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GTIN: A Unified Framework for Joint Event and Time Prediction in Temporal Graphs
Mohammad Ostadmohammadi, Sepehr Kazemi, Hamid R. Rabiee
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Abstract
Abstract:Temporal graphs are increasingly used to model dynamic systems in diverse domains such as social networks, financial networks, and traffic networks. Predicting both what the next event will be and when it will occur in these systems is crucial for understanding and anticipating complex behaviors, but has not been studied much. To address this gap, we propose a unified mathematical framework capable of capturing varying degrees of complexity across temporal graphs. Our framework is flexible and expressive enough to accommodate a wide range of network structures and temporal dynamics. Building upon this analysis, we introduce our novel approach for jointly predicting the next event and its occurrence time. Empirical evaluations across multiple datasets demonstrate that our method consistently outperforms existing techniques, particularly in scenarios involving irregular event patterns and complex temporal dependencies. These findings highlight the potential of our framework as a robust foundation for future research in temporal event prediction.
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- Source: https://arxiv.org/abs/2607.23556v1
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GTIN: A Unified Framework for Joint Event and Time Prediction in Temporal Graphs Mohammad Ostadmohammadi a , Sepehr Kazemi b and Hamid R. Rabiee a,β a Department of Computer Engineering, Sharif University of Technology, Azadi Ave., Tehran, Iran b Department of Electrical Engineering, Sharif University of Technology, Azadi Ave., Tehran, Iran A R T I C L E I N F O Keywords: Temporal events dynamic graphs message passing hierarchical clustering A B S T R A C T Temporal graphs are increasingly used to model dynamic systems in diverse domains such as social networks, financial networks, and traffic networks. Predicting both what the next event will be and when it will occur in these systems is crucial for understanding and anticipating complex behaviors, but has not been studied much. To address this gap, we propose a unified mathematical framework capable of capturing varying degrees of complexity across temporal graphs. Our framework is flexible and expressive enough to accommodate a wide range of network structures and temporal dynamics. Building upon this analysis, we introduce our novel approach for jointly predicting the next event and its occurrence time. Empirical evaluations across multiple datasets demonstrate that our method consistently outperforms existing techniques, particularly in scenarios involving irregular event patterns and complex temporal dependencies. These findings highlight the potential of our framework as a robust foundation for future research in temporal event prediction. 1. Introduction The prediction of future events in dynamic graphs has emerged as a central problem in modeling complex temporal systems. Applications such as communication networks, social media, transportation systems, and biological interactions depend on the ability to anticipate not only what event will occur next, but also when and where it will happen. Traditional static graph models fail to capture temporal dependencies, while existing temporal graph approaches often focus on localized subgraphs or are restricted in scalability. Recent developments in temporal graph neural networks (TGNNs) and temporal point process (TPP) models have made progress in understanding the temporal evolution of graph structures. However, most existing models operate within limited temporal windows or node neighborhoods, failing to utilize the global structural context. Furthermore, while several methods address link or event prediction, only a few explicitly model both the event occurrence time and its associated features. To address these limitations, we introduce a Global Temporal Interaction Network (GTIN), a unified framework capable of predicting the next eventβs time, location, and features by considering the graph as a whole. The model integrates hierarchical decomposition and temporal aggregation, enabling scalable training and inference even on large-scale temporal graphs. The main contributions of this work can be summarized as follows: β’ Global Event-Time Prediction Framework: We propose a model that jointly predicts the next eventβs location and timing by analyzing the entire graph structure rather than relying solely on local neighborhoods. β’ Hierarchical Scalability: The proposed model introduces a hierarchical graph decomposition mechanism that allows any graph, regardless of size, to be efficiently partitioned and processed while maintaining temporal dependencies. β’ Feature-Level Event Forecasting: Beyond structural prediction, the model is capable of estimating the attributes or features associated with the next event, providing more comprehensive predictive capability. m.ostadmohammadi91@sharif.edu (M. Ostadmohammadi); sepehr.kazemi99@sharif.edu (S. Kazemi); rabiee@sharif.edu (H.R. Rabiee) ORCID(s): 0000-0002-9835-4493 (H.R. Rabiee) M. Ostadmohammadi et al.: Preprint submitted to ElsevierPage 1 of 15 arXiv:2607.23556v1 [cs.SI] 26 Jul 2026 Joint Event and Time Prediction To validate the effectiveness and scalability of the proposed GTIN model, we conduct extensive experiments on multiple benchmark temporal graph datasets. The evaluation focuses on time estimation precision, comparing our approach against leading temporal graph baselines such as TGAT, TGN, and DyRep. We further assess scalability through hierarchical partitioning experiments on large graphs, demonstrating the modelβs efficiency and robustness under diverse dynamic environments. 