# socialaware_group_display_configuration_in_vr_conference__29e50e99.pdf Social-Aware Group Display Configuration in VR Conference Bay-Yuan Hsu,1 Chih-Ya Shen,2 Hao Shan Yuan,3 Wang-Chien Lee,4 De-Nian Yang5,6 1 Department of Industrial Engineering and Engineering Management, National Tsing Hua University 2 Department of Computer Science, National Tsing Hua University 3 Institute of Information Systems and Applications, National Tsing Hua University 4 Department of Computer Science and Engineering, Pennsylvania State University, USA 5 Institute of Information Science, Academia Sinica 6 Research Center of Information Technology Innovation, Academia Sinica byhsu@ie.nthu.edu.tw, chihya@cs.nthu.edu.tw, hsy110065507@gapp.nthu.edu.tw, wlee@cse.psu.edu, dnyang@iis.sinica.edu.tw Virtual Reality (VR) has emerged due to advancements in hardware and computer graphics. During the pandemic, conferences and exhibitions leveraging VR have gained attention. However, large-scale VR conferences, face a significant problem not yet studied in the literature displaying too many irrelevant users on the screen which may negatively impact the user experience. To address this issue, we formulate a new research problem, Social-Aware VR Conference Group Display Configuration (SVGD). Accordingly, we design the Social Utility-Aware VR Conference Group Formation (SVC) algorithm, which is a 2-approximation algorithm to SVGD. SVC iteratively selects either the P-Configuration or S-Configuration based on their effective ratios. This ensures that in each iteration, SVC identifies and chooses the solution with the highest current effectiveness. Experiments on real metaverse datasets show that the proposed SVC outperforms 11 baselines by 75% in terms of solution quality. Introduction Virtual Reality (VR) has experienced a surge in adoption as industries increasingly utilize it to promote products and enhance services (Mileva 2022; Ning et al. 2021). During the COVID-19 pandemic, VR conferences and exhibitions gained popularity as in-person events are constrained by travel restrictions and social distancing policies. To support virtual gatherings, various metaverse platforms have been exploited. For instance, the ACM SIGKDD 2020 conference used v Fair s 3D platform for a virtual conference. Meta Horizon Workrooms offers advanced features like persistent whiteboards and mixed-reality pass-through functions, enabling users to collaborate in a virtual space (Meta 2023). Spatial allows users to customize their 3D virtual space for immersive galleries or exhibitions (Spatial 2023). Mozilla Hubs provides spatial audio and media-sharing functions for socializing with custom avatars in a virtual space (Hubs 2023). Engage replicates face-to-face interactions, accommodating up to 5,000 live virtual reality users for events, training, and education (ENGAGE 2023). However, these platforms lack the support for personalized display, e.g., Copyright 2024, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. highlighting important nearby users in Head-Mounted Displays (HMDs), especially in crowded conference environments. Current VR social apps lack personalization, which displays all users uniformly in the virtual environment and neglects the potential benefits of customization. This curbs the social interactions in VR conferences and diminishes user satisfaction due to three drawbacks. D1) Obstructed View. In crowded virtual spaces, users often struggle to find friends or individuals of interest because nearby strangers obstruct their views. D2) Lack of Customization. The presence of undesirable strangers in proximity to a target user may lead to undesirable interactions. D3) Overload. In large-scale VR exhibitions/conferences, individuals may become overwhelmed by the abundance of participants for interactions or, conversely, become disinterested due to the lack of social connections. These three disadvantages, in combination, lead to VR-induced Social Isolation, where users may struggle to connect with individuals they are interested in or socially close to, resulting in reduced interactions and satisfaction. An effective solution to this issue is Personalized Display, which allows users to selectively enable or disable the rendering of other users. In VR environments, different users are not required to view the same set of individuals. Prior studies demonstrate that tailoring user displays in VR enhances social experiences (Pluto 2018). However, existing VR customization research mainly revolves around analyzing factory safety measures in computer vision and data mining, aiming to recommend items to users (Lacko 2020; Ko et al. 2020). These research overlook personalization factors among users. A personalized display enables the VR-based social metaverse with the following advantages. A1) Social Relationships. Enabling two users to see each other in the metaverse can enhance their satisfaction, fostering a feeling of shared presence (Bulu 2012). Thus, displaying a group of sociallyclose friends on an individual s VR