For undirected graphs, the weighted adjacency matrix (alias affinity matrix) is symmetrical, and thus, for each pair of nodes, the weight of the arcs joining them and/or the dissimilarity or distance between them are uniquely defined, since they have no dependence on the direction of the arc. The symmetry of the affinity matrix greatly simplifies the operations and computational procedures of the embedding methods. However, there is a significant amount of intrinsically asymmetric graph data, such as, for example, social networks, alignment scores between biological sequences, and citation data. As reported by Perrault-joncas and Meila (2011), a common strategy for this type of data is not to use as input for embedding procedures, the asymmetric affinity matrix W, but the matrices obtained from it as W+W^T or W^TW. In fact, suppose that

where a≠d and b≠c. Then,

and

are both symmetric matrices.

Already in the early 2000s, other approaches have been proposed to directly deal with the asymmetry of the affinity matrix (Zhou et al., 2004, 2005; Meilă and Pentney, 2007) or define directed Laplacian operators (Chung, 2006).

Interest in and the need to develop efficient ad hoc methods for embedding directed graphs have re-emerged very recently, and the attention of the community has focused mainly on embedding in non-Euclidean spaces. Indeed, it is becoming more and more apparent that Euclidean geometry cannot easily encode complex interactions on big graphs and is not flexible to handle edge directionality. Of interest, as being more functional for geometric deep learning, we mention the study by Sim et al. (2021). The authors of this study demonstrate that directed graphs can be efficiently represented by an embedding model that combines three elements, namely, a non-trivial global topology (directed graphs, eventually containing cycles), a pseudo-Riemannian metric structure, and an original likelihood function that explicitly takes into account a preferred direction in embedding space.

Pseudo-Riemannian manifolds are Riemannian manifold generalisations that relax the requirement of the non-degenerate metric tensor's positive definiteness. As comprehensively described in the study by Law and Lucas (2023), there are two categories in the machine learning literature on pseudo-Riemannian manifolds. The first category does not consider whether the manifold is time-oriented or not; instead, it concentrates on how to optimise a given function whose domain is a pseudo-Riemannian manifold (Law and Stam, 2020; Law, 2021). The second category makes the use of how general relativity interprets a particular family of pseudo-Riemannian manifolds known as “spacetime” (Clough and Evans, 2017; Sim et al., 2021). Spacetimes are actually linked time-oriented Lorentz manifolds. They inherently have a causal structure that shows whether or not there is a causal chain connecting occurrences at different positions on the manifold. In directed graphs, each node is an event, and the existence of a directed arc between two nodes depends on the causal characteristics of the curves connecting them (Bombelli et al., 1987). This causal structure has been used to depict these directed networks.

Sim et al. (2021), on the other hand, use three different spacetime types and suggest an ad hoc method introducing a time coordinate difference function, whose sign is then used to determine the orientation of edges. This has been an interesting and innovative approach that suggested further research and advancement as indicated by Law and Lucas (2023). In that study, Law et al. observe, regarding the study by Sim el al., that when the manifold is non-chronological and does not generalise to all spacetimes, the sign of such a function, for example, alternates periodically and is not always relevant. Additionally, when a geodesic cannot connect two points, their distance function remains constant.

In this brief subsection, we have emphasized embedding methods of directed graphs that adopt spacetime representations. Indeed, the wealth of information contained in spatiotemporal structures could inspire the design of a neural network that can learn most of it and can thus become as accurate a tool for geometric deep learning as possible. For the sake of completeness, however, we mention some other recent studies that propose other methods for embedding directed graphs, such as ShortWalk algorithm by Zhao et al. (2021), MagNet by Zhang et al. (2021a), and the study of Khosla et al. (2020). Less recent is the study of Chen et al. (2007) that proposed an approach taking into account the link structure of graphs to embed the vertices of a directed graph into a vector space.

The idea of ShortWalk is that long random walks may become stuck or stopped in a directed graph, producing embeddings of poor quality. To enhance the directed graph network embeddings, ShortWalk generates shorter traces by doing brief random walks that restart more frequently. MagNet uses a graph neural network for modelling directed graphs, exploiting a complex Hermitian matrix which encodes undirected geometric structure as the magnitude of the matrix entries and the direction of edges in their phase. Khosla et al. (2020) created role-specific vertex neighbourhoods and trained node embeddings in their associated source/target roles using alternating random walk technique, fully using the semantics of directed graphs. Alternating walk technique draws inspiration from SALSA (Lempel and Moran, 2001), a stochastic variant of the HITS (Kleinberg, 1999) algorithm that also recognises hubs and authorities as two different categories of significant nodes in a directed network. The pathways produced by alternating random walks between hubs (source nodes) and authorities (target nodes), sampling both nearby hubs and authorities in relation to an input node.

