Resilience in Knowledge Graph Embeddings

In recent years, there has been significant progress in the construction and application of knowledge graphs (KGs). Many KGs, including Freebase [14], DBpedia [4], YAGO [91], and NELL [20], have been developed and successfully implemented in various real-world applications. Due to their effectiveness in knowledge representation, KGs now find applications in domains such as information retrieval [32], question answering [42], and recommendation systems [99], amongst others. A KG serves as a structured depiction of knowledge, organized as a multi-relational graph where nodes denote entities or concepts, and edges signify relationships between them [53]. Knowledge therein is represented using assertions – model statements (which could in some cases be real-world facts) – in the form of triples denoted as $(h,r,t)$, where $h$ and $t$ correspond to the head and tail entities respectively, and $r$ represents the relationship between them. For instance, the fact “Biden is the president of USA” can be represented in a KG as $(\text{Biden},\text{presidentOf},\text{USA})$.

Knowledge Graph Embedding (KGE) involves transforming the entities and relations within a KG into vectors [15, 124, 35, 103, 121]. This transformation makes computational operations more feasible, allowing machine learning and deep learning techniques to be applied to extract insights from the KG. Consequently, an effective KGE model should aim to preserve the properties and semantics inherent in the original KG. Based on the type of KGE models, entities and relations are commonly embedded in $d$-dimensional vector spaces $𝕍$ such as ${ℝ}^d$ (real numbers) [15], ${ℂ}^d$ (complex numbers) [97], or even ${ℍ}^d$ (quaternions) [21].

Despite their effectiveness in capturing complex relationships between entities of KGs and facilitating various downstream tasks, KGE models can be vulnerable to adversarial manipulations [122, 80, 12, 11, 119, 108]. Since these models rely heavily on the observed connections in a given graph, noise or missing information can lead to sub-optimal embeddings and potentially incorrect inferences. For instance, the presence of incorrect triples (in the sense of non-conformity with an ontology, or wrong assertions) might lead to poorly performing KGE models on certain downstream tasks [129, 57]. To this end, a deliberate attack or the presence of noise can both equally degrade the performance. KGE models might also struggle to generalize to out-of-distribution or unseen data, e.g., when the underlying data distribution changes or when encountering new entities or relations. Since KGs often contain sensitive and critical information pertaining to individuals or organizations, this might give rise to potential security vulnerabilities. For instance, an attacker might subtly alter the relation between entities or introduce fictitious entities and relationships that distort the model’s understanding of the graph and make the KGE model learn poisoned embeddings. Such adversarial attacks on KGE models can take various forms, such as adding, deleting, or modifying triples within the knowledge graph, where such perturbations are often minimal and crafted to exploit vulnerabilities in the embedding process. Due to the usage of KGE models in various downstream tasks, such adversarial attacks can cause potential disruptions in these tasks, for instance, in

1. 1.

   Question answering, adversarial modifications can cause KGE models to produce incorrect or manipulated answers or fail to retrieve relevant information,

2. 2.

   Recommendation systems, the embeddings can be poisoned to promote certain items unfairly, leading to biased or irrelevant recommendations,

3. 3.

   Information extraction, adversarial perturbations can result in inaccurate extractions of facts, affecting downstream applications like content summarization or data integration, amongst others.

Therefore, to reliably use KGE models in downstream tasks, there is a need to develop models that can function without any potential disruption even in the presence of such adversarial conditions.

Although the aforementioned challenges pose potential threats to the use of KGE models in critical downstream tasks, current efforts to deal with these challenges still remain infancy. The existing literature mostly contains works addressing challenges related to noisy data, distribution shifts, and adversarial attacks in the context of graph neural networks [24, 127, 40]. So far, works considering KGE models mostly focused on performing adversarial attacks on them [122, 12, 11, 119]. The core idea behind these attacks is to target specific facts and modify the KGE model to either increase or decrease their plausibility scores. These scores reflect the likelihood of a fact being true: higher scores imply higher probability, while lower scores imply lower probability. For instance, if (Biden, PresidentOf, USA) is selected as the target triple, one type of adversarial attack would be to make the underlying KGE model assign a low plausibility score to it. In this case, such attacks are typically dealt with via a min-max optimization function, where the objective is to minimize the inclusion/deletion of adversarial/existing triples in/from the underlying KG [119]. Simultaneously, the attacker aims to maximize the objective function, which involves either increasing or decreasing the plausibility of a targeted fact being true.

Since KGs are used in many safety-critical environments, safeguarding sensitive information and preserving user privacy are paramount considerations in deploying KGE models in real-world settings. Furthermore, we need to enable KGE models to adapt to dynamic environments and evolving data distributions to enhance their resilience to concept drift and temporal changes. Therefore, in this work, we first of all propose the concept of resilience in the context of ML, and further extend the definition for KGE models. We aim to bridge the gap in resilience literature on KGE from a holistic perspective that considers the diverse facets of robustness, adaptability, distribution shift, and consistency, amongst others. By addressing these aspects comprehensively, researchers can propel the development of resilient KGE models that not only excel in performance metrics but also demonstrate stability and reliability in real-world applications. Note that, our resilience definition is quite generic, i.e., it does not depend on any specific application domain. Precisely, we give a generic formal definition of resilience in ML models considering (i) generalization consistency, (ii) distribution adaptation, (iii) in-distribution generalization, (iv) robustness, and (v) missing entry handling. We then discuss these aspects of resilience in the context of KGE models. To this end, we survey the works on KGE models considering the aforementioned aspects of resilience. Specifically, we provide a survey of works studying the resilience of KGE models in any of the aspects from (i)–(v). After discussing these works, we highlight possible challenges and suggest future work directions.

Note that this paper provides two-faceted contributions. After exploring existing literature on KGE models, we recognize the need for a holistic definition of resilience in embedding models. Therefore, in this work, we first introduce a formal definition of resilience, considering five aspects. Thereafter, we discuss the related works in this context. In this sense, our paper is not purely a survey, rather, it combines a conceptual framework that defines and evaluates resilience in KGE models with a survey of different notions of resilience. Additionally, we propose a comparative analysis of existing methods and explore potential challenges and future works.

This paper is organized as follows. Section 2 formalizes the notions of KGs and KGE models. The definition of resilience is given in Section 3. Section 4 describes the methodology regarding the collection of papers. Existing works discussing aspects of resilience are presented in Section 5. Section 6 presents and discusses different aspects of robustness. Section 10 highlights existing challenges and potential future work directions, and Section 11 concludes the paper.