screen could be advantageous (Pluto 2018). A2) Personal Preferences. Personalized displays, aligning with users preferences, enhance satisfaction; for example, some users prefer seeing prominent scholars or conference organizers at AAAI, while others anticipate connecting with colleagues sharing their re- The Thirty-Eighth AAAI Conference on Artificial Intelligence (AAAI-24) User Configuration Ranking ku Andy Personal preferences Bella Carl Faye Eddy Dora 1 Social utilities Eddy Faye Bella Dora Carl Bella Personal preferences Faye Carl Andy Eddy Dora 2 Social utilities Eddy Dora Andy Faye Carl Carl Personal preferences Andy Dora Eddy Bella Faye 1 Social utilities Faye Bella Eddy Andy Dora Dora Personal preferences Andy Eddy Bella Faye Carl 2 Social utilities Faye Bella Eddy Carl Andy Eddy Personal preferences Faye Carl Dora Bella Andy 2 Social utilities Andy Bella Carl Faye Dora Faye Personal preferences Dora Andy Carl Eddy Bella 2 Social utilities Dora Carl Bella Eddy Andy Table 1: Personal preferences ranking, social utilities ranking, and display limitation ku for each user search interests. A3) Limited Display Slots. To address issues D2 and D3, a limited number of users may be selected for VR displays based on personal preferences and social relationships. Users can specify their preferred number of display slots, enabling them to see those they are interested in or have close social relationships with. Allocating restricted display slots per user efficiently resolves issue D1 by enabling prioritization of interactions with preferred individuals, thus averting overcrowding. However, achieving a delicate balance between social connections and personal preferences while limiting the number of displayed individuals presents a challenge. Motivated by the advantages A1, A2, and A3, we propose a new approach, named Social-Aware VR Conference Group Display Configuration (SVGD), to address the issues D1, D2, and D3. Given users social networks, personal preferences, and social utilities (the likelihood of two users having active and joyful interactions.), SVGD seeks to identify the ideal personalized display configuration for each user within confined slots, maximizing overall user satisfaction. Example 1. (Motivating Example). Fig. 1 presents an illustrative example. Given a social network G = (V, E) of six VR users, as shown in Fig. 1(a), where the hollow user images next to each user indicate the number of slots in her VR display, and a solid user image represents an occupied slot. Table ?? presents the personal preferences rankings, social utilities rankings, and display limitations of each user. The User column lists the users. The Configuration column outlines the configuration factors, and the Ranking column indicates the priority of other users to each individual. The ku column denotes the number of slots available in each user s VR display. In Figs. 1(b), 1(c), and 1(d), we present three different approaches to configure each user s VR display (called configuration hereafter). i) Preference-based configuration (e.g., conventional friend recommendation). The top-ku users of interest to each user are configured based on personal preferences. Fig. 1(b) represents the sees-in-VR graph of this preferencebased configuration, where an arrow from user x to user y indicates that x sees y in her VR display. In an example, Andy has a display slot number ku of 1. Despite having a high social utility with Eddy, in the preference-based configuration, Andy sees Bella, who is ranked the highest by Andy s personal preferences. Similarly, Bella has a ku of 2 and sees Faye and Carl, the top-ranked individuals according to her personal preferences. In this configuration, each user sees ku users during the VR conference, but they do not see each other. This configuration only considers A2 (personal preferences) and neglects A1 (social relationships), leading to limited interaction among users during the VR conference and worsening the issue of VR-induced Social Isolation. ii) Social-based configuration (e.g., conventional cohesive group extraction). Fig. 1(c) represents the sees-in VR graph of this social-based configuration. A two-way arrow between two users indicates that they can see each other in the VR display, while a blue user picture indicates that both users have high social utilities. While this configuration promotes social interactions during the VR conference, it comes at the cost of individual preferences. For instance, Andy may desire to see Bella, but due to their significant social distance, Bella is not displayed, despite Andy s interest. Conversely, Bella, with a display slot count of 2 (ku), can see Dora and Eddy, who are ranked higher according to her social utilities. This social-based configuration primarily considers aspect A1 (social relationships) but overlooks aspect A2 (personal preferences), potentially leading to decreased user satisfaction. This reduces users satisfaction and neglects the crucial purpose of VR conferences: to meet and get acquainted themselves