2. Related Works Research on event prediction in temporal graphs can be broadly categorized into four main directions: (1) Temporal Point Process Models, (2) Temporal Graph Neural Networks, (3) Hybrid Temporal-Spatial Event-Time Models, and (4) Scalable and Hierarchical Graph Learning Frameworks. This section reviews representative works in each category and highlights their connections and limitations with respect to our study. 2.1. Temporal Point Process Models Temporal Point Processes (TPPs) provide a classical foundation for modeling event occurrences over continuous time. The Hawkes process Hawkes (1971), a self-exciting TPP, captures dependencies between past and future events through intensity functions. Neural extensions, such as the Neural Hawkes Process Mei and Eisner (2017), learn complex temporal patterns using neural representations. Although these methods accurately model event timing, they generally ignore the underlying graph structure, limiting their applicability to relational domains. 2.2. Temporal Graph Neural Networks Temporal Graph Neural Networks (TGNNs) extend static GNNs to dynamic settings, enabling the modeling of both structural and temporal dependencies. Models such as TGAT Xu, Ruan, Korpeoglu, Kumar and Achan (2020), TGN Rossi, Chamberlain, Frasca, Eynard, Monti and Bronstein (2020), and DyRep Trivedi, Farajtabar, Biswal and Zha (2019) leverage temporal message passing and node memory mechanisms to track evolving node states. More recent developments, including MTGN Liu, Tu, Xu and Wang (2023), HierTKG Almutairi, Aktas, Wali, Mitra and Zhou (2024), and RE-GAT Li, Feng, Shi, Zhou, Liao, Yang, Li, Yu and Shao (2023), have focused on temporal knowledge graphs, predicting future relations or interactions between entities. However, these models typically rely on local temporal neighborhoods and thus struggle to capture the global graph context required for comprehensive event forecasting. 2.3. Hybrid Temporal-Spatial and Event-Time Prediction Models Hybrid frameworks combine spatial, temporal, and event-level reasoning. For example, GSTPP Zhou, Kang, Liu, Zhang and Tian (2025) employs a graph spatio-temporal point process with a self-adaptive anchor graph to predict fine- grained event occurrences in both space and time. MTGN Liu et al. (2023) jointly models missing and observed events to improve prediction robustness under partial observability. Knowledge-enhanced approaches, such as CEEG Li, Jiang and Zhou (2025) and TPAR Chen, Wang, Li, Li, Yu and Song (2024), incorporate commonsense reasoning or interpolation-extrapolation mechanisms to improve long-term temporal predictions. Despite these advances, such models often suffer from high computational complexity and lack scalability to large graphs. 2.4. Scalable and Hierarchical Graph Learning Frameworks Scalability has become a key challenge in temporal graph learning. Approaches like TF-TGN Huang, Yan, Wang, Rao, Han, Fu, Zhang and Jiang (2024), SIMPLE Gao, Li, Zhang, Shen, Shao and Chen (2024), and TAP-GNN Zheng, Wang, Feng, Song, Hao, Song, Wang, Wang and Chen (2023) improve efficiency through temporal sampling, memory compression, and hierarchical training. While these models enhance scalability, they often operate on fixed-size graph partitions and fail to exploit the networkβs global temporal structure. In contrast, our proposed method introduces a hierarchical decomposition and aggregation strategy that maintains both temporal and structural dependencies across scales, enabling global-level event-time prediction while preserving computational efficiency. 2.5. Summary and Positioning In summary, prior research has achieved significant progress in modeling temporal dynamics and event prediction, but existing methods are typically constrained by one or more of the following: (1) absence of structural-temporal integration, (2) reliance on local graph neighborhoods, or (3) limited scalability on large graphs. Our work addresses M. Ostadmohammadi et al.: Preprint submitted to ElsevierPage 2 of 15 Joint Event and Time Prediction these limitations by presenting a global, hierarchical, and feature-aware event-time prediction model that simultaneously predicts when, where, and what the next event will be. 3. Preliminaries 3.1. Simple Point Process Consider the sequence,ν‘ 1 ,ν‘ 2 ,...,ν‘ ν . We defineν(ν‘|ξ΄ ν‘ ν ) as a conditional density function. The conditional intensity function for ν‘ > ν‘ ν , ν(ν‘) = ν(ν‘|ξ΄ ν‘ ν ) 1 β νΉ(ν‘|ξ΄ ν‘ ν ) is used more often than ν(ν‘|ξ΄ ν‘ ν ), because it considers the information up to time ν‘, which is larger than ν‘ ν . In other words, we can say that ν(ν‘) = ν(ν‘|ξ΄ ν‘ β ). From the definition, we have ν(ν‘|ξ΄ ν‘ ν ) = ν(ν‘)exp ( β β« ν‘ ν‘ ν ν(ν)νν ) The Log Likelihood of an observed sequence ν‘ 1 ,ν‘ 2 ,...,ν‘ ν in an interval [0,ν] is defined as: νΏ = log[ν(ν‘ 1 ,ν‘ 2 ,...,ν‘ ν )(1 β νΉ(ν |ξ΄ ν‘ ν ))] = log ( ν β ν=1 ν(ν‘ ν |ξ΄ ν‘ νβ1 )(1 β νΉ(ν |ξ΄ ν‘ ν )) ) = ( ν β ν=1 ννν(ν(ν‘ ν )) β β« ν‘ ν ν‘ νβ1 ν(ν)νν ) β β« ν ν‘ ν ν(ν)νν = ν β ν=1 ννν(ν(ν‘ ν )) β β« ν 0 ν(ν)νν 3.2. Marked Point Process Consider the sequence, (ν‘ 1 ,ν 1 ),(ν‘ 2 ,ν 2 ),...,(ν‘ ν ,ν ν ). We define ν(ν‘,ν|ξ΄ ν‘ β ) = ν(ν‘|ξ΄ ν‘ β )ν(ν|ν‘,ξ΄ ν‘ β ), where ν stands for both PMF and PDF. The conditional intensity function for ν‘ > ν‘ ν is defined as: ν(ν‘,ν) = ν(ν‘|ξ΄ ν‘ β )ν(ν|ν‘,ξ΄ ν‘ β ) = ν(ν‘)ν(ν|ν‘,ξ΄ ν‘ ν ) = ν(ν‘|ξ΄ ν‘ ν )ν(ν|ν‘,ξ΄ ν‘ ν ) 1 β νΉ(ν‘|ξ΄ ν‘ ν ) = ν(ν‘,ν|ξ΄ ν‘ ν ) 1 β νΉ(ν‘|ξ΄ ν‘ ν ) Also, we have: ν(ν‘,ν|ξ΄ ν‘ ν ) = ν(ν‘,ν)exp ( β β« ν‘ ν‘ ν ν(ν)νν ) Now, we derive the log-likelihood function for the marked sequence (ν‘ 1 ,ν 1 ),(ν‘ 2 ,ν 2 ),...,(ν‘ ν ,ν ν ) in an interval [0,ν]: νΏ = log [ ν(ν‘ 1 ,ν 1 ,ν‘ 2 ,ν 2 ,...,ν‘ ν ,ν ν )(1 β νΉ(ν |ξ΄ ν‘ ν )) ] = ν β ν=1 ννν(ν(ν‘ ν ,ν ν )) β β« ν 0 ν(ν)νν M. Ostadmohammadi et al.: Preprint submitted to ElsevierPage 3 of 15 Joint Event and Time Prediction 4. Materials and methods 4.1. Problem Formulation We define ξ΄ ν (ν‘) = (ν‘ (ν) ν ,ν (ν) ν ) | ν‘ (ν) ν β€ ν‘ , i.e., the history of the νth edge until time ν‘, where (ν‘ (ν) ν ,ν (ν) ν ) is the time and feature of theνth interaction. The features can be any type; also, for different timestamps, we can have different types of features. We define ξ΄ ν (ν‘,νΏ) = (ν‘ (ν) ν ,ν (ν) ν ) | ν‘ β€ ν‘ (ν) ν ,1 β€ ν β€ νΏ , i.e., the next νΏ interactions. Also we define ξ΄(ν‘) = (ν‘ (ν) ν ,ν (ν) ν ) β£ ν‘ (ν) ν β€ ν‘,1 β€ ν β€ ν , i.e., the history of the graph until time ν‘, and ξ΄(ν‘,νΏ) = (ν‘ (ν) ν ,ν (ν) ν ) β£ ν‘ β€ ν‘ (ν) ν ,1 β€ ν β€ νΏ,1 β€ ν β€ ν , i.e., the next νΏ interactions of the graph after time ν‘. Definition (Problem): Assume until time ν‘, we observe ξ΄(ν‘). Our objective is to estimate ξ΄(ν‘,1). You may wonder why we didnβt define our problem as predicting ξ΄(ν‘,νΏ). Actually, if we can predict ξ΄(ν‘,1), we can repeat this process νΏ times. The [1] considers the problem as predicting the next event time of each edge based on its history and its neighbors, not considering the whole graph structure. But in real-world applications, we need to estimate events in an interval considering all edges. In other words, in an interval, an edge may have many interactions, but some edges have no interactions. Also, the interaction on some edges changes the history of the graph, which can affect the prediction of other edges. 4.2. Objective To estimate the ξ΄(ν‘,1) as well as possible, we should maximize the following objective: ν(ξ΄(ν‘,1),ξ΄(ν‘)) = ν(ξ΄(ν‘))ν(ξ΄(ν‘,1)|ξ΄(ν‘)) For the first term, we should computeν(ξ΄(ν‘)). Without loss of generality, we writeξ΄(ν‘) = (ν‘ ν ,(ν ν ,ν ν ))|ν‘ ν β€ ν‘ νΎ ν=1 , where νΎ is the total number of events until time ν‘ and ν ν is an integer in range [1,ν] showing the edge that this (ν‘ ν ,ν ν ) belongs to. Also, ν‘ ν βs are ordered. Now we can write the first term as: ν(ξ΄(ν‘)) = ν(ν‘ 1 ,ν 1 ,ν 1 ,ν‘ 2 ,ν 2 ,ν 2 ,...,ν‘ ν ,ν ν ,ν νΎ ) = νΎ β ν=1 ν(ν‘ ν ,ν ν ,ν ν |ξ΄(ν‘ νβ1 )) = νΎ β ν=1 ν(ν ν |ξ΄(ν‘ νβ1 ))ν(ν‘ ν |ν ν ,ξ΄(ν‘ νβ1 ))ν(ν ν |ν‘ ν ,ν ν ,ξ΄(ν‘ νβ1 )) Please remember from section 2.2 that this is the likelihood of a marked point process where the marks are(ν ν ,ν ν )βs. Now, the probability of this likelihood can also be written as: ν(ξ΄(ν‘)) = νΎ β ν=1 ν(ν‘ ν ,ν ν ,ν ν )exp ( β β« ν‘ 0 ν(ν)νν ) Which format we write depends on our implementation, but working with density functions are easier because estimating the integral in intensity functions can lead to poor performance. Now we consider the second term, ν(ξ΄(ν‘,1)|ξ΄(ν‘)), where we want to predict the next interaction on the graph. So we have: ν(ξ΄(ν‘,1)|ξ΄(ν‘)) = ν( Μ ν‘, Μ ν , Μν|ξ΄(ν‘)) = ν(Μν|ξ΄(ν‘))ν( Μ ν‘|Μν,ξ΄(ν‘))ν( Μ ν | Μ ν‘, Μν,ξ΄(ν‘)) So we can write the final objective as: maximize νΎ β ν=1 logν(ν ν |ξ΄(ν‘ νβ1 )) + logν(ν‘ ν |ν ν ,ξ΄(ν‘ νβ1 )) + logν(ν ν |ν‘ ν ,ν ν ,ξ΄(ν‘ νβ1 )) M. Ostadmohammadi et al.: Preprint submitted to ElsevierPage 4 of 15 Joint Event and Time Prediction +logν(Μν|ξ΄(ν‘)) + logν( Μ ν‘|Μν,ξ΄(ν‘)) + logν( Μ ν | Μ ν‘, Μν,ξ΄(ν‘)) Or in terms of intensity function: maximize νΎ β ν=1 logν(ν‘ ν ,ν ν ,ν ν ) β β« ν‘ 0 ν(ν)νν +logν(Μν|ξ΄(ν‘)) + logν( Μ ν‘|Μν,ξ΄(ν‘)) + logν( Μ ν | Μ ν‘, Μν,ξ΄(ν‘)) 4.3. Model In this section, we introduce our implementation of the model. As mentioned previously, the computation of the integral of the conditional intensity function in the likelihood can require numerical approximation and may reduce the stability of the model. Therefore, instead of directly estimating the intensity function, our goal is to estimate the following conditional density function: ν(ν‘ ν ,ν ν ,ν ν |ξ΄(ν‘ νβ1 )) = ν(ν ν |ξ΄(ν‘ νβ1 ))ν(ν‘ ν |ν ν ,ξ΄(ν‘ νβ1 ))ν(ν ν |ν‘ ν ,ν ν ,ξ΄(ν‘ νβ1 )) This decomposition follows the natural order of the prediction problem. First, the model predicts the location of the next event, which is represented by ν ν . Then, conditioned on the predicted edge and the previous history, the model predicts the occurrence time ν‘ ν . Finally, conditioned on the predicted edge, the predicted time, and the previous graph history, the model predicts the event feature ν ν . Therefore, the model jointly answers three questions: where the next event happens, when it happens, and what its feature value is. Now we take care of ξ΄(ν‘ ν ). For computing this, we consider that each edge has an embedding β ν (ν‘) up to some time ν‘. Similar to the previous section, each edge in the original graph is treated as a node in the transformed graph, and neighboring edge-nodes are those that share a common endpoint in the original graph. So we use the Message Passing Framework as follows: β ν (ν‘) = COMBAGGΞ¦(ν‘ ν ν ,ν ν ν )|ν‘ ν ν β€ ν‘,ν β ξΊ(ν),Ξ¦(ν‘ ν ν ,ν ν ν )|ν‘ ν ν β€ ν‘ In this equation, ξΊ(ν) is the set of neighbors of edge ν. The first part aggregates the temporal information coming from neighboring edges, while the second part keeps the temporal information of the edge itself. Therefore, the embedding β ν (ν‘) is not only based on the past events of edge ν, but also on the past events of its neighboring edges. One can stack multiple layers to consider more hops and to propagate temporal information to a larger neighborhood of the graph. The function Ξ¦(β ) converts each temporal interaction into a vector representation. Since time is a continuous variable, using the raw timestamp alone is not enough to represent complex temporal patterns. Therefore, following the time conversion strategy used in GNPP Xia, Li and Li (2022), we encode each timestamp using a harmonic time encoder: ν β (ν‘) = β 1 ν ν [cos(ν 1 ν‘),sin(ν 1 ν‘),...,cos(ν ν ν ν‘),sin(ν ν ν ν‘)] where ν 1 ,ν 2 ,...,ν ν ν are learnable frequency parameters. This transformation maps each continuous timestamp into a high-dimensional vector. The advantage of this conversion is that the model can capture different temporal patterns, such as short-term recency effects, long-term dependencies, and possible periodic behavior, without discretizing the temporal graph into fixed snapshots. Then, the encoded time is combined with the feature of the event: Ξ¦(ν‘ ν ν ,ν ν ν ) = ν ( ν Ξ¦ [ν β (ν‘ ν ν )βν(ν ν ν )] + ν Ξ¦ ) where β denotes concatenation, ν(ν ν ν ) is a feature encoder for the event attribute, ν Ξ¦ and ν Ξ¦ are trainable parameters, and ν(β ) is a nonlinear activation function. If an event does not have explicit attributes, ν(ν ν ν ) can be replaced by