A Fuzzy-SNA Computational Framework for Quantifying Intimate Relationship Stability and Social Network Threats

Abstract

Intimate relationship stability is fundamental to human wellbeing, yet its quantitative assessment faces dual challenges: the inherent subjectivity of psychological constructs and the complexity of social ecosystems. Symmetry, as a fundamental structural feature of social interaction, plays a pivotal role in shaping relational dynamics. To address these limitations, this study proposes an innovative computational framework that integrates Fuzzy Set Theory with Social Network Analysis (SNA). The framework consists of two complementary components: (1) a psychologically grounded fuzzy assessment model that employs differentiated membership functions to transform discrete subjective ratings into continuous and interpretable relationship quality indices and (2) an enhanced Fuzzy C-Means (FCM) threat detection model that utilizes Weighted Mahalanobis Distance to accurately identify and cluster potential interference sources within social networks. Empirical validation using a simulated dataset—comprising typical characteristic samples from 10 couples—demonstrates that the proposed framework not only generates interpretable relationship diagnostics by correcting biases associated with traditional averaging methods, but also achieves high precision in threat identification. The results indicate that stable relationships exhibit greater symmetry in partner interactions, whereas threatened nodes display structural and behavioural asymmetry. This study establishes a rigorous mathematical paradigm—“Subjective Fuzzification → Multidimensional Feature Engineering → Intelligent Clustering”—for relationship science, thereby advancing the field from descriptive analysis toward data-driven, quantitative evaluation and laying a foundation for systematic assessment of relational health.

1. Introduction

The stability of intimate relationships forms a crucial foundation for individual well-being and social functioning, profoundly impacting personal, physical, and mental health [1,2]. However, maintaining stable long-term intimate relationships faces dual challenges from both internal and external sources. Internal dynamics include the constant evolution of individual needs and the shifting patterns of interpersonal communication. External pressures stem from professional demands and increasingly complex social environments [3]. Therefore, understanding relationship stability and its scientific quantification are crucial for promoting healthy intimate relationships.

However, current research faces significant limitations that manifest themselves primarily in three bottlenecks. First, ambiguity bottleneck—mainstream methods rely on Likert scales to forcibly discretize continuous indicators such as trust, commitment, and satisfaction in graded scores, resulting in loss of information granularity and hindering the capture of the actual continuous spectrum within relationships [4,5]. Second, complexity bottleneck—while acknowledging environmental influences, classical theories often oversimplify operational aspects. They do not reflect the structural characteristics of support networks or depict the transmission pathways of social influences within relationships, making it difficult to effectively identify and quantify subtle risks embedded within social networks [6]. Finally, the static bottleneck—most models remain confined to cross-sectional static depictions—failing to capture the real-time interactions and strategy adjustments that co-evolve between partners over time. This limits their ability to explain and predict relationship dynamics [7,8].

Moreover, symmetry represents a fundamental yet under-explored dimension within intimate relationships. Within dyadic bonds, symmetry manifests itself as balanced emotional investment, reciprocal trust, and two-way communication—hallmarks of relational health. In contrast, asymmetry frequently signals underlying tensions or external changes. At the network level, symmetrical connections foster relational stability, whereas asymmetric threat propagation may undermine this equilibrium. Despite the significance of symmetry, current computational models seldom incorporate explicit symmetry metrics, thus limiting their capacity to capture the nuanced interplay between relational equilibrium and threat dynamics.

To address the first two bottlenecks, this study proposes a computational framework that integrates fuzzy set theory with social network analysis (SNA). Crucially, we incorporate symmetry considerations into this unified framework, aiming to bridge the aforementioned gaps and provide a more comprehensive, symmetry-aware approach to quantifying relational stability and detecting latent threats. In this manner, not only are we able to describe the fuzzy attributes and complex structures within relationships, but we are also able to capture the inherent bidirectionality and equilibrium of relational systems. The core contributions of this paper are primarily in the following four aspects:

1.

We propose a paradigm for constructing differentiated fuzzy membership functions based on psychological constructs. This approach transcends the conventional homogeneous function framework by adaptively designing membership function shapes—such as trapezoidal, S-curves, and Gaussian—to match the intrinsic dynamic characteristics of distinct psychological constructs like trust, satisfaction, and communication.

2.

We propose a displayed partner symmetry factor, distinct from methods that implicitly assume perceived symmetry through arithmetic averaging. This factor quantifies and corrects for asymmetry between partners, ultimately generating a system relationship score that more accurately reflects bilateral realities.

3.

We propose a weighted Mahalanobis fuzzy C-means (WM-FCM) algorithm. By assigning feature weights based on information gain ratios and optimizing distance metrics through covariance matrix adjustments, it substantially enhances the detection of potential threats within social networks. Compared with FCM, this approach demonstrates superior robustness and accuracy, achieving a paradigm shift from merely describing network structures to actively detecting threats.

4.

We have constructed a closed-loop, symmetry-aware computational paradigm for relational science, progressing from fuzziness metrics to feature space sets and ultimately to fuzzy clustering. This drives the field from descriptive analysis towards data-driven, quantitative systems diagnostics, providing a generalizable template for evaluating relational ecosystems.

The remaining sections of this paper are organized as follows: Section 2 presents a review of the relevant literature; Section 3 describes the proposed fuzzy-social network analysis framework; Section 4 illustrative application and numerical analysis; Section 5 discusses the experiments and provides a comparative analysis; Section 6 summarizes the entire article and outlines future research directions.

