SentiRank: A Novel Approach to Sentiment Leader Identification in Social Networks Based on the D-TFRank Model

Abstract

With the rapid evolution of social computing, online sentiments have become valuable information for analyzing the latent structure of social networks. Sentiment leaders in social networks are those who bring in new information, ideas, and innovations, disseminate them to the masses, and thus influence the opinions and sentiment of others. Identifying sentiment leaders can help businesses predict marketing campaigns, adjust marketing strategies, maintain their partnerships, and improve their products’ reputations. However, capturing the complex sentiment dynamics from multi-hop interactions and trust/distrust relationships, as well as identifying leaders within sentiment-aligned communities while maximizing sentiment spread efficiently through both direct and indirect paths, is a significant challenge to be addressed. This paper pioneers a challenging and important problem of sentiment leader identification in social networks. To this end, we propose an original solution framework called “SentiRank” and develop the associated algorithms to identify sentiment leaders. SentiRank contains three key technical steps: (1) constructing a sentiment graph from a social network; (2) detecting sentiment communities; (3) ranking the nodes on the selected sentiment communities to identify sentiment leaders. Extensive experimental results based on the real-world datasets demonstrate that the proposed framework and algorithms outperform the existing algorithms in terms of both one-step sentiment coverage and all-path sentiment coverage. Furthermore, the proposed algorithm performs around 6.5 times better than the random approach in terms of sentiment coverage maximization.

1. Introduction

With the booming emergence and rapid proliferation of social computing applications, users prefer to express their sentiments and opinions on products, services, or political subjects through various online channels, including E-commerce websites, forums, blogs, wikis, etc. In such distributed social networking systems, users interact with each other to maximize information sharing and recommendation based on their social relationships [1,2]. Social networks further accelerate the generation and spreading of users’ sentiments within shared communities, which significantly affects consumers’ buying decisions, images of enterprisers, and public opinions. Unlike influential users in social networks who can drive behavioral changes through their reach and authority, sentiment leaders shape public emotional trends and opinions through emotional resonance. That is to say that sentiment leaders in social networks are those who bring in new information, ideas, and innovations and disseminate them to the masses. Their sentiments are often more representative and authoritative, and thus, other members are easily influenced by sentimental leaders. Here is a natural question: “who are the sentiment leaders that disseminate their sentiments to others?” The identification of sentiment leaders in social networks is very challenging and has much commercial value; it can help businesses predict marketing campaigns, adjust marketing strategies, maintain their partnerships, and improve their products’ reputations.

Sentiment analysis and opinion mining that aim to identify the opinions, judgments, and information related to the feelings and attitudes expressed in natural language texts have been recently studied in the context of social networks [3,4]. Sentiment analysis can be used as a means of automatically handling customer feedback, as a basis for targeting advertisements, and as a tool to assist in analyzing consumer tendencies. Singh et al. [5] conducted the sentiment analysis based on BERT model over Twitter for understanding the mental state of users during the period of the pandemic. Fu et al. [6] firstly discussed the selection strategy of multi-source domains to optimize data quality and further proposed a domain adapter and discriminator for sentiment transfer.

As a newly emerging research direction, sentiment propagation in social networks has been investigated recently [7,8,9,10]. Liu et al. [11] conducted text modeling with words’ co-occurrence based on the topic model. Additionally, sentiment computing and the time factor were incorporated to construct the dynamic topic–sentiment mixture model. An et al. [12] measured the topical influence and sentiment contagion of public event stakeholders and proposed four indicators. By quantifying and comparing the topical and emotional influence of different stakeholders, deep insights into the public opinion propagation pattern in social media contexts were presented. Aiming to infer the sentiment expressed within a document, Dragoni [13] developed an approach that relies on evolutionary algorithms and exploiting semantic relationships to estimate domain-dependent polarities of opinion concepts. Wang et al. [14] investigated an interesting problem concerning sentiment spreading in text-based social networks. Concretely, a unified probabilistic framework was formalized for this problem as a topic-enhanced sentiment spreading model that can predict users’ sentimental statuses based on their historical emotional status, topic distributions in tweets, and social structures.

Although sentiment analysis and propagation have attracted much research interest, sentiment leader identification, as a new research domain, has not been thoroughly investigated yet. Specifically, the existing studies have not considered the fact that sentiment propagation can reach maximal sentiment coverage through some key sentiment leaders. Therefore, there exists an important gap between sentiment computing, business intelligence, and distributed social networking services. To fill in this gap, a new research issue–sentiment leader identification in social networks—is pioneered in this study, aiming to identify the sentiment leaders who initiate sentiments and influence other users in social networks. Although sentiment polarity is indeed related to opinions, sentiment leaders play a unique role in shaping public emotional trends and opinions through emotional resonance. Unlike opinion leaders, who primarily focus on disseminating specific viewpoints [15,16], sentiment leaders influence the emotional climate surrounding a topic. This distinction makes sentiment leader identification a novel and challenging task that extends beyond traditional opinion leader identification. Technically, capturing the complex sentiment dynamics shaped via multi-hop interactions and trust/distrust relationships across distributed networks poses a challenge, as sentiments propagate through intricate social ties. To address these challenges, we develop a novel sentiment leader identification approach, SentiRank. The major contributions of this paper are summarized as follows.

• To obtain a comprehensive understanding of the information diffusion over social networks, we formalize the problem of sentiment leader identification, which aims to identify sentiment leaders who can propagate their sentiment to other users and maximize the sentiment coverage in social networks.
• We propose an original solution framework, named “SentiRank”, in order to solve the problem of sentiment leader identification. In this proposed framework, (1) we initially construct the sentiment graph using a social network with trust and distrust relationships, (2) we consider the sentiment communities in both the positive and negative sentiment systems and the five-star scale sentiment system, and we transform the sentiment community detection problem in to a semi-definite programming (SDP) problem and solve it using an SDP-based random rounding method, (3) and we investigate the importance metric D-TFRank of a node in dynamic social networks that is used to rank nodes in a selected sentiment community.
• Corresponding to the proposed solution framework, efficient algorithms of sentiment leader identification based on a sentiment community with respect to two types of sentiment representations are proposed. Extensive experiments based on real-world datasets are conducted to evaluate the performance and effectiveness of the proposed framework and associated algorithms. The results demonstrate their efficiency in terms of both one-step sentiment coverage and all-path sentiment coverage. It is worth noting that the proposed algorithm performs around 6.5 times better than the random approach in terms of sentiment coverage maximization.