with new people. iii) Preferences and socially balanced configuration. This configuration s sees-in-VR graph is presented in Fig. 1(d). For instance, Dora sees Eddy due to her high personal preferences for Eddy, and she also sees Faye since they have a high social utility ranking between them. Likewise, Bella sees Faye as she has a strong personal preference for her, while Bella and Andy see each other due to their relatively high social utility and personal preferences rankings. This configuration, which takes into account A1, A2, and A3 (limited display slots), enhances user satisfaction and mitigates VR-induced Social Isolation. The SVGD problem is distinct from conventional friend recommendation and personalized recommendation approaches (Cheng et al. 2019; Chen et al. 2017; Zhao et al. 2016; Bagci and Karagoz 2016; Lin et al. 2017), as well as cohesive group extraction in social networks (Lu et al. 2022; Al-Baghdadi and Lian 2020; Ma et al. 2022; Sanei-Mehri et al. 2021; Dong et al. 2021; Yang et al. 2021, 2012a). Conventional friend recommendation focuses on suggesting potential friends based on preferences, without considering the impact of other users configurations in a VR setting. Cohesive group extraction in social networks aims to identify socially-close friends but does not incorporate the inclusion of strangers based on user preferences or interests, which could hinder the formation of new friendships. More importantly, the SVGD problem allows the rendering of different users on individual VR displays, setting it apart from social/preferences group queries and group recommendations. In summary, conventional research does not consider sev- The Thirty-Eighth AAAI Conference on Artificial Intelligence (AAAI-24) Empty ku slots in the input data User connections (a) Input social network Viewing preferred individuals Fulfilling with the ku slots (b) Personal preferences-only Reciprocated viewing of individuals Both users have high social utility (c) Social relationships-only Socializing with preferred users Both users have high social utility (d) Our idea Figure 1: An illustrative example of preference-based, social-based, and balanced configuration eral factors particularly important in VR conferences, including i) personal preferences, ii) the impact of other users configurations on user satisfaction in VR, iii) co-display among users (two users appear in each other s VR displays), iv) handling over-crowding in VR events, and v) different users to be rendered on different users VR displays. For instance, the user recommendation approaches maximize individual users personal preferences and satisfaction, but it neglects ii), iii), iv) . Similarly, group formation and recommendation tend to find socially-close friends, but they do not consider the inclusion of strangers and overlook i), ii), iv), and v). Therefore, previous approaches cannot be applied directly to the SVGD problem studied in this paper. In this paper, we prove the NP-hardness of the SVGD problem and propose a 2-approximation algorithm, named Social Utility-Aware VR Conference Group Formation (SVC). This algorithm addresses the challenges posed by personal preferences and social relationships within limited VR display slots. We introduce the concept of SVD utility to quantify user satisfaction and propose the SVD kuconfiguration, where ku represents the maximum number of users displayable in a user s VR setup. We evaluate the SVC algorithm on real datasets. The results demonstrate the effectiveness of our approach. The paper s contributions are summarized as follows. We present the new notion of SVD ku-configuration under the context of VR conferences and formulate a new research problem, Social-Aware VR Conference Group Display Configuration (SVGD). SVGD aims to identify an SVD ku-configuration that facilitates social interactions without sacrificing users individual preferences. We analyze the NP-hardness of SVGD and propose a 2-approximation algorithm, named Social Utility-Aware VR Conference Group Formation (SVC). We conducted extensive experiments on 5 real datasets. The results indicate that SVC significantly outperforms other baselines in terms of solution quality and efficiency. Related Works VR applications. A wide spectrum of VR applications have emerged recently, such as online VR shopping (Ko et al. 2020), friend-making (Raber, Schommer, and Kr uger 2019), social interactions in VR (Mc Veigh-Schultz, Kolesnichenko, and Isbister 2019), and social VR in edge computing (Wang et al. 2018a). We envisage that the proposed SVGD problem helps users obtain a better VR group conference experience by selecting a suitable set of users for their VR display with the maximum personal preferences and social utilities. To the best of our knowledge, similar functions are not currently available in VR conference products on the market (Meta 2023; Spatial 2023). Dense subgraph extraction. Extracting dense subgraphs in social networks has been actively studied for decades, e.g., (Shen et al. 2015; Lu et al. 