2. Literature Review

Traditional studies on relationship stability predominantly employ investment models [9] and vulnerability-stress-coping frameworks [8]. While these theories effectively identify key indicators such as trust and satisfaction, their quantification suffers from two primary shortcomings: First, complex and continuous emotional experiences are forcibly compressed into discrete rating scales, resulting in the loss of nuanced “fuzzy” information [10]; Second, existing scales struggle to capture contradictory states within relationships, such as the complex paradox of “feeling highly emotionally satisfied with one’s partner while simultaneously experiencing extreme exhaustion due to objective stressors both within and outside the relationship” [11]. Recent research indicates that single-point ratings are insufficient to depict and reflect the complex, multifaceted nature of genuine contradictions [12].

Fuzzy mathematics has significant advantages in quantifying subjective concepts [13], but its application in intimate relationship research remains constrained. Current studies predominantly lack systematic fuzzy quantification of multidimensional constructs like trust. As a core relational element, trust itself possesses multidimensional attributes [4], that encompass cognitive, emotional, and behavioral dimensions [14]. However, existing studies are predominantly based on single-dimensional measurements using discrete scales, failing to capture the inherent fuzziness of “trust levels.” A more significant shortcoming is the lack of differentiated membership functions designed for the conceptual characteristics of relationships. In particular, emerging mathematical tools such as fuzzy sets and Gaussian aggregation operators [15] are gradually evolving. Although they show significant potential for precise quantification [16], their application in this field remains underdeveloped.

Meanwhile, social network analysis (SNA), a classic method for examining the external structure of relationships, exhibits significant limitations in intimate relationship research [6]. First, representational ambiguity: traditional SNA struggles to capture ambiguous relationship states such as “close but not deeply intimate” [17]. Second, methodological constraint: while SNA excels in describing overall network structures, it lacks built-in algorithmic frameworks, making it incapable of automatically diagnosing specific risks within relationships [18]. In particular, research in fake news detection has demonstrated that integrating internal traits with external network characteristics significantly enhances analytical efficacy [19], offering crucial methodological insights for intimate relationship studies. Meanwhile, review studies in the SNA field indicate that inadequate dynamic modeling is one of the current main bottlenecks [20]. These findings provide theoretical justification for this study’s focus on “static snapshot” analysis and also point the way for future dynamic model research.

Regarding clustering algorithms, traditional methods show significant limitations when handling mixed-type, heterogeneous feature data, such as intimate relationships. Recent research demonstrates that Mahalanobis distance has notable advantages in processing correlated feature data [21], particularly in multi-view clustering, where the MMA method research proves that maximizing Mahalanobis distance can significantly enhance cluster discriminability. These findings provide a solid theoretical and methodological foundation for our improvement of the FCM algorithm, allowing us to more effectively identify potential threat patterns in relationship networks.

Through a systematic review of the literature, this study identifies two key research gaps. In terms of fuzzy quantification, multidimensional constructs in intimate relationships lack systematic fuzzy mathematical quantification methods and differentiated membership functions designed for relationship characteristics. In terms of systematic threat identification, SNA lacks algorithmic frameworks for automatic diagnosis and prediction of relationship risks. Interdisciplinary research cases provide important information: Just as data integration frameworks drive breakthroughs in biomedicine [22], and AI models demonstrate enormous potential to assess social situations [23], this study offers an innovative way to solve the classic socio-psychological problem of intimate relationships by constructing an interdisciplinary computational framework integrating fuzzy mathematics and SNA.

3. The Proposed Fuzzy-SNA Framework

3.1. Overview of the Framework

The stability of a dyadic system is conceptualized as emerging from the continuous interaction between the internal subjective states of individuals and the external structural environment of their shared social network. To model this integration scientifically, we propose a novel computational framework that synergistically combines fuzzy mathematics and SNA. The framework comprises two computationally parallel, task-independent, yet complementary models:

1.

The Internal-State Fuzzy Assessment Model: This model performs a parallel determination of the uncertain, subjective perceptions of nodes within the social network through a hierarchical fuzzy logic system.

2.

The External-Network Risk Screening Model: This model enables the proactive identification and categorization of nodes that are likely to pose risks to dyadic stability, utilizing an enhanced clustering algorithm.

To systematically simulate the interaction between the state of the internal relationship and the impact of the external network, we proposed a dual-path computing framework, as shown in Figure 1. This integrated architecture handles subjective perception and network risks in parallel, thereby realizing the overall diagnosis of relationship health.

The following sections elaborate on the general standardized mathematical expression of each model, aiming to establish it as a reusable methodological system, to complete the construction of the theoretical framework before the specific application and verification in Section 4.

3.2. Quantification of Fuzzy Indicators

A hierarchical, two-layer fuzzy aggregation model is proposed to convert discrete subjective ratings (such as Likert-scale surveys) into a continuous measure of relationship quality. This approach provides a transparent and psychologically plausible framework for the conversion.

Convert discrete survey results into continuous relational quality indicators, following a structured two-layer fuzzy aggregation process. Figure 2 details the flow chart, from the original input blurring to the final blurring score.

3.2.1. Indicator System and Fuzzification

The model obtains the original numerical score from five basic sub-dimensions: Cognitive Trust ($T_1$), Affective Trust ($T_2$), Behavioral Trust ($T_3$), Satisfaction (S) and Communication Quality (C). Set x to represent the original brittleness score of any of these subdimensions. Each fraction is converted into a fuzzy set, which is defined as a language variable (Low (L), Medium (M), High (H)) by its specific affiliation function.