In the study by Chen et al. (2007), the main goal of the authors was to keep vertices on the locality property of a directed graph in the embedded space. To assess this locality quality, they combined the transition probability with the stationary distribution of Markov random walks. It turns out that they obtained an ideal vector space embedding that maintains the local affinity that is inherent in the directed graph by utilising random walks to explore the directed links of the graph.

Finally, it is worth mentioning that most of the directed graph embedding techniques mentioned in this section are also appropriate for embedding bipartite graphs. A comprehensive survey on bipartite graph embedding can be found in the study by Giamphy et al. (2023), and a short review is also presented in the next sub-section.

Since bipartite graphs are frequently utilised in many different application domains [many of them in biology and medicine (Pavlopoulos et al., 2018; Ma et al., 2023) and drug discovery an repurposing (Zheng et al., 2018; Manoochehri and Nourani, 2020; Hostallero et al., 2022; Yu et al., 2022)], bipartite graph embedding has recently received a lot of interest. The majority of earlier techniques, which use random walk- or reconstruction-based objectives, are usually successful at learning local graph topologies. However, the general characteristics of the bipartite network, such as the long-range dependencies of heterogeneous nodes and the community structures of homogeneous nodes, are not effectively retained. To circumvent these constraints, Cao et al. (2021) developed BiGI, a bipartite graph embedding that captures such global features by introducing a novel local-global infomax objective. BiGI generates a global representation initially, which is made up of two prototype representations. The suggested subgraph-level attention method is then used by BiGI to encode sampled edges as local representations. BiGI makes nodes in a bipartite graph globally relevant by maximising mutual information between local and global representations.

Also the Yang et al. (2022b) proposal for efficient embedding of bipartite graphs of big size is of interest. Yang et al. pointed out that existing solutions are rarely scalable to vast bipartite graphs, and they frequently produce subpar results. The study by Yang et al. (2022b) introduced Generic Bipartite Network Embedding (GEBE), a general bipartite network embedding (BNE) framework, with four core algorithmic designs that achieves state-of-the-art performance on very large bipartite graphs. First, GEBE provides two generic measures that may be instantiated using three popular probability distributions, such as Poisson, Geometric, and Uniform distributions, to capture multi-hop similarity (MHS)/ multi-hop proximity (MHP) between homogeneous/heterogeneous nodes. Second, GEBE develops a unified BNE goal to preserve the two measurements of all feasible node pairs. Third, GEBE includes a number of efficiency strategies for obtaining high-quality embeddings on large graphs. Finally, according to the study by Yang et al. (2022b), GEBE performs best when MHS and MHP are instantiated using a Poisson distribution, therefore they continued to build GEBEp based on Poisson-instantiated MHS and MHP with challenging efficiency improvements.

A network whose links are not continuously active is referred to as a temporal network or a time-varying network (Holme and Saramäki, 2012). Each connection includes its active time as well as any further details that may apply, including its weight. Traditional network embedding methods are created for static structures, frequently taken into account nodes, and face significant difficulties when the network is changing over time. Various methods for embedding temporal networks have been proposed in the last years, e.g., Stwalk (Pandhre et al., 2018), tNodeEmbed (Singer et al., 2019), Online Node2Vec (Béres et al., 2019), and the Dynamic Bayesian Knowledge Graphs Embedding model of Liao et al. (2021). In many methods of temporal network embedding, the traditional representation of the temporal network is often modified, whether it takes the form of a list of events, a tensor [as in the method by Gauvin et al. (2014)], or a supra-adjacency matrix [as in DyANE algorithm by Sato et al. (2021)]. Each of these approaches, such as DyANE, Online Node2Vec, STWalk, and tNodeEmbed, share the common goal of resolving the node embedding issue by locally sampling the temporal–structural neighbourhood of nodes to produce contexts, which they then feed into a Skip-Gram learning architecture adapted from the text representation literature. They have been used in chemical and biological research studies, e.g. Kim et al. (2018), Öztürk et al. (2020), Fang et al. (2021), and Gharavi et al. (2021) and in biomedical language processing (Zhang et al., 2019b).