2022; Al-Baghdadi and Lian 2020; Ma et al. 2022; Chen et al. 2018a; Shen et al. 2017; Hsu, Shen, and Yan 2019). However, they cannot be applied directly to our VR conference scenario due to the following reasons. i) The limited number of slots in the VR display (allowed to vary for each user) is not considered, and different users can be rendered on different users VR displays. ii) The important notion of co-display is not incorporated, and many users might suffer from VR-induced Social Isolation. iii) The users personal preferences and social utilities are not jointly examined, causing poor interactions between the users in the VR conference. Personalized recommendation. Personalized recommendations are widely used in E-commerce, suggesting products based on user preferences and browsing history (Chen et al. 2017; Liao et al. 2018; Zhao et al. 2022). However, these approaches fail to jointly consider the personal preferences and social interactions. In SVGD, both personal preferences and social interactions are crucially considered, accounting for diverse users displayed on different VR screens. This distinction sets the problem apart from social/preferences group queries (which identify identical users for the entire group)(Yang et al. 2012a) and group recommendations (which suggest the same items for all users). Problem Formulation and Hardness Result Given a directed social network G = (V, E), where V represents the set of users and edge set E specifies their social relationships, we first introduce Social-Aware VR Conference Display with ku-configuration (SVD ku-configuration), which configures the limited display slots of users in a largescale VR conference event. Definition 1. Social-Aware VR Conference Display with ku-configuration (SVD ku-configuration). Given ku display slots specified for user u to display other users in a VR The Thirty-Eighth AAAI Conference on Artificial Intelligence (AAAI-24) conference, an SVD ku-configuration is a collection of sets A = {Au| u V }, where a set Au, corresponding to user u, contains at most ku other users that appear in user u s VR display. For a large-scale VR conference, each user u sees at most ku other users to avoid the overcrowded view. As a consequence, some users may not be rendered in a specific user s VR display due to the limited display slots. With the definition of SVD ku-configuration in hand, we now introduce co-display as follows. Definition 2. Co-display (u v) and I(u, v). Let u v denote that users u and v appear in each other s VR display, i.e., v Au and u Av. In this case, u v is referred to as a co-display. We employ a binary indicator function I(u, v) = 1 to indicate this co-display relationship, i.e., I(u, v) = 1 if u v, and I(u, v) = 0, otherwise. Since the VR display slot is limited for each user, users u and v might not see each other, i.e., I(u, v) = 0, and they cannot have social interactions. Previous studies (Wang et al. 2018b; Gao et al. 2018) indicate that a user s satisfaction in a group activity is affected by two important factors: personal preferences and social utilities. Therefore, to address the above issue and enhance social interactions, it is important to consider personal preferences and social utilities jointly in our problem formulation. Specifically, given a pair of users u and v in a social network G = (V, E) with (u, v) E, let p(u, v) 0 denote the personal preference of u on v when v is rendered in u s VR display, and let τ(u, v) 0 denote the social utility for u against v, i.e., how likely u believes that u and v would have active and joyful interactions (Lai et al. 2019; Shuai et al. 2013). Please note that p(u, v) (τ(u, v)) and p(v, u) (τ(v, u)) may be different. The assignment of personal preferences and social utilities can be done either by the user themselves, obtained through the use of social-aware recommendation models (Sankar et al. 2021; Fan et al. 2019), or inferred by event recommendation models (Liao et al. 2018; Yang et al. 2023). Given the above definitions and following (Ko et al. 2020; Wang et al. 2018b; Chen and Yang 2022; Tong, Meng, and She 2015), we define the SVD utility, which integrates personal preferences and social utilities and acts as a metric for a proper configuration of other users appearing on each user s VR display. Definition 3. SVD utility (w Au(u, v)). Given an SVD kuconfiguration A = {A1, A2, ..., A|V |}, the SVD utility of user u on user v Au combines personal preference and social utility, w Au(u, v) = (1 λ) p(u, v) + λ τ(u, v) I(u, v), (1) where λ [0, 1] is a weighting factor, which can be directly set by a user or implicitly learned from existing models (Zhao, Mc Auley, and King 2014; Liao et al. 2018). An alternative approach for incorporating preferential and social factors is to employ an end-to-end machine learning approach, generating user and item representations and using a neural network aggregator to compute overall user satisfaction (Cao et al. 2018). However, this approach demands an algorithm to generate potential configurations for