a zero vector or omitted. In this way, Ξ¦(β ) jointly represents both the time and the feature of an observed interaction. Finally, the embedding of the graph up to time ν‘ is the sum of all of its edge embeddings: M. Ostadmohammadi et al.: Preprint submitted to ElsevierPage 5 of 15 Joint Event and Time Prediction β νΊ (ν‘) = ν β ν=1 ν€ ν β ν (ν‘) where ν is the number of edges in the graph and ν€ ν is the weight of edge ν. This weight can be fixed, learned, or computed by an attention mechanism. The graph embedding β νΊ (ν‘) is used as the representation of the whole history ξ΄(ν‘). This is important because the next event is not only affected by the history of one edge, but also by the global temporal state of the whole graph. Now we explain the three terms of the likelihood in more detail. The first term ν(ν ν |ξ΄(ν‘ νβ1 )) is a classification problem, because ν ν is a discrete variable showing the edge on which the next event occurs. Therefore, we use a softmax function over all candidate edges: ν(ν ν = ν|ξ΄(ν‘ νβ1 )) = softmax(ν ν β νΊ (ν‘ νβ1 ) + ν ν ) ν This term gives the probability that edge ν will be the location of the next event. It forces the model to learn which parts of the graph are more likely to become active based on the previous temporal history. The second term ν(ν‘ ν |ν ν ,ξ΄(ν‘ νβ1 )) is a regression problem. Since the occurrence time is continuous, we consider a Gaussian distribution for this term. Therefore, the model has to predict both the mean and the variance of the time distribution. For this purpose, we use both the global graph embedding and the embedding of the selected edge: ν§ ν‘ ν = [β νΊ (ν‘ νβ1 )ββ ν ν (ν‘ νβ1 )] Then, the parameters of the time distribution are obtained by a neural network: ν ν‘ ν ,ν ν‘ ν = MLP ν‘ (ν§ ν‘ ν ) ν ν‘ ν = softplus(ν ν‘ ν ) + ν where ν is a small positive value added for numerical stability. Therefore, the time density is written as: ν(ν‘ ν |ν ν ,ξ΄(ν‘ νβ1 )) = ξΊ(ν‘ ν ;ν ν‘ ν ,(ν ν‘ ν ) 2 ) In implementation, this distribution can be applied either to normalized timestamps or to normalized inter-event times. The mean ν ν‘ ν represents the expected occurrence time of the next event, while the variance (ν ν‘ ν ) 2 represents the uncertainty of this prediction. Thus, the model is not only penalized for predicting an inaccurate time, but also for assigning an unsuitable uncertainty to the prediction. The final term ν(ν ν |ν‘ ν ,ν ν ,ξ΄(ν‘ νβ1 )) can have different interpretations depending on the type of event feature. Here, we convert our attributes to real numbers, so we use a regressor whose output distribution is a multivariate normal distribution. Since the feature of an event depends on both where and when the event occurs, we condition this term on the graph embedding, the selected edge embedding, and the encoded event time: ν§ ν ν = [β νΊ (ν‘ νβ1 )ββ ν ν (ν‘ νβ1 )βν β (ν‘ ν )] The parameters of the feature distribution are computed as: ν ν ν ,ν ν ν = MLP ν (ν§ ν ν ) ν ν ν = softplus(ν ν ν ) + ν Therefore, the conditional distribution of the event feature is: ν(ν ν |ν‘ ν ,ν ν ,ξ΄(ν‘ νβ1 )) = ξΊ(ν ν ;ν ν ν , diag((ν ν ν ) 2 )) where diag((ν ν ν ) 2 ) shows that we use a diagonal covariance matrix. If the event feature is categorical, this part can be replaced by a categorical distribution. Based on the three terms above, the negative log-likelihood loss for a sequence of νΎ events is: M. Ostadmohammadi et al.: Preprint submitted to ElsevierPage 6 of 15 Joint Event and Time Prediction νΎ β ν=1 [logν(ν ν |ξ΄(ν‘ νβ1 )) + logν(ν‘ ν |ν ν ,ξ΄(ν‘ νβ1 )) + logν(ν ν |ν‘ ν ,ν ν ,ξ΄(ν‘ νβ1 ))] This loss can also be interpreted as the sum of three different parts: ξΈν + ν ν‘ ξΈν‘ + ν ν ξΈ ν where ξΈν is the classification loss for predicting the next edge, ξΈν‘ is the time prediction loss, and ξΈ ν is the feature prediction loss. The parameters ν ν‘ and ν ν control the importance of the time and feature terms. This decomposition makes the training objective more interpretable, because each part corresponds to one aspect of the next-event prediction problem. In training, we use a teacher forcing approach. At each step ν, the model receives the true history ξ΄(ν‘ νβ1 ), and the likelihood is computed using the true event triplet (ν‘ ν ,ν ν ,ν ν ) . After the loss is computed, the true event is added to the history, and the model moves to the next step. This helps the model learn from correct historical contexts and prevents error accumulation during training. In inference, the model receives the observed historyξ΄(ν‘) and first computes β νΊ (ν‘) and the edge embeddings β ν (ν‘). Then, it computes ν(ν|ξ΄(ν‘)) for candidate edges. For each candidate edge ν, the model predicts the mean of the time distribution: Μ ν‘ ν = ν ν‘ ν Then, conditioned on Μ ν‘ ν , ν, and ξ΄(ν‘), the model predicts the expected feature vector: Μ ν ν = ν ν ν Finally, the model selects the triplet with the maximum joint probability: arg max ν‘,ν ,ν [logν(ν|ξ΄(ν‘)) + logν(ν‘|ν,ξ΄(ν‘)) + logν(ν |ν‘,ν,ξ΄(ν‘))] 5. Experiments and Results In this section, we will show our experiments and results related to. We have used three real-world datasets to compare the results with baseline models. 5.1. Baseline methods In this section, we discuss baselines we used in experiments, including three neural point process models, i.e., Transformer Hawkes Process (THP) Zuo, Jiang, Li, Zhao and Zha (2020), Self-Attentive Hawkes Process (SAHP) Zhang, Lipani, Kirnap and Yilmaz (2020), and Neural Network Point Process (NNPP) Omi, Aihara et al. (2019), and seven graph neural models, i.e., TGAT Xu et al. (2020), TGN Rossi et al. (2020), GHNN Han, Ma, Wang, GΓΌnnemann and Tresp (2020), HTNE Zuo, Liu, Lin, Guo, Hu and Wu (2018), DySAT Sankar, Wu, Gou, Zhang and Yang (2020), CTDNE Nguyen, Lee, Rossi, Ahmed, Koh and Kim (2018), and GAT-LSTM. GAT-LSTM is a GAT model equipped with the harmonic encoder and an LSTM module as a naive GNN extended baseline, while others are originally devised for temporal graphs and GNPP model Xia et al. (2022). 