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|pdfUrl=https://ceur-ws.org/Vol-3318/paper2.pdf
|volume=Vol-3318
|authors=Michail Karavokyris,Spyros Sioutas
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Graph Neural Networks For Affective Social Media: A Comprehensive Overview Michail Karavokyrisβ , Spyros Sioutas Computer Engineering and Informatics Department, University of Patras, Patras 26504, Hellas Abstract Social media have become the main platforms for expressing and supplementing nuanced human activity such as engaging in public and private conversations, creating and sharing multimedia content, participating to digital culture events, and recently describing emotions about events, places, or even products. In this survey, we provide a comprehensive overview of graph mining and machine learning on affective social media through graph neural networks (GNNs). The latter are capable of performing a variety of tasks, such as graph and vertex classification, link prediction, and graph clustering using vertex information, edge information, and topological structure. These capabilities are critical in harnessing the vast emotional information available in social media in order to generate meaningful and scalable affective analytics. Keywords graph neural networks, distributed computation, graph mining, graph convolution, network topology, convergence, link prediction, label prediction, community discovery, affective computing, PyTorch, 1. Introduction class of neural network architectures depending strongly on information propagation mechanisms such as mes- Currently social media are widely considered to be the sage passing between graph nodes or attention functions digital reflection, or even the digital twin in certain cases, between network layers to encapsulate the higher order of individuals and groups. Among the prime informa- communication flow and interplay inherent in graphs. tion found in social media are affective indicators such Although their functionality may resemble that of other as the emotional polarity of posts or reactions to them. architectures like the established multilayer perceptrons This is especially true in Twitter which abounds with (MLPs) found in many machine learning (ML) applica- long conversations full with emotionally charged replies tions, it is fundamentally different mainly because the [1][2], whereas Facebook [3] and LinkedIn [4][5] have role of higher order patterns is more intense. dedicated emotional reaction buttons for each post. Even The primary research objective of this conference pa- Instagram contains images which have been reported to per is the presentation of the predominant GNN architec- elicit emotional responses [6]. tures and their primary properties as well as how they Typically, in deep learning applications, such as fraud can be applied to basic tasks related to affective social detection, natural language processing (NLP), biomedical network analysis. This will give the interested reader a image processing, and computer vision, the datasets are brief yet concise view of the research landscape of a field represented as manifolds in the Euclidean space. How- which is the focus of intense interdisciplinary research. ever, recently the number of engineering scenarios requir- ing non-Euclidean data and instead rely on graphs has Table 1 been rising. Therein topological relations and intercon- Notation Summary nectivity play a major role. Graphs enable the modeling of important problems in various scientific fields includ- Symbol Meaning First in ing complex systems, social networks, protein-protein = β³ Equality by definition Eq. (1) interaction networks, logistics and long supply chains, xΜ First vector derivative Eq. (4) transportation networks, knowledge graphs, and others. tanh (β ) Hyperbolic tangent Eq. (8) Graph Neural Networks (GNNs) constitute a broad deg (π£) Degree of vertex π£ Eq. (1) diag [π1,1 , β¦ , ππ,π ] Diagonal matrix Eq. (1) CIKMβ22: 31st ACM International Conference on Information and Iπ π Γ π identity matrix Eq. (2) Knowledge Management (companion volume), October 17β21, 2022, Atlanta, GA β Corresponding author. The remainder of this work is structured as follows. In Envelope-Open karavokyrism@gmail.com (M. Karavokyris); section 2 the recent scientific literature regarding GNNs, sioutas@ceid.upatras.gr (S. Sioutas) affective social media, and graph mining is overviewed. Orcid 0000-0002-1263-0785 (M. Karavokyris); 0000-0003-1825-5565 Then in section 3 the primary properties of GNNs are enu- (S. Sioutas) Β© 2022 Copyright for this paper by its authors. Use permitted under Creative Commons License merated in detail, whereas in section 4 the applications of Attribution 4.0 International (CC BY 4.0). CEUR Workshop Proceedings http://ceur-ws.org ISSN 1613-0073 CEUR Workshop Proceedings (CEUR-WS.org) GNNs to affective social network analysis are presented. 