The differentiated membership functions constructed for different psychological constructs are based on a comprehensive consideration of their measurement characteristics and psychological theories:

• Cognitive trust ($T_1$): It involves explicit rational judgements, with evaluations typically possessing clear threshold intervals such as the boundary between ‘basic trust’ and ‘complete trust’ [13,24]. The plateau region of the trapezoidal function effectively represents this relatively stable intermediate state, aligning with the ‘ambiguous yet stable’ psychological response pattern corresponding to mid-range scores on Likert scales.

$${\mu }_{T_1\left(L\right)}\left(x\right)=\left\{\begin{array}{cc}1\hfill & x\in [1,2]\hfill \\ {\displaystyle \frac{3-x}{1}}\hfill & x\in (2,3)\hfill \\ 0\hfill & x\in [3,7]\hfill \end{array}\right.$$

(1)

$${\mu }_{T_1\left(M\right)}\left(x\right)=\left\{\begin{array}{cc}0\hfill & x\in [1,2]\cup [6,7]\hfill \\ {\displaystyle \frac{x-2}{1}}\hfill & x\in (2,3)\hfill \\ 1\hfill & x\in [3,5]\hfill \\ {\displaystyle \frac{6-x}{1}}\hfill & x\in (5,6)\hfill \end{array}\right.$$

(2)

$${\mu }_{T_1\left(H\right)}\left(x\right)=\left\{\begin{array}{cc}0\hfill & x\in [1,5]\hfill \\ {\displaystyle \frac{x-5}{1}}\hfill & x\in (5,6)\hfill \\ 1\hfill & x\in [6,7]\hfill \end{array}\right.$$

(3)

• Affective trust ($T_2$): It often exhibits clustered distributions among populations, with few extremes and most individuals occupying intermediate levels. The piecewise linear function fits this centrally clustered distribution with gradual attenuation at both ends, consistent with the linear gradation characteristic of emotional evaluations [4].

$${\mu }_{T_2\left(L\right)}\left(x\right)=\left\{\begin{array}{cc}{\displaystyle \frac{3-x}{2}}\hfill & x\in [1,3]\hfill \\ 0\hfill & x\in (3,7]\hfill \end{array}\right.$$

(4)

$${\mu }_{T_2\left(M\right)}\left(x\right)=\left\{\begin{array}{cc}0\hfill & x\in [1,2]\cup [6,7]\hfill \\ {\displaystyle \frac{x-2}{2}}\hfill & x\in (2,4)\hfill \\ {\displaystyle \frac{6-x}{2}}\hfill & x\in [4,6)\hfill \end{array}\right.$$

(5)

$${\mu }_{T_2\left(H\right)}\left(x\right)=\left\{\begin{array}{cc}0\hfill & x\in [1,5]\hfill \\ {\displaystyle \frac{x-5}{2}}\hfill & x\in (5,7]\hfill \end{array}\right.$$

(6)

• Behavioral trust ($T_3$): It exhibits a characteristic ‘threshold effect,’ where minor cumulative behaviors trigger a qualitative shift in trust. The sigmoid function’s gradual progression followed by abrupt change precisely models this non-linear dynamic [14].

$$\begin{array}{cc}\hfill {\mu }_{T_3\left(L\right)}\left(x\right)& ={\displaystyle \frac{1}{1+e^{1.5(x-2)}}}\hfill \\ \hfill {\mu }_{T_3\left(M\right)}\left(x\right)& ={\displaystyle \frac{1}{1+e^{-1.5(x-2)}}}-{\displaystyle \frac{1}{1+e^{1.5(x-5)}}}\hfill \\ \hfill {\mu }_{T_3\left(H\right)}\left(x\right)& =1-{\displaystyle \frac{1}{1+e^{-1.5(x-5)}}}\hfill \end{array}$$

(7)

• Satisfaction (S): As a continuous affective state, satisfaction typically undergoes gradual, smooth transitions [11]. The bell-shaped curve of the Gaussian function naturally represents the gradual attenuation around the ‘peak satisfaction point,’ avoiding the abrupt boundaries of trapezoidal functions and better reflecting the inherently fuzzy, gradual nature of subjective satisfaction.

$$\begin{array}{cc}\hfill {\mu }_{S\left(L\right)}\left(x\right)& =e^{-{\left({\displaystyle \frac{x-2}{1.5}}\right)}^2},\hfill \\ \hfill {\mu }_{S\left(M\right)}\left(x\right)& =e^{-{\left({\displaystyle \frac{x-4}{1.5}}\right)}^2},\hfill \\ \hfill {\mu }_{S\left(H\right)}\left(x\right)& =e^{-{\left({\displaystyle \frac{x-6}{1.5}}\right)}^2}\hfill \end{array}$$

(8)

• Communication Quality (C): It is often categorized into finite, relatively well-defined categories such as ‘conflict’, “neutral”, and ‘harmony’ [25]. This psychological tendency towards classification inherently endows perceptions of communication quality with interval characteristics. The trapezoidal function clearly represents both the plateau regions within these categories and the gradient zones between them.