5.2. Datasets 5.2.1. Real-World datasets Wikipedia is a temporal graph containing edited pages and users as nodes. Each edition between a user and an page is recorded with a timestamp. This temporal graph contains about 9300 nodes and 160000 temporal edges. Kumar, Zhang and Leskovec (2019). Reddit is an online community that records interactions between active users and their posts under subreddits. It has about 11000 nodes and 700000 temporal edges. Kumar et al. (2019). M. Ostadmohammadi et al.: Preprint submitted to ElsevierPage 7 of 15 Joint Event and Time Prediction CollegeMsg is a communication graph that records private Email interactions of an online society at the University of California, Irvine. This temporal graph contains about 2000 users and 60000 temporal edges Panzarasa, Opsahl and Carley (2009). 5.2.2. Synthetic datasets These datasets are generated as mentioned in Xia et al. (2022).In this paper, firstly a random WattsβStrogatz small- world graph is generated with 500 nodes and 1000 edges. Secondly, using Temporal Point Processes edge events are generated by Hawkes-negative, Hawkes-positive and Poisson process. 5.3. Evaluation Metrics We evaluate GTIN from three complementary perspectives: time prediction accuracy, next-event location predic- tion, and joint eventβtime probabilistic performance. Let ξ΄(ν‘ νβ1 ) denote the observed graph history before the νth test event, let ν ν denote the edge on which the event occurs, and let ν‘ ν denote its occurrence time. All predictions are made chronologically using only events observed before ν‘ ν . For time prediction, we retain the root mean squared error: RMSE = β β β β 1 νΎ νΎ β ν=1 ( ν‘ ν β Μ ν‘ ν ) 2 , where νΎ is the number of test events and Μ ν‘ ν is the predicted occurrence time. For location prediction, the model assigns a probability to every candidate edge: ν ( ν ν = ν β£ ξ΄(ν‘ νβ1 ) ) , ν β ξ― ν , where ξ― ν is the candidate edge set for event ν. The rank of the true edge is denoted by ν ν . We report Hits@ν and mean reciprocal rank (MRR), defined as Hits@ν = 1 νΎ νΎ β ν=1 ν [ ν ν β€ ν ] , and MRR = 1 νΎ νΎ β ν=1 1 ν ν . Higher Hits@ν and MRR values indicate better next-event location predictions. 5.4. Results on real-world datasets As shown in Table 1, our Global Temporal Interaction Network (GTIN) achieves the best overall performance across all three real-world benchmarksβWikipedia, Reddit, and CollegeMsg. On the Wikipedia dataset, GTIN attains an RMSE of 10.6, surpassing GNPP (11.14) and significantly outperforming traditional temporal graph baselines such as TGAT (37.66) and TGN (23.81). This improvement demonstrates the modelβs ability to capture the irregular yet correlated editing patterns typical of collaborative systems, where both short-term bursts and long inactivity intervals coexist. For the Reddit dataset, which features a denser temporal structure and frequent userβpost interactions, GTIN achieves an RMSE of 0.51, nearly halving the error compared to GNPP (1.08). The strong performance in this high- activity environment underscores GTINβs capacity to leverage temporal dependencies through its message-passing mechanism, effectively weighting recent interactions more heavily when forecasting upcoming events. Meanwhile, in the CollegeMsg datasetβcharacterized by sparser and more irregular communicationβthe performance gain is smaller but still notable, with GTIN (27.12) outperforming GNPP (29.75). This indicates that while the modelβs advantage is most pronounced in dense event streams, it maintains robust generalization even under limited temporal context. Overall, these findings highlight that GTIN adapts effectively to varying network dynamics. Its strength lies particularly in modeling complex, high-frequency interactions, where integrating structural and temporal cues jointly provides substantial predictive gains over existing temporal GNN and point process models. M. Ostadmohammadi et al.: Preprint submitted to ElsevierPage 8 of 15 Joint Event and Time Prediction Table 1 Implementation results comparison for real world datasets ModelWikipediaRedditCollegMsg THP17.133.7782.18 SAHP361.07427.2540.62 NNPP31.472.67105.8 GHNN116.264.47240.47 TGAT37.661.41188.36 TGN23.811.46170.91 HTNE12.721.61665.95 GAT-LSTM60.822.13406.65 DySAT22.790.86318.51 CTDNE22.510.82320.78 GNPP11.141.0829.75 Our Model10.60.5127.12 Table 2 Implementation results comparison for synthetic datasets ModelHawkes-negHawkes-posPoisson THP24.6121.797.54 SAHP11.622.8612.89 NNPP59.2614.909.86 GHNN38.2737.4318.04 TGAT35.9633.513.19 TGN12.0111.3612.51 HTNE16.0415.8728.55 GAT-LSTM85.5292.8485.40 DySAT9.2059.0315.57 CTDNE9.118.7915.79 GNPP2.428.212.50 Our Model2.212.21.9 5.5. Results on synthetic datasets Table 2 summarizes the results on three synthetic datasetsβHawkes-negative, Hawkes-positive, and Poissonβeach simulating distinct temporal dynamics. GTIN consistently achieves the lowest RMSE values across all settings, recording 2.21, 2.20, and 1.90, respectively. Compared to the strongest baseline, GNPP, our model reduces error rates by 8β70%, with the largest improvement observed in the Hawkes-positive dataset, where event dependencies are strongest. This illustrates GTINβs enhanced ability to model self-exciting behaviors and long-range temporal correlations. The superior results on the Poisson dataset, where events occur more independently, further demonstrate the flexibility of our framework. Unlike previous approaches that rely heavily on explicit event triggering assumptions, GTIN dynamically adapts to both dependent and independent temporal processes through its probabilistic formulation and joint modeling of structure and time. Collectively, the synthetic results confirm that GTIN generalizes across a spectrum of temporal dynamics, maintaining stability under both stochastic and bursty event generation processes, and offering a more unified approach to temporal graph prediction than existing methods. Overall, our results indicate that our model provides more accurate predictions and is particularly effective in environments with high event frequency. 