1 Michail Karavokyris et al. CEUR Workshop Proceedings 1β10 Future research directions are given in section 5. Capital [44], approximating directed graphs with undirected ones boldface letters denote matrices, small boldface vectors, based on enegry criteria [45], managing graph streams and normal small scalars. Acronyms are explained the with relational algebra [46], computing graph topological first time they are encountered in the text. Additionally, correlation [47], efficiently inferring graph isomorphism the terms vertex and node are used interchangeably in [48] and performing generic pattern search [49] with this work. The same holds true for the terms edge and GNNs on graphs, and massive graph visualization with link. In function definitions parameters follow the re- feedback for graph matching [50]. Applications of graph spective arguments after a semicolon. Finally, in table 1 mining include among others co-author recommenda- the notation used in this work is summarized. tion [51], efficient new drug discovery [52], consensus protocols in blockchains [53], and energy management in smart power grids [54]. Other considerations include 2. Related Work fairness [55], explainability and automation [48], and application to the emerging field microservices [56]. As stated earlier GNNs are neural networks tailored for Social network analysis, although it relies heavily on natively handling graphs or any kind of linked data for graph mining [57], it is a distinct field since it also fo- that matter [7]. Techniques for doing so include graph cuses on social media functionality [58], which includes embedding [8], message passing [9], and attention mech- posts [59], conversations [60], and even digital trust as a anisms [10], the latter primarily in the form of graph conditional extension of the one found in the real world attention networks (GATs) [11]. The current state of [61, 62]. Moreover, psychological aspects such as self- the art in GNNs allows them to perform link prediction esteem [63] and cognitive ones like consumer engage- [12], graph convolution [13], semi-supervised [14] and ment and online time [64] play a central role. Among the unsupervised [15] graph clustering, and node classifica- numerous social media applications can be found stock tion [16]. Regarding applications, GNNs have been used market trend prediction [65], the acceleration under suit- to evaluate the affective coherence of ordinary [17] and able conditions of open innovation [66], the selection fuzzy [18] Twitter graphs, to perform content filtering database architecture according to social queries regard- [19], to yield social recommendations [20], to compute ing Twitter account influence [67], the alteration of the recommendations at large scale systems [21], to perform value of NFTs depending on the Twitter influence of the image classification [22], to do vertex classification based respective holder [68], and the data-driven deployment of on their susceptibility in SIS-type propagation models digital marketing [69]. Reviews of the field include [70] [23], for fake news discovery [24], and for rumor tracing which places special emphasis on community structure [25]. Comprehensive field reviews regarding GNNs can discovery, [71] which explores the dynamics of academic be found in [26] and also in [27]. social networks and online communities, and [72] where Neural network architectures are ubiquitous in ML collaborative innovation processes are explored. [28, 29], especially in conjunction with low rank tensor approximation [30], and signal processing [31]. Bayesian neural networks stem directly from non-classical signal 3. Graph Neural Networks estimation theory [32]. Convolutional neural networks (CNNs) are extensively used in image processing [33]. Re- 3.1. Overview cently deep neural networks have been trained to obey physical laws [34]. In [35] a sequence of social graphs In this section first the most frequent tasks performed is compressed with the two dimensional discrete cosine by the GNN architectures are described. Then, the most transform (DCT2) but expanded with a tensor stack net- prominent GNN types and their properties are presented. work (TSN) trained with information from the entire sequence. Moreover, TSNs have been used for sound 3.2. GNN Tasks classification [36] and large scale urban network speed prediction [37]. Self organizing maps (SOMs) for cultural Typically, every application for affective social media fits content recommendation are described in [38]. Recent into one of the following basic tasks: and extensive reviews on neural network architectures β’ Node classification: The goal is to predict miss- include [39] and [40], where an extended and neural net- ing node labels in a social network using the la- work taxonomy is described as well. bels of the neighbor nodes. For example, the emo- Graph mining aims at locating and extracting latent tional state of a user can be predicted as a function and non-trivial knowledge from graphs such as cycle of the attributes of that user and of its neighbours. lengths in massive graphs [41], higher order spatiotempo- β’ Link prediction: In this scenario the objective ral patterns [42], and triangles [43]. Techniques include is to predict the link between various entities in a employing intelligent agents for autonomous mining 2 Michail Karavokyris et al. CEUR Workshop Proceedings 1β10 network by utilizing a partial or otherwise incom- There are two different types of graph convolution plete adjacency matrix. This task is frequently operations, which in turn determine the domain a given used in social network settings because it can pre- GCN is defined on: dict whether any two vertices, which may well β’ Spatial convolution: These GCNs operate di- be accounts, pages, or even entire communities, rectly on the graph adjacency matrix as if were a are likely to be