Regarding these studies, Torricelli et al. (2020) highlighted that as a workaround these methods construct a series of correlated/updated embeddings of network snapshots that take into account the network's recent past. Torricelli et al. stated that the main drawback of these approaches is that it might be challenging to control a large number of hyper-parameters for the sampling random walk process and the embedding itself. The embedding of nodes may, however, fail to capture the dynamic shifts in temporal interconnections. The performance of the prediction might be severely hindered by simply considering past and current interactions in the embedding, whereas bringing future occurrences into account can greatly enhance this task. Torricelli et al. (2020) developed weg2vec to overcome these limitations. Weg2vec learns a low-dimensional representation of a temporal network based on the temporal and structural similarity of occurrences. A higher order static representation of temporal networks, sampled locally by weg2vec, codes the intricate patterns that define the structure and dynamics of real-world networks. This method of unsupervised representation learning can concurrently take into account an event's past and future contexts. It samples without the use of dynamical processes, making it controllable by a few hyper-parameters. It finds similarities between various events or nodes that may be active at various periods but have an impact on a group of nodes that are similar in future.

Finally, the studies by Yang et al. (2022a) proposed a hyperbolic temporal graph network on the Poincaré ball model of hyperbolic space; the temporal network embedding using graph attention network by Mohan and Pramod (2021); the embedding based on a variational autoencoder able to capture the evolution of temporal networks by Jiao et al. (2022), ConMNCI by Liu et al. (2022) that inductively mines local and communal influences. The authors of ConMNCI suggested an aggregator function that combines local and global influences to produce node embeddings at any time and presented the concept of continuous learning to strengthen inductive learning; the continuous-time dynamic network embeddings by Nguyen et al. (2018), the causal anonymous walk representations for temporal network embedding by Makarov et al. (2022); and TempNodeEmb by Abbas et al. (2023). To extract node orientation using angle approach, the methodology by Abbas et al. considered a three-layer graph neural network at each time step, taking advantage of the networks' ability to evolve.

Finally, we mention a study aimed at the possibility of using embedding methodologies in practice for networks of degree sizes, i.e., the proposal for parallelising temporal network embedding procedure by Xie et al. (2022).

The graph convolution network (GCN) is a widely-used method to embed multi-label graphs (Ye and Wang, 2022). However, Gao et al. (2019), pointed out that for multi-label learning problems, the supervision component of GCN just minimises the cross-entropy loss between the last layer outputs and the ground-truth label distribution, which often misses important information such as label correlations and prevents obtaining high performance. In this study, the authors proposed ML-GCN, a semi-supervised learning approach for Multi-Label classification based on GCN. ML-GCN first makes the use of a GCN before including the node attributes and graph topological data. Then, a label matrix is generated at random, with each row (or label vector) denoting a different type of label. Before the most recent convolution operation performed by GCN, the label vector's dimension was the same as the node vector's. In other words, every label and node is contained within a constant vector space. The label vectors and node vectors are finally concatenated during the ML-GCN model training to serve as the inputs of the relaxed Skip-Gram model to identify the node-label correlation and the label-label correlation.

Another study by Shi et al. (2020), to learn feature representation for networked multi-label instances, presented an interesting multi-label network embedding (MLNE) method. The key to MLNE learning is to combine node topological structures, node content, and multi-label correlations. To couple information for successful learning, the authors developed a two-layer network embedding approach. To capture higher order label correlations, the authors employed labels to construct a high-level label-label network on top of a low-level node-node network, with the label network interacting with the node network via multi-labelling interactions. Latent label-specific characteristics from a high-level label network with well-captured high-order correlations between labels are used to improve the low-level node-node network. In MLNE, both node and label representations are forced to be optimised in the same low-dimensional latent space to enable multi-label informed network embedding.

Recent successful applications of the embedding for multi-label classification in medical domain is the medical term semantic type classification (Yue et al., 2019) and the knowledge graph embedding to profile transcription factors and their target genes interaction (Wu et al., 2023).

To conclude this section on graph embedding, we would like to provide some information about the complexity of the embedding.

Due to the high dimensionality and heterogeneity of real-world size networks, classical adjacency matrix-based techniques suffer from high computational costs and prohibitive memory needs. The computational complexity of adjacency matrix approaches is at best O(n²) (where n is the number of nods of the graph).

