Time-decay and frequency-domain transformation for temporal knowledge graph completion

Abstract

Temporal knowledge graph completion aims to predict missing entities and relationships over time. However, traditional methods often fail to effectively capture temporal dynamics and frequency-domain features. They struggle to properly weigh historical information across varying time intervals and overlook periodic patterns inherent in the data. To address these shortcomings, we introduce TDFT, a novel model that integrates time decay factors and frequency-domain transformations. TDFT incorporates a time decay mechanism to prioritize relevant historical information and uses frequency-domain transformations to uncover latent periodic features. These features are learned via a frequency-domain neural network, improving the model’s ability to model periodic fluctuations in entities and relationships. Experimental results show that TDFT outperforms existing methods on multiple benchmark datasets.

Keywords

temporal knowledge graph time decay factor frequency domain transformation

1. Introduction

Knowledge graphs (KGs) are structured representations of knowledge that have been widely applied in various domains, such as recommendation systems^1–3, search engines,^4,5 and question-answering systems.^6–8 Each fact in a knowledge graph is represented as a triple ( $e_{s}$ , $r$ , $e_{o}$ ), where $e_{s}$ (subject entity) and $e_{o}$ (object entity) are the nodes (entities), and $r$ represents the edge (relationship) connecting es and $e_{o}$ , thereby storing and representing knowledge in a graph-based structure. With the rapid advancement of artificial intelligence technologies, knowledge graphs are playing an increasingly critical role in enhancing machine understanding, reasoning, and decision-making capabilities.

However, knowledge graphs in the real world are often incomplete due to manual or automatic construction methods, with missing or erroneous entities and relationships. This incompleteness adversely impacts the effectiveness of knowledge reasoning and application. As a result, knowledge graph completion (KGC) has emerged as a vital area of research. The goal of KGC is to infer and supplement missing entity or relationship information using the existing partial knowledge, thereby improving the completeness and accuracy of the knowledge graph.

In the study of KGC, traditional methods predominantly rely on static graph structure information to capture the relationships between entities, typically using embedding techniques or graph neural networks (GNNs). However, these methods fail to account for the dynamic nature of knowledge graphs, especially in time-sensitive domains, limiting their performance. Over time, both entities and relationships evolve. For instance, the statement that Angela Merkel was the German Chancellor holds true only between 2005 and 2021. Such temporal changes place higher demands on knowledge graph completion techniques.

To address this issue, temporal knowledge graph (TKG) has been developed. Temporal knowledge graphs not only capture static entity and relationship information but also incorporate a time dimension, representing the dynamic evolution of entities and relationships over time. In TKG, each static knowledge graph triple ( $e_{s}$ , $r$ , $e_{o}$ ) is augmented with a timestamp to form a quadruple ( $e_{s}$ , $r$ , $e_{o}$ , $t$ ), where t denotes the timestamp, indicating the specific period when the fact holds true. Unlike traditional static knowledge graphs, TKG must consider the impact of time on entities and relationships while completing missing elements.

Despite the progress in TKG, existing methods still have several limitations. One major issue is the excessive influence of historical information. In temporal knowledge graph completion, the status of entities and relationships evolves over time. A significant challenge in model design is balancing the weight of historical versus current information to avoid the detrimental effects of outdated information on current predictions. Traditional temporal models, such as RNNs and LSTMs, may overlook this issue, resulting in either an overemphasis on historical data or a diminishing influence of earlier information, both of which hinder the model’s performance.

For instance, as shown in Figure 1, users tend to make purchases on items they have recently clicked, while items viewed a long time ago are less likely to be purchased. Consequently, shopping platforms should prioritize recommending recently clicked items.

Figure 1.

An example of a temporal knowledge graph.

Furthermore, while existing methods often focus on time-domain modeling, they tend to neglect the potential impact of frequency-domain features. Periodic or complex changes often manifest as frequency-domain features, which are typically difficult for conventional temporal models to capture directly. These periodic fluctuations, sudden events, and other nonlinear patterns are better modeled through frequency-domain analysis.For example, in Figure 1, users regularly purchase daily consumables such as shampoo and bread. The periodic nature of these purchases is better captured in the frequency domain.

To address these challenges, this paper proposes a temporal knowledge graph completion framework that integrates a time decay factor with a frequency-domain module. Specifically, the relationship time decay encoder assigns different weights to relationship dynamics at various timestamps through a time decay mechanism, while the entity time decay encoder captures the temporal evolution of entity characteristics. In addition, the frequency-domain transform decoder leverages the frequency-domain representation of temporal data to generate final prediction scores by combining historical time series scores via complex transformation and decoding operations. Our contributions are summarized as follows:

Time decay factor: To mitigate the problem of excessive or insufficient influence from historical data on current predictions, we design a dynamic time-decay mechanism. This mechanism adjusts the weight of historical information according to the characteristics of different time steps, thereby preventing over-reliance on outdated data. The time decay factor enables the model to dynamically adjust the significance of historical information, improving the performance of the model in temporal knowledge graphs.

Frequency domain feature modeling: By incorporating a frequency domain module, we enhance the model’s ability to capture periodic features and complex evolutionary patterns in temporal data. This approach improves the model’s capacity to model relationship evolution by effectively capturing periodic changes and other nonlinear trends, which are crucial for temporal knowledge graph completion.

Although time-decay mechanisms and frequency-domain transformations have been independently investigated in previous temporal modeling studies, their joint application to temporal knowledge graph completion remains largely unexplored. Our contribution lies not only in exploring novel decay and transform strategies, but also in the synergistic integration of these two paradigms—wherein the time-decay encoder adaptively regulates temporal influence, while the frequency-domain decoder effectively captures periodic regularities. This joint design enables the model to capture multi-scale temporal dynamics more effectively than either component alone.

The rest of the paper is organized as follows: Section 2 reviews related work. Section 3 presents the notation and details of our method. Experimental setup and results are provided in Section 4. Finally, Section 5 concludes the paper and outlines future research directions.

2. Related work

2.1. Static knowledge graph completion

Static Knowledge Graph Completion aims to predict the missing triples in a knowledge graph, typically leveraging graph embedding methods that map entities and relations into a low-dimensional space. Major approaches for static knowledge graph completion include entity representation learning, relation representation learning, and graph convolutional networks.