connected. Moreover, in certain grid but with additional constraints. Thus, con- cases and depending on the available features, the volution is performed in a way similar to images strength of this link may be estimated as well. by using spatial features learned from the graph. β’ Community detection: The case here is to allo- This is the equivalent to the time domain filtering. cate nodes into clusters whose size is unknown be- β’ Spectral convolution: These GCNs utilize the forehand, namely it is a clustering problem. This eigendecomposition of the graph Laplacian ma- can be done by partitioning the vertex sex based trix in order to propagate information across on edge features like weights or, alternatively, nodes. Therefore, processing takes place in the by viewing the nodes as items and by grouping two-dimensional spatial frequency domain akin together items with comparable properties. For to the transform domain adaptive algorithms. instance, community detection can be used on affective social media analysis to locate commu- Recall that the graph Laplacian of equation (2) can be nities with similar emotional characteristics. defined based on the graph degree matrix of equation (1). Observe that nodes of zero degree essentially do not 3.3. Architectures contribute to the overall graph structure and thus are considered to have been removed during a preprocessing GNNs constitute a class of neural networks based on the stage. Therefore, matrix D is always invertible. dependence between the elements of the graph. The term β³ GNN does not refer to a single algorithm or architecture D = diag [deg (π£1 ), β¦ , deg (π£π )] (1) but rather to a plethora of distinct algorithms. The com- mon denominator for each GNN is the ability to exploit With this knowledge the graph Laplacian matrix can the information inherent in graph topology in order to then be constructed from the respective adjacency matrix compute a global steady state. This is more evident in A as shown in equation (2). The eigenexpansion of L is the message passing architectures, but this can also be the graph spectrum on the corresponding basis. seen in some other of the most common GNN architec- β³ tures that have been developed in recent years like graph L = Iπ β Dβ1 A (2) convolutional networks (GCNs) and graph attention net- Although spectral GCNs can construct powerful graph works (GATs). In table 2 the architectures examined here representations and act as convolutional filters for graph and their main properties are presented. classification with considerable accuracy, they fail to uti- lize feature locality commonly found in most graphs. Ad- 3.3.1. Graph Convolutional Networks ditionally, spectral GCNs come with great computational Graph convolutional networks (GCNs), which seek to cost, especially for large networks. imitate the functionality of ordinary CNNs, are currently In order to address the issues of locality and computa- the prime candidate architectures for most real life ap- tional complexity, ChebNets were developed in order to plications. Specifically, the main idea behind GCNs is combine CNNs with the spectral networks theory. Thus, to adapt CNNs to natively handle linked data, namely in ChebNets the representation of any feature vector graphs. CNNs in order to create highly expressive repre- should only be influenced by the π-hop neighbors. There- sentations can extract multiscale localized spatial infor- fore, ChebNets provide the essential algorithmic foun- mation and combine it in order to yield the final result. In dation and effective schemes since the convolution is this sense, they exploit the higher order patterns inherent computed using Chebyshev polynomials instead of the in graphs. Since CNNs are able to capture meaningful eigenvectors of the Laplacian matrix. Therefore, spectral features across the entire data sets, GCNs adjust the oper- GCNs can be considered as ChebNets where the neigh- ation of convolution from grid data to graph data. Graph borhood depth equals one. The objective of this model is convolution uses the features of the neighbors of a given to learn a function of features which operates on a graph node to make predictions by transforming the features of πΊ represented as in equation (3): a node in a latent space. The objective for these models is β³ to train a function of features on a graph where the input πΊ = (π , πΈ) (3) is a set of nodes and edges which are described from a Specifically, a ChebNet is designed to build an π Γ πΉ feature vector that contains their attributes. output matrix where πΉ is the number of output attributes 3 Michail Karavokyris et al. CEUR Workshop Proceedings 1β10 and π is the number of vertices. Said matrix is iteratively normalised coefficients from the unnormalized ones. Typ- constructed given the following graph input. ically, the softmax function is the key to normalizing these coefficients as it can convert a set of raw scores to β’ Feature description vectors, one for each of the an exponentially weighted distribution. π nodes, are stacked are form a π Γ π· feature matrix where π· denotes the number of features. 