The remainder of this paper is organized as follows. Section 2 presents the problem definition and the solution framework. The proposed identification algorithms are presented in Section 3. Section 4 shows the experimental results and evaluates the proposed algorithm. Finally, Section 5 concludes this paper.

2. Related Work

This section mainly analyzes the literature related to the topic of this study. Therefore, we conducted a literature review involving the following aspects: sentiment propagation and semi-definite programming.

2.1. Sentiment Propagation on Social Networks

The recent literature on sentiment propagation in social networks highlights the dynamic interplay of user behavior, platform algorithms, and contextual factors. Chen et al. [17] proposed a dynamic model integrating time-decay and emotion-exciting mechanisms, demonstrating how sentiment evolves over extended diffusion periods. Their model, validated on real-world datasets like Bitcoin-OTC, showed improved F1-scores by 3.5% and reduced MAE, emphasizing the role of opinion leaders in convergence over time. Hung et al. [18] proposed a cognitive emotional contagion model (CECM), which combines the model of individual characteristics, the topology of a social network, and the changing process, as well as the evolution, of emotion contagion. Yin et al. [19] established an emotional Susceptible-Forwarding-Immune (E-SFI) propagation dynamics model that integrates the classification of emotions (positive, neutral, and negative) with community emotional selection mechanisms, investigating their combined impact on information propagation and subsequent public sentiment formation. Ban et al. [20] presented innovative methodologies that automate the construction of comprehensive sentiment lexicons by integrating traditional linguistic approaches with state-of-the-art artificial intelligence technologies.

It is noted that sentiment propagation is also important for sentiment community detection. Wang et al. [21] firstly introduced the concept of a “sentiment community“ and further proposed two methods for discovering sentiment communities by adopting the optimization models of SDP.

2.2. Semi-Definite Programming

Semi-definite programming (SDP) has emerged as a fundamental tool in modern optimization, extending linear and convex programming to matrix spaces. Over the past decade, SDP has seen significant theoretical refinements, algorithmic improvements, and novel applications in machine learning [22], quantum computing [23], and large-scale data analysis [24].

First of all, many research efforts have involved algorithmic innovations in SDP. Shinde et al. [25] developed a randomized algorithm that returns a random vector whose covariance matrix is near-feasible and near-optimal for the SDP that can be extended for dealing with a broader range of structured convex optimization problems. Han et al. [26] presented a new first-order method for solving general SDP problems, based on the alternating direction method of multipliers (ADMM) and a matrix-splitting technique. Fernando et al. [27] designed their algorithm by “quantizing” classical SDP solvers based on the matrix multiplicative weight method.

Additionally, cutting-edge applications based on SDP are emerging in the fields of machine learning and AI, quantum information, and control and robotics. Zhang et al. [28] proposed a geometric technique to determine the exactness of SDP certificates for least-squares restricted adversarial attack problems, showing that the certificate is exact when the projection of a point onto a hyperbola lies on its major axis, proving exactness for single hidden layers under mild assumptions, and explaining conservativeness for multiple hidden layers. Yau et al. [29] developed graph neural network architectures that capture optimal approximation algorithms for a large class of combinatorial optimization problems, using powerful algorithmic tools from SDP. Wu et al. [30] proposed a novel control protocol and an implementation algorithm that enables leaders to achieve desired formations via SDP techniques.

3. Problem Definition

This section presents the definitions of sentiment graph, sentiment community, and sentiment coverage, and then it formulates the problem of sentiment leader identification. In addition, the proposed solution framework SentiRank is described.

3.1. Problem Definition

Definition 1

(Sentiment Graph). A sentiment graph is represented as a directed graph, $G=(V,E,S)$, where $V=\{v_1,v_2,\dots ,v_n\}$ denotes its node (i.e., user) set, E indicates its edge set with the edge $(u,v)\in E$, representing a sentiment orientation from v (sentiment sender) to u (sentiment receiver), and $S:\{s_1,s_1,\dots ,s_n\}$ is the set of sentiment values held by a user in set V; for example, user $v_i$ holds a certain sentiment, $s_i$, towards a particular product or topic. The value of $s_i$ can be defined either as $\{positive,negative\}$ in the positive and negative sentiment system or represented by the number of stars in the 5-star scale sentiment system (1 star means the most negative, and 5 stars means the most positive).

Definition 2

(Sentiment Community).  A sentiment graph, G, can be divided into a number of clusters (i.e., various sentiment communities), $\{C_1,C_2,\dots ,C_m\}$, in terms of different sentiments, such that the following applies:

• Each node, $v_i$, belongs to one cluster, $C_k$, ($1\le k\le m$);
• Each cluster, $C_k$, is a connected subgraph in G;
• Nodes within a cluster should share similar sentiments, and nodes in different clusters should have different sentiments.

Definition 3

(Sentiment Coverage).  In a sentiment graph, G, let T represent the subset of nodes selected to initiate the sentiment propagation in a sentiment community, $C_i$, which is called the seed set. The sentiment coverage, $\sigma \left(T\right)$, is defined as the expected number of nodes infected with sentiment $s_i$.

Based on the above definitions, the problem of sentiment leader identification is stated as follows:

Given a sentiment graph, $G=(V,E,S)$, and an integer, $K\le \left|V\right|$, sentiment leader identification is conducted to find a set, $T\subseteq V$, $\left|T\right|=K$, such that the sentiment coverage, $\sigma \left(T\right)$, is the maximum.