$${\mu }_{C\left(L\right)}\left(x\right)=\left\{\begin{array}{cc}1\hfill & x\in [1,2]\hfill \\ {\displaystyle \frac{3-x}{1}}\hfill & x\in (2,3)\hfill \\ 0\hfill & x\in [3,7]\hfill \end{array}\right.$$

(9)

$${\mu }_{C\left(M\right)}\left(x\right)=\left\{\begin{array}{cc}0\hfill & x\in [1,2]\cup [6,7]\hfill \\ {\displaystyle \frac{x-2}{1}}\hfill & x\in (2,3)\hfill \\ 1\hfill & x\in [3,5]\hfill \\ {\displaystyle \frac{6-x}{1}}\hfill & x\in (5,6)\hfill \end{array}\right.$$

(10)

$${\mu }_{C\left(H\right)}\left(x\right)=\left\{\begin{array}{cc}0\hfill & x\in [1,5]\hfill \\ {\displaystyle \frac{x-5}{1}}\hfill & x\in (5,6)\hfill \\ 1\hfill & x\in [6,7]\hfill \end{array}\right.$$

(11)

3.2.2. Two-Layer Fuzzy Aggregation

• First-Layer Aggregation: Computing Comprehensive Trust ($\tilde{\mathit{T}}$)
  Gather three trust sub-dimensions into a composite trust fuzzy set. The weighting is based on Rempel’s multidimensional theory of trust [4]. This model posits that trust within intimate relationships comprises three distinct yet interrelated components. Rempel et al. propose that while all three are indispensable, the affective dimension typically constitutes the core of intimate trust and is more directly linked to relational well-being. Consequently, we assign it a slightly higher weighting $w_2=0.4$. While cognitive and behavioral dimensions are crucial, they are given equal weightings $w_1=w_3=0.3$ to reflect their complementary roles in forming the overall trust judgment, satisfying $w_1+w_2+w_3=1$. The aggregation uses the fuzzy weighted average (FWA) operator for each linguistic term $k\in \{L,M,H\}$:

  $${\mu }_{T\left(k\right)}=w_1·{\mu }_{T_1\left(k\right)}+w_2·{\mu }_{T_2\left(k\right)}+w_3·{\mu }_{T_3\left(k\right)}(k=L,M,H)$$

  (12)
  The output $\tilde{T}$ is the fuzzy set $\{{\mu }_{T\left(L\right)},{\mu }_{T\left(M\right)},{\mu }_{T\left(H\right)}\}$
• Second-Layer Aggregation: Computing Relationship Quality ($\tilde{\mathit{Q}}$)
  Intermediate indicators—Composite Trust ($\tilde{T}$), Satisfaction ($\tilde{S}$), and Communication Quality ($\tilde{C}$)-are summarized in the fuzzy set of the final quality of the relationship. The weighting is based on a mature three-dimensional model of relationship quality [11], $w_T=0.4$, $w_S=0.3$, $w_C=0.3$, satisfying $w_T+w_S+w_C=1$, which considers trust as the foundational pillar, while satisfaction and communication serve as key reinforcing elements. This model emphasizes that although satisfaction reflects immediate emotional rewards and communication ensures day-to-day functioning, trust is the cornerstone of long-term stability in intimate relationships. Consequently, composite trust is assigned the highest weighting. Satisfaction and communication quality are equally weighted, reflecting their complementary and interdependent roles in sustaining relationship wellbeing.

3.2.3. Defuzzification

The mass center method converts the final fuzzy set ($\tilde{Q}$) into a single explainable, clear fraction (Q) on a continuous scale to provide a quantitative measure of the health of the overall relationship. This study employs the centroid method for defuzzification, a standard strategy within fuzzy systems. This approach delivers smooth, continuous output whilst comprehensively incorporating the overall distribution information of fuzzy sets.

$$Q=\mathrm{Defuzz}\left(\tilde{Q}\right)={\displaystyle \frac{{\nu }_L·{\mu }_{Q\left(L\right)}+{\nu }_M·{\mu }_{Q\left(M\right)}+{\nu }_H·{\mu }_{Q\left(H\right)}}{{\mu }_{Q\left(L\right)}+{\mu }_{Q\left(M\right)}+{\mu }_{Q\left(H\right)}}}$$

(13)

Here ${\nu }_L=1,{\nu }_M=4,{\nu }_H=7$ are the representative values for the low, medium and high linguistic terms, corresponding to the 7-point scale.

3.2.4. Dyadic Symmetry Adjustment

To transform individual assessments into estimates of the dyadic system, we introduce a symmetry adjustment factor based on data from both partners. For each dimension d, the symmetry factor is as follows:

$${\sigma }_d^{\mathrm{sum}}=1-{\displaystyle \frac{|S_A^d-S_B^d|}{max(S_A^d,S_B^d)-min(S_A^d,S_B^d)+ϵ}}$$

(14)

where $S_A^d$ and $S_B^d$ are partners’ scores (1–7 scale), and $ϵ={10}^{-5}$. The overall dyadic symmetry is as follows:

$$\overline{\sigma }={\displaystyle \frac{1}{5}}\sum _{d=1}^5{\sigma }_d^{sym}$$

(15)

The symmetry-adjusted system-level relationship quality is then the following:

$$Q_{system}^{sym}={\displaystyle \frac{Q_A\times \overline{\sigma }+Q_B\times \overline{\sigma }}{2}}=\overline{\sigma }\times {\displaystyle \frac{Q_A+Q_B}{2}}$$

(16)

where $Q_A$ and $Q_B$ are the defuzzified relationship quality scores for each calculated partner.