A knowledge graph is a collection of assertions that describe a domain of interest. In this paper, we consider knowledge graphs composed of triples $(h,r,t)\in ℰ\times ℛ\times ℰ$, where $ℰ$ is a discrete set of entities and $ℛ$ is a discrete set of relations. Therefore, KGs are representations of information in a discrete space. More formally, a KG is defined as a set of triples $𝒢:=\{(h,r,t)\in ℰ\times ℛ\times ℰ\}$, where $ℰ$ and $ℛ$ stand for a set of entities and a set of relations [35, 7]. To facilitate downstream applications, KGE algorithms have been developed to represent a KG in a continuous, low-dimensional vector space. Given a KG $𝒢\subseteq ℰ\times ℛ\times ℰ$, the goal of a KGE model is to learn continuous vector representations for entities and relation types in $𝒢$ such that these representations can be used to recover all the facts in $𝒢$. Most KGE approaches are tailored towards link prediction [21, 53], i.e., their scoring function is ${\varphi }_{\mathrm{\Theta }}:ℰ\times ℛ\times ℰ\to ℝ$, where $\mathrm{\Theta }$ denotes parameters and often comprises $𝐄$, $𝐑$, and additional parameters (e.g., affine transformations, batch normalizations, convolutions). Given an assertion in the form of a triple $(h,r,t)\in ℰ\times ℛ\times ℰ$, a prediction $\widehat{y}:={\varphi }_{\mathrm{\Theta }}(h,r,t)$ signals the likelihood of $(h,r,t)$ being true [35, 103, 121]. Therefore, KGE models are learned in such a way that the scoring function assigns a higher score to the triples that exist in the KG compared to the non-existing ones.

Let $𝕍$ denote a normed-division algebra, e.g. $ℝ,ℂ,ℍ$, or $𝕆$ [6, 34, 116, 97, 125]. A KGE model of a KG comprises entity embeddings $𝐄\in {𝕍}^{|ℰ|\times d_e}$ and relation embeddings $𝐑\in {𝕍}^{|ℛ|\times d_r}$, where $d_e$ and $d_r$ are the size of the embedding vectors. Note that some KGE models represent entities or relations as matrices or higher-dimensional tensors, e.g., RESCAL [76, 65]. Throughout this paper, we will focus on vector representations for entities and relations and denote embedding vectors with bold fonts, for instance, the embedding of $h$, $r$, and $t$ will be denoted as $𝐡$, $𝐫$, and $𝐭$, respectively. Since KGs contain triples which represent the existing facts only, to learn a KGE model effectively, non-existing facts, i.e., negative facts often need to be incorporated into the learning process. For that, a technique called negative sampling is used to generate a number of false facts or negative triples. To this end, Bordes et al. [15] proposed a negative sampling technique by perturbing an entity in a randomly sampled triple from the KG. In this setting, a triple $(h,r,t)\in 𝒢$ is considered as a positive example, whilst $\{(h,r,x)|x\in ℰ\wedge (h,r,x)\notin 𝒢\}\cup \{(x,r,t)|x\in ℰ\wedge (x,r,t)\notin 𝒢\}$ is regarded as the set of possible candidate negative triples corresponding to $(h,r,t)$. During training, $k$ negative triples are constructed for every correct triple.

As mentioned beforehand, resilience is a term that is frequently used when engineering systems, more specifically in the context of building fault-tolerant systems [90]. In those systems, resilience refers to the ability of a system to maintain its functionality and performance in the face of faults, failures, disruptions, or adverse conditions. In other words, a resilient system is capable of detecting, mitigating, and recovering from faults or failures, ensuring continuous operation and minimal impact on its overall performance and availability. Therefore, in [10], the authors defined the resilience of a system using

1. 1.

   availability, i.e., the readiness for correct service,

2. 2.

   reliability, i.e., the probability of performing correctly for a period of time,

3. 3.

   safety, i.e., the robustness against adversarial manipulations,

4. 4.

   integrity, i.e., the absence of improper system altercation, and

5. 5.

   maintainability, i.e., the ability to undergo modifications and repairs.

Note that, due to the lack of resilience definition in the ML literature, we use this as our starting point. While the typical definition of resilience in fault-tolerant systems provides a useful help for understanding resilience in the machine learning domain, it needs to be extended and adapted to account for the unique characteristics, challenges, and considerations inherent in machine learning models and systems. More specifically, for ML models, resilience cannot be defined by using these parameters directly since they do not capture the typical data-driven workflow that is used in ML. For this, we need to consider other factors such as consistent performance in a distribution, or when a distribution shift occurs, robustness, stability, amongst others.

This paper makes a two-fold contribution in addressing this gap. First, after analyzing existing works on KGE models, we identify the necessity for a holistic definition of resilience tailored to embedding models. To this end, we propose a formal definition of resilience, considering aspects that encapsulate the ability of a model to generalize, adapt, and maintain stable performance under varying conditions. Second, we provide a structured discussion of related works that align with these resilience principles, offering insights into existing approaches and their limitations. As mentioned beforehand, our work is not purely a survey; rather, it combines theoretical formalization with a comprehensive review, bridging the gap between conceptual understanding and practical advancements in resilient knowledge representation.

To define resilience, we start with some basic formalization. Let us consider an ML model as a function $f$ which takes as input $x$ coming from a specific distribution $𝒟$. We define two types of distributions from where the data might come, the source distribution which is defined as ${𝒟}_s$ from where the training data comes, and the target distribution ${𝒟}_t$ on which the model would typically operate. The sets of values corresponding to the distributions ${𝒟}_s$ and ${𝒟}_t$ can be defined as ${𝒳}_s$ and ${𝒳}_t$, respectively. $\mathscr{H}({𝒟}_s,{𝒟}_t)$ defines a divergence measure between the two distributions ${𝒟}_s$ and ${𝒟}_t$. Furthermore, we define ${ℒ}_f$ as the loss function of the model measuring the model’s performance on a set of data instances. Note that, the loss function can be of any type, however, our definition is independent of it. An ML model $f$ is said to be resilient if it conforms to the following constraints:

Generalization consistency

corresponds to the ability of the model to generalize consistently across different distributions of data. This can be formally defined as

$$\exists ϵ>0\text{ s.t. }\forall {𝒟}_s,{𝒟}_t,|{𝔼}_{{𝒟}_s}({ℒ}_f)-{𝔼}_{{𝒟}_t}({ℒ}_f)|\le ϵ,$$

(1)

where $ϵ$ defines a threshold that basically bounds the difference between the average losses on the training data distribution ${𝒟}_s$ and the target data distribution ${𝒟}_t$. Xu et al. [112] defined this as the robustness property of the learning algorithms where they argued that a robust algorithm should achieve similar performance on the training and testing data that are close in some sense; which basically corresponds to the robust optimization problem. However, in connection to resilience, we define this as the consistency property over the generalization of the model $f$. To this end, we simply say that the loss occurring on the data instances taken from the target distribution might differ only by a threshold $ϵ$ from the loss occurring on the instances of the source distribution. Note that, in that sense, this definition could also be termed as out of distribution generalization, since this captures how well a model can perform when the inputs are out-of-distribution compared to the training dataset. Furthermore, in this definition, we are not concerned with whether a model $f$ achieves high accuracy or low loss on the training data; with generalization consistency, we aim to signify that the model’s performance should not vary drastically between the training and test data distributions.

In the context of knowledge graph embedding models, generalization consistency refers to the model’s ability to meaningfully construct embeddings for unseen entities or relations, and accurately predict missing links between entities based on the learned patterns from the training data. Assuming ${ℒ}_{{\varphi }_{\mathrm{\Theta }}}$ is a loss function which can be used to train the parameterized embedding model ${\varphi }_{\mathrm{\Theta }}$, generalization consistency can be defined as

$$\exists ϵ>0\text{ s.t. }\forall {𝒟}_{𝒢},{𝒟}_{{𝒢}^{\prime }},|{𝔼}_{𝒢}({ℒ}_{{\varphi }_{\mathrm{\Theta }}})-{𝔼}_{{𝒢}^{\prime }}({ℒ}_{{\varphi }_{\mathrm{\Theta }}})|\le ϵ,$$

(2)

where ${𝒟}_{𝒢}$ and ${𝒟}_{{𝒢}^{\prime }}$ refer to the distribution of the training knowledge graph’s data and that of the test knowledge graph’s data, respectively.