ranking and relies heavily on substantial training data to fine-tune the aggregator s parameters. In contrast, prior research (Wang et al. 2018b; Liao et al. 2018; Zhao, Mc Auley, and King 2014) has shown that a blend of preferential and social factors, combined with assigned or learned weights, can effectively evaluate user satisfaction. Various objective functions, such as w Au(u, v) = min(p(u, v), τ(u, v)), w Au(u, v) = max(p(u, v), τ(u, v)), and w Au(u, v) = p(u, v) τ(u, v), yield subpar solutions and user satisfaction. In specific contexts, min and max solely consider personal or social factors. Moreover, when either min or product is exceedingly low for either p or τ, the outcomes might not favor a particular individual despite deserving consideration in real scenarios. Hence, akin to (Wang et al. 2018b; Ko et al. 2020), we frame the SVD utility as a weighted fusion of aggregated personal preferences and social utilities, governed by the parameter λ. Here, w Au(u, v) is a directional utility from user u to a user v in her configuration, u V . The users selected in user u s configuration appear in user u s VR display, but user u may not be in their configurations. Hence, SVD utility w Au(u, v) is a directional utility. The SVD utility w Au(u, v) incorporates both the personal preference p(u, v) and social utility τ(u, v). The social utility τ(u, v) takes effect only when the co-display condition holds, i.e., u v holds (I(u, v) = 1). This is because users u and v can interact only when they see each other in their own VR displays. In this paper, we formulate the Social-Aware VR Conference Group Display Configuration (SVGD) problem to configure suitable users in every user s VR display by identifying the best SVD ku-configuration. Here, SVGD includes two additional constraints: i) Personal preference constraint θ, which requires that for a user u, any user v rendered in u s VR display must have a personal preference at least θ, i.e., p(u, v) θ, u V, v Au. Following (Hsu, Shen, and Chang 2020; Hsu, Lan, and Shen 2018), the above constraint aims to meet the minimum required personal preferences to prevent users from becoming extremely dissatisfied. In SVGD, this personal preference constraint (referred to as preference constraint hereafter) avoids rendering users that receive a low personal preference; ii) Display slot ku for all u V , which specifies the maximum number of other users that can be rendered on user u s VR display, to prevent too many people being rendered on users VR displays, making the displays too crowded1. Here, even if two users social utilities are low, e.g., previously unknown to each other, they may still want to see each other on their VR displays if they have a high preference. Therefore, we do not set the social utility constraint in our problem. Please note that SVGD allows each user to see different surrounding 1The constraint ku can be modified to a weighted constraint to account for users varying weights based on their proximity to each other such that the sum of the weights of all the users in a given user s configuration does not exceed ku. The Thirty-Eighth AAAI Conference on Artificial Intelligence (AAAI-24) users, which makes the problem different from traditional group query and group recommendation problems. As mentioned in Section , there are three disadvantages, D1, D2, and D3 in current VR, and it is not feasible to let everyone see each other. Therefore, we employ A1 and A2 to customize the display for each user (addressing D2 and D3), and use A3 to avoid the user s display being overcrowded (addressing D1). Moreover, we define the total SVD utility of A as P u V P v Au w Au(u, v), which is the summation of the utility values of each user u and the other users in u s VR display, i.e., Au. Here, w Au(u, v) is the SVD utility defined in Definition 3. Specifically, the SVGD problem is formulated as follows. Problem.: Social-Aware VR Conference Group Display Configuration (SVGD). Given: A social network G = (V, E), personal preference p(u, v) for all u, v V , social utility τ(u, v) for each directed edge (u, v) E, weighting parameter λ, personal preference constraint θ, and display slot ku, u V . Objective: To find an SVD ku-configuration A that maximizes the total SVD utility: P u V P v Au w Au(u, v), where each set Au A contains at most ku other users (ku can vary for each individual), and for each v in Au, its personal preference for u is at least θ. An alternative objective is to maximize users exposure, aiming for each user to appear in the displays of the largest possible number of other users on average. However, this objective does not consider the preferences factor, leading to poor potential interactions. Another potential problem formulation is to maximize the social utilities while requiring the personal preferences to be at least θ (the personal preference constraint). However, this formulation lacks the flexibility to cover various VR conference scenarios and cater to varying user intentions. Our problem formulation permits