Below is the Algorithm 1 flow chart, which fuzzifies individual scores and aggregates them through two layers to obtain a system-level relationship quality score.

Algorithm 1 Fuzzy-SNA Relationship Quality Assessment Model
Require: Individual scores for partners A and B:
1:    $T^A=[T_1^A,T_2^A,T_3^A]$, $T^B=[T_1^B,T_2^B,T_3^B]$	▹ Trust dimensions
2:    $S^A$, $S^B$	▹ Satisfaction scores
3:    $C^A$, $C^B$	▹ Communication scores
Ensure: System-level relationship quality score: $Q_{system}$    4: Step 1: Fuzzification for each parameter    5: for each partner $P\in \{A,B\}$ do
6:     Apply trapezoidal MF to $T_1^P$, $T_2^P$, $T_3^P$	▹ Trust parameters
7:     Apply Gaussian MF to $S^P$	▹ Satisfaction parameter
8:     Apply S-shaped MF to $C^P$	▹ Communication parameter
9:     Obtain fuzzy membership vectors: $F_T^P$, $F_S^P$, $F_C^P$   10: end for   11: Step 2: First-layer aggregation (Rempel’s theory)   12: Apply weights $w_1=0.3$, $w_2=0.4$, $w_3=0.3$ for trust dimensions   13: $Q_A^T\leftarrow w_1·F_{T1}^A+w_2·F_{T2}^A+w_3·F_{T3}^A$   14: $Q_B^T\leftarrow w_1·F_{T1}^B+w_2·F_{T2}^B+w_3·F_{T3}^B$
15: $Q_A^S\leftarrow F_S^A$, $Q_B^S\leftarrow F_S^B$	▹ Satisfaction aggregation
16: $Q_A^C\leftarrow F_C^A$, $Q_B^C\leftarrow F_C^C$	▹ Communication aggregation
17: Step 3: Second-layer aggregation (cross-dimension)   18: Apply dimension weights: $w_T=0.4$, $w_S=0.3$, $w_C=0.3$   19: $Q_A\leftarrow w_T·Q_A^T+w_S·Q_A^S+w_C·Q_A^C$   20: $Q_B\leftarrow w_T·Q_B^T+w_S·Q_B^S+w_C·Q_B^C$   21: Step 4: Dyadic symmetry adjustment   22: for each dimension $d\in \{T,S,C\}$ do   23:     Compute symmetry factor ${\sigma }_d^{sym}$:   24:     ${\sigma }_d^{sym}=1-{\displaystyle \frac{|Q_A^d-Q_B^d|}{max(Q_A^d,Q_B^d)-min(Q_A^d,Q_B^d)+ϵ}}$   25:     where $ϵ={10}^{-5}$   26:     Compute adjusted quality: $Q_{sym}^d\leftarrow {\displaystyle \frac{Q_A^d+Q_B^d}{2}}\times {\sigma }_d^{sym}$   27: end for   28: Step 5: System-level integration   29: $Q_{system}\leftarrow {\alpha }_T·Q_{sym}^T+{\alpha }_S·Q_{sym}^S+{\alpha }_C·Q_{sym}^C$   30: Step 6: Defuzzification (centroid method)   31: $Q_{system}\leftarrow \mathrm{CentroidDefuzzify}\left(Q_{system}\right)$   32: return$Q_{system}$

3.3. Identification of Potential Threats via FCM

To capture the inherent ambiguity in social threat assessment, we employ Fuzzy C-Means clustering. Unlike hard clustering, FCM assigns soft membership degrees, making it particularly suitable for modeling psychological and behavioral data with ill-defined boundaries [26].

3.3.1. Multimodal Feature Engineering Framework

The social network is represented by graph $G=(V,E)$, where V is the set of nodes (individuals) and E is the set of edges (relations). For each node $i\in V$, we construct a contour using the characteristic vector ${\mathbf{x}}_i\in {\mathbb{R}}^d$ derived from four orthogonal dimensions. Feature dimensions include network structure, interaction behavior, sentiment orientation, and interference sources.

The features of d were normalized to the $[0,1]$ range by the min-max normalization method. For feature j of node i:

$$x_{ij}^{\prime }={\displaystyle \frac{x_{ij}-min\left(x_j\right)}{max\left(x_j\right)-min\left(x_j\right)}}$$

(17)

Here $min\left(x_j\right)$ and $max\left(x_j\right)$ denote the minimum and maximum values of the feature j at all nodes, respectively.

3.3.2. Two-Stage Feature Selection

Not all social network features contribute equally to threat identification. To ensure that more discriminative features exert a greater influence on clustering outcomes, we introduce a feature weighting mechanism. This approach builds upon the growing body of research on feature-weighted clustering, which has demonstrated its advantages in enhancing clustering compactness and interpretability [27]. We compute the information gain ratio for each feature, a well-established metric in feature selection that mitigates bias towards multi-valued features [28]. The IGR measures the reduction in entropy regarding threat levels given a feature, normalized by the feature’s inherent entropy.

1.

Redundancy Removal: We compute the Pearson correlation coefficient matrix for all input features d, denoted as $R=\left[r_{mn}\right]$. Typically, $\theta =0.8$ is set. If the absolute correlation coefficient $|r_{mn}|$ between any feature pair m and n exceeds $\theta$, the less important feature is discarded. Subsequently, $d^{\prime }$ denotes the number of features retained following this procedure.