Distribution adaption

corresponds to the model’s ability to adapt to a target domain (i.e., test data distribution) without significantly compromising its performance as achieved on the source domain (i.e., training data distribution). This can be defined as follows

$$\forall ϵ>0,\exists \delta >0\text{ s.t. }\forall {𝒟}_s,{𝒟}_t,\mathscr{H}({𝒟}_s,{𝒟}_t)\le \delta \Rightarrow |{𝔼}_{{𝒟}_s}({ℒ}_f)-{𝔼}_{{𝒟}_t}({ℒ}_f)|\le ϵ,$$

(3)

where $\mathscr{H}({𝒟}_s,{𝒟}_t)$ defines any divergence measure such as maximum mean discrepancy(MMD) [89], Kullback-Leibler (KL) divergence [59], or Wasserstein distance [106]. Informally, if the distributions ${𝒟}_s$ and ${𝒟}_t$ are different with a bound $\delta$, then the average prediction losses on the data instances in these distributions must not differ more than $ϵ$. Note that, the distributional mismatch between the training and test data has been studied in many settings, for instance, in [82, 88, 54, 13, 39, 115] and as pointed out by the authors in [1] most of these works assume the covariate shift where only the distribution of class labels differs between the training and test distributions. There exist some works such as [9, 41, 29] which consider shift of generic data distributions, however, none of them consider this as part of the resilience of ML models. For KGE models, distribution adaptation refers to a model’s ability to adjust its parameters to account for changes in a given knowledge graph. When new entities, relation types, or new links are added to (or removed from) a given knowledge graph, the resulting graph data distribution might deviate from the initial one. In this case, the KGE model’s adaptation to this distribution change can be formally defined as follows.

$$\forall ϵ>0,\exists \delta >0\text{ s.t. }\forall {𝒟}_{𝒢},{𝒟}_{{𝒢}^{\prime }},\mathscr{H}({𝒟}_{𝒢},{𝒟}_{{𝒢}^{\prime }})\le \delta \Rightarrow |{𝔼}_{{𝒟}_{𝒢}}({ℒ}_{{\varphi }_{\mathrm{\Theta }}})-{𝔼}_{{𝒟}_{{𝒢}^{\prime }}}({ℒ}_{{\varphi }_{\mathrm{\Theta }}})|\le ϵ.$$

(4)

Unlike generalization consistency, which assumes stable data conditions, distribution adaptation ensures that the model can adjust to new distributions without significant performance degradation. In other words, generalization consistency ensures stability across similar distributions, while distribution adaptation guarantees stability on dynamic or shifted distributions.

Note that, in the context of graphs, a distribution shift refers to a change in the statistical distribution of the graph data. This can manifest in different ways, such as

1. 1.

   Node feature distribution shift which occurs when the distribution of node attributes or features changes over time or across different subsets of the graph. For example, in a knowledge graph representing entities and their attributes (e.g., people and their professions), a node attribute shift could involve changes in the distribution of professions among individuals over time or across different subsets of the graph. Nodes may furthermore be added to or removed from the knowledge graph, leading to changes in the overall node distribution. This could happen, for instance, when new entities are discovered or when outdated entities are removed from the knowledge graph.

2. 2.

   Node degree shift which happens when some relationships between entities are removed (e.g., two entities that were previously friends are no longer friends) or added, e.g., (an entity gets married to another entity). It could also be the case that new entities are introduced but with little to zero links to other entities in the graph. When such changes in relationships between entities are significant, the average degree of nodes in the considered knowledge graph might also shift.

3. 3.

   Edge feature distribution shift which refers to changes in the properties or attributes associated with the relationships (edges) between nodes in the knowledge graph. For example, in a knowledge graph representing relationships between entities (e.g., co-authorship relationships between researchers), an edge attribute shift could involve changes in the publication venues or collaboration patterns over time. New relationships may further be established or existing relationships may be removed from the knowledge graph, leading to changes in the edge distribution. This could occur due to the emergence of new relationships or the obsolescence of existing ones.

4. 4.

   Graph structure shift which involves alterations in the overall structure or topology of the knowledge graph, including changes in connectivity patterns between nodes, changes in node/edge attributes (e.g., many entities and relationships in the reference knowledge graph now have textual descriptions), and changes in entity type hierarchies. For example, in a knowledge graph representing hierarchical relationships (e.g., taxonomy or ontology), changes in the hierarchy or the addition of new branches can lead to structural shifts. Changes to the schema or ontology of the knowledge graph, such as the addition, modification, or removal of entity types, relationship types, or property types, can also constitute graph structure shifts. These changes may reflect updates in domain knowledge or evolving data modeling requirements.

To make the distinction between generalization consistency and distribution adaption more concrete in the context of KGE models, we consider the following example. Consider a recommendation system based on a knowledge graph. Generalization consistency would ensure that the embeddings trained on historical data remain effective for predicting new links in the same dataset. However, distribution adaptation would be required if the dataset undergoes significant changes, such as the inclusion of new user demographics or shifting product categories.

In-distribution generalization

corresponds to the model’s ability to perform consistently across different instances or subsets of a data distribution. Typically, this distribution could be the target distribution ${𝒟}_t$ where the data instances come from the model deployment phase. Formally, consistency can be defined as follows,

$$\exists ϵ>0\text{ s.t. }\forall 𝒮,{𝒮}^{\prime }\subseteq {𝒟}_{𝒢},|{𝔼}_{𝒮}({ℒ}_f)-{𝔼}_{{𝒮}^{\prime }}({ℒ}_f)|\le ϵ.$$

(5)

Here we enforce that, for any two non-empty subsets $𝒮$ and ${𝒮}^{\prime }$ from the distribution ${𝒟}_{𝒢}$, the expected losses achieved on the two sets differ at most only by some parameter $ϵ$ ¹¹1Note that, here we have considered a strong notion of consistency, however, a weaker notion can also be chosen where the subsets must follow some specific rules.. If the observed differences between the losses are statistically significant (e.g., greater than $ϵ$), it indicates that the model’s performance varies consistently across different subsets of the data, suggesting potential limitations or biases in the model. On the other hand, if the observed differences are not statistically significant, it suggests that the model’s performance remains consistent across subsets, providing greater confidence in its resilience. Furthermore, this measure of consistency is different from the generalization consistency in the sense that herein we consider uniform performance across different sub-spaces of the same distribution space, whereas in case of generalization consistency, two different distributions are considered.

In-distribution generalization for the KGE models refers to the model’s ability to maintain consistent performance across different instances or subsets of the knowledge graph data distribution, particularly when deployed in real-world applications where the distribution of incoming triples may vary. In other words, the KGE model should demonstrate resilience to variations in the distribution of knowledge graph data encountered during deployment, ensuring that its performance remains reliable and predictable across different scenarios. This consistency is crucial for maintaining the effectiveness and reliability of the model in real-world applications where the knowledge graph would evolve over time or across different contexts.