Static knowledge graph completion methods primarily focus on entity representation learning, such as TransE,⁹ TransH,¹⁰ and TransR.¹¹ These methods optimize entity and relation embeddings by vectorizing their representations using an objective function that minimizes the distance between true triples and maximizes the distance between false triples in the embedding space. However, these methods assume that relations are simple linear mappings, which limits their ability to capture more complex relationships. To better model intricate relations, subsequent studies introduced relation-based embedding techniques. For example, DistMult¹² represents relations as diagonal matrices, while ComplEx¹³ uses complex-valued embeddings to efficiently capture asymmetric relations. In this approach, both entities and relations are embedded in a complex space rather than the real space.

Recently, graph convolutional networks (GNNs) and tensor decomposition methods have been incorporated into knowledge graph completion. R-GCN¹⁴ was the first to apply a GCN model to knowledge graph completion, capturing nonlinear dependencies between entities and relations through convolutional operations. It incorporates graph convolution of multi-relationship types, allowing it to better handle multi-relational static graphs. In contrast, tensor decomposition methods decompose high-dimensional data into low-rank matrices or vectors to uncover underlying patterns and relationships. For instance, TuckER¹⁵ uses Tucker decomposition to model the knowledge graph as a three-dimensional tensor, extracting embedded representations of entities and relations through tensor decomposition.

2.2. Temporal knowledge graph completion

Research in Temporal Knowledge Graph Completion largely focuses on several approaches that combine the modeling strategies of static knowledge graphs with the dynamic properties of temporal information. These models aim to enhance the ability to represent and reason about dynamic changes over time.

TTransE¹⁶ extends the static TransE model by incorporating temporal information, enabling it to capture the dynamic properties of facts at different time points by adding an additional temporal dimension to the embedding space. HyTE¹⁷ uses low-dimensional embeddings for entities and relations, projecting facts onto a temporal hyperplane to model time-dependent changes by adjusting the embeddings through temporal constraints. Similarly, DE-SimplE¹⁸ augments the static SimplE¹⁹ model with diachronic embedding functions, enabling it to learn the dynamic properties of entities over time. TNTComplEx²⁰ extends the third-order tensor of ComplEx to a fourth-order tensor, generating time-dependent embeddings by decomposing the tensor.

TA-DistMult²¹ is a temporal version of DistMult, focusing on modeling temporal knowledge graphs through time-aware embeddings. It retains the simplicity of DistMult while capturing the temporal correlations of entities and relations. The RE-Net²² model encodes historical facts using a cyclic event encoder and aggregates neighboring information for entities in the current time slice through a neighborhood aggregator. RE-GCN²³ comprehensively models historical information by capturing both structural dependencies and sequential patterns of neighboring facts. The TANGO²⁴ model encodes both temporal and structural information of a historical knowledge subgraph using differential equations, obtaining continuous time embeddings while capturing structural dependencies with multi-relational graph convolution.

Furthermore, some models focus on capturing the uncertainty inherent in temporal data and embeddings. For example, ATiSE²⁵ applies an additive temporal decomposition approach to break down the temporal into trending, periodic, and irregular components, injecting temporal information into entity and relation embeddings. It also models embeddings using Gaussian distributions to account for dynamic changes and uncertainties in the temporal knowledge graph. BoxTE,²⁶ derived from BoxE,²⁷ employs multidimensional hyper-rectangles to represent entities and relations. By modeling the range of attributes and temporal correlations with box centers and boundary lengths, it captures the uncertainty of facts and relationships through the intersection of boxes.

In recent years, research has explored temporal knowledge graph completion approaches that transcend traditional embedding-based models to enhance interpretability and adaptability.

One prominent research direction focuses on reinforcement learning (RL)–based methods, such as RLPR,²⁸ LogiRL,²⁹ and RTTI.³⁰ These methods formulate temporal reasoning as a sequential decision-making problem, where an agent interacts with the graph environment to explore multi-hop relational paths constrained by temporal order and receive reward signals based on prediction accuracy.

Another emerging line of research centers on large language model (LLM)–based methods, including TV-LLM,³¹ LLM-DA,³² and GenTKG.³³ These methods leverage the semantic comprehension and contextual generalization capabilities of LLMs, integrating them with temporal knowledge graphs for language-guided temporal reasoning.

Existing temporal knowledge graph completion methods typically incorporate temporal information into the embedding space via additional temporal dimensions or projection functions. RL-based models treat temporal reasoning as sequential decision-making but require sophisticated reward design and extensive path exploration, leading to high computational costs. LLM-based methods rely heavily on external textual semantics and often suffer from poor structural consistency in graph representations. In contrast, our proposed TDFT model integrates adaptive time-decay encoding with frequency-domain reasoning, providing a complementary perspective that unifies temporal adaptation and frequency-aware relational modeling.

Despite these advances, the aforementioned approaches still face several challenges:

Insufficient temporal characterization: Many methods only model temporal changes by simply appending a time dimension, which fails to accurately capture information decay, periodic fluctuations, or abrupt changes.

Inadequate handling of dynamic dependencies: Although models like RE-Net²² and RE-GCN²³ integrate historical information with the current state, they often fall short in modeling long-term dependencies and periodic patterns.

Neglect of frequency domain information: Most approaches focus solely on the time domain, overlooking the spectral features that are crucial for capturing complex dynamic behaviors.

To overcome these limitations, our proposed algorithm introduces two key innovations. First, we incorporate a time decay factor that applies exponential weighting to historical information at each time step, allowing the model to adaptively balance long-term memory with recent observations, thereby better reflecting dynamic changes. Second, we introduce a frequency domain transformation module that uses sine functions to non-linearly map temporal information, effectively capturing periodic patterns and spectral features. With these two designs, our algorithm addresses the shortcomings of existing methods in modeling temporal dependencies and periodic dynamics, leading to a more efficient and robust framework for dynamic knowledge representation and prediction.

3. The proposed model

3.1. Model overview

Figure 2 illustrates the overall architecture of our model, which consists of two main components: the Time Decay Encoder and the Frequency Domain Transform Decoder. The Time Decay Encoder captures the dynamic evolution of relationship and entity representations by employing gated recurrent units (GRU) alongside a time decay factor that integrates historical context with current observations.