adjusting these two trade-off parameters to fit different scenarios. Theorem 1. SVGD is NP-hard. Proof. We prove this theorem by reducing it from the Exact Cover by Three Sets problem (Cormen et al. 2022). The complete proof is shown in Appendix A. Approximation Algorithm For SVGD In this section, we present the Social Utility-Aware VR Conference Group Formation (SVC) algorithm, a 2approximation algorithm for SVGD. A simple greedy approach falls short of achieving a guaranteed performance bound. For instance, the preference-based configuration (illustrated in Fig. 1) solely accounts for A2 (personal preferences), while the social-based setup only considers A1 (social relationships), leading to compromised user satisfaction. The SVC algorithm efficiently tackles the SVGD problem by assigning near-optimal configurations to users to maximize SVD utility. It employs a Quantified Social Benefit Network (QSB network) to accommodate two selection strategies: Personal Preference Attention Configuration (PConfiguration) and Social utilities Aware Configuration (SConfiguration). During the configuration, the QSB network integrates social utilities and personal preferences into the original graph, using directed edges for P-Configurations and undirected edges for S-Configurations. Next, in the Configuration Comparison stage, the effective SVD utility ratio between the two configurations is compared in order to determine the sequence of users to be included in the configuration. A superlative sequence is created for each user s configuration, ensuring that the utility obtained by each user s choice of the configuration decreases iteratively. This ensures a better utility to be extracted earlier. Detailed Algorithm Design Effective SVD utility with P/S-Configuration. To facilitate the effective design of P-Configuration and S-Configuration, we first extend the SVD utility defined in Equation (1), and propose the concept of effective SVD utility to capture the increment of SVD utility when executing P-Configuration or S-Configuration. For a user u in a P-Configuration, when user v is added to Au, the effective SVD utility with PConfiguration of user u is denoted as (u v). Similarly, the concept of effective SVD utility with S-Configuration of users u and v is denoted as (u v). The S-Configuration involves two users u and v, i.e., SVC selects user u into Av and selects user v into Au simultaneously. Effective SVD utility ratio with P/S-Configuration (Effective P/S ratio). To enable SVC to identify appropriate users to include in a user u s VR display set Au, we first present the effective SVD utility mentioned above. We define the effective SVD utility ratio with P/S-Configuration, the effective SVD utility normalized by the number of selected users whose ku are not full, in order to find the effective SVD utility increment when adding a user to another user s display slots under both configurations. The effective SVD utility ratio with P-Configuration (referred to as effective P-Configuration ratio for short) of users u on v is denoted as ρ(u v). The effective SVD utility ratio with SConfiguration (referred to as effective S-Configuration ratio for short) of users u and v is denoted as ρ(u v). The two effective ratios are critical because the display slots are limited. By measuring the normalized increment of SVD utility in user selections, SVC effectively identifies the candidates that result in better solutions with limited display slots. Construction of Quantified Social Benefit Network. Given the input social network G = (V, E), with i) personal preference constraint θ, ii) display slot ku, u V , SVC embeds the social utilities and personal preferences into the original graph and constructs the QSB network. The SVD utility containing only personal preferences is embedded in the two weighted directed edges between each pair of users, and the sum of SVD utilities between two users with personal preferences and social utilities is embedded as a weighted undirected edge between users. We present an example in Appendix C. Upon performing the configuration, the edge weights between the selected users are updated to preserve the approximation ratio. The Thirty-Eighth AAAI Conference on Artificial Intelligence (AAAI-24) SVC maintains a collection S = {Su | u V } of user sets. Each set Su S represents the users selected by SVC to appear in the VR display of user u. Initially, SVC generates a collection Aθ = {Aθ u | u V } of potential sets. Each set Aθ u Aθ contains only the users who satisfy the preference constraint, i.e., Aθ u = {v | p(u, v) θ, v V }, to filter out impossible user selections. Subsequently, SVC iteratively selects the P-Configuration or S-Configuration with the greater effective ratio. This approach ensures that the user selected in each iteration represents the current best solution. After each selection, SVC updates the corresponding ρ( ) accordingly. Specifically, if ρ(x y) < ρ(u v), SVC performs a P-Configuration and identifies the user v Aθ u and the corresponding Su