Robustness

focuses on the model’s stability with respect to some small changes in the input. In the literature, two versions of robustness are generally considered, namely local and global robustness [47, 85]. Informally, local robustness corresponds to a single point $x$, and requires any points within a specific distance $\mathrm{\Delta }$ to $x$ to be classified as the same as the former. More formally, this can be defined with respect to a data point $x$ as

$$\exists ϵ>0\text{ s.t. }\forall x^{\prime },{\Vert x-x^{\prime }\Vert }_p\le \mathrm{\Delta }\Rightarrow {\Vert f(x)-f(x^{\prime })\Vert }_p\le ϵ.$$

(6)

On the other hand, Seshia et al. [85] defined global robustness considering all the points within a specific distribution $𝒟$. In other words, for every point $x$ within a considered distribution, any other point $x^{\prime }$ which is within $\mathrm{\Delta }$ distance from $x$ should be classified as the same class as $x$. This can be formally defined as

$$\exists ϵ>0\text{ s.t. }\forall x,x^{\prime }\in 𝒟,{\Vert x-x^{\prime }\Vert }_p\le \mathrm{\Delta }\Rightarrow \Vert f(x)-f(x^{\prime })\Vert \le ϵ.$$

(7)

Note that, in the literature, robustness is more often associated with the idea of local robustness for a single point or a set of points. Thus, in defining resilience, we would primarily consider the local robustness property of ML models. Note that, herein, $f(x)$ could be a single integer or could also be a probability. This would depend on the type of the underlying model.

In the context of KGE models, we can adapt the concept of local robustness to refer to the model’s ability to produce consistent embeddings for entities or relations that are similar in the graph structure. Informally, local robustness in this context would correspond to: any entity (respectively, relation) within a specific neighborhood of an entity $h$ (respectively a relation $r$), defined by a distance metric, should have an embedding that is similar to that of $h$ (respectively $r$). More formally, for an entity or a relation $x$ in the knowledge graph, and for any other entity or relation $x^{\prime }$ within a specific distance $\mathrm{\Delta }$ of $x$ in the graph structure, the embeddings produced by the KGE model, say $𝐱$ and ${𝐱}^{\prime }$ should be similar, with their distance in the embedding space bounded by $ϵ$. Given a knowledge graph $𝒢$, this idea of robustness can be formally defined as

$$\exists ϵ>0\text{ s.t. }\forall x,x^{\prime }\in 𝒢,d_{𝒢}(x,x^{\prime })\le \mathrm{\Delta }\Rightarrow d_{Emb}(𝐱,{𝐱}^{\prime })\le ϵ,$$

(8)

where $d_{𝒢}:𝒢\times 𝒢\to {ℝ}_{+}$ is a distance on the graph $𝒢$, e.g., Adamic-Adar index, Katz similarity, or Common Neighbors, and $d_{Emb}$ a distance function in the embedding space, e.g., the Euclidean distance. $ϵ$ is a threshold that limits the allowable difference between embeddings to ensure local robustness.

The above definition of robustness concerns the functionality of the embedding models in generating robust embeddings. However, we require further robustness notion encompassing the KGE model as well as the scoring function together. To this end, first of all, we define the adversarial robustness.

Adversarial robustness for KGE models refers to a model’s ability to maintain its performance and produce reliable predictions in the presence of worst-case perturbations intentionally crafted to degrade its functionality. Herein, we define the robustness property considering the KGE model+scoring function whereas the previous robustness definition (Equation 8) considers solely the KGE models. The perturbations considered can be applied to the symbolic KG or directly to the embedding space and are designed to maximize the model’s predictive errors. Formally, adversarial robustness by considering the symbolic KG can be described as

$$\exists ϵ>0\text{ s.t. }\forall 𝒢,{𝒢}^{\prime },\mathrm{\Psi }(𝒢,{𝒢}^{\prime })\le \mathrm{\Delta }⟹\frac{{𝔼}_{𝒢}({ℒ}_{{\varphi }_{\mathrm{\Theta }}})}{{𝔼}_{{𝒢}^{\prime }}({ℒ}_{{\varphi }_{\mathrm{\Theta }}})+\eta }\approx ϵ,$$

(9)

where $\mathrm{\Psi }(𝒢,{𝒢}^{\prime })$ denotes the structural similarity of two graphs, ${𝔼}_{𝒢}({ℒ}_{{\varphi }_{\mathrm{\Theta }}})$, ${𝔼}_{{𝒢}^{\prime }}({ℒ}_{{\varphi }_{\mathrm{\Theta }}})$ denote the expected loss of the embedding model ${\varphi }_{\mathrm{\Theta }}$ on the graphs $𝒢$ and ${𝒢}^{\prime }$, respectively, and $\eta$ is an infinitesimal number (e.g., $\eta ={10}^{-8}$, i.e., a small but non-zero scalar value). $ϵ$ in the ideal case should be close to 1. $\mathrm{\Psi }(𝒢,{𝒢}^{\prime })$ could be defined using graph isomorphism [49], or sub-graph similarity [81] matching technique such as graphlet similarity, frequent subgraph mining, or global graph similarity techniques. Such a measure could be decided based on the specific domain. Note that, the above definition considers changes in the KG, however, this could be extended considering perturbations performed on the embedding. Formally, adversarial robustness by considering the embedding can be described as

$$\exists ϵ>0\text{ s.t. }\forall 𝐄,{𝐄}^{\prime }:\mathrm{\Psi }(𝐄,{𝐄}^{\prime })\le \mathrm{\Delta }⟹\frac{{𝔼}_{𝐄}({ℒ}_{{\varphi }_{\mathrm{\Theta }}})}{{𝔼}_{{𝐄}^{\prime }}({ℒ}_{{\varphi }_{\mathrm{\Theta }}})+\eta }\approx ϵ.$$

(10)

Here $𝐄$ and ${𝐄}^{\prime }$ represent the original and perturbed embeddings, respectively. $\mathrm{\Psi }(\text{E},{\text{E}}^{\prime })$ measures the similarity or distance between E and ${\text{E}}^{\prime }$. This can be defined as ${\Vert 𝐄-{𝐄}^{\prime }\Vert }_p$ (e.g., $p=1$ for the L1 norm and $p=2$ for the L2 norm) or $\frac{𝐄\cdot {𝐄}^{\prime }}{\Vert 𝐄\Vert ,\Vert {𝐄}^{\prime }\Vert }$ (cosine similarity) or ${\sum }_i{𝐄}_i\log \left(\frac{{𝐄}_i}{{𝐄}_i^{\prime }}\right)$ (KL divergence).

Note that Equations 8, 9, and 10 give a generic notion of adversarial robustness which can be extended by considering the case where the aim is to degrade the score of a specific triple, i.e., ${\varphi }_{\theta }(h,r,t)$ by doing $\delta$ changes on the KG $𝒢$ or on the embedding space $𝐄$. The existing works on adversarial robustness of KGE models, while lacking a formal definition, focus on this specific notion [111, 86, 87, 75, 128].