The problem of sentiment leader identification aims to maximize sentiment coverage by identifying influential users who can propagate their sentiments to others. This differs from opinion leader identification, which focuses on spreading specific viewpoints. It must account for complex sentiment dynamics shaped by multi-hop interactions and trust/distrust relationships, making it a distinct and challenging problem.

For the sake of understanding, the above problem can be further described using the example of a social marketing application. The working process of the sentiment leader identification in social marketing is as follows: a company usually selects only some customers with positive sentiment on their products and plans advertisements targeting these positive customers. Once these potential customers accept the content of the given advertisements and love the advertised products, they start influencing their friends in distributed social networks through sentiment contagion. Their friends further influence their friends’ friends, and so on. Thus, a large population in the social network adopts the advertisement through both information and sentiment propagation.

3.2. Solution Framework

Different from influence propagation in the influence maximization problem, sentiment leader identification seeks to propagate sentiment in order to maximize sentiment coverage eventually. Therefore, this process can be viewed as a special case of the influence maximization problem, which has been proven to be NP-hard [31,32]. As a result, the sentiment leader identification problem is also NP-hard, and it is very challenging to solve this problem. Furthermore, as social networks are large-scale, involve complex connection structures, and are very dynamic, the solution to this problem should be efficient and scalable.

To solve the addressed problem, a solution framework called SentiRank is proposed in this paper. The following section presents the proposed solution framework of SentiRank and the related algorithms in detail. SentiRank contains three key technical steps: (1) constructing the sentiment graph from a social network; (2) detecting the sentiment communities; (3) ranking the nodes on selected sentiment communities to determine sentiment leaders.

4. Identifying Sentiment Leaders in Social Networks

SentiRank is elaborated in this section as a solution for the problem of sentiment leader identification. In what follows, we will present the key issues of SentiRank.

4.1. Framework of SentiRank

Figure 1 presents a framework of SentiRank.

Apparently, the input of SentiRank is a social network with trust/distrust relationships, and then the sentiment graph can be constructed (Step 1). After that, an SDP-based random rounding method is adopted for detecting corresponding sentiment communities (Step 2). Within sentiment communities, a novel ranking model called D-TFRank is proposed by considering the global importance and local importance of users in dynamical social networks. Ultimately, sentiment leaders can be obtained by ranking the nodes in the selected sentiment communities (Step 3).

4.2. Construction of the Sentiment Graph

To construct the sentiment graph, the sentiment value of each user in social networks should be calculated. The sentiment variable is used to describe the emotion of users towards a certain product or opinion. As illustrated in Figure 2a, there exist certain trust/distrust relationships between users in social networks. Thus, sentiment distributions can be described using the trust/distrust relationships. Given that the initial sentiment of user $v_4$ is “positive” and $v_1$ trusts $v_4$ (as shown in Figure 2a), the sentiment of user $v_1$ should be “positive”, too (Figure 2b). On the contrary, user $v_5$ holds the negative sentiment because user $v_1$ distrusts $v_5$.

The working process of sentiment graph construction is described as follows. With $v_4$ as the seed node having an initial “Positive” sentiment, sentiment propagation follows defined rules: trust relationships make a node adopt the sentiment of the trusted node, while distrust leads to adopting the opposite. Starting from $v_4$, $v_1$ trusts $v_4$ (via the + edge), so $v_1$ becomes “Positive”. Then, $v_2$ trusts $v_1$ ( + edge) and takes “Positive”, while $v_3$ and $v_5$ distrust $v_1$ ( − edges) and take “Negative”, resulting in a sentiment graph where $v_4$ (seed), $v_1$, $v_2$ are “Positive”, and $v_3$ and $v_5$ are “Negative”—the rules can be refined for complexity, but this traceable process builds sentiment assignments.

Eventually, a social network with trust/distrust relationships is converted into a sentiment graph (Figure 2b). It is worth noting that the sentiment graph is a directed graph in which each node holds sentiments. However, a social network with trust/distrust relationships is a weighted graph in which each edge exhibits the trust or distrust relationship.

4.3. Detection of the Sentiment Communities

We study the detection of sentiment communities with two typical cases: (1) simple sentiment labels (positive and negative) and (2) five-star rating scale.

4.3.1. Case 1

In this case, users have either positive or negative sentiments on a certain item in sentiment graph, G. For two linked users, $v_i$ and $v_j$, in a sentiment graph, G, if they have the same sentiment polarity, the link is assigned with a weight, $w_{ij}=+1$; otherwise, $w_{ij}=-1$. The objective of sentiment community detection is converted to maximize the agreement on users’ sentiments within clusters. It is equal to maximizing the number of +1 edges within clusters, plus the number of −1 edges between clusters. Clearly, this is a typical correlation clustering problem and is NP-hard. Since addressing the correlation clustering can be viewed as a convex constrained optimization process, we transform it into a semi-definite programming (SDP) problem because semi-definite programs are important tools for developing approximation algorithms for NP-hard maximization problems. Then, we solve this problem using an SDP-based random rounding method [33,34].

The membership of node $v_i$ to a cluster can be represented as a vector, $x_i=(x_{i1},x_{i2},\dots ,x_{im})$, where $x_{ij}=1$ if $v_i$ belongs to $C_j$, and $x_{ij}=0$ otherwise. The objective function is formalized as follows:

$$\begin{gathered} max\left\{\sum _{\forall w_{ij}=+1}{X}_i•X_j+\sum _{\forall w_{ij}=-1}(1-X_i•X_j)\right\} \\ s.t. \\ {X}_i•X_i=1(\forall v_i\in V), \\ {X}_i•X_j\ge 0(\forall v_i,v_j\in V,i\ne j). \end{gathered}$$

(1)

Now, solving the above SDP problem becomes a key step for detecting sentiment communities. Algorithm 1 presents the implementation codes using MATLAB to solve this problem. The proposed algorithm aims to partition users in a sentiment graph into communities to maximize internal sentiment consistency, transforming the NP-hard correlation clustering problem into an SDP problem. The objective function balances maximizing +1 edges within clusters and −1 edges between clusters, with constraints like the unit vector and non-negative inner product. After solving the SDP via relaxation, random rounding converts continuous vectors to discrete cluster assignments. Specifically, Lines 4–6 indicate a basic constraint. Lines 7–9 limit the range of the value of the sentiment vector. It should be (1,0) (positive sentiment) or (0, 1) (negative sentiment). Lines 10–14 refer to another inequality constraint in the SDP problem, while Lines 15–23 denote the objective function. The MATLAB2024a tool box–YALMIP [35] is called to solve this problem and obtain the sentiment communities assignment (Line 25).