Random walk-based methods, on the other hand, proved as more efficient in terms of both space and time requirements than both matrix factorisation and BFS/DFS-based methods. For example in node2vec (Grover and Leskovec, 2016), the space complexity for storing each node's closest neighbours is O(e) (where e is the number of edges). The links between each node's neighbours need to be stored for 2nd order random walks, incurring a space complexity of O(d²n) (here d is the average degree of the graph, and n the number of nodes). Random walks have a temporal complexity advantage over conventional search-based sampling techniques too. Random walks, in particular, offer a straightforward way to boost the effective sampling rate by reusing samples across several source nodes by imposing network connectedness in the sample production process.

The increasing of methods for graph embedding obviously prevents us from making generalisations or categorisations regarding the complexity of the various methods. Here, we wanted to report the information in this regard that is confirmed in the literature. At the moment of writing, and to the best of our knowledge, systematic studies on estimating the complexity of each proposed method are very limited, if any, part of the literature on the subject.

To conclude, we mention the studies of Archdeacon (1990) and Chen et al. (1993) on the complexity of graph embedding procedures. It is well-known that embedding a graph G into a surface of minimum genus γ_min(G) is NP-hard whereas embedding a graph G into a surface of maximum genus γ_max(G) can be done in polynomial time. Chen et al. demonstrated that the problem of embedding a graph G with n vertices into a surface with genus at most γ_min(G)+f(n) remains NP-hard, but there is a linear time algorithm that approximates the minimum genus embedding either within a constant ratio or within a difference O(n) for any function f(n^ϵ) = O(n) where 0 ≤ ϵ ≤ 1.

In Table 1, we summarize the main categories of the geometric deep learning and the current main applications of it. Various applications of geometric deep learning to the study of biological systems, especially at the level of dynamic interactome, are beginning to emerge, and we will make special mention of these in this review.

Pineda et al. (2023) apply geometric deep learning to the analysis of dynamical process. For their mechanical interpretation and connection to biological functions, dynamical processes in living systems can be characterised in order to gain valuable information. It is now possible to capture the movements of cells, organelles, and individual molecules at several spatiotemporal scales in physiological settings to recent advancements in microscopy techniques. However, the capture of microscopic image sequences still lags behind the automated analysis of dynamics happening in crowded and complicated situations. The authors in the study mentioned in the reference (Pineda et al., 2023) offer a methodology built on geometric deep learning that successfully estimates dynamical properties with accuracy in a variety of biologically relevant settings. This deep learning strategy makes the use of an attention-based graph neural network (see in Table 1 for some literature reference on attention graphs). This network can do a variety of things, such as to convert coordinates into trajectories and infer local and global dynamic attributes, by processing object features with geometric priors.

Drug repositioning is another growing application domain of geometric deep learning on graphs. Currently, drug repositioning uses artificial intelligence tools to find new markers of authorised medications. The non-Euclidean nature of biological network data, however, is not well accounted for the majority of drug repositioning computational approaches. To solve this issue, Zhao et al. (2022) developed a deep learning system called DDAGDL. It uses geometric deep learning over heterogeneous information networks to predict drug-drug associations. By cleverly projecting drugs and diseases that include geometric prior knowledge of network topology in a non-Euclidean domain onto a latent feature space, DDAGDL can take advantage of complex biological information to learn the feature representations of pharmaceuticals and disorders. The authors in the study by Zhao et al. (2022) showed that according to experimental findings, DDAGDL may recognise high-quality candidates for breast neoplasms and Alzheimer's dementia that have already been described by published studies.

Another application of considerable interest in the medical and biological fields of geometric deep learning was proposed by Das et al. (2022). The authors start from the consideration that drug-virus interactions should be investigated in order to prepare for potential new forms of viruses and variants and rapidly generate medications or vaccinations against potential viruses. Despite expensive and time-consuming experimental procedures, geometric deep learning is a way that can be utilised to make this process faster and cheaper. Das et al. (2022) offered a new model based on geometric deep learning for predicting drug-virus interactions against COVID-19. First, in the SMILES molecular structure representation (Weininger, 1988), Das et al. employ antiviral medication data to generate features and better define the structure of chemical species. Then, the data are turned into a molecular representation, which is subsequently translated into a graphical structure that the geometrical deep learning model can understand.