2.

Importance Assessment & Weighting: We calculate the information gain ratio for $d^{\prime }$. Let us consider a feature A. Its values are partitioned into bins v. The information gain ratio for feature A is defined by the following formula:

$$\mathrm{GainRatio}\left(A\right)={\displaystyle \frac{\mathrm{Gain}\left(A\right)}{\mathrm{SplitInfo}\left(A\right)}}={\displaystyle \frac{\mathrm{Ent}\left(D\right)-{\sum }_{s=1}^v\frac{|D_s|}{\left|D\right|}\mathrm{Ent}\left(D_s\right)}{{\sum }_{s=1}^v\frac{|D_s|}{\left|D\right|}{log}_2\frac{|D_s|}{\left|D\right|}}}$$

(18)

$$w_j={\displaystyle \frac{\mathrm{GainRatio}\left(A_j\right)}{{\sum }_{n=1}^{d^{\prime }}\mathrm{GainRatio}\left(A_n\right)}}$$

(19)

These weights form the diagonal weight matrix $W=\mathrm{diag}(w_1,w_2,\dots ,w_{d^{\prime }})$.

3.3.3. The Weighted Mahalanobis FCM (WM-FCM) Algorithm

The objective function of the algorithm is defined as

$$J_{\mathrm{WM}-\mathrm{FCM}}=\sum _{i=1}^n\sum _{j=1}^c{\left(u_{ij}\right)}^m{\left(d_{ij}^{\mathrm{WM}}\right)}^2$$

(20)

The fuzzifier parameter $m>1$ controls the degree of overlap between clusters. In this study, it was set to $m=2$ a conventional choice that provides a good balance between the fuzziness of the membership assignments and the computational efficiency of the algorithm [29].

Step 1: Data Standardization & Feature Weight Matrix W

Standardize the $d^{\prime }$-dimensional feature vectors $\mathbf{X}=\{x_1^{\prime },x_2^{\prime },\dots ,x_n^{\prime }\}$ for n nodes to eliminate scale differences. Construct the diagonal feature weight matrix $W=\mathrm{diag}(w_1,w_2,\dots ,w_{d^{\prime }})$, where each weight $w_j$ is derived from the Two-Stage Feature Selection protocol.

Step 2: Initialize Cluster Centers via the Max-Min Distance Method

Initialize c cluster centers $V=\{v_1,v_2,\dots ,v_c\}$ using the deterministic Max-Min Distance Method to ensure robust convergence and mitigate poor local optima.

1.

Select the first initial center $v_1^{\left(0\right)}$:

$$v_1^{\left(0\right)}=x_p^{\prime }$$

(21)

The sample $x_p^{\prime }$ that is predefined as a “threat” and exhibits extreme feature values—such as lying near the $[1,1,\dots ,1]$ boundary in the standardized space—is chosen.

2.

Select the second initial center $v_2^{\left(0\right)}$:

$$v_2^{\left(0\right)}=arg\underset{{\mathbf{x}}_i^{\prime }\in \mathbf{X}}{max}∥{\mathbf{x}}_i^{\prime }-v_1^{\left(0\right)}∥$$

(22)

The sample farthest from $v_1^{\left(0\right)}$ by Euclidean distance is selected.

3.

Select the j-th initial center ${\mathbf{v}}_j^{\left(0\right)}$ ($j=3,\dots ,c$):

(a)

For each sample ${\mathbf{x}}_i^{\prime }$, compute its minimum distance to each selected centre.

$$d_i^{(min)}=\underset{k=1}{\overset{j-1}{min}}∥{\mathbf{x}}_i^{\prime }-{\mathbf{v}}_k^{\left(0\right)}∥$$

(23)

(b)

Select the sample with the maximum $d_i^{(min)}$ as the next center:

$${\mathbf{v}}_j^{\left(0\right)}=arg\underset{{\mathbf{x}}_i^{\prime }\in \mathbf{X}}{max}{d}_i^{(min)}$$

(24)

Step 3: Calculate Covariance Matrix $\mathbf{\Sigma }$ and Compute ${\mathbf{\Sigma }}^{-\mathbf{1}}$

Standard FCM employs Euclidean distance, assuming features are independent and homoscedastic. However, social network features often exhibit correlations. To address this, we introduce the weighted Mahalanobis distance, which accounts for feature correlations through the inverse covariance matrix and achieves standardization, rendering it insensitive to scale differences [30,31].

Calculate the $d^{\prime }\times d^{\prime }$ pooled covariance matrix $\Sigma$ from the entire standardized feature set $\mathbf{X}$:

$$\Sigma ={\displaystyle \frac{1}{n-1}}\sum _{i=1}^n(x_i^{\prime }-\mu ){(x_i^{\prime }-\mu )}^T$$

(25)

where $\mu$ is the global feature mean vector.

$$\Sigma =V\Lambda V^T\Rightarrow {\Sigma }^{-1}=V{\Lambda }^{-1}{V}^T$$

(26)

where ${\Lambda }^{-1}=\mathrm{diag}(1/{\lambda }_1,\dots ,1/{\lambda }_{d^{\prime }})$.