Non-adversarial robustness corresponds to the ability of a KGE model (including its scoring function) to be invariant to a certain level of noise present in a KG. More specifically, the performance of a robust KGE model should not degrade considerably when noise is prevalent in KG. Consider $𝒢$ as a clean KG and ${𝒢}^{\prime }$ as a noisy KG, where the latter is obtained by adding $\delta$ amount of noise to the KG $𝒢$, i.e., ${𝒢}^{\prime }=𝒢+\delta$. ²²2Here, $+$ is not the usual addition, but a perturbation operator instead. Then the robustness can be defined as

$$\frac{{𝔼}_{𝒢}({ℒ}_{{\varphi }_{\mathrm{\Theta }}})}{{𝔼}_{{𝒢}^{\prime }}({ℒ}_{{\varphi }_{\mathrm{\Theta }}})+\eta }\approx 1,$$

(11)

where ${𝔼}_{𝒢}({ℒ}_{{\varphi }_{\mathrm{\Theta }}})$, ${𝔼}_{{𝒢}^{\prime }}({ℒ}_{{\varphi }_{\mathrm{\Theta }}})$ denote the expected loss of the embedding model ${\varphi }_{\mathrm{\Theta }}$ on the graphs $𝒢$ and $𝒢$, respectively, and $\eta$ is an infinitesimal number. This implies that the performance of the KGE models should remain almost the same even when $\delta$ amount of noise is present in $𝒢$. Note that we assume the expected loss not to be zero, as it is often the case in most machine learning tasks. Note that the primary difference between adversarial and non-adversarial robustness lies in the nature of the perturbations. Adversarial perturbations are crafted with intent and precision, targeting the model’s weaknesses, while non-adversarial perturbations are accidental and random, reflecting real-world data imperfections. Additionally, adversarial robustness is critical for security-focused applications to protect against malicious attacks, whereas non-adversarial robustness is essential for ensuring reliability in a real-world environment.

Stability w.r.t. incomplete input

deals with the model’s ability to handle missing values, more specifically, maintain accurate predictions despite the presence of missing values in the input features. We can express this as follows:

$$\exists ϵ>0\text{ s.t. }\forall x,x^{\ast },|x|>|x^{\ast }|\wedge \text{Sim}(x,x^{\ast })\le \delta \Rightarrow \Vert f(x)-f(x^{\prime })\Vert \ge ϵ.$$

(12)

Herein $\text{Sim}(x,x^{\ast })$ measures the similarity between two vectors $x$ and $x^{\ast }$ with unequal numbers of elements (i.e., $|x|>|x^{\ast }|$). One such similarity measure could be the cosine similarity, which is often used for comparing the similarity between vectors in high-dimensional spaces. The cosine similarity measures the cosine of the angle between two vectors and is defined as the dot product of the vectors divided by the product of their magnitudes. Elements that are missing in one vector but present in the other are effectively treated as zeros in the dot product. Another approach is to use measures that explicitly handle missing values, such as the Jaccard similarity or the Pearson correlation coefficient, with the imputation of missing values.

In the context of KGs, this aspect of resilience deals with the ability of a KGE model to maintain stable predictions despite missing nodes, edges, or attributes in the input knowledge graph. To this end, we define stability to incomplete input formally as follows:

$$\exists ϵ>0\text{ s.t. }\forall 𝒢,{𝒢}^{\ast },|𝒢|>|{𝒢}^{\ast }|\wedge \text{Sim}(𝒢,{𝒢}^{\ast })\le \delta \Rightarrow \Vert {𝔼}_{𝒢}({ℒ}_{{\varphi }_{\mathrm{\Theta }}})-{𝔼}_{{𝒢}^{\prime }}({ℒ}_{{\varphi }_{\mathrm{\Theta }}})\Vert \le ϵ.$$

(13)

where $𝒢$ is the original, complete knowledge graph, and ${𝒢}^{\prime }$ is the incomplete KG with missing nodes, edges, or attributes. $\text{Sim}(𝒢,{𝒢}^{\ast })$ measures the structural similarity between the two graphs. For KGs, Sim() could be graph edit distance, i.e., number of node/edge insertions, deletions, or modifications required to transform $𝒢$ into ${𝒢}^{\prime }$, or Jaccard similarity over entity/relation sets, amongst others. Note that, this definition implies that the expected loss of the embedding model ${\varphi }_{\mathrm{\Theta }}$ on the graphs $𝒢$ and $𝒢$, ${𝔼}_{𝒢}({ℒ}_{{\varphi }_{\mathrm{\Theta }}})$, ${𝔼}_{{𝒢}^{\prime }}({ℒ}_{{\varphi }_{\mathrm{\Theta }}})$ respectively, should not change more than a fixed threshold $ϵ$. The similarity measure can be quite flexible and will potentially depend on the domain (for e.g., image classification, graph data, and others), and hence, we do not fix the Sim() function. Depending on this function and the domain of the application, the bound $\delta$ will also change, however, not drastically.

To discuss resilience in KGs and KGE models, in this work, we further review existing works in this domain. While doing a literature survey of such works, we adhere to specific inclusion criteria for compiling papers for our review. If a paper satisfies any or many of the following criteria, it is considered for inclusion:

1. 1.

   the paper introduces or discusses the overarching concept of any related aspect outlined in Section 3.

2. 2.

   the paper proposes an approach, study, or tool/framework aimed at developing resilient or robust KGE models.

3. 3.

   the paper introduces a set of measurement criteria applicable for defining resilience of KGE models or KGs.

We briefly discuss some papers focusing solely on using KGs to make resilient systems, however, we do not delve into detail on this. To comprehensively gather papers across various research domains, we initiated our search process by employing precise keyword queries on prominent scientific databases such as Google Scholar, DBLP, and arXiv. The keywords that we searched for are detailed in the Keywords column in Table 1. We conducted searches across the three repositories until 23.09.2024, aiming to encompass a broad spectrum of literature. The specifics of the paper collection outcomes are outlined in Table 1. It is observed that the papers obtained from Google Scholar and arXiv were subsets of those gathered from DBLP. Therefore, we solely present the results obtained from DBLP. Furthermore note that apart from the papers that discuss resilience in KGE models, or in KGs, we also report the results here where any of the aspects of resilience (as described in Section 3) are discussed in the body of the paper.

Note that there exist a number of surveys discussing primarily two aspects of resilience proposed in this paper, such as robustness of deep learning models [113, 74, 28], language models [48, 27]; distribution adaption [64]. However, none of them give a definition of resilience considering the notions that we describe above and discuss the related works encompassing these aspects individually. Moreover, such a study is not done considering the KGE models. The closest work to ours are the works related to GNNs. Existing literature considering GNNs is quite vast and furthermore, there already exist surveys discussing the robustness [114, 31] and some other aspects of resilience of graph neural networks [131, 123]. Discussing the works related to the resilience of GNNs would extend this paper to a much greater extend. Therefore, we do not consider the GNNs in this paper.

The existing works considering resilience in KGE models mainly focus on a specific aspect, that is building KGE models that are resilient against the noise present in the KGs. To this end, there exist a number of such contributions [111, 86, 87, 75, 128]. These works consider if the performance of the model does not degrade with noise present in the KG, then the underlying KGE model is said to be resilient. This, however, is not resilience based on the definition provided in Section 3 where we defined resilience as a multi-faceted term that takes into account many aspects. Based on our definition of resilience, the work on resilience to safeguard against noises in the KGs mostly aligns with the definition of robustness, more specifically, the non-adversarial robustness (as defined in Equation 11). Therefore, we categorize this line of work related to resilience against noise as part of the non-adversarial robustness.