Algorithm 1: Sentiment community detection using SDP.

Require:  $G=(V,E,S)$; Number of nodes N; Number of +1 edges $e_1$; Number of -1 edges $e_2$; Ensure:  Sentiment community assignment vector of nodes x; 1: t = sdpvar[1] 2: $obj$ = t; 3: x = sdpvar[N,1, ‘full’]; 4:    for $i=1$ to N do 5:       $a=[1^{x\left(i\right)}\ast 1^{x\left(i\right)}]==1]$; 6:    end 7:    for $i=1$ to N do 8:       $b=[0\le x(i)\le 1]$; 9:    end 10:    for $i=1$ to N do 11:       for $j=i+1$ to N do 12:          $c=\left[x\right(i)\ast x(j)\ge 0]$; 13:       end 14:    end 15:    for $t_1=1$to$e_1$ do 16:       for $t_2$ = $t_1+1$to $e_2$ do 17:          for $t_3=1$to$e_1$ do 18:             for $t_4$ = $t_3+1$to$e_2$ do 19:          $obj$ = [sum($x\left(t_1\right)$*$x\left(t_2\right)$)] + [sum(1−$x\left(t_3\right)$*$x\left(t_4\right)$)]; 20:             end 21:          end 22:       end 23:    end 24: $constraint$ = $[a,b,c]$; 25: slovesdp($constraint$,-$obj$); 26: $int\left(x\right)$;

As a powerful optimization toolbox, YALMIP helps formulate the SDP problem with high-level syntax, interfaces with solvers to find optimal solutions for the relaxed problem, and supports result extraction for post-processing. This approach ensures computational tractability, theoretical guarantees, and practical efficiency, balancing rigor and applicability for real-world sentiment community detection.

4.3.2. Case 2

In this case, the sentiments of users are denoted in a five-star rating scale. The sentiment difference between users $v_i$ and $v_j$ fall into $[0,4]$, i.e., $|s_i-s_j|\in [0,4]$. Then, $(2-|s_i-s_j|\in [-2,2])$. In the clustering scenario, the transformation from sentiment difference [0,4] to [−2,2] via (2 − |$s_i$−$s_j$|) serves to map similarity/dissimilarity in a symmetric range of around zero, where positive values indicate similarity, and negative values indicate dissimilarity. If two users share the same sentiment, $(2-|s_i-s_j\left|\right)=2$, then they should be within the same cluster, i.e., $X_i•X_j=1$; otherwise, $(2-|s_i-s_j\left|\right)=-2$, i.e., $X_i•X_j=0$.

Regarding $X_i•X_j$: here, $X_i$ and $X_j$ are likely cluster assignment vectors (not sentiment values), where $X_i•X_j$ represents the dot product of binary vectors indicating a cluster membership. If two users are in the same cluster, their vectors have 1s in the same position, leading to a dot product of 1 (e.g., $X_i$ = [1,0], $X_j$ = [1,0], dot product = 1 × 1 + 0 × 0 = 1). If in different clusters, their vectors have 1s in different positions, resulting in a dot product of 0 (e.g., $X_i$ = [1,0], $X_j$ = [0,1], dot product = 0).

Therefore, the task of grouping linked users with similar sentiments into the same clusters while separating those with opposite sentiments into different clusters can be formalized as an optimization problem as follows:

$$\begin{gathered} max\sum _{\forall e_{ij}\in E}(2-|s_i-s_j\left|\right)X_i•X_j \\ s.t. \\ {X}_i•X_i=1(\forall v_i\in V), \\ {X}_i•X_j\ge 0(\forall v_i,v_j\in V,i\ne j). \end{gathered}$$

(2)

Hence, this problem can be solved using the SDP-based rounding methods as well.

4.4. Ranking Model

Sentiment leaders are sentiment initiators, and they influence the sentiments of other users. Thus, both the interactions between social sentiments and the reputation of users are essential characteristics for determining their sentimental leadership. In this section, we discuss how to rank users in terms of sentiment in a selected sentiment community, and then we propose our ranking model—SentiRank.

The proposed ranking model is an extended version of our previous work TFRank [36], namely D-TFRank. D-TFRank is a novel evaluation approach to user importance that takes into account the global importance (fractal importance) and local importance (topological importance) of users in a dynamic social network. Basically, D-TFRank works with a social graph as an input; after the evaluation of fractal importance value and topological importance value, the TFRank values are dynamically estimated with the considerations of nodes’ insertion and nodes’ deletion.

4.4.1. Fractal Importance of Nodes

As a global node importance evaluation approach, the fractal importance of nodes considers the fractal values of each child nodes in the weighted tree converted from a social network.

Definition 4

(Fractal Importance of Node [36]). Let T be a tree, with A indicating the root node of T; the fractal importance of the node is defined as follows:

$${\tilde{F}}_A=\sum _{i=1}^L\sum _{j=1}^{N\left(L_i\right)}\left(F_{L_i}^j\right)\ast {\left({\displaystyle \frac{1}{2}}\right)}^i$$

(3)

where $L_i$ indicates the $i_{th}$ level in T, $N\left(L_i\right)$ is the number of nodes in the $i_{th}$ level of T, L is the total number of levels of T, and the fractal value of the $j_{th}$ node in the $i_{th}$ level $F_i^j$ can be calculated with the approach in [37].