Figure 2.

Illustration of the proposed TDFT model.

For relationships, the GRU fuses current relational data with the previous state to update the relational embedding. For entities, a relational graph convolutional network (RGCN) aggregates neighborhood information, which is then combined with the prior state via a GRU to update the entity representation. In both cases, the time decay factor modulates the influence of historical data, ensuring that older events are gradually attenuated while recent events are prioritized.

In the Frequency Domain Transform Decoder, time-domain features are first converted to the frequency domain using the Fourier transform, processed by a dedicated neural network, and then transformed back via the inverse Fourier transform to yield the final predictions.

By integrating the time decay mechanism in both relational and entity updates, the Time Decay Encoder leverages dynamic graph modeling and time-series analysis to enhance representation quality, adaptability, and predictive performance on evolving temporal knowledge graphs.

3.2. Time decay encoder

In dynamic knowledge graphs, both relationships and entities evolve over time. Inspired by the success of temporal decay functions in time-series analysis and the recent advances in dynamic graph neural networks, we propose a unified temporal decay encoder that simultaneously captures the evolution of relational and entity representations. The central idea is to integrate a time decay mechanism into the update processes of both components, thereby ensuring that the influence of past events gradually diminishes while preserving critical historical context.

3.2.1. Time decay factor

To reflect the intuition that temporally distant events should exert a weaker influence on current representations, we design a learnable Gaussian-based decay function to adaptively modulate the temporal attenuation process:

α_{t} = \exp (- \frac{(t^{'} - μ)^{2}}{2 σ^{2}}), α_{t} \in [0.1, 0.9]

(1)

Inspired by Gaussian attenuation in physical and statistical systems, this factor serves as a tunable weighting term that governs the contribution of past events.

Here, $t^{'}$ represents a temporal scaling variable, which does not denote physical time but functions as a control term that adaptively regulates the decay intensity across different time intervals. Meanwhile, $μ$ and $σ$ denote the mean and standard deviation controlling the center and spread of the decay, respectively. A larger $σ$ results in slower temporal attenuation, whereas $μ$ determines the peak position of temporal influence.

The scaled temporal interval $t^{'}$ is further defined as:

t^{'} = τ \cdot i \cdot sequence\_len,

(2)

where

i

is the current time step,

sequence\_len

denotes the total number of time steps within the sequence, and

τ

is a learnable scaling parameter that adaptively adjusts the decay curve’s sensitivity to temporal intervals. This formulation enables the model to dynamically stretch or compress temporal influence according to the characteristics of the data distribution.

By constraining $α_{t}$ within $[0.1, 0.9]$ , he model achieves a balance between short-term responsiveness and long-term retention. An excessively large decay factor would overemphasize obsolete information, whereas an excessively small one might disregard valuable historical context.

We further compared several alternative decay functions, including exponential, sigmoid, polynomial, inverse, arctangent, and MLP-based variants. Among them, the Gaussian decay achieved the best trade-off among smoothness, adaptivity, and predictive accuracy.

Moreover, we observed that the proposed Gaussian-based time-decay factor inherently mitigates extreme temporal noise by constraining the effective temporal range within a bounded window. This design allows the model to maintain stable representations even under irregular or abnormal time intervals, thereby highlighting its robustness against noisy temporal inputs.

3.2.2. Relational time decay encoder

The relational embedding is updated at each time step using recurrent units to capture evolving temporal dependencies among relations. After comparing different recurrent structures, including GRU, LSTM, BiGRU, BiLSTM, and Transformer-based encoders, we found that GRU achieves the most stable convergence and best predictive performance. Hence, we adopt GRU as the final choice due to its efficiency and ability to mitigate vanishing gradients in long sequences.

The update formula for the relational embedding is as follows:

R_{t} = (1 - α_{t}) \cdot G R U_{r} (R_{t}^{'}, R_{t - 1}) + α_{t} \cdot R_{e m b}

(3)

Where

G R U_{r}

denotes the gated recurrent unit for dynamic relationship updating,

R_{e m b}

represents the initial embedding of the relationship,

R_{t}^{'}

denotes the aggregated feature vectors for each relationship type, and

α_{t}

represents the time decay factor. This design enables the model to continuously adapt to new relational information while maintaining context from prior states.

3.2.3. Entity time decay encoder

Entities in a Temporal knowledge graph also exhibit temporal evolution. To capture this, we integrate a relational graph convolution network (RGCN) with GRU units. The RGCN first aggregates information from neighbors and relations:

h_{o, t}^{l + 1} = φ (\sum_{(s, r, o) \in ρ_{t}} \frac{1}{c_{o}} W_{r}^{l} (ζ (h_{s, t}^{l}, r_{t})) + W_{o}^{l} h_{o, t}^{l})

(4)

Where

h_{s, t}^{l}, h_{o, t}^{l}

denote the

l

th level embedding of entity

s, o

t

W_{r}^{l}, W_{o}^{l}

denote the learnable weights,

W_{r}^{l}

is relation-specific,

c_{o}

is the incidence with normalization factor equal to

o

φ

is the RReLU activation function, and

ζ

is the one-dimensional convolution operator. The entity embeddings are then updated via a GRU unit that integrates the RGCN output with the previous entity state:

H_{t} = (1 - α_{t}) \cdot G R U_{e} (H_{t - 1}^{G C N}, H_{t - 1}) + α_{t} \cdot H_{static}

(5)

Where

G R U_{e}

is the gated recurrent unit for entity update,

H_{static}

represents the static graph embedding of entities,

H_{t - 1}^{G C N}

is the entity embedding processed by the RGCN at time

t - 1

, and

α_{t}

represents the time decay factor.

3.3. Frequency domain transform decoder

The frequency domain transform decoder enhances the representation of time series data by incorporating frequency-domain features. The decoder utilizes a frequency domain transform model to process temporal features, which includes time embedding, frequency domain transformation, historical time processing, and prediction generation. The time information is initially converted into discrete time indices using the following formula:

T_{t}^{i d x} = ⌊ \frac{T_{t}^{r a w}}{Δ t} ⌋

(6)

Where

T_{r a w}

is the original timestamp and

Δ t

is the time interval (e.g., days, hours, etc.). By applying this formula, the timestamp is converted into a discrete time index.