3.3.3. Message Passing Neural Networks β’ The π Γ π graph adjacency matrix. Therein are contained all local patterns and its powers encode Message passing neural networks (MPNNs) are decentral- all higher order ones. ized architectures which rely heavily on message passing in order to perform a given computation. Such communi- Each network layer has a nonlinear function which cation may take place synchronously or asynchronously. acts as the ChebNet propagation rule. Based in the choice Each node starts with a local ground truth vector and of the propagation rule and the numbers this is succes- progressively based on input from neighboring vertices sively applied models may vary. The most common prop- evolves into a steady state vector. Although initially the agation rule is ReLU operating on a linear combination information exchanged between vertices may be inaccu- of the outputs of the previous layers. The features pro- rate, this is remedied at later stages, provided the update cessed at each layer are aggregated to form the attributes mechanisms are designed to do so. This is by no means of the following layer. This implies that each node in a trivial task as essentially this is a decentralized non- the π-th layer will collect information from their π-hop linear control problem. Therefore, extended care must neighbors. It has been observed that a small number of be taken beforehand in order to avoid effects such as layers, typically at most four, suffices. Witsenhausenβs counterexample [73]. Since in this model the aggregated representation of In contrast to other neural network architectures, each vertex includes only local features, namely those MPNNs have a flat architecture in the sense that there of its neighbors, this has to be taken into consideration are no layers. This implies that the diameter of the net- in the structure of the adjacency matrix. This is done in work plays a crucial role as it represents the maximum two ways, by adding the identity matrix to it to allow amount of time, measured in the number of hops, which the construction of its powers and also by normalizing it is necessary in order for a given piece of information to similarly to the graph Laplacian of (2). So when GCNs be transmitted across the MPNN. Related metrics such and ChebNets are trained by stochastic gradient descent as the effective diameter reveal the links necessary for a algorithms, which tend to be sensitive to the scale of input considerable segment of the graph to be reached. Strong features, there are no vanishing or exploding gradients locality, expressed in the number of triangles or equiva- which frequently delay or even derail training. lently in the clustering coefficient, contributes to quick It should be also mentioned that GCNs are mainly propagation. On the contrary, bridges may be congestion used for semi-supervised node classification, whether points. In any case, topology is central in MPNNs and its binary or multi-class by adding a softmax layer at the end. effects are more intense compared to other GNN types. Also by combining graph convolution layers with graph In table 2 are listed some of the most representative pooling layers the GCN model will be able to predict the convergence schemes proposed in the bibliography. class labels for an entire graph. Table 2 3.3.2. Graph Attention Networks GNN Architectures Analogous to GCNs, GATs average hidden attributes on GNN architecture Description a local level. But unlike GCN, which compute the prop- Message passing Communication with messages agation weights explicitly during training, GATs define Graph convolution Aggregation of hidden features them implicitly. This is accomplished by the attention ChebNet Aggregation of attributes mechanism, namely a learnable function to re-weight Graph attention Self attention mechanism synapses between neurons as a function of the values of the hidden features. In this way, the significance of each node can be specified by utilizing more information than the structure of the graph and the connectivity patterns 3.4. Convergence contained in the latter. However, this local aggregation 3.4.1. State Vectors has to be eventually compensated for when values are propagated to other layers and this is in fact one of the Convergence is a major topic since GNNs are distributed factors differentiating GATs. and, hence, there is not a single point of centralized con- In particular, the synaptic weights are computed as a trol. As such, various techniques based on traditional result of an attention mechanism which computes the control equations such as those describing continuous, 4 Michail Karavokyris et al. CEUR Workshop Proceedings 1β10 linear, and time invariant systems as in equation (4) do MPNNs which employ with proper scaling the sigmoid not directly apply. Therein A is the system plant, b is the or hyperbolic function as activation function as shown input distribution vector, and x is the state vector. in equation (8). β³ xΜ = Ax + bπ’, A β βπΓπ , b β βπΓ1 (4) β³ β³ π π½0 π β π βπ½0 π π(π ; πΌ0 , π½0 ) = πΌ0 tanh (π½0 π ) = πΌ0 (8) π π½0 π + π βπ½0 π In equation (4) xΜ is defined as the column vector con- taining the first time derivatives of the control variables As stated earlier, topology plays a central role in con- π₯ [1] to π₯ [π] as shown in equation (5). The selection of vergence, since it determines the average and maximum these variables essentially determines the graph model. rate of spatial information propagation in terms of the number of links between any two processing vertices. β³ ππ₯ [1] ππ₯ [2] ππ₯ [π] π πΓ1 In table 3 are listed some of the most representative xΜ = [ β¦ ] ββ (5) ππ‘ ππ‘ ππ‘ convergence schemes proposed in the bibliography. Another control model based also on the concept of the Table 3 state vector which is more general but at the same time Graph Neural Network Convergence Criteria less tractable is that of the nonlinear control model of equation (6). In the latter π(β ) is a nonlinear differentiable Type Description vector valued function codifying network dynamics. BFPL Based on continuous maps β³ State convergence Aggregation of local convergence xΜ = π(x, π’), π βΆ β(π+1)Γ1 β βπΓ1 (6) Although the nonlinear control model of (6) covers more cases than that of its linear counterpart of (4), there 3.5. Learning Tasks are less analytical tools to explore and handle it. More- over, many control related results depend heavily on the Irrespective of their architectural classification GNNs properties of π(β ). On the contrary, the control model are called to perform the following fundamental algo- of (4) is appealing for a number of reasons including rithmic tasks across a broad spectrum of applications. tractability and explainability. To this end, often many These include discovering graph community structure, instances of (6) are linearlized with various methods to a setting a message passing mechanism, performing vertex time varying version of equation (4) where the properties classification, and doing graph convolution. of the latter hold true locally. 3.4.2. Browerβs Fixed Point Lemma For most message passing architectures an alternative methodology to monitor convergence lies in the Browerβs fixed point lemma (BFPL). The latter states that any con- tinuous function π(β ) mapping any interval πΌ0 to itself has at least one fixed point π 0 β πΌ0 as shown in (7). π 0 = π(π 0 ), π βΆ πΌ0 β πΌ 0 (7) The existence of the fixed point π 0 guarantees that the MPNN cannot escape from it and as such it is in one of the Figure 1: Graph community discovery. potentially many steady states. However, that requires that a significant number of neurons reach that state before they start propagating it to their neighbors. More- Graph community structure discovery is paramount over, methodologies based on the BFPL are considered to in graph mining as it reveals latent dynamics as shown be indirect in the sense that they monitor the output of in figure 1. Still, in the scientific literature there is more each node π and not their internal state vector s as before. than one definition of what makes a community as this Therefore, the global convergence is tracked through may well depend on the semantics of the underlying individual vertices. Still, they have been applied success- domain. For instance, graphs may be weighted, signed, fully, especially when the processing involves smooth or undirected. Each such property adds constraints to functions, in cases where the local computation is yields community discovery. Moreover, since this task relies a single scalar. For instance, BFPL has been applied to on higher order patterns, it is also computationally chal- 5 Michail Karavokyris et al. CEUR Workshop Proceedings 1β10 lenging. Consequently, a number of diverse heuristics have been developed for it. Message passing mechanisms are crucial in most engi- neering scenarios involving graphs, even indirectly since most networks are set up in order to achieve coherency and communication. Especially in MPNNs selecting the attributes represented in the ground truth vector of each vertex is of paramount importance since that determines what is exchanged during communication. A static snap- shot of message passing is shown in figure 2. Figure 3: Node classification. Figure 2: GNN message passing. Node classification is another important task where Figure 4: Graph convolution. each vertex is assigned one out of many possible labels drawn out of a finite label set based on a decision rule. This functionality is shown in figure 3. Labels may be repeated and, depending on the problem, some vertices may already have a label. Moreover, this task has close ties with the community discovery task, although in clas- sification nonadjacent nodes may have the same label. More recently ML models which can utilize structural and functional attributes, whenever the latter are avail- able, have been proposed in the literature. It should be noted though that functional features depend heavily on the underlying domain, whereas structural attributes can be applied to any scenario. Graph convolution is an operation involving a pair of graphs and yields a larger one whose topology depends Figure 5: Link prediction. on theirs. This allows the efficient discovery of local patterns and, depending on how convolution is defined, even their variants or incomplete ones. This operation two given nodes exists. In order to determine whether initially appeared in the field of computer vision and has such link should be added to the graph, a segment of the found numerous applications in social media analysis graph considered as ground truth is used along with the and ML. Figure 4 shows an instance of this operation. assumption that scale free graphs exhibit self-similarity Finally in figure 5 the task of link prediction task is in many levels. Alternatively, state vectors in every ver- shown. It is an important task where given a partial tex or structural patterns may be used to train an ML graph or an evolving one and a decision rule must be