Exploiting the idea that a molecule can be represented as a graph whose vertices correspond to the atoms of the compound and edges correspond to chemical bonds, geometric deep learning has been also applied for the prediction of molecular structure in structural chemistry and drug design (Hop et al., 2018; David et al., 2020; Atz et al., 2021). Geometric deep learning has been also applied to macromolecular structure (i.e., molecular graph) in structure-based drug design. Structure-based drug design identifies appropriate ligands by utilising the three-dimensional geometric information of macromolecules such as proteins or nucleic acids. An overview of recent geometric deep learning applications in bioorganic and medicinal chemistry, emphasising its promise for structure-based drug discovery and design, is presented by Isert et al. (2023). The focus of this overview is on predicting molecular properties, ligand binding site and posture prediction, and structure-based de novo molecular design. Similarly, geometric deep learning has been applied on molecular graph for molecular crystal structure prediction in mentioned in the study by Kilgour et al. (2023) and the RNA molecular structure prediction mentioned in the study by Townshend et al. (2021).

Although not directly relevant to the conception of the structure of a molecule as a graph, an interesting recent application of geometric deep learning to biomolecular data, which contributes to create knowledge about the interaction networks between proteins and other biomolecules, is the study by Gainza et al. (2019). The rationale of the authors in this study is that it is still difficult in biology to predict interactions between proteins and other biomolecules merely based on the structure. On the basis of their study, the experimental well-consolidated knowledge is that the molecular surface, a high-level illustration of protein structure, shows the patterns of chemical and geometric properties that uniquely identify a protein's modes of interaction with other biomolecules. Consequently, Gainza et al. hypothesized that independent of their evolutionary background, proteins involved in related interactions may have comparable signatures. Although fingerprints might be challenging to understand visually, they can be learned from big datasets. To identify fingerprints that are crucial for particular biomolecular interactions, the authors introduced MaSIF (molecular surface interaction fingerprinting), a conceptual framework based on a geometric deep learning algorithm. MaSIF can predict protein pockets with ligands, protein-protein interaction sites, and rapid scanning of protein surfaces for protein-protein complex prediction.

Townshend et al. (2021) pointed out that the intricate three-dimensional configurations that RNA molecules take on are challenging to measure experimentally or anticipate computationally. The analysis of the RNA structure is crucial to find medications to treat diseases that are currently incurable. A machine-learning technique proposed by Townshend et al. developed a geometric deep learning technique to enhance the prediction of RNA structures. Interestingly, the authors highlighted an important feature of geometric deep learning, namely, the flexibility of this approach to be suitably adapted to work with a smaller amount of training data than that required by traditional deep learning techniques.

Finally, we mention the use of geometric deep learning techniques in neuroimaging and the study of the brain connectome (Gurbuz and Rekik, 2020; Huang et al., 2021; Williams et al., 2021), as well as the study on (i) the relationship of human brain structure to cognitive function (Wu et al., 2022), (ii) the topographic heterogeneity of cortical organisation as a necessary step toward precision modelling of neuropsychiatric disorders (Williams et al., 2021), (iii) brain aging (Besson et al., 2022).

To compare graphs, it is necessary to define a distance metric between graphs. This is a very complex undertaking that necessitates balancing interpretability, computational efficiency, and outcome effectiveness - all of which are frequently dependent on the particular application area. It should come as no surprise that there is a tonne of literature on this subject and that many various approaches have been suggested (Wills and Meyer, 2020). Of these three categories of methods for comparing graphs, the most closely related to latent geometry is the category of methods based on spectral analysis of the weighted adjacency matrix or Laplacian matrix, since the latent geometry of the network is derived from this. The rationale behind the use of spectral methods for graph comparison is that by comparing spectra provide metrics for comparing networks since the spectrum of a network's representation matrix (an adjacency or Laplacian matrix) contains information about its structure. In particular, the authors of NetSLD made explicit the relationship between weighted graph adjacency or Laplacian matrix and latent geometry through metaphors introduced by Günthard and Primas (1956) and Kac (1966). Spectral graph theory is effective in the comparison of 3D objects—said Tsitsulin et al. (2018)—but graphs lack a fixed form, yet 3D objects have an exact low-dimensional shape. However, a graph can be thought of as a geometric entity. Günthard and Primas (1956) posed the initial inquiry, “Can one hear the shape of a drum?” Kac (1966) elegantly posed the same query “To what extent may a graph (or, in general, a manifold) be determined by its spectrum?”. Since then, research has revealed that certain graphs are dictated by their spectrum and that isospectral graphs typically are isometric. Consequently, spectral graph theory provides a foundation for graph comparison.