Definition 4 is suitable for measuring local importance in social networks because it incorporates the fractal characteristics of child nodes within the hierarchical structure, enabling a fine-grained analysis of node influence at the local level. In social networks, the importance of a node depends not only on its global hierarchical position but also on the complexity and connectivity of its local neighborhood. The formula considers the fractal value ($F_{L_i}^j$) of each node at level $L_i$, where the fractal value likely reflects the local structural complexity or connection density of the node (e.g., the number of neighbors, connection patterns, or information aggregation capabilities). By multiplying the fractal value with the exponentially decreasing weight ${{\displaystyle \frac{1}{2}}}^i$, the definition emphasizes that the local characteristics of nodes at closer hierarchical levels have a more direct impact on the root node’s importance—this aligns with the local interaction mechanisms in social networks, where the influence of nearby nodes (e.g., direct friends or close collaborators) is often more immediate and significant. Moreover, the double summation over levels and nodes at each level allows for the capture of diverse local structural features: even if nodes are at the same level, their different fractal values can reflect variations in local importance (such as key nodes in a community versus ordinary members). This approach thus quantifies the local importance of nodes by integrating hierarchical depth and local fractal properties, providing a basis for identifying core nodes within local communities, analyzing local information aggregation effects, or detecting tightly connected substructures in social networks.

4.4.2. Topological Importance of Nodes

During the sentiment propagation, the evaluation of the local node importance of the social graph is very important to dominate the sentiment leaders.

Definition 5

(Topological Importance of Node [36]). Let T be a tree, with A indicating the root node of T; the topological importance of the node is defined as follows:

$${\tilde{T}}_A=\sum _{i=1}^{L}N\left(L_i\right)\ast {\left({\displaystyle \frac{1}{2}}\right)}^i$$

(4)

where $L_i$ indicates the $i_{th}$ level in T, $N\left(L_i\right)$ is the number of nodes in the $i_{th}$ level of T, and L is the total number of levels of T.

Obviously, the topological importance of a node is not related to fractal values according to Equation (4). It is just calculated through the structure of the tree. This definition of topological importance for nodes can be applied to measure node importance in social networks due to its ability to comprehensively consider both the hierarchical structure of the network and the distribution of node connectivity. In social networks, nodes (such as users) at different hierarchical levels often have varying influences: root nodes (e.g., core users or sentiment leaders) located at the top of the hierarchy may have broader reach, while the importance of nodes at lower levels may depend on their positional depth and the number of connected neighbors. The formula assigns exponentially decreasing weights (i.e., ${{\displaystyle \frac{1}{2}}}^i$) to nodes at different levels, emphasizing that higher-level nodes have a more significant impact on overall network topology—this aligns with the observation that core nodes in social networks can spread information or influence more efficiently across levels. Meanwhile, $N\left(L_i\right)$ captures the density of nodes at each level: a higher node density at a specific level may indicate a more fragmented or diversified structural feature at that layer, which can be used to assess the contribution of nodes at that level to network connectivity or information diffusion. By integrating level depth and node quantity, this definition quantifies the topological importance of nodes, providing a basis for identifying key nodes (e.g., influencers or bridge nodes) in social networks and analyzing information dissemination paths or structural stability.

Once a node is given, two types of importance of nodes, (1) the topological importance of the node and (2) the fractal importance of the node, can be obtained easily via Equation (3) and Equation (4), respectively. They yield different measurement results in terms of the local importance and global importance of nodes. Therefore, we attempt to combine these two types of importance of nodes together and propose the D-TFRank Algorithm for the evaluation of node importance in dynamic social networks.

4.4.3. The Decision Level of a Tree

Since the real-life social network is very complex, the weighted tree converted from the social network has many levels. Intuitively, users can only infect the sentiments of those who are near to them during the procedure of sentiment propagation; it is unnecessary to consider all the nodes at different levels in a tree. The decision level of a tree, $L^{TF}$, must be determined, and it can be calculated as follows.  

$$\begin{array}{c}\hfill L^{TF}=\left\{\begin{array}{cc}L,\hfill & L\le 3\hfill \\ \lfloor ln(10L+20)\rfloor ,\hfill & otherwise.\hfill \end{array}\right.\end{array}$$

(5)

where L refers to the total level number in T, and $L^{TF}$ is the number of levels in the tree we should consider when calibrating the fractal importance and topological importance.

Figure 3 illustrates the correlation between L and $L^{TF}$ based on the simulated data. Obviously, as L increases, $L^{TF}$ falls into a certain range due to our given assumption, i.e., users only infect the sentiments of those who are close to them within a given range. Actually, our assumption reduces the complexity when the scale of the social network increases.

4.4.4. D-TFRank Evaluation

During the sentiment propagation, the structure of a social network is dynamically changing as time elapses. For example, a certain user can join or leave the current social network, which is equivalent to the process of node insertion or node deletion from a dynamic social graph.

A dynamic social network is a time-varying social network; it can be represented with a graph, $G_t=(V_t,E_t)$. Here, $V_t$ is the set of vertices that represent users, $U_t$, that are a part of the network at time t, and $E_t$ is the set of all edges created up to time t. We assume edges and nodes are not only added to the graph but also deleted from the graph. Therefore, we focus on how to estimate the D-TFRank values when nodes are added to or deleted from the graph in this section. The proposed D-TFRank ranking algorithm is presented as follows.

Algorithm 2 describes the calculation of the D-TFRank value of node $v_i$. The D-TFRank algorithm is working as follows: (1) A social network $G_t$ at time t is converted to a weighted tree with approach in [38] to generate the tree structure via finding the shortest paths from the focus to every other node in the network by using the shortest path algorithm, such as the Dijkstra and Floyd algorithms (Lines 3–4); (2) The fractal importance and topological importance of each node are calculated according to Equations (3) and (4) (Lines 11–12); (3) Line 13 is intended to estimate the D-TFRank value of each node by multiplying the fractal importance and topological importance.