Next, the discrete time index is mapped to the frequency domain space via the frequency domain transform, where a neural network processes the time index to extract frequency domain information. The process proceeds as follows:

First, the time series is transformed into the frequency domain using the Fast Fourier Transform (FFT):

T_{f} = F F T (T_{t}^{i d x})

(7)

Where

T_{f}

represents the frequency domain signal, and FFT denotes the Fast Fourier Transform operation.

The frequency domain results are normalized by subtracting the mean and dividing by the standard deviation:

\begin{aligned} T_{mean} & = \frac{1}{L} \sum_{i}^{L} T_{f}^{i} \end{aligned}

(8)

\begin{aligned} T_{stdev} & = \sqrt{\frac{1}{L} \sum_{i = 1}^{L} {(| T_{f}^{i} | - T_{mean})}^{2}} \end{aligned}

(9)

\begin{aligned} T_{f}^{norm} & = \frac{T_{f} - T_{mean}}{T_{stdev}} \end{aligned}

(10)

Where

T_{mean}

is the mean of the frequency domain signal and

T_{stdev}

is the standard deviation of the frequency domain signal.

The normalized frequency domain signal is then fed into a neural network and processed through multiple layers of complex encoders with the aim of learning patterns in the frequency domain signal to extract frequency domain information:

{\hat{T}}_{f}^{norm} = N N (T_{f}^{norm})

(11)

The predictions are then denormalized to restore them to their original scale:

T_{f}^{'} = {\hat{T}}_{f}^{norm} \cdot T_{stdev} + T_{mean}

(12)

Finally, the inverse Fast Fourier Transform (IFFT) is applied to convert the frequency domain signal back to the time domain, retaining only the predicted portion of the signal.

{\hat{T}}_{t}^{idx} = I F F T (T_{f}^{'})

(13)

After being processed by the frequency domain transform, the processed time indexes are then converted into linear time feature vectors and cyclic time feature vectors, respectively, by the time embedding function:

\begin{aligned} T_{t}^{l t} & = W_{1} \cdot {\hat{T}}_{t}^{idx} + b_{1} \end{aligned}

(14)

\begin{aligned} T_{t}^{c t} & = \sin (W_{2} \cdot {\hat{T}}_{t}^{idx} + b_{2}) \end{aligned}

(15)

Where

W_{1}, W_{2}

and

b_{1}, b_{2}

are the weights and biases obtained during training. After obtaining the time vectors, we modeled entities, relationships, and temporal embeddings using Time-ConvTransE and Time-ConvTransR³⁴ and scored them according to the following equation

\begin{aligned} P_{score}^{o} & = Φ (TimeConvTransE (h_{t}^{s}, r_{t}, T_{t}^{l t}, T_{t}^{c t})) \end{aligned}

(16)

\begin{aligned} P_{score}^{r} & = Φ (TimeConvTransR (h_{t}^{s}, h_{t}^{o}, T_{t}^{l t}, T_{t}^{c t})) \end{aligned}

(17)

Where

Φ

represents the softmax function,

h_{t}^{s}, r_{t}, h_{t}^{o}

represent subject entity embedding, relation embedding, and object entity embedding respectively,

T_{t}^{l t}, T_{t}^{c t}

represent linear time vector embedding and cyclic time vector embedding.

The scores for entities and relationships are then calculated by summing the raw scores $P_{score}^{o}, P_{score}^{r}$ and historical scores $P_{history}^{o}, P_{history}^{r}$ through the variable $β, γ$ control ratio:

\begin{aligned} P_{total}^{o} & = β \cdot P_{score}^{o} + (1 - β) \cdot P_{history}^{o} \end{aligned}

(18)

\begin{aligned} P_{total}^{r} & = γ \cdot P_{score}^{r} + (1 - γ) \cdot P_{history}^{r} \end{aligned}

(19)

Finally, the entity and relationship scores are combined by weighting and summing them with a variable $λ$ to obtain the total score:

P_{total} = λ \cdot P_{total}^{o} + (1 - λ) \cdot P_{total}^{r}

(20)

We treat the entity and relationship prediction problem as a multi-label learning task, where the total loss functions for entity and relationship prediction are given by:

L = L_{e} + L_{r} = \sum y_{t}^{e} \log P_{total}^{o} + \sum y_{t}^{r} \log P_{total}^{r}

(21)

Where

P_{total}^{o}

and

P_{total}^{r}

are the final scores for entity prediction and relationship prediction respectively,

y_{t}^{e}

and

y_{t}^{r}

are the label vectors for entity prediction task and relationship prediction task, and the elements of

y_{t}^{e}

and

y_{t}^{r}

are 1 when the fact occurs, and 0 when it does not.

Although the proposed frequency-domain transformation enhances periodic pattern modeling, it inherently assumes local temporal smoothness. In practice, temporal knowledge graphs often exhibit non-stationary dynamics, where event intensity and relation occurrence vary significantly over time. To mitigate potential spectral distortion under such conditions, our model applies normalization and adaptive time-decay weighting before transformation, ensuring that temporal signals are stabilized prior to frequency mapping.

The specific process steps of the algorithm are shown below:

4. Experiments

4.1. Dataset

We utilize three distinct temporal knowledge graph (TKG) datasets to evaluate the model’s performance in the temporal knowledge graph completion task. These datasets include ICEWS14,³⁶ ICEWS18,²² and WIKI.¹⁶

ICEWS14: This dataset is derived from the Integrated Crisis Early Warning System (ICEWS) and records global political events from 2014. With a time granularity of days, the dataset comprises 6,869 entities and 230 relationships.

ICEWS18: Also sourced from ICEWS, this dataset documents global political events from 2018, with a daily time granularity. It includes 23,033 entities and 256 relationships.

WIKI: This dataset, drawn from Wikipedia, Wikimedia Commons, and other Wikimedia projects, supports a wide range of granularities. It contains 12,554 entities and 24 relationships. These datasets span different domains and time periods, providing a diverse context to assess the model’s performance.

