Abstract Opinion maximization has attracted much attention in viral marketing. It selects an initial seed user set to disseminate user opinions on the target product and finally produces more positive opinions in social networks. In earlier studies, a critical but not studied problem is the fairness of information dissemination in groups with sensitive characteristic (such as age or race). People prefer to promote products for target users (majority groups) rather than in sensitive characteristic groups (minority groups). That leads to the differences in information dissemination between minority groups. In addition, social networks have the in-depth structural information. Therefore, in this paper, we design opinion maximization based on fairness framework (OMBF) using graph attention networks (GAT) to exploit more network information, and only consider the fairness in minority groups. OMBF composes of three parts: (1) the determination of candidate nodes according to node representations, (2) the dynamic changes in opinions, (3) the selection of final seed nodes. Firstly, we utilize GAT to obtain node representations and to determinate candidate nodes and design a node opinion formation model to model the dynamic changes in opinions. Then, we use the fair constraint value to ensure the fairness in the information dissemination process of minority groups. Based on above, final seed nodes are selected. We conduct experiments on synthetic and real-world datasets to show the effectiveness of our approach. The results indicate that the total opinions of active nodes in all nodes and fair values in minority groups are better than the chosen state-of-the-art benchmarks.

Radius-bounded$k$-core queries (RB-$k$-core queries) in geo-social networks aim to identify all$k$-cores containing a given query vertex$q$, where all vertices in each$k$-core fall within a circle defined by a specified query radius$r$. These queries are widely used in applications such as team formation and event organization. However, specifying query parameters$k$and$r$can be challenging for users without domain expertise, often resulting in misaligned query results. Specifically, some expected vertices may be missing, while unexpected vertices may appear in the results. To address this issue, we investigate the problem ofexploringoptimalrefinedparametersforexpected(EOPE) andunexpected(EOPU) results in RB-$k$-core queries. The goal is to explore optimal parameters that ensure the expected vertex$\omega$(or unexpected vertex$\psi$) and query vertex$q$appear (or do not appear) in the same RB-$k$-core. For the EOPE problem, we first propose two baseline algorithms:PriorityRandHybridR. To improve efficiency, we develop two more advanced algorithms:PriorityKandHybridK. Additionally, we introduce a novel index calledHCR-Tree, based on hierarchical coreness of vertices and R-Tree, to enhance exploration efficiency. For the EOPU problem, we begin with a basic solution (BS) and then design the segmentation algorithmSA, which incorporates effective pruning and termination strategies. We conduct extensive experiments on five real-world geo-social network datasets. The results demonstrate that our proposed algorithms effectively explore optimal parameters. Among them,HybridKproves most effective for EOPE, whileSAperforms best for EOPU. Furthermore,HCR-Treeoutperforms R-Tree for both EOPE and EOPU problems.

Radius-bounded$k$-core queries (RB-$k$-core queries) in geo-social networks aim to find all$k$-cores containing a given query vertex$q$while all vertices in each$k$-core fall into a circle under a given query radius$r$, which is widely used in many applications, such as team formulation and event organization. However, the query parameters$k$and$r$are hard to specify by the users without any background knowledge, which means the query results often do not meet the users' requirements, i.e., some expected vertices are missed in the query results. To tackle this issue, we investigate the problem of exploring optimal refined parameters (EOP) for expected results on RB­$k$-core queries, which aims to explore the optimal parameters that make the expected vertex$\omega$and query vertex$q$appear in the same RB-$k$-core. To address the EOP problem, we first propose two baseline algorithms, namely PriorityR and HybridR, which refine the parameters$k$and$r$simultaneously based on the effective bounds of the refined$r^{\prime}$• To enhance the efficiency of exploring optimal parameters, we develop two efficient al-gorithms. The first algorithm, Priority K, simultaneously refines both parameters based on the effective bound of the refined$k$• The second algorithm, HybridK, explores the optimal parameters using the continuous convergence bounds of the refined$k^{\prime}$and$r$• Furthermore, to enhance exploration efficiency, we develop a novel index, called HCR-Tree, based on the hierarchical coreness of vertices and R- Tree. This index accelerates the verification of whether the coreness of a vertex in any sub graph exceeds$k$in the above algorithms. Finally, we conduct extensive experiments using five real geo-social network datasets, which show that the optimal parameters can be explored effectively by the algorithms, and HybridK is the most effective. Meanwhile, the HCR- Tree performs better than the R- Tree for the EOP problem.