Algorithm 2: D-TFRank algorithm

1: Input: Node $V_t$, Social Graph $G_t=(V_t,E_t)$ at time t 2: Output: D-TFRank Value $DTF\left(v_i\right)$ 3: for each node $v_i$ in $V_t$ 4:    $Floyd(G_t,v_i)$ 5:    $T=Tree(G_t,i)$ 6:    $L=GetLevel\left(T\right)$; 7:    $L^{TF}=Calculate\left(L\right)$; 8:    for each node $v_i$ in T 9:       $F\left(v_i\right)$; 10:    end 11:    ${\tilde{F}}_A\left(v_i\right)={\sum }_{i=1}^{L^{TF}}{\sum }_{j=1}^{N\left(L_i\right)}\left(F_{ChildNode{s}_i^j}\right)\ast {\left({\displaystyle \frac{1}{2}}\right)}^i$ 12:    ${\tilde{T}}_A\left(v_i\right)={\sum }_{i=1}^{L^{TF}}N\left(L_i\right)\ast {\left({\displaystyle \frac{1}{2}}\right)}^i$ 13:    $DTF\left(v_i\right)={\tilde{T}}_A\left(v_i\right)\times {\tilde{F}}_A\left(v_i\right)$ 14: end

Definition 6

(Sentiment leader identification). Given a sentiment community, $C_i$, and a number, K, the top-Ksentiment leader identification is represented as follows:

$$U(C_i,K):=arg\underset{v_i\in C_i}{\overset{K}{max}}DTF\left(v_i\right)$$

(6)

$U(C_i,K)$ is intended to return the top-K users with maximum D-TFRank value by invoking Algorithm 2 as the sentiment leaders in sentiment community $C_i$.

4.5. Algorithm Description for SentiRank

The proposed SentiRank algorithm consists of the detection of the sentiment communities and the identification of the sentiment leaders in the corresponding sentiment community. Algorithm 3 shows the pseudocode of SentiRank.

SentiRank first detects sentiment communities under two cases of sentiment representation (Lines 1–9), and then it initializes the set of sentiment leaders discovered in the whole network (T) and in each sentiment community $(T_1,\dots ,T_m)$ as the empty set (Line 11). After the initialization, SentiRank performs the sentiment leader identification and returns the top-K users (Lines 12–14). Finally, the output result, T, can be represented as a set with the elements $(T_1,\dots ,T_m)$ (Line 15).

Algorithm 3: SentiRank

Require:  Sentiment graph at time t: $G_t=(V_t,E_t,S)$; Number of sentiment leaders K; Sentiment communities detection cases ID Z 1: positive and negative sentiment system; 2: 5-star sentiment rating system; Ensure:  T: set of top-K sentiment leaders in the detected sentiment community C; 1: begin 2:    switch(Z) 3:       case 1: 4:       $C\leftarrow$ detect sentiment communities using Alg. 1; 5:       break; 6:       case 2: 7:       $C\leftarrow$ detect sentiment communities in $G_t$; 8:       break; 9:       end 10:    $m=\left|C\right|$; 11:    $T=T_1=T_2=\dots =T_m=\varnothing$; 12:    for i=1 to m do 13:       $T_i\leftarrow U(C_i,K)$; 14:    end 15:    $T\leftarrow \{T_1,T_2,\dots ,T_m\}$; 16: end

Sentiment leader identification in social networks is an NP-hard problem, reducible from influence maximization, meaning that finding exact solutions for large networks with n nodes and m edges is unfeasible. Key components of the problem involve varying computational complexities. Sentiment propagation models, like iterative SentiRank, have a complexity of $O(m·log(1/ϵ\left)\right)$ for sparse graphs, where $ϵ$ is a small positive parameter defining the convergence threshold. Leader identification algorithms range from greedy approaches with $O(k·n·m)$ complexity (where k is the number of leaders to identify) to heuristic centrality metrics with complexities from $O(n+m)$ (degree centrality) to $O\left(n\right(m+nlogn\left)\right)$ (betweenness centrality). The preprocessing of sentiment and trust data adds $O(L·P+m)$ complexity, with L being the number of sentiment labels and P the number of sentiment-labeled posts. NP—hardness forces the use of approximation algorithms, and real-world scalability depends on graph sparsity, with sparse networks allowing for more efficient linear—scaled solutions through methods like distributed computing. Overall, balancing theoretical intractability and practical scalability is crucial in designing effective algorithms for this task.

5. Experiments and Evaluation

This section evaluates the proposed sentiment leader identification algorithm SentiRank through extensive experiments with the goal of validating its effectiveness for maximizing the sentiment coverage in social networks.

5.1. Datasets

We conducted the experiments of sentiment leader identification in positive/negative sentiment systems using the data collected from Slashdot (http://snap.stanford.edu/data/soc-sign-Slashdot090216.html, accessed on 8 July 2025), which is a technology-related news website known for its specific user community. Its news content covers a wide range of technological fields, including hardware, software, the internet, and more. What is special about Slashdot is that all the news on the website is voluntarily submitted by users, and after being reviewed and selected by editors, it is published on the homepage. Each news story is accompanied by a comments section, where users can actively participate in discussions and express their views. That is to say that Slashdot allows users to tag each other as friends or foes. The network contains friend/foe links between the users of Slashdot. We firstly built the social networks based on the trust relationships between users. Then, we considered their attitudes in the system regarding news in order to determine a user’s sentiment That is to say that they both share the same sentiment if he/she trusts another user’s review. Otherwise, they are in different sentiment communities.

Table 1. Data structure of DataSets.

UserID	UserID	Trust(+1)/Distrust(−1)
1	112	+1
1	163	−1
1	522	−1
1	604	+1
1	605	+1
522	483	+1

illustrates the data storage representation of a dataset based on the trust relationships between users.

Table 1 contains three columns (userIDs and the trust/distrust relationship). Each row denotes the trust/distrust relationship between two users. If two users trust reciprocally, they both are considered to share the same sentiments. Otherwise, they have different sentiments. For example, users “1”, “112”, “604”, and “605” share the same sentiment. They should be in the same sentiment community (for example, $C_1$). Since user “1” distrusts user “163” and “522”, they should be in different sentiment communities. Hence, users “163” and “522” are in another sentiment community (for example, $C_2$). Thus, user “483” and user “522” are in sentiment community $C_2$.