5.6. Next-Event Location Prediction Table 3 reports the performance of GTIN in predicting the edge on which the next event occurs. This experiment evaluates the βwhereβ component of the proposed prediction task, which is not captured by the time RMSE. M. Ostadmohammadi et al.: Preprint submitted to ElsevierPage 9 of 15 Joint Event and Time Prediction Table 3 Next-event location prediction results. Higher values are better. DatasetHits@1Hits@10MRR Wikipedia 0.17 0.42 0.21 Reddit 0.09 0.27 0.13 CollegeMsg 0.22 0.44 0.28 Hawkes-negative 0.18 0.39 0.22 Hawkes-positive 0.19 0.37 0.28 Poisson 0.13 0.32 0.21 5.7. Analysis of the Source of Performance Improvements To investigate the source of GTINβs performance improvement, we analyze the results according to the amount of recent temporal activity around the true event edge. This analysis is motivated by the main difference between GTIN and GNPP. GTIN constructs a global temporal representation by propagating information between structurally related edges, whereas GNPP relies more strongly on the history of the target edge and its restricted neighborhood. Therefore, if cross-edge temporal information is responsible for the observed improvement, the performance difference should be more pronounced for events surrounded by lower neighboring-edge activity. For every test event (ν‘ ν ,ν ν ), we define the recent neighboring-edge activity as ν΄ ν (Ξ) = β νβξΊ(ν ν ) | | | | ν‘ (ν) ν | | | ν‘ ν β Ξ < ν‘ (ν) ν < ν‘ ν | | | | ,(1) where ξΊ(ν ν ) is the set of edges that share at least one endpoint with edge ν ν , and Ξ is the length of the historical observation window. We divide the test events into four equally sized groups according to ν΄ ν (Ξ). The first quartile, Q1, contains events with the highest neighboring activity, while Q4 contains events with the lowest neighboring activity. As shown in Fig. 1, the increasing performance difference across the activity groups provides insight into the source of GTINβs advantage. The larger improvement in low-activity groups indicates that events occurring on global edges contain useful information for predicting both the location and occurrence time of the next event. GTIN can incorporate this information through temporal message passing and its global graph representation, while GNPP cannot exploit it to the same extent. 6. Extension via Deep Graph Infomax and Balanced Subgraph Clustering To enrich our temporal event prediction framework, we introduce an unsupervised representation learning extension using Deep Graph Infomax (DGI) VeliΔkoviΔ, Fedus, Hamilton, LiΓ², Bengio and Hjelm (2018), followed by subgraph generation via balanced k-means clustering. This extension is motivated by the need to capture high- quality structural features and improve scalability when operating on large temporal graphs. By embedding nodes into a latent space that captures both local and global structure and partitioning the graph into approximately equal-sized subgraphs, we enable more efficient, localized temporal modeling. 6.1. Motivation and Overview The main limitation of our approach is the time complexity of training and inference. To overcome this issue, we use a combined approach: we construct balanced-size subgraphs of the main graph and apply the main method to each subgraph separately. Our approach addresses this by combining: β’ Deep Graph Infomax (DGI): to learn unsupervised node embeddings that preserve the mutual information between individual nodes and the overall graph context. β’ Balanced k-means clustering: to partition nodes into structurally coherent and size-constrained subgraphs that support parallel processing and localized prediction tasks. This modular pipeline allows us to generalize the learning process across large-scale temporal graphs while maintaining robustness and prediction fidelity. M. Ostadmohammadi et al.: Preprint submitted to ElsevierPage 10 of 15 Joint Event and Time Prediction 6.2. Graph Representation Learning via DGI Let ξ³ = (ξ,ξ±) denote the input graph, where ξ is the set of ν nodes, and ξ± is the set of edges. Each node ν£ ν β ξ is associated with a feature vector ν± ν β β ν . Let ν β β νΓν denote the full node feature matrix, and ν β β νΓν the adjacency matrix. DGI uses a graph encoder ξ± ν , typically implemented as a Graph Convolutional Network (GCN), to produce embeddings ν β β νΓν β² : ν = ξ± ν (ν, ν), where each row ν‘ ν represents the embedding of node ν£ ν . To encourage the encoder to retain informative and discriminative features, DGI introduces a discriminator that maximizes mutual information between the patch (node-level) representations and a global summary vector ν¬. The summary vector is obtained via a readout function: ν¬ = READOUT(ν) = ν ( 1 ν ν β ν=1 ν‘ ν ) , where ν(β ) is a non-linear activation function. DGI optimizes the following binary classification objective: ξΈ DGI = β ν β ν=1 [ logν(ν‘ β€ ν νν¬) + log(1 β ν( Μ ν‘ β€ ν νν¬)) ] , where Μ ν is a corrupted version of the node embeddings (e.g., via feature shuffling), and ν is a trainable weight matrix of the discriminator. The loss encourages real node embeddings to align with the global context while forcing corrupted ones to be distinguishable. 6.3. Balanced K-means for Subgraph Generation After obtaining the node embeddings ν, we aim to cluster them into ν groups of approximately equal size, producing node partitions νΆ 1 ,νΆ 2 ,...,νΆ ν such that β ν νΆ ν = ξ and νΆ ν β© νΆ ν = β for ν β ν. To preserve computational balance across partitions and ensure uniform coverage of the graph structure, we formulate a balanced k-means objective: min νΆ ν ν ν=1 ν β ν=1 β ν‘ ν βνΆ ν βν‘ ν βν ν β 2 , subject to | | | | |νΆ ν | β ν ν | | | | β€ ν, βν β 1,...,ν whereν ν = 1 |νΆ ν | β ν‘ ν βνΆ ν ν‘ ν is the centroid of cluster νΆ ν , and ν is a user-defined size tolerance. This constrained optimization can be implemented via heuristic search, integer programming, or recent approxi- mations using projection-based or regularized variants of standard k-means. 