Learning the latent geometry of a network also proves to be very useful in graph comparison because it can help address some of the challenges that this task presents. We summarise the main ones here and then explain how learning latent geometry can prove useful. First, graph comparison must not care about the order in which nodes are displayed; this is known as permutation -invariance. Second, a good method of graph comparison would make it possible to compare graphs both locally (representing, for example, the variations in atomic bonds between chemical compounds) and globally or communally (recording, for example, the various topologies of social networks). This ability is referred to as scale-adaptivity. Third, it would identify structural similarities regardless of network size (e.g., determining the similarity of two criminal networks of various sizes). This ability is referred to as size-invariance.

The research in graph comparison is very active today [see the recent review by Tantardini et al. (2019)], because—as Tsitsulin et al. (2018) highlights—there is not a method for graph comparison that meets all three of these criteria. In addition to these standards of quality, a workable method of graph comparison should be effectively computable. After preprocessing, it is suitable to do graph analytics jobs that frequently need pairwise graph comparisons inside a large collection of graphs in constant time. Unfortunately, current techniques perform considerably worse in this regard. Graph edit distance (GED) is a widely used graph distance; machine learning has utilised graph edit distance to compare objects when the objects are represented as attributed graphs rather than vectors. The GED is typically used in these situations to determine the distance between attributed graphs. GED is defined as the smallest number of edit operations (such as the deletion, insertion, and replacement of nodes and edges) required to change one graph into another (Serratosa, 2021). GED is NP-hard and APX-hard to compute (Lin, 1994); significant research in GED-based graph comparison has not been able to ignore this fact [see a concise but comprehensive state of the art on this in the study by Tsitsulin et al. (2018)]. Similar to GED, graph kernel approaches (Borgwardt et al., 2020; Kriege et al., 2020) do not explicitly represent the graph and and they do not scale well (Tsitsulin et al., 2018). Identifying the challenges in meeting the requirements of efficiency, scalability, size, and permutation invariance, Tsitsulin et al. moved the issue to the spectral realm and provides an evocative metaphor to clarify their process. Heating the nodes of the graph and tracking the heat spread over time is a metaphor. The idea of a system of masses for the graph's nodes and springs for its edges is another helpful metaphor. The authors (Tsitsulin et al., 2018) claimed that in both scenarios, the entire procedure embodies more global information with time and defines the graph in an efficient permutation-invariant way. They developed the NetLSD that summarises the features of the (undirected, unweighted) graph by a vector derived from the solution of the “heat equation” ∂u_t/∂ = −Lu_t, where u_t is an n-dimensional vector and = I−D^−1/2AD^−1/2 is the normalised Laplacian matrix. Since L is symmetric, it can be written as L = ΦΛΦ^⊤, where Λ is a diagonal matrix of sorted eigenvalues λ₁ ≤ λ₂ ≤ ⋯ ≤ λ_n of which ϕ₁, ϕ₂, …, ϕ_n are the corresponding eigenvectors. Hence, the closed-form solution is given by the n×n “heat kernel” matrix

whose entry (_Ht)ij is the heat transferred from node i to j at time t.

NetLSD summarises the graph representation in the heat trace signature as follows:

Finally, the continuous-time function h_t is converted into a finite-dimensional vector by sampling over a suitable time interval, and the distance between two networks G₁, G₂ is taken as the L₂-norm of the vector difference between h(G₁) and h(G₂). The time complexity of NetLSD is O(n³), if the full eigen-decomposition of the Laplacian is carried out. Specifically, the heat or wave kernel of the Laplacian spectrum is inherited by the compact signature that NetLSD derives, thus “it hears the shape of a graph”—said Tsitsulin et al. (2018).

At the conclusion of this section, some general comments on the spectral methods are made. Despite their use of ease and rigorous theoretical foundation, these methods exhibit some limitations, such as cospectrality between graphs, reliance on matrix representation, and abnormal sensitivity, wherein slight alterations in the graph's structure can result in significant changes in the spectrum (Tantardini et al., 2019) and vice versa. However, it is reasonable to think that strategies to manage the sensitivity to noise or perturbations will not be long in coming, given the numerous advantages that spectral methods offer compared with their limitations, first of all the possibility that these methods offer to efficiently satisfy the three main desiderate of graph comparison as NetLSD has demonstrated.