Moreover, there exist some works which concentrate on constructing resilient systems leveraging KGs [117, 58, 3, 30]. For instance, the works in [117] focus on building a KG-based risk assessment framework to improve the resiliency of supply chain management. A KG is built in [58] from the natural disaster data to improve the disaster management department’s resilience towards such incidents. A similar sort of study is done in [3] to employ a resilient management system in case a crisis happens in a city. To assess the resiliency of the cyber-physical system for a water management system, Dagnas et al. [30] utilized the KG as a modeling graph.

Apart from the works mentioned earlier, no other works could be found that consider resilience (as we defined in this work) as part of KGs or KGE models, therefore, in this work, we further survey the existing literature by considering individual aspects of resilience as described in Equation (1)–(13). Figure 1 shows the categorization of the works found corresponding to the aspects of resilience that we described in Section 3. There exist works that focus on improving the generalizability of the KGE models [50, 62, 55, 107], or focusing on the logical consistency of the ontological rules [84, 33, 51, 38, 37], however, no work exists discussing generalization consistency, distribution adaption, and stability to incomplete input of KGE models. A very recent work by Zhu et al. [133] discusses the in-distribution generalization aspect and this is the only work that we could find related to this aspect of resilience.

More importantly, all the works found in regards to resilience in KGE can be distributed along two fields of robustness, namely adversarial and non-adversarial. There exists work such as by Zhu et al. [132] where they proposed to use KG to tackle the distribution shift problem for the few-shot learning approach. More specifically, by using KGs, the aim is to capture the semantic relationship between different categories of instances. Despite the data samples originating from diverse distributions, they frequently possess shared auxiliary knowledge, along with prior semantic relationships between classes. For this, KG can be used to find out when such a distribution shift occurs and help the underlying model to adapt.

Note that there exists temporal KG which dynamically evolves over time [56]. The idea therein is to model the temporal information in KGs to keep track of how different assertions/facts evolve over time. For this, the KG is defined as ${𝒢}_t:=\{(h,r,t,t^{\prime })\in ℰ\times ℛ\times ℰ\times 𝒯\}$, where $t^{\prime }$ basically points to the timestamp for a specific fact. For instance, “(Barack Obama, President_of, USA, [2009-2017])” is an assertion associated with a timestamp [2009-2017] for which it is true. Temporal KGs are called dynamic KGs since they are not static and evolve with the addition of new timestamps corresponding to assertions, and there are KGE models that attempt to learn embeddings for such graphs, such as [2, 94, 52, 92, 120]. Although such KGE models learn to map entities and relations into an embedding space that changes over time, we do not expect the learned embeddings to adapt to the distribution shift of the KG. This is because such KGE approaches assume that, alike typical KGs, the distribution of temporal KGs does not change and only a new timestamp is added by replacing the old timestamp, for instance, $(h,r,t,t^{\prime })$ is replaced by $(h,r,t,t^{\ast })$. The underlying KGE models therefore only need to adapt based on this newly added timestamp. Therefore, we do not consider such works under distribution adaption aspect of resilience, and a survey on temporal KGE models can be found here [101].

Since there do not exist any works considering the aspects of resilience apart from robustness and in-distribution generalization, in the following, we first describe the related works encompassing the two areas that exist regarding the resilience of KGE.

The concept of robustness in KGE models, as discussed earlier, can be divided into two main types: adversarial and non-adversarial robustness. As mentioned in Section 3, adversarial robustness concerns the model’s ability to withstand intentional attacks, where malicious entities modify the knowledge graph (KG) to compromise the KGE model’s performance. In contrast, non-adversarial robustness deals with the model’s resilience against noise and inconsistencies naturally present in KGs, without any malicious intent. In the following subsections, we provide a detailed survey of the current research related to these two areas of robustness in KGE models. We begin with adversarial robustness, followed by an exploration of non-adversarial robustness techniques.

There are some works that do not directly discuss the robustness of the KGE models; however, they consider the KG-driven systems, such as entity linking [72], cross-lingual entity alignment [78], knowledge-grounded dialogue system [100], improving the robustness of the facts of KG [109].

Mao et al. [72] propose a robust entity linking method that tackles 3 aspects, namely, inefficient graph encoders, the need for negative sampling, and catastrophic forgetting in semi-supervised learning. To improve the graph encoders therein, they use relational attention to update the entity features. Furthermore, the authors prove that negative samples are unnecessary in entity linking. It adopts a symmetric negative-free alignment loss to align entity pairs without generating negative samples thereby removing the need for negative samples, and aligning entity pairs with the loss function. Finally, to mitigate catastrophic forgetting, the approach stores previously learned embeddings and selectively reviews them during each training iteration. This approach allows the model to maintain alignment accuracy without retraining on all previous data. The evaluation has shown state-of-the-art results with improved robustness.

Pei et al. [78] propose REA (Robust Entity Alignment), a method for cross-lingual entity alignment between noisy knowledge graphs (KGs). Existing entity alignment models assume clean labeled data, but in real-world scenarios, labeled entity pairs often contain errors that degrade the alignment quality. REA first encodes the structure of knowledge graphs using a Graph Neural Network (GNN). The GNN processes entities and their relationships within each knowledge graph, learning meaningful embeddings that capture the structural similarities between entities, even if they exist in different languages. For example, if “Eiffel Tower” in an English knowledge graph has the same connections as “Tour d’Eiffel” in a French knowledge graph, their embeddings should be similar. The approach further introduces a trust score for each labeled entity pair. The trust score acts as a measure of confidence, determining how reliable a given entity alignment is. REA also uses a margin-based ranking loss function. This function ensures that correctly aligned entity pairs have their embeddings placed closer together, while incorrect pairs are pushed further apart in the learned space. The noise detection module within REA, which operates using an adversarial training framework, continuously updates the trust scores based on newly identified errors. In turn, the noise-aware entity alignment module adjusts the entity embeddings based on these refined trust scores. This iterative learning process ensures that the model becomes increasingly accurate, filtering out noise while improving the quality of entity alignment. An extensive evaluation on real-world multilingual knowledge graph datasets such as DBP15K, DWY100K [93] show that REA outperforms state-of-the-art methods (e.g., GCN-Align [104], MuGNN [19]) in noisy settings. REA provides a robust approach for integrating multilingual knowledge graphs, ensuring high-quality entity alignment despite label noise.

Wang et al. [100] introduce an entity-based contrastive learning framework, named EnCo, to enhance the robustness of knowledge-grounded dialogue (KGD) systems. Given a dialogue context $C=\{u_1,u_2,\mathrm{\dots },u_{n-1}\}$ consisting of utterances $u_i$ and an external knowledge set $K=\{(h_1,r_1,t_1),\mathrm{\dots },(h_m,r_m,t_m)\}$ comprising knowledge triples where $h_i$, $r_i$, and $t_i$ represent the head entity, relation, and tail entity respectively, the goal of a KGD system is to generate a response $u_n$ based on $C$ and $K$. The authors aim to enhance the robustness of KGD models to handle real-world perturbations, including semantic-irrelevant (e.g., misspellings, paraphrasing) and semantic-relevant (e.g., incorrect entity replacements) perturbations. To this end, they leverage contrastive learning to improve robustness by constructing positive and negative samples and training the model to recognize semantic similarities and differences.

