popularity over all items. For a dataset of M items, the MSLE is defined as:

(4) $$MSLE=\frac{1}{M}\sum _{i=1}^M{\left(\log \mathrm{\Delta }{\hat{N}}_T^i-\log \mathrm{\Delta }{N}_T^i\right)}^2.$$

The MRLE (Eq. (5)) expresses how large the absolute error is compared with the ground truth, which is defined as:

(5) $$MRLE=\frac{1}{M}\sum _{i=1}^M\left[\left(\log \mathrm{\Delta }{\hat{N}}_T^i-\log \mathrm{\Delta }{N}_T^i\right)/\log \mathrm{\Delta }{N}_T^i\right].$$

Running environment and parameter settings

All experiments were done on a Ubuntu 20.04 server with Intel^® Core™ i9-10850K CPU @ 2.60 GHz and a RAM of size 128 Gb. Besides, We use an NVIDIA 3080Ti GPU.

For hyper parameters, the mini-batch size of the stochastic gradient descent is set as 32 and 128 in Sina Weibo and APS dataset respectively. We select the initial learning rate from $\{0.1,0.05,0.001,\dots ,{10}^{-5}\}$ and weight decay from $\{0,0.01,0.005,\dots ,{10}^{-4}\}$. For Node2Vec (Grover & Leskovec, 2016), $p$, $q$ are chosen from {0.25, 0.5, 1, 2, 4}, the length of walk is chosen from {10, 25, 50, 80, 100}, and the candidate embedding size from {32, 64, 128}. For mini-batch k-means++ (Arthur & Vassilvitskii, 2007), batch size is selected from {1,000, 5,000, 10,000, 50,000} and the number of clusters from {32, 64, 128, 256}.

Prediction performance

As shown in Table 2, the overall prediction performance of all competing methods across two datasets is displayed. Except for the prediction based on 3 year observation on APS dataset, which our model performs on par with the state-of-the-art method, the proposed CST-Net outperforms all four baseline methods on both scenarios. Because of the richer information within a longer observation time window, we can find that the longer the observation time, the better the prediction performance.

We noticed that the model of Feature-linear and Feature-deep perform not well in popularity prediction since it is difficult to extract all of the effective hand-crafted features of a cascade. Within the short observation time, the experimental result of DeepCas is better than Feature-linear and Feature-deep. Nevertheless, when the observation time being longer, Feature-linear and Feature-deep outperform DeepCas significantly. One possible explanation is that DeepCas focuses on sampling sequences via random walks while ignores the rich structural and temporal information of a cascade.

On Sina Weibo dataset, our model outperforms all of the four baseline methods within any observation window. However, we can see that CST-Net is slightly worse than DeepHawkes when the observation time window is set to 3 years on APS dataset. One possible explanation is that the community interaction matrix would be sparse because the life cycle of articles is significantly longer than that of microblogs. Therefore, it is difficult for CST-Net to capture the dynamics of citations. We leave exploring the sparsity problem to our future work.

It is worth mentioning that we have also investigated some poor prediction cases. Surprisingly, we found that there is an interesting number of cascades which do not get much retweets/citations for a period of time after being published, but then suddenly start getting popular heavily. This phenomenon is called “sleeping beauties” which has raise much attention in other fields such as social computational science. However, how to effectively model and predict this kind of cascades is another line of research, which is beyond the focus of this manuscript and we will try to solve them in our future work.

In addition, comparing the experimental results in two scenarios, we can find that the CST-Net exhibits a better performance in Sina Weibo than in APS. We show in Fig. 6 the number of articles published in each year in the APS dataset. The inset of Fig. 6 gives a cumulative view, i.e., total number of articles published up to a certain year, on a log-linear scale. Figure 6 is in good agreement with previous findings (Wang, Song & Barabási, 2012) that the number of articles published each year increases exponentially, indicating that the cascading pattern of articles is quite different from that of microblogs. Therefore, the model trained in the early stage would be inappropriate for the prediction in 20 years, leading to the slightly poor performance in APS dataset.

Ablation study

In order to demonstrated the effectiveness of the components in the proposed CST-Net, a detailed analysis is given in this section. For comparison, three simplified variants of CST-Net, denoted as CST-basic, CST-feature and CST-community are represented, where one or two components are removed from the complete CST-Net model.

CST-basic model randomly arrange communities in the interaction matrix, in which the similarity between adjacent communities is uncertain and neglected. The hand-crafted temporal and structural features are also discarded. This version is constructed to offer a basic model for comparison.

CST-feature model includes the hand-crafted features which implicate the structural and temporal characteristics of a cascade on the basis of CST-basic. This version is used to evaluate whether extracted features are effective for popularity prediction.

CST-community model uses Algorithm 1 to construct community interaction matrix on the basis of CST-basic. We use this version to verify the effectiveness of community interaction matrix construction.

The prediction results of these simplified versions of CST-Net are shown in Table 3. Comparing to the complete version of CST-Net model, all these three variants lead to certain degradation of performance in different extent. Firstly, we can observe that CST-basic and CST-feature have similar prediction performance, and CST-feature is slightly better than CST-basic. Accordingly, these hand-crafted features play a positive role on popularity prediction. Secondly, the version of CST-community outperforms both CST-basic and CST-feature with a significant reduction of prediction error, especially within short observation time window. Since the detected communities in CST-basic and CST-feature are arranged randomly, the local similarity between adjacent communities is uncertain and neglected. However, in order to feed into convolutional neural networks, the local similarity in the input matrix is required. Thus, the community interaction matrix construction contributes to learn the deep features of cascade among communities. For simplicity, in this article, we utilize the historic interactions among users to compute the community similarity and then assemble them orderly in the community interaction matrix. In addtion, we also extended our ablation study to more precisely isolate the impact of each key component of CST-Net. Specifically, we conducted additional experiments where we remove the convolutional architecture (CST-Net w/o Conv) and residual units (CST-Net w/o SE-ResUnit) from CST-Net, respectively. The extended ablation study results are also summarized in Table 3. These results further demonstrate the effectiveness of each component to the model’s performance. Besides, we can also observe that the convolutional architecture performs more importance than the residual units in the model.

Moreover, in order to investigate the ability of the proposed CST-Net to predict popularity dynamics, we set the training period, i.e., T, as 2 h and 6 years for Sina Weibo and APS dataset respectively, and then predict the incremental popularity every 3 h or 2 years on both scenarios. As shown in Fig. 7, we can observe that with the increase of prediction time, the performance of all four versions of CST-Net decreases.