4.2. Temporal encoder architecture

We evaluated several temporal sequence modeling architectures under consistent experimental settings, including GRU, BiGRU, LSTM, BiLSTM, and Transformer networks. As shown in Figure 3(left 1), the GRU-based encoder consistently achieved the best overall performance. Compared with LSTM and BiLSTM, GRU achieved higher accuracy while requiring fewer parameters, demonstrating its efficiency in capturing temporal dependencies within dynamic graphs. Although Transformer-based models performed competitively on large-scale datasets, their higher computational cost and weaker generalization on smaller datasets led us to adopt GRU as the temporal encoder in the final framework.

Figure 3.

Performance (in percentage) of various temporal Encoder Architecture and time Decay Functions with ICEWS14.

4.3. Time decay functions

To further analyze the influence of various temporal decay mechanisms, we compared several functional forms, including exponential, linear, polynomial, and Gaussian decay. As shown in Figure 3(right 1), the Gaussian decay consistently achieves the best overall performance, showing more stable convergence and stronger generalization capability. This superiority arises from its smooth weighting properties, which effectively prevent abrupt truncation of historical information and enhance temporal continuity. Therefore, Gaussian decay is adopted as the default temporal decay mechanism in the final TDFT model.

4.4. Evaluation metrics

In the temporal knowledge graph reasoning task, key evaluation metrics include Mean Reciprocal Rank MRR,³⁷ Hits@1, Hits@3, and Hits@10,³⁷ which are used to assess the accuracy of entity and relationship predictions. MRR calculates the average of the ranked inverse of correct answers across all queries, with higher scores indicating better ranking of correct answers. Hits@K (where K can be 1, 3, or 10) measures the percentage of correct answers within the top K predicted results, offering a direct assessment of the model’s accuracy at high ranks. In our experiments, we use filter MRR, Hits@1, Hits@3, and Hits@10 as evaluation metrics.

4.5. Baseline models

We perform a comprehensive comparison of our model’s performance with several classical models from recent years, spanning three categories: static TKG inference models, interpolated TKG inference models, and extrapolated TKG inference models.

Interpolated TKG inference models: We select state-of-the-art models, including HyTE,¹⁶ TTransE,¹⁵ TA-DistMult,²¹ DE-SimplE,¹⁸ and TNTComplEx.²⁰

Extrapolated TKG inference models: We include cutting-edge models such as RGCRN,³⁵ RE-NET,²² CyGNet,³⁸ TANGO,²⁴ xERTE,³⁹ RE-GCN,²³ TITer,⁴⁰ TECHS,⁴¹ BH-TDEN,⁴² TIRGN,³⁴ PPT,⁴³ STE-GCN⁴⁴ and CEGRL-TKGR.⁴⁵

4.6. Hyperparameters

For all datasets, we set the embedding dimension to 200. The history lengths was set to 9, 10, and 1 for ICEWS14, ICEWS18, and WIKI, respectively. The Gaussian decay parameter $σ$ was set to 0.1, 0.1, and 0.3 correspondingly. The dropout rate for each layer was set to 0.2, and the number of GCN layers was fixed at 2. For the decoder, we set the number of convolutional kernels to 50 and their size to $4 \times 3$ . The model parameters were optimized using the Adam³⁵ optimizer, with a learning rate of 0.001.The hardware setup is supported by an A100 and seven 4060ti graphics cards.

4.7. Entity prediction results

Table 1 present the performance of various methods in the entity prediction task. Optimal results are highlighted in bold, and sub-optimal results are underlined. Our proposed method consistently outperforms all baseline models across the three benchmark datasets. Notably, compared to models that only account for temporal features or simplistic relationship modeling (such as xERTE, RGCRN, TANGO, TITer, and TECHS), our model captures both temporal and frequency domain features through frequency domain transformations, leading to superior performance.

Table 1.
Performance (in percentage) for the entity prediction task on ICEWS18, ICEWS14 and ICEWS05-15 with raw metrics.

ICE14 ICE18 WIKI

Model MRR H@1 H@3 H@10 MRR H@1 H@3 H@10 MRR H@1 H@3 H@10

TTransE $^{1}$ [2018b] 13.43 3.11 17.32 34.55 8.31 1.92 8.56 21.89 29.27 21.67 34.43 42.39

TA-DistMult $^{1}$ [2018] 26.47 17.09 30.22 45.41 16.75 8.61 18.41 33.59 44.53 39.92 48.73 51.71

DE-SimplE $^{1}$ [2020] 32.67 24.43 35.69 49.11 19.30 11.53 21.86 34.80 45.43 42.6 47.71 49.55

TNTComplEx $^{1}$ [2020] 32.12 23.35 36.03 49.13 27.54 19.52 30.80 42.86 45.03 40.04 49.31 52.03

RGCRN $^{2}$ [2018] 38.48 28.52 42.85 58.10 28.02 18.62 31.59 46.44 65.79 61.66 68.17 72.99

RE-NET $^{2}$ [2020] 39.86 30.11 44.02 58.21 29.78 19.73 32.55 48.46 58.32 50.01 61.23 73.57

CyGNet $^{2}$ [2021] 37.65 27.43 42.63 57.90 27.12 17.21 30.97 46.85 58.78 47.89 66.44 78.70

TANGO $^{2}$ [2021b] – – – – 28.97 19.51 32.61 47.51 53.04 51.52 53.84 55.46

xERTE $^{2}$ [2021a] 40.79 32.70 45.67 57.30 29.31 21.03 33.51 46.48 73.60 69.05 78.03 79.73

RE-GCN $^{2}$ [2021b] 42.00 31.63 47.20 61.65 32.62 22.39 36.79 52.68 78.53 74.50 81.59 84.70

TITer $^{2}$ [2021] 41.73 32.74 46.46 58.44 29.98 22.05 33.46 44.83 73.91 71.70 75.41 76.96

TiRGN $^{2}$ [2022] 44.04 33.83 48.95 63.84 33.66 23.19 37.99 54.22 81.65 77.77 85.12 87.08