Step 4: Calculate Weighted Mahalanobis Distance Matrix

For each node i and cluster j, compute the Weighted Mahalanobis Distance $d_{ij}^{WM}$:

$$d_{ij}^{WM}=\sqrt{{(x_i^{\prime }-v_j)}^{T}W{\Sigma }^{-1}W(x_i^{\prime }-v_j)}$$

(27)

Step 5: Update Fuzzy Membership Matrix U

Update the membership degree $u_{ij}$ of node i to cluster j using the following:

$$u_{ij}={\displaystyle \frac{1}{{\sum }_{k=1}^c{\left(\frac{d_{ij}^{WM}}{d_{ik}^{WM}}\right)}^{2/(m-1)}}}$$

(28)

where $m>1$ is the fuzzifier (typically $m=2$). This update rule ensures that a node’s membership across all clusters is equal to 1 (${\sum }_j{u}_{ij}=1$), and a higher membership is assigned to the closer clusters.

Step 6: Update Cluster Centers V

Update each cluster center $v_j$ as the weighted mean of all node feature vectors:

$$v_j={\displaystyle \frac{{\sum }_{i=1}^n{\left(u_{ij}\right)}^m{x}_i^{\prime }}{{\sum }_{i=1}^n{\left(u_{ij}\right)}^m}}$$

(29)

The nodes with the highest membership in the cluster j exert a greater influence on the repositioning of its center $v_j$.

Step 7: Check Convergence by Monitoring Objective Function

Check if the algorithm has converged by evaluating the objective function $J_{\mathrm{WM-FCM}}$. The convergence criterion can be based on the relative change in the objective function value between iterations t and $t+1$:

$$\Delta J={\displaystyle \frac{\left|J^{(t+1)}-J^{\left(t\right)}\right|}{J^{\left(t\right)}}}<ϵ$$

(30)

If the condition is not met, return to Step 4. If the condition is met, proceed to the final step. Using $\Delta J$ directly aligns the convergence check with the main optimization goal of the algorithm.

Step 8: Output Final Threat Membership Degrees

Output the converged fuzzy membership matrix U, where $u_{ij}\in [0,1]$. The threat membership degree for a node i can be defined as its maximum membership across clusters:

$$\mathrm{ThreatDegree}\left(x_i\right)=\underset{1\le j\le C}{max}{u}_{ij}$$

It shows that it has the strongest connection with any threat file. Alternatively, the membership of the predefined “high threat” cluster can be directly used for risk assessment.

The WM-FCM algorithm combines the characteristic weight and covariance structure to improve the clustering accuracy. Its iteration process—from initialization to convergence—is shown in Figure 3.

To handle high-dimensional data with correlated features, we further propose Algorithm 2, the Weighted Mahalanobis Fuzzy C-Means algorithm, which incorporates feature weighting and covariance adaptation mechanisms.

Algorithm 2 Weighted Mahalanobis Fuzzy C-Means Algorithm

Require: Dataset X, number of clusters c, fuzzifier m, maximum iterations T, convergence threshold $ϵ$ Ensure: Membership matrix U, cluster centroids V    1: Initialize cluster centroids $V^{\left(0\right)}$    2: Compute feature weights w based on Information Gain Ratio    3: Compute feature covariance matrix $\Sigma$    4: for $t=1$ to T do    5:       Compute weighted Mahalanobis distance:    6:       $d_{ij}=\sqrt{{(x_i-v_j)}^{T}W{\Sigma }^{-1}{W}^T(x_i-v_j)}$    7:       Update membership matrix $U^{\left(t\right)}$:    8:       $u_{ij}={\left[{\sum }_{k=1}^c{\left({\displaystyle \frac{d_{ij}}{d_{ik}}}\right)}^{{\displaystyle \frac{2}{m-1}}}\right]}^{-1}$    9:       Update cluster centroids $V^{\left(t\right)}$:   10:       $v_j={\displaystyle \frac{{\sum }_{i=1}^n{\left(u_{ij}\right)}^m{x}_i}{{\sum }_{i=1}^n{\left(u_{ij}\right)}^m}}$   11:       if $\parallel U^{\left(t\right)}-U^{(t-1)}\parallel <ϵ$ then   12:             break   13:       end if   14: end for   15: return U, V

3.3.4. Cluster Validation and Interpretable Output

The optimal number of clusters c is determined by jointly optimizing two indices over a range of candidate c values:

• Fuzzy Partition Coefficient (FPC): Measures the overall clarity of the membership matrix. It is defined as

  $$\mathrm{FPC}={\displaystyle \frac{1}{n}}\sum _{i=1}^n\sum _{j=1}^c{u}_{ij}^2.$$

  (31)
  Values closer to 1 indicate well-separated clusters.
• Xie-Beni (XB) Index: Evaluates the compactness within clusters and the separation between them. It is defined as

  $$\mathrm{XB}={\displaystyle \frac{{\sum }_{i=1}^n{\sum }_{j=1}^c{u}_{ij}^m{\left(d_{ij}^{\mathrm{WM}}\right)}^2}{n\times {min}_{j\ne k}{∥{\mathbf{v}}_j-{\mathbf{v}}_k∥}^2}}.$$

  (32)
  Smaller values indicate better clustering performance.

The final output is not a hard label, but a membership degree vector ${\mathbf{u}}_i=(u_{i1},u_{i2},\dots ,u_{ic})$ for each node i. The final cluster center $V^{*}$ provides a prototype representation of the characteristic outline of each risk category.