that maximizes the effective PConfiguration ratio ρ(u v). SVC then adds user v to Su. To find ρ(u v), i) if user u is not in Av, then (u v) = (1 λ) p(u, v), because in this case I(u, v) = 0, indicating that user u does not appear in user v s VR display. ii) Otherwise, if user u is already in Av, then (u v) = (1 λ) p(u, v) + λ (τ(u, v) + τ(v, u)), because users u and v establish the co-display relationship and I(u, v) = 1, and thus λ (τ(u, v) + τ(v, u)) is included in (u v). Please note that p(v, u) is excluded here as it has previously been introduced in a P-Configuration (v u). Then, the effective P-Configuration ratio ρ(u v), which is (u v) 1 , where the 1 in the denominator indicates that only 1 user, i.e., user v, is selected at this iteration. Next, after a P-Configuration that adds v to Su, SVC updates ρ(v u) to (1 λ) p(v, u)+λ (τ(u, v)+τ(v, u)) and sets ρ(u v) to 0, because the SVD utility of v on u increases and v can no longer be selected for u by an S-Configuration. In previous iterations, a user may have chosen another user based on a high effective P-Configuration ratio. However, it is possible that the social utilities between these two users are greater than other users P-Configuration. Therefore, the social utilities between the users who have previously made a P-Configuration needs to be examined to ensure that the current selection is still the best option, i.e., the largest ratio. In addition, the process of edge update is also important, since it involves one-way and opposite selections to partition the globally optimal solution. One-way selection refers to the situation whether a user u chooses another user v, whereas the opposite selection refers to user v selecting user u. With one-way selection, we partition the global optimal solution into two segments, with one segment chosen by SVC, while the other is not. Similarly, to carry out an S-Configuration, SVC identifies two users x and y that maximize the effective SConfiguration ratio ρ(x y) so as to add users x and y to Sy and Sx, respectively. The calculation of ρ(x y) is based on the effective SVD utility with S-Configuration (u v), which is the SVD utility increment when user u is selected into Av and user v is selected into Au simultaneously. Therefore, (u v) = (1 λ) (p(u, v)+p(v, u))+ λ ((τ(u, v) + τ(v, u))). In this case, I(u, v) = 1, and the co-display relationship between users u and v thereby holds. Then, ρ(u v) = (u v) 2 because 2 users (i.e., users u and v) are chosen for Av and Au, respectively. After performing an S-Configuration of x and y, SVC also needs to update the effective SVD utility. In this case, SVC sets ρ(x y) and ρ(y x) to 0, as x and y can no longer be selected for each other. The updating of edge weights in the S-Configuration is used to prevent redundant selection. Since x and y are already in the configuration of each other, if the weights are not updated in time, redundant P-Configurations may be executed in the future, which impacts the efficiency. Next, in each iteration, SVC adds new users to the collection S by performing either a P-Configuration or SConfiguration with the greater effective ratio. If the number of users in a set Su for a user u reaches the limit of ku display slots, SVC stops adding users to that set. If all the sets in S are full or no more users can be selected, SVC stops and returns S as the final solution A . We present a running example and the pseudocode in Appendix D. Analysis of Approximation Ratio of SVC In this section, we prove that SVC is a 2-approximation algorithm. The core idea is that, given any optimal solution O, the selected users can be viewed as a sequence of singleuser selections with a descending order based on the effective SVD utility, i.e., {u O 1 , u O 2 , ...}. This is because the optimal solution is the result with the highest total SVD utility, the configuration with a high SVD utility must be selected firstly. For the proposed SVC, it also selects a sequence of users {u1, u2, ...} with P/S-Configurations. We prove that the effective SVD utility of u O i being less or equal to two times the effective SVD utility for each user ui must hold. In other words, the effective SVD utility ratio with P/SConfiguration is bounded by two on the selected users of SVC. The time complexity of SVC is O(max u(ku) |V | log(|E|)). Theorem 2. SVC is a 2-approximation algorithm to SVGD with a time complexity O(max u(ku)|V | log(|E|)). Proof. The detailed proof is presented in Appendix E. In real applications, ku is usually small, e.g., ku < 100, enabling SVC to find the solution efficiently. Moreover, the size of users V in the VR scenarios is much smaller than the number of users V in each dataset in Section , because it only encompasses the individuals participating in the VR event. Therefore, SVC is very efficient in practical scenarios. Experimental Results Performance Evaluation The detailed experiment setup is presented in Appendix I. To evaluate the effectiveness and the efficiency of the proposed SVC, we compare SVC with 12 various baselines