PPT $^{3}$ [2023] 38.42 28.94 – 57.01 26.63 16.94 – 45.43 – – – –

TECHS [2023] 43.88 34.59 49.36 61.95 30.85 21.81 35.39 49.82 75.98 – – 82.39

BH-TDEN [2024] 39.30 29.70 44.00 58.10 28.90 19.10 32.60 48.70 – – – –

STE-GCN $^{3}$ [2024] 43.26 33.71 – 61.47 32.67 22.80 – 51.60 – – – –

CEGRL-TKGR [2025] 42.74 32.32 48.05 62.68 32.90 22.70 36.91 52.95 79.66 75.73 82.83 85.59

TDFT 44.41 34.14 49.48 64.14 33.96 23.45 38.41 54.47 81.93 78.20 85.38 86.96

	ICE14	ICE18	WIKI
TTransE $^{1}$ [2018b]	13.43	3.11	17.32	34.55	8.31	1.92	8.56	21.89	29.27	21.67	34.43	42.39
TA-DistMult $^{1}$ [2018]	26.47	17.09	30.22	45.41	16.75	8.61	18.41	33.59	44.53	39.92	48.73	51.71
DE-SimplE $^{1}$ [2020]	32.67	24.43	35.69	49.11	19.30	11.53	21.86	34.80	45.43	42.6	47.71	49.55
TNTComplEx $^{1}$ [2020]	32.12	23.35	36.03	49.13	27.54	19.52	30.80	42.86	45.03	40.04	49.31	52.03
RGCRN $^{2}$ [2018]	38.48	28.52	42.85	58.10	28.02	18.62	31.59	46.44	65.79	61.66	68.17	72.99
RE-NET $^{2}$ [2020]	39.86	30.11	44.02	58.21	29.78	19.73	32.55	48.46	58.32	50.01	61.23	73.57
CyGNet $^{2}$ [2021]	37.65	27.43	42.63	57.90	27.12	17.21	30.97	46.85	58.78	47.89	66.44	78.70
TANGO $^{2}$ [2021b]	–	–	–	–	28.97	19.51	32.61	47.51	53.04	51.52	53.84	55.46
xERTE $^{2}$ [2021a]	40.79	32.70	45.67	57.30	29.31	21.03	33.51	46.48	73.60	69.05	78.03	79.73
RE-GCN $^{2}$ [2021b]	42.00	31.63	47.20	61.65	32.62	22.39	36.79	52.68	78.53	74.50	81.59	84.70
TITer $^{2}$ [2021]	41.73	32.74	46.46	58.44	29.98	22.05	33.46	44.83	73.91	71.70	75.41	76.96
TiRGN $^{2}$ [2022]	44.04	33.83	48.95	63.84	33.66	23.19	37.99	54.22	81.65	77.77	85.12	87.08
PPT $^{3}$ [2023]	38.42	28.94	–	57.01	26.63	16.94	–	45.43	–	–	–	–
TECHS [2023]	43.88	34.59	49.36	61.95	30.85	21.81	35.39	49.82	75.98	–	–	82.39
BH-TDEN [2024]	39.30	29.70	44.00	58.10	28.90	19.10	32.60	48.70	–	–	–	–
STE-GCN $^{3}$ [2024]	43.26	33.71	–	61.47	32.67	22.80	–	51.60	–	–	–	–
CEGRL-TKGR [2025]	42.74	32.32	48.05	62.68	32.90	22.70	36.91	52.95	79.66	75.73	82.83	85.59
TDFT	44.41	34.14	49.48	64.14	33.96	23.45	38.41	54.47	81.93	78.20	85.38	86.96

$^{1}$ The results are taken from.⁴⁰ $^{2}$ The results are taken from.³⁴ $^{3}$ The results are taken from.⁴⁶ Others are taken from the original paper.

RE-NET uses recurrent neural networks to model temporally adjacent entity relationships, while our model employs an adaptive time-gated network junction unit, which allows the model to flexibly capture feature dependencies across different time steps.

Compared to classical models like RE-GCN and TIRGN, our model enhances temporal evolution modeling by incorporating a time decay factor. This factor enables the model to weight events based on the duration between them, effectively capturing both long-term and short-term dynamic features.

BH-TDEN, which learns temporal embeddings via autoregressive time gates and a neighborhood relation updating mechanism, is less effective in capturing global periodic patterns. In contrast, our model’s frequency domain transformation module excels at identifying both periodic and global patterns in time-series data, significantly improving prediction accuracy and robustness.

The proposed TDFT model consistently outperforms all baseline models across multiple benchmark datasets. Although the magnitude of improvement varies across datasets, the performance gains remain stable and reproducible under varying random seeds and hyperparameter settings, demonstrating that the observed improvements are both reliable and statistically significant. Furthermore, these results align with the theoretical expectation that the joint use of adaptive time-decay encoding and frequency-domain reasoning enhances temporal reasoning capabilities.

Compared to the suboptimal TIRGN model, the TDFT model’s MMR index was 0.82%, 0.45%, and 0.39% higher on the ICEWS14, ICEWS18, and WIKI datasets, respectively. Compared to the latest CEGR-TKGR model, TDFT achieved MMR improvements of 3.88%, 2.77%, and 2.90% on the same three datasets, respectively. These results demonstrate the effectiveness and superiority of the TDFT model.

4.8. Relationship prediction results

Because some models are not specifically designed for relationship prediction, we selected temporal models that are suitable for this task. As shown in Table 2, our model outperforms all baseline models in relationship prediction. The integration of the time decay factor and the frequency domain transformation module enables more effective learning of temporal effects on relationship evolution. The performance gap between RE-GCN and other baselines is smaller in the relationship prediction task compared to entity prediction. However, since the number of relationships is much smaller than the number of entities, relationship prediction is inherently easier, resulting in smaller performance differences between models.

Table 2.
Performance (in percentage) for relation prediction task with time-aware MRR.

Model ICE14 ICE18 WIKI

RGCRN 41.92 42.19 96.47

RE-GCN 45.29 45.66 98.90

TiRGN 47.71 46.64 99.04

TDFT 47.81 46.96 99.18

Model	ICE14	ICE18	WIKI
RGCRN	41.92	42.19	96.47
RE-GCN	45.29	45.66	98.90
TiRGN	47.71	46.64	99.04
TDFT	47.81	46.96	99.18

Therefore, in relation prediction, TDFT improved the MMR metric by 0.04%, 0.38%, and 0.07% over the suboptimal TIRGN model on ICEWS14, ICEWS18, and WIKI, respectively.