6.4. Benefits and Integration The combination of DGI and balanced k-means brings several advantages to our framework: β’ Scalability: The graph is decomposed into balanced subgraphs, allowing parallel temporal modeling and faster training. β’ Representation richness: DGI captures complex dependencies and patterns in a task-agnostic fashion. In the next stage of our pipeline, each subgraph is fed into the event prediction module, where both the next event type and its occurrence time are jointly predicted using our temporal model described. M. Ostadmohammadi et al.: Preprint submitted to ElsevierPage 11 of 15 Joint Event and Time Prediction Table 4 Runtime comparison of GTIN with and without DGI Model/MetricWikipediaRedditCollegeMsg GTIN training (s)1680.007350.00620.00 GTIN + DGI training (s)655.202352.00297.60 Training reduction61%68%52% Training speedup2.56Γ3.13Γ2.08Γ GTIN inference (s)145.00530.0052.00 GTIN + DGI inference (s)60.90180.2026.00 Inference reduction58%66%50% Inference speedup2.38Γ2.94Γ2.00Γ 6.5. Results for the added pipeline As illustrated in 2, incorporating the Deep Graph Infomax (DGI) and balanced k-means clustering pipeline yields a favorable trade-off between predictive accuracy and computational efficiency. While the Root Mean Square Error (RMSE) slightly increases compared to the original GTIN, the performance remains superior to all baseline methods, confirming that the pipeline successfully preserves the modelβs predictive strength while significantly improving scalability. The marginal rise in RMSE is offset by substantial reductions in both training and inference time, attributed to the parallelized processing of balanced subgraphs. A closer inspection of the results in 2 shows that the proposed pipeline maintains high predictive fidelity across all evaluated datasets, with particularly stable performance on denser graphs such as Reddit and Wikipedia. In these datasets, subgraph partitioning helps isolate localized temporal dependencies, allowing the model to process smaller and structurally coherent components without losing global contextual information. Conversely, in sparser datasets such as CollegeMsg, the gain in computational efficiency is more pronounced, albeit with a minor increase in error, likely due to reduced cross-subgraph interaction information. To evaluate the computational benefit of the proposed DGI-based pipeline more explicitly, we compare the runtime of the original GTIN model with the runtime of GTIN after adding Deep Graph Infomax and balanced k-means clustering. The comparison is performed for both training and inference, because the DGI-based pipeline introduces an additional representation-learning and clustering stage, but reduces the cost of the main temporal prediction module by applying it to smaller balanced subgraphs instead of the entire graph. Table 4 reports the runtime comparison on the three real-world datasets. The runtime values are reported in seconds. Since the datasets have different sizes, the original GTIN runtime is different for each dataset. Reddit has the largest runtime because it contains the largest number of temporal edges, followed by Wikipedia and CollegeMsg. The speedup is computed as follows: Speedup = Runtime of GTIN Runtime of GTIN + DGI As shown in Table 4, the DGI-based pipeline consistently reduces the runtime of GTIN across all datasets. The improvement is more evident on larger datasets such as Reddit and Wikipedia, where applying temporal message passing and candidate event scoring to the whole graph is more computationally expensive. In contrast, CollegeMsg is smaller, and therefore the relative reduction is slightly lower. Overall, these results demonstrate that the addition of the DGI and balanced clustering stages effectively extends GTIN to large-scale temporal graphs. The new pipeline achieves a balanced compromise between accuracy and efficiency, making the framework more practical for real-world scenarios where computational resources or latency constraints are critical. This enhancement underscores the versatility of GTIN as a unified temporal modeling architecture capable of maintaining robust predictive power even under partitioned or distributed graph processing. 6.6. Other methods for clustering In addition to our primary approach using Deep Graph Infomax (DGI) with balanced k-means, we consider two alternative methods to generate clusters of roughly β ν nodes for comparison. DeepWalk with balanced k-means learns low-dimensional node embeddings by simulating truncated random walks on the graph and applying a Skip-gram model, capturing both local and global structural patterns. The embeddings are M. Ostadmohammadi et al.: Preprint submitted to ElsevierPage 12 of 15 Joint Event and Time Prediction then clustered using balanced k-means, ensuring approximately equal cluster sizes while preserving meaningful graph structure. Randomized graph partitioning serves as a simple baseline, where nodes are initially assigned to clusters at random, and assignments are iteratively adjusted using the graphβs edge information to minimize inter-cluster connectivity. This approach naturally enforces the desired cluster size of β ν nodes and provides a computationally efficient contrast to embedding-based methods, enabling evaluation of the benefits of learned representations versus random partitioning. Results for the two new clustering methods with comparison to DGI is as shown in 3: It is obvious that any partitioning method that preserve the intrinsic clustering of the main graph is more beneficial in this manner. This may be a future direction to find the best possible clustering method. It is worthwhile to mention that omitting the restriction of equal cluster size and soften it may help to improve the efficiency as well. 6.7. Effect of ignoring attention-based aggregation We performed an ablation study to evaluate the contribution of the attention mechanism in our graph neural network model. In this experiment, we replaced the attention-based neighbor aggregation with a simple mean aggregation, where all neighboring nodes contribute equally to the updated