Xiao et al. [109] address the problem of evaluating the robustness of outstanding facts (OFs) derived from KGs. An OF is defined as a statement highlighting how an entity stands out based on specific attributes when compared to its peers. Consider a KG $𝒢$ containing information about universities and their employees, including attributes like gender. An OF from this KG might state: “At the American Council on Education (ACE), only 31% of the employees are male.” This statement could suggest a notable gender disparity at ACE-affiliated institutions. However, the robustness of this fact needs to be evaluated by considering the broader context and possible data variations. To formalize this, Xiao et al. introduce the concept of robustness by analyzing how the “strikingness” of an OF changes under various perturbations. Let $𝒮(f)$ denote the strikingness of an OF $f$ in a given context. The goal is to ensure that $𝒮(f)$ remains consistent even when the context or data changes slightly. The authors propose two types of perturbations to evaluate this:

• $\mathrm{■}$

  Entity perturbation. It assesses the robustness of an OF by replacing its context entity $c$ with a similar entity $c^{\prime }$. Formally, let $c$ represent the context entity in the OF $f$. We replace $c$ with $c^{\prime }$, where $c^{\prime }$ is chosen based on its similarity to $c$. The similarity between entities $c$ and $c^{\prime }$ is computed as

  $$\text{Sim}(c,c^{\prime })=\frac{|N(c)\cap N(c^{\prime })|}{|N(c)\cup N(c^{\prime })|},$$

  where $N(c)$ and $N(c^{\prime })$ are the sets of neighbors of $c$ and $c^{\prime }$, respectively.

• $\mathrm{■}$

  Data perturbation. It involves modifying the KG by adding or altering edges, thereby changing the peer entity set of the OF. Formally, the relevance of a data perturbation is quantified using a head-tail relevance function, which measures the semantic connection of the newly added edges to the original fact. Given an added edge $(h^{\prime },r^{\prime },t^{\prime })$, the head-tail relevance function $ℛ(h^{\prime },r^{\prime },t^{\prime })$ evaluates whether the modification preserves the context’s semantic integrity.

The robustness of an OF is then defined by the expected strikingness ${𝔼}_{p(𝒫)}[𝒮(f)]$ over a perturbation relevance distribution (PRD) $p(𝒫)$. This method of evaluating robustness relates to earlier discussions in the literature on robustness, specifically, similar to our proposed robustness formalization in Equation 10. Much like ensuring that KGE models are resilient against adversarial attacks and noise (e.g., as described in works like Xie et al. [111] and Shan et al. [86]), evaluating OFs for robustness ensures that their interpretations remain valid across different contexts.

Note that, similar to using KGs to improve the resilience of several systems, there also exist a number of works that use KGs [118, 83] and KGE models [61] to improve the robustness of ML models. However, the notion of robustness therein pertains to the effectiveness of performing the underlying tasks. Below we describe some of them.

The work by Zhu et al. [133] is the only work that contributes to this aspect of resilience. Therein, they define this as predictive multiplicity, a phenomenon where multiple models with similar accuracy make conflicting predictions for the same query. The authors conduct an empirical study on multiple KGE models and datasets to measure predictive multiplicity. For each KGE algorithm, they train multiple models with different random initializations and hyperparameters. They then select a set of “competing” models – those whose link prediction performance is virtually the same as a best baseline model (within a small tolerance $ϵ$, e.g. 1% difference in Hits@K). Using this set of models, the authors evaluate how often their predictions diverge. For each test query (a partially specified triple such as $(h,r,\mathrm{?})$), they compare the model’s top-ranked results. If one model’s top answer is different from another’s, that query is counted as a conflicting case. The ambiguity metric is computed as the percentage of test queries with any such conflict, and discrepancy reflects the maximum disagreement rate among the models. These metrics provide a quantitative measure of predictive multiplicity for the link prediction task.

After measuring the extent of conflicting predictions, the authors apply ensemble voting methods to combine model outputs. Each model in the competing set produces a ranked list of candidate entities for a query. The authors apply three voting schemes to aggregate these rankings into one result,

1. 1.

   majority voting which picks the candidate that appears as the top choice for most models,

2. 2.

   borda voting, which assigns points based on rank position (e.g., a candidate gets more points for being ranked 1st, slightly fewer for 2nd, and so on, across all models) and then selects the candidate with the highest total points, and

3. 3.

   range voting, which uses the actual prediction scores from each model (rescaled to a common range) and sums them up for each candidate.

These methods generate an aggregated ranking intended to reflect a consensus. The impact of aggregation is assessed by recomputing the ambiguity and discrepancy metrics on the combined ranking, and by checking the standard accuracy metrics (Hits@K) to ensure that the ensemble prediction is still performing well.

Future research in the domain of resilience on knowledge graphs and KGE models presents a number of possibilities to improve different aspects of resilience that we defined in this work. We can envisage works aiming at developing KGE models considering generalization consistency, distribution adaptation, and in-distribution generalization, amongst others. We describe future work directions in more detail in the following.

Generalization under Distribution Shift.

One promising avenue for future work is the development of resilient KGE models that can adaptively adjust to changes in the underlying data or graph structure. Traditional KGE models often assume static or stationary environments, which may not hold in dynamic or evolving KGs. Future research could explore dynamic embedding techniques that continuously update entity and relation embeddings to capture temporal or contextual changes, for instance, when new entities or relations are added to the KG. Additionally, integrating uncertainty modeling and probabilistic reasoning mechanisms into KGE models could enhance their resilience to noisy or uncertain data. Some existing works attempt to quantify uncertainty in KGE models, such as probabilistic soft logic-based methods in [23] and confidence-aware embedding techniques in [66]. The work in [23] employs probabilistic soft logic to generate confidence scores capturing structural and assertional uncertainties, enabling the model to provide confidence-based predictions when new entities and relations are introduced. Similarly, [66] defines a KG as uncertain when each assertion is associated with a confidence score, which is integrated into the KGE learning process to adjust predictions dynamically. These studies establish an important foundation for making KGE models aware of distribution shifts by incorporating uncertainty estimation.

To further enhance robustness against distribution shifts, conformal prediction could be incorporated into KGE models. Conformal prediction provides a mathematically sound framework for quantifying the uncertainty of model predictions by constructing prediction sets that offer guaranteed coverage probabilities [44]. Instead of generating point estimates, KGE models could output prediction intervals for link prediction tasks, ensuring that the true answer is included within a certain confidence level. Note that there already exist some works incorporating this technique in KGE models, such as [134] where the authors apply conformal prediction theory, which enables uncertainty-aware answer set prediction by ensuring that the correct answer is included within a generated answer set with probabilistic guarantees. This shows the potential of utilizing conformal prediction in KGE models in dealing with resilience. For instance, adaptive conformal prediction techniques could be applied to KG completion tasks, where the KGE model dynamically updates its uncertainty estimates as new data arrives. When distribution shifts occur, such as new entities being introduced or relationships evolving, the conformal predictor could adjust its confidence intervals accordingly. This is particularly useful in real-world applications, where decision-making systems rely on KGE models and require calibrated confidence scores for each prediction.

Besides uncertainty-aware learning and conformal prediction, other techniques could be explored to increase the robustness of KGE models in dynamic environments such as:

Bayesian knowledge graph embeddings. Instead of learning fixed embeddings, a Bayesian approach would model entity and relation embeddings as probability distributions (e.g., using Gaussian distributions) [96], allowing the model to express uncertainty in predictions explicitly. This would be particularly effective in scenarios where distribution shifts occur.