4.9. Ablation study

To analyze the individual contributions of different components, we conducted an ablation study on the ICEWS14, ICEWS18, and WIKI datasets. Table 3 presents the results of systematically removing key modules from our model.

Table 3.
Results (in percentage) of ablation studies with time-aware MRR.

Model ICE14 ICE18 WIKI

Base 44.04 33.66 81.65

TDF 44.15 33.72 81.78

FT 44.33 33.68 81.77

TDFT 44.41 33.97 81.93

Model	ICE14	ICE18	WIKI
Base	44.04	33.66	81.65
TDF	44.15	33.72	81.78
FT	44.33	33.68	81.77
TDFT	44.41	33.97	81.93

Note: Base represents the unmodified base model, TDF is for entity time decay facter, FT is for frequency domain transform decoder.

The time decay factor (TDF) had the largest impact, with its removal leading to a substantial drop in performance across all datasets. This demonstrates that the TDF is essential for capturing the temporal relevance of events, preventing outdated events from being treated equally, and improving prediction accuracy.

The Frequency Domain Transformation (FT) Module also contributed positively, as removing it reduced model performance. Its effect can be interpreted as enhancing the model’s ability to capture complex temporal patterns beyond short-term changes, though the specific contribution is complementary to the TDF.

We evaluated the model using only the TDF or only the FT module. Using only TDF, the MMR metric increased by 0.24%, 0.17%, and 0.16% on ICEWS14, ICEWS18, and WIKI, respectively. Using only FT, the MMR metric improved by 0.66%, 0.06%, and 0.15%. When both TDF and FT were applied together, the improvements were 0.84%, 0.92%, and 0.34%, respectively. These results indicate that while each module contributes independently to model performance, their combination provides a modest but consistent benefit, highlighting the complementary nature of temporal decay and frequency-based feature extraction.

4.10. Sensitivity analysis

4.10.1. Embedded dimension

To assess the adaptability of TDFT across various dataset categories, we conducted sensitivity tests on its key hyperparameters. By tuning these parameters, we analyzed the model’s performance variation across three datasets.

Specifically, to evaluate the impact of embedding dimensionality on model performance, we tested different embedding dimensions on the ICEWS14 dataset, where $n \in {100, 200, 300, 400, 500}$ . As shown in Figure 4, the model maintains strong performance across both low and high dimensions. The performance improves as the number of embedding dimensions increases; however, after reaching a critical point, the performance begins to degrade. The optimal value for Mean Reciprocal Rank (MRR) occurs when the embedding dimension is set to 200. This suggests that while higher embedding dimensions can enhance performance, excessively large dimensions introduce additional computational costs without significant performance improvement. Similarly, very small embedding dimensions lead to a noticeable decrease in model performance.

Figure 4.

Performance (in percentage) of various embedding dimensions with ICEWS14.

Although a direct visualization of embedding trajectories is not included due to high dimensional complexity, consistent temporal drift patterns were observed during training. Specifically, entity embeddings associated with long-term relations exhibited stable and gradual transitions, whereas those linked to short-lived events underwent more rapid updates. This behavior verifies that the Gaussian decay mechanism effectively regulates the temporal influence of historical information, thereby ensuring a smooth and interpretable evolution of both entity and relation representations.

4.10.2. History length

In this test, we explored the effect of history length on model performance by testing history lengths ranging from 1 to 12. As seen in Figure 5, the model’s overall performance improves as the history length increases. This indicates that performance is closely related to the event history length, and selecting an appropriate history length is crucial for accurately modeling temporal relationships. However, performance declines when the history length exceeds a critical value. This could be due to the inclusion of irrelevant redundant information from extended history, which negatively impacts the model’s learning process.

Figure 5.

Performance (in percentage) of various history lengths with ICEWS14.

4.10.3. Gaussian decay parameter

To further investigate the effect of the Gaussian decay parameter $σ$ on model performance, we conducted a sensitivity analysis on the ICEWS14 dataset. Specifically, $σ$ was varied from 0 to 0.5 with an interval of 0.05. The experimental results, shown in Figure 6, indicate that the Mean Reciprocal Rank (MRR) gradually increases as $σ$ grows, reaching the best performance when $σ = 0.1$ , and then slightly declines beyond this point. This observation suggests that an overly small $σ$ restricts the temporal receptive field, resulting in insufficient utilization of historical information, while an excessively large $σ$ causes over-smoothing and weakens temporal discrimination. Therefore, setting $σ = 0.1$ provides the optimal trade-off between stability and accuracy, validating the robustness and adaptiveness of the proposed Gaussian decay mechanism.

Figure 6.

Performance (in percentage) of various $σ$ values with ICEWS14.

4.11. Computational efficiency analysis

To evaluate the computational efficiency of different model variants, we measured the average training time per epoch across three benchmark datasets, as illustrated in Figure 7. The results reveal that the incorporation of temporal-decay and frequency-domain transformation modules leads to a slight increase in training time due to additional temporal-weighting and spectral-mapping operations. Nevertheless, the overall increase remains moderate, with the full TDFT model being only about 10–15% slower than its base variant. These findings indicate that the proposed enhancements effectively improve temporal reasoning and frequency-domain representation while maintaining acceptable computational overhead.

Figure 7.

The average training time per epoch (in seconds) on the ICEWS14, ICEWS18, and WIKI datasets.

5. Conclusion

This paper introduces a Time–Decay and Frequency–Transformation (TDFT) model designed for temporal knowledge graph completion. By incorporating gated recurrent units (GRU) and adaptive time–decay factors into both entity and relation representations, the model enables dynamic modeling of temporal evolution, prioritizing recent events and effectively balancing historical and current information. Furthermore, the model projects temporal signals into the frequency domain and extracts discriminative spectral representations via Fourier and inverse Fourier transformations, thereby providing a deeper characterization of temporal patterns. Experimental results on three benchmark datasets demonstrate that the proposed TDFT model achieves consistent superiority and robustness in both entity and relation completion tasks.