node representation. This approach eliminates the modelβs ability to assign varying levels of importance to different neighbors based on their features or structural relevance. By doing so, we aim to quantify the impact of attention on capturing nuanced relationships within the graph and to establish a baseline for comparison against the full attention-based model. This setup allows us to examine whether the learned weighting of neighbors significantly influences the quality of the node embeddings and the downstream task performance. Our results (4) show that the simple averaging approach leads to a consistent drop in performance. This indicates that the attention mechanism plays a key role in selectively amplifying the contributions of more informative or relevant neighbors, thereby improving the representational capacity of the model. Moreover, the degradation in performance is particularly pronounced for nodes in denser or more heterogeneous regions of the graph, suggesting that attention helps navigate complex local structures that uniform aggregation cannot capture. These findings underscore the importance of attention in graph neural networks and provide empirical evidence supporting its use for effectively modeling relational dependencies in graph-structured data. 7. Discussion and Limitations Our proposed Global Temporal Interaction Network (GTIN) introduces a novel approach to predicting temporal events across complex, dynamic networks. By incorporating a message-passing framework alongside temporal point processes, our model captures interactions at a global scale rather than focusing solely on individual edges. The experimental results demonstrate that GTIN achieves superior predictive accuracy compared to existing models, particularly in datasets with high event frequency. However, while our model presents several advantages, it also has certain limitations that should be addressed in future research. 7.1. Homogeneity Assumption in Edge Representation One of the primary limitations of our work is the assumption of homogeneity across edges. Our message-passing aggregation function treats all edges as equivalent, disregarding their specific types or contextual relationships. In real- world applications, however, interactions in temporal graphs are often heterogeneousβdifferent types of connections (e.g., social interactions, financial transactions, or communication exchanges) exhibit distinct dynamics. By failing to differentiate between edge types, our model may overlook critical structural properties, leading to loss of contextual information and potential inaccuracies in prediction. 7.1.1. Future Directions: β’ Incorporating edge-type embeddings or relational graph neural networks (RGNNs) could help distinguish between different edge interactions. β’ Implementing an attention mechanism to weigh edges differently based on their significance. 7.2. Exclusion of Node Features Currently, our model does not incorporate node-specific attributes such as user demographics, entity properties, or contextual embeddings. Many real-world datasets include rich metadata associated with nodes, which could provide M. Ostadmohammadi et al.: Preprint submitted to ElsevierPage 13 of 15 Joint Event and Time Prediction valuable insights into interaction patterns. The omission of this information results in potential data loss, as certain node features may be strong predictors of future interactions. 7.2.1. Future Directions: β’ Extending our framework to incorporate node feature embeddings into the message-passing function. β’ Exploring a hybrid architecture that combines graph-based structural learning with node-level feature learning. 8. Conclusion In this work, we proposed the Global Temporal Interaction Network (GTIN), a novel framework for predicting temporal events across dynamic graphs. Unlike previous studies that focus primarily on event prediction at the edge level, our approach captures interactions occurring anywhere in the graph, leading to improved accuracy in predicting event timing. By integrating a message-passing framework with temporal point processes, our model efficiently propagates information across the graph, enhancing event-prediction capabilities. Experimental results demonstrate that GTIN consistently outperforms existing methods, particularly in datasets with high event frequency, reinforcing its effectiveness in modeling complex temporal dependencies. Despite these advances, our work assumes homogeneous edge representations, which may not fully capture the diversity of interactions in real-world networks. Future improvements should incorporate heterogeneous edge types and node-specific attributes to further refine prediction accuracy. Additionally, while our model excels in dense event environments, its performance in sparse interaction settings could be enhanced by integrating self-supervised learning techniques and data augmentation strategies. Moreover, addressing computational scalability will be crucial for applying GTIN to large-scale networks. Future research should explore distributed computing approaches and adaptive learning mechanisms to extend the applicability of the model. 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Embedding temporal network via neighborhood formation, in: Proceedings of the 24th ACM SIGKDD international conference on knowledge discovery & data mining, p. 2857β2866. M. Ostadmohammadi et al.: Preprint submitted to ElsevierPage 15 of 15 Joint Event and Time Prediction Figure 1: Performance (RMSE) of GTIN and GNPP according to recent neighboring-edge activity in the Hawkes-pos dataset. Test events are divided into quartiles based on the number of events observed on neighboring edges during the preceding interval of length Ξ. Figure 2: Comparison of RMSE after adding the pipeline M. Ostadmohammadi et al.: Preprint submitted to ElsevierPage 16 of 15 Joint Event and Time Prediction Figure 3: Different clustering methods Figure 4: Effect of removing attention M. Ostadmohammadi et al.: Preprint submitted to ElsevierPage 17 of 15