For the positive/negative sentiment representation system of Case 1,

Table 2. Dataset I Statistics.

	Nodes	Edges	Trust	Distrust
Statistics	81,871	545,671	422,349	123,322

presents the statistics of Dataset I which contains 81,871 nodes and 545,671 edges, with 422,349 trust relationships and 123,322 distrust relationships between users.

Figure 4 shows the visualization of this social network. Obviously, the positive sentiment dominates the whole social network, rather than the negative sentiment.

For the five-star rating sentiment representation system of Case 2, we collect the product rating dataset (Dataset II) from TaoBao.com (www.taobao.com, accessed on 8 July 2025), which is the largest E-commerce social networking site with the query keywords of the latest smartphone made by Apple Corporation “Apple iPhone 3GS White (16 GB) Smart phone”. Consequently, we obtained 110 reviews made by 110 customers. Consequently, Figure 5 shows the visualization of this social network.

5.2. Detection of Sentiment Communities

The problem of detecting the sentiment communities is solved using the aforementioned SDP optimization methodology for two types of sentiment representation systems. Let us take Table 1 as an example and obtain the two sentiment communities (positive and negative sentiment communities) under Case 1.

• $C_1=\{1,112,604,605\}$
• $C_2=\{163,522,483\}$

We run the SDP optimization approach of Case 1 on the Slashdot social network dataset to get the positive and negative sentiment communities. Figure 6a shows the positive sentiment community, including 422,349 positive edges; i.e., the consumers have positive sentiment toward a product. On the contrary, Figure 6b shows the negative sentiment community, including 123,320 negative edges; i.e., the consumers have a negative sentiment on the product.

As shown in Figure 6, the negative sentiment community is quite sparse compared to the positive sentiment community. Hence, this is consistent with real-life sentiment distribution on a product; i.e., most customers hold a positive sentiment on a new product, while a small fraction of customers hold a negative sentiment on the product.

Figure 7 reports the detection results of sentiment communities of Dataset II. We run the SDP optimization approach of Case 2 on Dataset II and obtain five sentiment communities in terms of rating on “Apple iPhone 3GS White (16 GB) Smart phone".

5.3. Identification of Sentiment Leaders

This step is based on the results of sentiment community detection. Therefore, we run various related algorithms of ranking nodes, as well as the SentiRank algorithm, to mine the sentiment leaders from the sentiment communities. The description of the related algorithms of ranking nodes and SentiRank are listed as follows:

• SentiRank: This is the proposed sentiment community-based identification algorithm for sentiment leader identification.
• Degree: As a comparison, we provide a simple heuristic algorithm that selects the k vertices with the largest degrees; this heuristic was also evaluated in [39,40]. Formally, the degree-based sentiment leader identification is described as follows.

  $$U(C_i,K):=arg\underset{v_i\in C_i}{\overset{K}{max}}deg\left(v_i\right)$$

  (7)
• Closeness Centrality: The main idea of this algorithm is that the member takes the central position if it can quickly contact other members in the network. The closeness centrality ($CC$) of member v closely depends on the geodesic distance, i.e., the shortest paths from member v to all other people in the social network [41] and is calculated as follows:

  $$CC\left(v\right)={\displaystyle \frac{N-1}{{\sum }_{u\ne v,u\in V}d(u,v)}}$$

  (8)

  where $d(u,v)$ is a function describing the distance between nodes u and v, and N is the number of nodes in a network. Formally, the closeness centrality-based sentiment leader identification is described as follows.

  $$U(C_i,K):=arg\underset{v_i\in C_i}{\overset{K}{max}}CC\left(v_i\right)$$

  (9)
• Betweenness Centrality: The betweenness centrality ($BC$) of member v pinpoints the extent to which v is between other members. Members with a high $BC$ are very important to the network because other members can connect with each other only through them. The $BC$ of member v is the sum of all the outcomes [42,43]:

  $$BC\left(v\right)={\displaystyle \frac{{\sum }_{i\ne v\ne j,i,j\in V}{s}_{ij}\left(v\right)}{s_{ij}}}$$

  (10)

  where $s_{ij}\left(v\right)$ is the number of the shortest paths from i to j that pass through v; $s_{ij}$ is the number of all the shortest paths between i and j. N is the number of nodes in a network. Formally, the betweenness centrality-based sentiment leader identification is described as follows.

  $$U(C_i,K):=arg\underset{v_i\in C_i}{\overset{K}{max}}BC\left(v_i\right)$$

  (11)
• Random: As a baseline comparison, we simply select k random nodes as the sentiment leaders in the graph; this method was also adopted in [44].

5.4. Evaluation Metrics

To evaluate the performance of the proposed SentiRank algorithm from perspectives on the link structure and content, the one-step sentiment coverage and all-path sentiment coverage are introduced.

One-step sentiment coverage: One-step sentiment coverage is defined as the number of nodes that are directly infected via this set of nodes. This is a special case of sentiment coverage.

All-path sentiment coverage: All-path sentiment coverage is defined as the number of nodes that are directly or indirectly infected via this set of nodes. This is a general concept of sentiment coverage.

To obtain the sentiment coverage of the above algorithms, we incorporate the topological information of the social network; e.g., if two nodes are connected with a link and have the same sentiment, there exists a sentiment interaction/sentiment propagation between them. However, in the sentiment communities, this type of tropology is broken and divided into some substructures of the social network. For all these algorithms, we compare the sentiment coverage at a given time is subjected to different seed sets with the size ranging from 1 to 46 and with the step width = 5 in Dataset I. For Dataset II, we compare the sentiment coverage subject to different seed sets with the size ranging from 1 to 5 and with the step width = 1. Then, we evaluate the sentiment coverage of SentiRank with respect to sentiment communities for two datasets.

5.5. Experimental Results and Analysis

We performed independent t-tests to compare the mean sentiment coverage between SentiRank and each baseline method. The results are summarized in

Table 3. Statistical analysis for experimental results of Figure 7 ($\alpha$ = 0.05).