model. Either case may require a considerable amount devised which can predict whether a link between any 6 Michail Karavokyris et al. CEUR Workshop Proceedings 1β10 Table 4 Computational Tasks For Each Affective Computing Task Affective task Computational tasks Node affective state Graph attention, node classification Edge emotional potential Node classification, message passing, graph attention Post emotional potential Node classification, link prediction, graph convolution Node affective influence Message passing, link prediction, node classification Affective communities Community discovery, node classification, link prediction of computational resources, depending on the algorithm. tive communities in case of a bridge, it also depends on its functionality. As such, in addition to node classifica- tion and graph attention analysis pertaining to message 4. Affective Social Media Analysis passing should be employed. Tracing the emotional effect of a post is more challeng- Affective computing is a recent field which extends the ing since a number of interconnected instances of the existing knowledge in social network analysis with emo- previous problem should be studied as a post propagates tional attributes and their study. It has already bore fruits through a graph. Moreover, possible variations of or in- [5, 4] and its prospects look bright with the advent of tentional modification to the latter should be also taken sophisticated DL techniques such as the GNN architec- into consideration as well as the overall information con- tures described earlier but also like autoencoders, graph text of the adjacent edges and vertices. Consequently, the adversarial networks (GANs), and CNNs. All these mod- entire route of a post should be analyzed in this case us- els operate on a plethora of affective attributes including ing graph convolutions and node classification, whereas among others word length and polarity, number of sen- certain propagation patterns of important posts may be tences, use of punctuation, mentions, and words having explained with link prediction techniques. special meaning such as modifiers, negations, and of con- The affective influence of a node can be considered as a siderable emotional weight. generalization of a potentially nonlinear combination of As stated above, affective social media analysis places determining the emotional state of a number of vertices emphasis on the emotional state of social media accounts with evaluating the impact of the posts of the node under through their posts as well as through the interactions be- consideration. This happens as influence is frequently tween them. The methodologies most commonly found taken to be a function of the topological properties of its in the scientific literature can be broadly divided into the high order neighborhood and of the emotional potential following categories. Furthermore, in table 4 is shown of its post. In order to evaluate said affective influence, how each of the affective applications presented in this node classification techniques, message passing, and link section can take advantage of the potential offered by the prediction are frequently employed. learning tasks of GNNs. Finally, affective community discovery is perhaps the The determination of the affective state of a node or a most challenging of the tasks commonly encountered in group of nodes is paramount as it allows, among others, affective social media analysis since it entails the compu- for locating potential starting points for various online tation of various higher order influence metrics. There- digital campaigns with political, commercial, or social fore, a considerable portion of or even the entire graph topics. Moreover, it determines which sort or messages topology and, depending on the problem perhaps the are appropriate for a given node given its affective state. associated functionality, must be factored in. However, a To this end, a number of node classification techniques or, far more accurate insight into the total network dynam- more recently graph attention-based mechanisms, can be ics is obtained. Therefore, approximate analysis of an applied. Given the phenomenon of homophily in social evolving network for a number of steps can take place media stating that nodes with similar behavior eventu- before such a computation can be performed again. ally tend to connect with each other, the neighborhood of the vertex under consideration may as well provide additional affective attributes. 5. Conclusions In a sense the dual problem of the above is finding out the affective potential of an edge as the latter is primarily This conference paper focuses on a comprehensive pre- a function of the affective state of its endpoints. How- sentation of a large number of graph neural network ever, since links in a network may accommodate other architectures tailored for performing affective analysis communication needs, for instance that of the respec- on social media. The latter abound with heterogeneous 7 Michail Karavokyris et al. CEUR Workshop Proceedings 1β10 human emotional information coming from sources so embedding and graph neural network, Information diverse as text, music, images, and even direct emo- Sciences 607 (2022) 1617β1636. tional markings. Therefore, there is more than sufficient [9] F. Gao, J. Zhang, Y. 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