Despite these advantages, the model still has several limitations. While the FFT module effectively captures periodic and spectral patterns, it may introduce aliasing artifacts when processing non-stationary temporal signals. The current implementation employs a fixed transformation window without adopting adaptive windowing strategies that could further improve frequency resolution. Moreover, the sustained performance improvements observed on longer-duration datasets (e.g., ICEWS14) suggest that the time–frequency fusion mechanism strengthens the modeling of long-range dependencies and complements the short-term dynamics captured by decay-based encoding. However, dedicated quantitative analyses of the model’s ability to capture such dependencies remain to be conducted. Finally, despite its promising predictive performance, the proposed TDFT model currently lacks an explicit interpretability mechanism to assess the contribution of each component during temporal reasoning.

In future work, we aim to improve the computational efficiency of the frequency-domain module, refine the adaptive parameter optimization strategy, and explore adaptive and multi-scale frequency decomposition techniques. Additionally, we plan to incorporate interpretable temporal reasoning analyses to more precisely quantify each module’s contribution. These directions will further reinforce the contributions outlined in the introduction—to establish a robust, interpretable, and scalable framework for temporal reasoning in dynamic knowledge graphs.

Footnotes

Acknowledgements

This research received funding from the National Natural Science Foundation of China (Grants 62466016 and 62241202), and the Hainan Provincial Natural Science Foundation of China (626MS0094).

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article.

Conflicts of interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

ORCID iDs

Siling Feng

Yujie Zheng

Qian Liu

Peng Xu

Housheng Lu

Bolin Chen

Mengxing Huang

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	ICE14				ICE18				WIKI
Model	MRR	H@1	H@3	H@10	MRR	H@1	H@3	H@10	MRR	H@1	H@3	H@10
TTransE $^{1}$ [2018b]	13.43	3.11	17.32	34.55	8.31	1.92	8.56	21.89	29.27	21.67	34.43	42.39
TA-DistMult $^{1}$ [2018]	26.47	17.09	30.22	45.41	16.75	8.61	18.41	33.59	44.53	39.92	48.73	51.71
DE-SimplE $^{1}$ [2020]	32.67	24.43	35.69	49.11	19.30	11.53	21.86	34.80	45.43	42.6	47.71	49.55
TNTComplEx $^{1}$ [2020]	32.12	23.35	36.03	49.13	27.54	19.52	30.80	42.86	45.03	40.04	49.31	52.03
RGCRN $^{2}$ [2018]	38.48	28.52	42.85	58.10	28.02	18.62	31.59	46.44	65.79	61.66	68.17	72.99
RE-NET $^{2}$ [2020]	39.86	30.11	44.02	58.21	29.78	19.73	32.55	48.46	58.32	50.01	61.23	73.57
CyGNet $^{2}$ [2021]	37.65	27.43	42.63	57.90	27.12	17.21	30.97	46.85	58.78	47.89	66.44	78.70
TANGO $^{2}$ [2021b]	–	–	–	–	28.97	19.51	32.61	47.51	53.04	51.52	53.84	55.46
xERTE $^{2}$ [2021a]	40.79	32.70	45.67	57.30	29.31	21.03	33.51	46.48	73.60	69.05	78.03	79.73
RE-GCN $^{2}$ [2021b]	42.00	31.63	47.20	61.65	32.62	22.39	36.79	52.68	78.53	74.50	81.59	84.70
TITer $^{2}$ [2021]	41.73	32.74	46.46	58.44	29.98	22.05	33.46	44.83	73.91	71.70	75.41	76.96
TiRGN $^{2}$ [2022]	44.04	33.83	48.95	63.84	33.66	23.19	37.99	54.22	81.65	77.77	85.12	87.08
PPT $^{3}$ [2023]	38.42	28.94	–	57.01	26.63	16.94	–	45.43	–	–	–	–
TECHS [2023]	43.88	34.59	49.36	61.95	30.85	21.81	35.39	49.82	75.98	–	–	82.39
BH-TDEN [2024]	39.30	29.70	44.00	58.10	28.90	19.10	32.60	48.70	–	–	–	–
STE-GCN $^{3}$ [2024]	43.26	33.71	–	61.47	32.67	22.80	–	51.60	–	–	–	–
CEGRL-TKGR [2025]	42.74	32.32	48.05	62.68	32.90	22.70	36.91	52.95	79.66	75.73	82.83	85.59
TDFT	44.41	34.14	49.48	64.14	33.96	23.45	38.41	54.47	81.93	78.20	85.38	86.96

Time-decay and frequency-domain transformation for temporal knowledge graph completion

Abstract

Keywords

1. Introduction

2.1. Static knowledge graph completion

2.2. Temporal knowledge graph completion

3. The proposed model

3.1. Model overview

3.2.1. Time decay factor

4.1. Dataset

4.2. Temporal encoder architecture

4.4. Evaluation metrics

4.5. Baseline models

4.6. Hyperparameters

4.7. Entity prediction results

Table 2. Performance (in percentage) for relation prediction task with time-aware MRR. Model ICE14 ICE18 WIKI RGCRN 41.92 42.19 96.47 RE-GCN 45.29 45.66 98.90 TiRGN 47.71 46.64 99.04 TDFT 47.81 46.96 99.18

Table 3. Results (in percentage) of ablation studies with time-aware MRR. Model ICE14 ICE18 WIKI Base 44.04 33.66 81.65 TDF 44.15 33.72 81.78 FT 44.33 33.68 81.77 TDFT 44.41 33.97 81.93

4.10.1. Embedded dimension

Footnotes

Acknowledgements

Funding

Conflicts of interests

ORCID iDs

References

Table 2.
Performance (in percentage) for relation prediction task with time-aware MRR.

Model ICE14 ICE18 WIKI

RGCRN 41.92 42.19 96.47

RE-GCN 45.29 45.66 98.90

TiRGN 47.71 46.64 99.04

TDFT 47.81 46.96 99.18

Table 3.
Results (in percentage) of ablation studies with time-aware MRR.

Model ICE14 ICE18 WIKI

Base 44.04 33.66 81.65

TDF 44.15 33.72 81.78

FT 44.33 33.68 81.77

TDFT 44.41 33.97 81.93