on 5 real datasets. i) Timik (Jankowski, Michalski, and Br odka 2017), ii) Pokec (Takac and Zabovsky 2012), iii) Youtube (Yang and Leskovec 2012), iv) SMMnet (Moraes and Cordeiro 2019), and v) Facebook (Mc Auley and Leskovec 2012). The specifics of these datasets are described in Appendix I. While there is no existing algorithm for the SVGD problem, we implement 12 baseline approaches. i) PER (Yang The Thirty-Eighth AAAI Conference on Artificial Intelligence (AAAI-24) Timik Youtube Pokec SMMnet Total SVD Utility (ku = 25, λ = 0.7, θ = 0.1) SVC (Ours) PER SOC RAND MAXD (a) Total SVD utility on diff. datasets Timik Youtube Pokec SMMnet (ku = 25, λ = 0.7, θ = 0.1) SVC (Ours) PER SOC RAND MAXD (b) Time on diff. datasets Figure 2: Results on four large-scale datasets et al. 2012b), ii) SOC (Yang et al. 2012b), iii) RAND, iv) MAXD (Behnezhad and Derakhshan 2020), v) KOAVG (Ko et al. 2020), vi) SSGQ (Chen et al. 2018b), vii) BCC (Dong et al. 2021), viii) COMUR (Chen and Yang 2022), ix) Gra Frank (Sankar et al. 2021), x) FSGSel (Shen et al. 2022), xi) MAXGF (Shen et al. 2020), and xii) BF, which enumerates all combinations to find the optimal solution. The baselines are introduced in Appendix H. We evaluate them by the following metrics: i) total SVD utility, and ii) total execution time. All algorithms are implemented in a server with an Intel Xeon W-2245 CPU with 128 GB RAM. Personal preferences and social utilities are assigned according to (Ko et al. 2020; Chen and Yang 2022). Sensitivity tests and comparisons with optimal solutions on the small dataset. To understand the performance gap between the proposed approach and the optimal solution, we first compare the results on the small dataset, Facebook. The results demonstrate that our algorithm outperforms other baselines. Moreover, we conduct additional sensitivity tests on dataset Facebook with different λ, θ, ku, and different inputs generated by Random, Rev GNN (Li et al. 2021), Graph Rec (Fan et al. 2019), and DGRec (Yang et al. 2023). Details of the experiments are shown in Appendix J. Scalability and sensitivity tests on large networks. Fig. 2 compares the total SVD utility on the four large social networks, Timik, Youtube, Pokec, and SMMnet. We set λ = 0.7, θ = 0.1, and all users display slots ku = 25 by default. Figs. 2(a) and 2(b) present the objective values and execution time of the proposed SVC and other baselines. Because the numbers of users in these datasets are different, we performed Z-Score Normalization (Patro and Sahu 2015) on the results. The results illustrate that the proposed SVC outperforms the other baselines in terms of solution quality and efficiency. Fig. 3 presents the sensitivity tests on the 3D VR metaverse social network dataset, Timik. In Figs. 3(a) and 3(b), we compare the results with different values of ku = {23, 25, 28, 30} and values of λ = {0.5, 0.6, 0.7, 0.8}. SVC achieves the best performance over all the baselines. PER and SOC do not perform well because they consider either personal or social aspect only. MAXD employs the degree of edges to create the configuration and thereby is difficult to ensure that the chosen edges have a higher personal preference or social utility. KOAVG is designed to recommend items to a group of users, and thereby ignoring users per- 23 25 28 30 Total SVD Utility ku (λ = 0.7, θ = 0.5) SVC (Ours) PER SOC RAND MAXD (a) Total SVD utility on diff. ku SVC (Ours) PER SOC RAND MAXD KOAVG SSGQ BCC MAXGF COMUR Gra Frank FSGSel Total SVD Utility λ (ku = 25, θ = 0.5) Personal preference Social utility 0.6 0.7 0.5 0.8 (b) Total SVD utility on diff. λ 23 25 28 30 ku (λ = 0.7, θ = 0.5) SVC (Ours) PER SOC RAND MAXD (c) Time on diff. ku 0.5 0.6 0.7 0.8 λ (ku = 25 , θ = 0.5) SVC (Ours) PER SOC RAND MAXD (d) Time on diff. λ Figure 3: Sensitivity tests on the large-scale Timik dataset sonal preferences. BCC and SSGQ do not pay special attention to personal preferences and do not guarantee that users can see each other, leading to high social utility limitations. Gra Frank, COMUR, FSGSel, and MAXGF establish configurations based on different factors, which might disregard the important personal preferences and social utilities in the process. Moreover, SVC outperforms other baselines by 75% on average, while the p-values of SVC to the baselines are all less than 0.05, indicating that the SVD utility of SVC is statistically greater than other baselines. Figs. 3(c) and 3(d) present the execution time of the sensitivity test. The value of λ only influences the calculation of the total SVD utility, does not affect the execution time. Conclusion This paper explores the new research problem, SVGD, to configure display slots for users in VR conferences by jointly considering three important factors. We analyze the hardness of SVGD and propose a 2-approximation algorithm, named SVC, to tackle SVGD. In our experiments with 5 real datasets manifest that SVC surpasses other baseline methods in both solution quality and efficiency. 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