Meta-learning for KGE adaptation. A meta-learning framework could be designed to quickly adapt KGE models when distribution shifts occur. Few-shot learning techniques, such as Model-Agnostic Meta-Learning (MAML) [43], could be used to train KGE models to generalize across different graph structures with minimal re-training. This would be beneficial in dynamic knowledge graphs, where new domains or unseen entities frequently appear.

Contrastive learning for distribution shift detection. Contrastive learning techniques could be integrated to detect and quantify shifts in graph structures [45]. By learning embedding distances between past and present snapshots of a KG, models can determine when a significant shift has occurred and retrain embeddings accordingly. This approach could also be combined with self-supervised learning, enabling KGE models to update embeddings without requiring extensive labeled data.

Incorporating conformal prediction, Bayesian embeddings, meta-learning, and contrastive learning into KGE models could significantly enhance their ability to handle distribution shifts and noisy data. Note that, the list of approaches mentioned here is not exhaustive, and there could be further techniques to tackle the aforementioned problems. Future research should delve into exploring more such approaches and find out explore how these techniques can be efficiently integrated into KGE pipelines while ensuring computational scalability and real-time adaptation capabilities.

Adversarial and non-adversarial robustness.

Another possible research direction is that of resilience of KGE models against adversarial attacks and manipulations, i.e., developing KGE models that are adversarially robust. As mentioned beforehand, real-world KGs might suffer from adversarial attacks where adversaries may attempt to exploit vulnerabilities in the KG or KGE models to inject false information, manipulate inference results, or disrupt system functionality. There have already been some works to this end, however, all of them focus on developing methods to perform targeted attacks, i.e., considering a specific fact to add or remove from the KG and thereby making the KGE model learn based on the attackers’ goal. To this end, only the KG has been considered as a possible attack surface. However, there can be other possibilities, for instance, the parameters of the already trained KGE model can be attacked. Such kind of attacks are often prevalent in the ML domain and termed Trojan attacks [67, 68, 102] where the attacker aim to make the model learn their objective either by generating inputs with certain triggers, or by changing the already trained model’s parameters. For KGE models, such Trojan attacks could correspond to the modification of the entries of learned embedding vectors so as to achieve a specific attacker’s objective. Apart from considering targeted attacks by taking into account different attack surfaces, it would also be needed to consider performing non-targeted attacks [57], where the idea is to simply disrupt the performance of the underlying KGE models by introducing noise in the KGs or in the KGE models. As mentioned previously, there are already some works which considered such kind of attacks. However, more sophisticated attack approaches to this end could be explored.

Additionally, the targeted attacks so far have been considered only for a specific type of task, namely link prediction tasks. KGE models are used in many critical downstream application tasks [32, 42, 99], and hence, more research is needed to understand how to perform adversarial attacks on such KGE-based tasks. This basically opens up a number of different attack surfaces along with the need to explore different attack dimensions, including non-targeted attacks.

While several works considered adversarial attacks on KGE models, a much-needed direction to be focused on is the development of defence mechanisms against such attacks that can detect and mitigate adversarial attacks in real-time, thereby enhancing the overall robustness of KG-based applications. This would include developing graph-based anomaly detection algorithms to identify and mitigate adversarial attacks or abnormal patterns in the KG, performing adversarial training of KGE algorithms, developing certified guaranteed methods to build robust KGE models, and so on. Furthermore, defence mechanisms should be extended to combat non-targeted attacks, effectively addressing noise, or incompleteness inherent in KGs. This entails the creation of robust data integration and ensemble algorithms capable of handling diverse and noisy information from various sources. Moreover, exploring techniques for automated error detection, correction, and data validation within KGs could significantly enhance their quality and reliability over time.

Recently, the works to combine large language models (LLMs) with KGEs are gaining popularity [77]. There is a potential that by augmenting LLMs to KGEs, one could achieve improved robustness. Leveraging the semantic richness of natural language representations encoded in LLMs, such as BERT [36] or GPT [17], may enhance the understanding and representation of entities and relations in the KG. This integration could potentially mitigate the impact of noisy or incomplete KGs on downstream tasks.

Stability to incomplete inputs.

Existing works primarily focus on handling missing data through imputation or data augmentation rather than explicitly ensuring robustness against missingness. The lack of standardized evaluation benchmarks and theoretical formulations of stability further hinders progress in this area. Therefore, to this end, we first of all require a benchmark to evaluate different approaches. Afterward, we require suitable graph similarity metrics depending on different domains, moving beyond standard metrics like graph edit distance or Jaccard similarity. Furthermore, we can envisage using adversarial training [18] where missing elements are simulated during training to improve resilience. To this end, dropouts for KGEs, randomly removing nodes/edges during training could also help. Knowledge distillation technique is quite useful in learning where a model trained on complete KGs transfers knowledge to one dealing with incomplete KGs. Apart from the training techniques, adapting loss functions could also be considered. For instance, a stability-aware loss function could be designed that explicitly penalize drastic embedding changes due to missing data, ensuring bounded divergence within $ϵ$. To this end, furthermore, a consistency regularization technique, where models minimize differences in predictions from full vs. incomplete graphs could also be used.

Finally, resilience has already been vastly explored in fault-tolerant systems, therefore, interdisciplinary approaches that draw insights from fields such as network science, complex systems theory, and resilience engineering could provide valuable perspectives and methodologies for enhancing the resilience of KGs and KGE models. By leveraging principles from these domains, researchers can develop holistic, multi-faceted strategies for improving the reliability and robustness of KG-based systems in diverse application domains.

In this work, we explored the resilience of knowledge graph embedding models, addressing their ability to withstand and adapt to various challenges such as noise, adversarial attacks, and dynamic changes in the underlying knowledge graphs. While significant research has been conducted on robustness, particularly adversarial robustness, there is a pressing need to consider a more comprehensive notion of resilience. This broader understanding includes aspects such as generalization consistency, distribution adaption, and performance stability under diverse real-world conditions. A key finding of this survey is that while adversarial robustness has received considerable attention, with various strategies to perform attacks on KGE models, resilience in non-adversarial contexts is equally critical. Models must not only defend against malicious interventions but also maintain their reliability in the presence of natural noise and inconsistencies prevalent in real-world KGs. The surveyed works on non-adversarial robustness primarily focus on mitigating the effects of noise by incorporating confidence-aware learning and enhanced negative sampling strategies. However, these approaches often overlook the dynamic nature of KGs, particularly temporal and evolving KGs, where distribution shifts are inevitable. Addressing such shifts through adaptive retraining mechanisms remains an open challenge. Moreover, ensuring in-distribution generalization across diverse application domains is essential. KGE models must be able to operate effectively even with incomplete input data, which is a common scenario in real-world applications. Achieving this consistency demands that future research goes beyond traditional robustness frameworks, integrating novel methodologies from graph neural networks, reinforcement learning, and explainable AI to enhance both the adaptability and transparency of KGE models. In conclusion, while much progress has been made in improving the robustness of KGE models, a more holistic approach to resilience – incorporating adaptability, consistency, and robustness in the face of both adversarial and natural challenges – will be key to unlocking the full potential of these models in real-world, dynamic, and noisy environments.