Baselines	Homogeneity of Var. (p)	t-Statistic	p-Value	Effect Size (d)	Signf.
Degree	0.9887	+1.1004	0.2857	+0.4921
Closeness	0.9441	+2.1566	0.0448	+0.9644	*
Betweenness	0.9437	+1.7450	0.0980	+0.7804
Random	0.0339	+7.0184	0.0000	+3.1387	*

Remark: * denotes that the p-value is less than 0.05, indicating a significant impact.

.

For the experimental results presented in Table 3, statistical analyses were conducted across multiple baselines. The homogeneity of variance (p-values for homogeneity: 0.9887 for degree, 0.9441 for closeness, 0.9437 for betweenness, and 0.0339 for random) varied among the baselines. The T-statistics were +1.1004 (degree), +2.1566 (closeness), +1.7450 (betweenness), and +7.0184 (random). Corresponding p-values were 0.2857 (degree, non-significant), 0.0448 (closeness, significant at conventional levels), 0.0980 (betweenness, marginally non-significant), and 0.0000 (random, highly significant). Effect sizes (d) were +0.4921 (degree), +0.9644 (closeness), +0.7804 (betweenness), and +3.1387 (random). Overall, closeness and random baselines demonstrated significant effects (*), while degree did not reach significance, and betweenness was marginally non-significant, indicating differential impacts across these network-related baselines in the experiment.

Table 4. Statistical analysis for experimental results of Figure 8 ($\alpha$ = 0.05).

Baselines	Homogeneity of Var. (p)	t-Statistic	p-Value	Effect Size (d)	Signf.
Degree	0.5237	+0.9342	0.3775	+0.5908
Closeness	0.5237	+0.9342	0.3775	+0.5908
Betweenness	0.6163	+0.9717	0.3597	+0.6145
Random	0.4088	+4.5596	0.0019	+2.8838	*

Remark: * denotes that the p-value is less than 0.05, indicating a significant impact.

and

Table 5. Statistical analysis for experimental results of Figure 9 ($\alpha$ = 0.05).

Baselines	Homogeneity of Var. (p)	t-Statistic	p-Value	Effect Size (d)	Signf.
Degree	0.9548	+0.3560	0.7310	+0.2252
Closeness	0.8640	+0.8639	0.4128	+0.5464
Betweenness	0.9541	+0.4778	0.6455	+0.3022
Random	0.4616	+3.3265	0.0104	+2.1039	*

Remark: * denotes that the p-value is less than 0.05, indicating a significant impact.

present statistical analyses of experimental results for different baselines (degree, closeness, betweenness, and random) under $\alpha$ = 0.05. In Table 4, for Figure 8, the random baseline shows a significant result (marked with *) due to a low p-value (0.0019), large t-statistic (+4.5596), and high effect size (+2.8838), while others have non-significant p-values. In Table 5, for Figure 9, the random baseline also has a significant result (p-value = 0.0104) with a notable t-statistic (+3.3265) and effect size (+2.1039), and the other baselines have higher p-values, indicating non-significance. Overall, the random baseline demonstrates significant effects in both analyses compared to the other baselines.

We denote the sentiment community with five stars as $C_{5-★}$. The top five sentiment leaders are identified via SentiRank. $U(C_{5-★},5)=\{616,620,645,670,692\}$. Figure 9 and Figure 10 show the one-step sentiment coverage and all-path sentiment coverage of different algorithms in terms of various K values. In both cases, the performance of our proposed algorithm SentiRank is the same as the degree-based algorithm [44]. However, the closeness-based algorithm [45] is worse than the degree-based algorithm and SentiRank.

As shown in Table 4 and Table 5, the statistical analysis shows that the inter-group differences between the SentiRank and random methods are statistically significant (p < 0.05) and have a large effect size, indicating the high reliability of the results. Therefore, their practical significance can be discussed in detail.

In the sentiment communities with three stars $C_{3-★}$ and with one star $C_{1-★}$, there are 6 and 16 users within the corresponding sentiment community, respectively. In both cases, this customer is the sentiment leader in the corresponding sentiment community.

The problem addressed in this paper and the proposed solution are benefit to social marketing and social advertising. Identifying the sentiment leaders in social networks can help businesses understand what consumers are thinking about their products (http://venturebeat.com/2011/03/20/why-sentiment-analysis-is-the-future-of-ad-optimization/, accessed on 10 March 2025). Nowadays, social measurement for companies is becoming pervasive in marketing organizations. Identifying the sentiment leaders can help businesses track in real time how a new product has been spreading and decide what opinions to re-promote in order to grease the wheels of message spread. By analyzing the sentiment of users and sentiment leaders, marketers are able to predict exactly how a marketing campaign will perform and, in real time, incorporate strategies into the campaign.

In addition, exploring whether sentiment leaders predominantly share positive or negative sentiments, and how their influence correlates with specific topics (e.g., politics, culture, or consumer trends), would indeed enrich the analysis. This could reveal whether their impact stems from reinforcing existing emotional biases or challenging them and whether certain themes inherently attract more emotional leadership. Such depth would clarify whether these leaders operate as niche emotional curators or cross-topic influencers, enhancing the understanding of their role in shaping network-wide emotional dynamics.

6. Conclusions

Online sentiments have become a valuable source of information for analyzing the latent structure of social networks. In this paper, we have presented the concepts of the sentiment graph and a sentiment community, and we have further proposed a sentiment leader identification framework and associated algorithms of SentiRank to identify sentiment leaders in sentiment communities. The proposed framework consists of sentiment community detection and ranking the nodes on the selected sentiment communities to identify the sentiment leaders. For sentiment community detection, we have studied the detection algorithms with two sentiment representation systems through the SDP optimization approach. To rank the nodes in sentiment communities, we have proposed a D-TFRank-based ranking model. The experimental results demonstrate that SentiRank outperforms the other existing algorithms in terms of one-step sentiment coverage and all-path sentiment coverage. Investigating the sentiment leader identification problem can bring various benefits to social marketing or advertising, such as adjusting and improving advertising strategies, predicting exactly how a marketing campaign will perform, and improving the reputation of products.