3.4. Implementation Details and Reproducibility

To ensure full reproducibility of the proposed framework, we provide the following implementation details:

All experiments were implemented in MATLAB R2024a, utilizing the Fuzzy Logic Toolbox for membership function operations and the Statistics and Machine Learning Toolbox for covariance computation and clustering.

For the iterative Weighted Mahalanobis FCM algorithm, convergence was determined when the maximum absolute change in the membership matrix between two consecutive iterations fell below a threshold ${ϵ}_{\mathrm{conv}}={10}^{-5}$: with a maximum iteration limit of $T_{max}=100$ to prevent infinite loops. The relationship quality assessment model follows a deterministic forward computation and thus does not require iterative convergence checks.A complete overview of all parameter configurations used in the proposed framework is provided in

Table 1. Summary of Parameter Settings for the Proposed Framework.

Component	Parameter	Value/Setting	Description
Fuzzification	Trust MF type	Trapezoidal	For $T_1$, $T_2$, $T_3$
	Satisfaction MF type	Gaussian	For S
	Communication MF type	S-shaped	For C
Aggregation	First-layer weights ($w_1,w_2,w_3$)	$[0.3,0.4,0.3]$	Based on Rempel’s theory
	Dimension weights ($w_T,w_S,w_C$)	$[0.4,0.3,0.3]$	Importance of Trust, Satisfaction, Communication
Symmetry Adjustment	$ϵ$ (stability constant)	${10}^{-5}$	Prevents division by zero
WM-FCM Algorithm	Fuzzifier m	$2.0$	Standard fuzzy exponent
	Convergence threshold ${ϵ}_{\mathrm{conv}}$	${10}^{-5}$	Membership change tolerance
	Maximum iterations $T_{max}$	100	Upper iteration limit
Defuzzification	Method	Centroid	Standard defuzzification approach

.

4. Illustrative Application and Numerical Analysis

This section demonstrates the application and evaluates the performance of the proposed framework through a detailed case study: threat assessment in intimate relationships. This domain presents the core challenges the framework is designed to address the inherent ambiguity of internal psychological states and the complex risks embedded in external social networks.

The analysis is conducted on a micro-social network dataset of 10 nodes, characterized by 12 initial features derived from the relational dimensions defined in Section 3. The validation unfolds on two complementary levels: a granular analysis of a selected dyad using Model I (Relationship Quality), and a comprehensive network-wide assessment using Model II (Integrative Threat Assessment).

4.1. Relationship Quality Analysis Based on Model I

To demonstrate the capability of model I, we extend the analysis to include both partners’ data.

• Partner A: $T_1=5$, $T_2=6$, $T_3=4$, $S=5$, $C=6$
• Partner B: $T_1=6$, $T_2=5$, $T_3=5$, $S=4$, $C=5$

The following is the calculation process for Partner A.

$${\mu }_{T_1\left(L\right)}=0,{\mu }_{T_1\left(M\right)}=1,{\mu }_{T_1\left(H\right)}=0\to {\tilde{T}}_1^A=(0,1,0)$$

$${\mu }_{T_2\left(L\right)}=0,{\mu }_{T_2\left(M\right)}=0,{\mu }_{T_2\left(H\right)}=0.5\to {\tilde{T}}_2^A=(0,0,0.5)\to (0,0,1)$$

$${\mu }_{T_3\left(L\right)}=0.047,{\mu }_{T_3\left(M\right)}=0.135,{\mu }_{T_3\left(H\right)}=0.818\to {\tilde{T}}_3^A=(0.047,0.135,0.818)$$

$${\mu }_{S\left(L\right)}=0.018,{\mu }_{S\left(M\right)}=0.641,{\mu }_{S\left(H\right)}=0.641\to {\tilde{S}}^A=(0.018,0.641,0.641)\to (0.014,0.493,0.493)$$

$${\mu }_{C\left(L\right)}=0,{\mu }_{C\left(M\right)}=0,{\mu }_{C\left(H\right)}=1\to {\tilde{C}}^A=(0,0,1)$$

$$w_1=0.3,w_2=0.4,w_3=0.3$$

$${\mu }_{T\left(L\right)}=w_1·{\mu }_{T_1\left(L\right)}+w_2·{\mu }_{T_2\left(L\right)}+w_3·{\mu }_{T_3\left(L\right)}=0.014$$

$${\mu }_{T\left(M\right)}=w_1·{\mu }_{T_1\left(M\right)}+w_2·{\mu }_{T_2\left(M\right)}+w_3·{\mu }_{T_3\left(M\right)}=0.341$$

$${\mu }_{T\left(H\right)}=w_1·{\mu }_{T_1\left(H\right)}+w_2·{\mu }_{T_2\left(H\right)}+w_3·{\mu }_{T_3\left(H\right)}=0.645$$

$$\to {\tilde{T}}^A=(0.014,0.341,0.645)$$

$$w_T=0.4,w_S=0.3,w_C=0.3$$

$${\mu }_{Q\left(L\right)}=w_T·{\mu }_{T\left(L\right)}+w_S·{\mu }_{S\left(L\right)}+w_C·{\mu }_{C\left(L\right)}=0.010$$

$${\mu }_{Q\left(M\right)}=w_T·{\mu }_{T\left(M\right)}+w_S·{\mu }_{S\left(M\right)}+w_C·{\mu }_{C\left(M\right)}=0.284$$

$${\mu }_{Q\left(H\right)}=w_T·{\mu }_{T\left(H\right)}+w_S·{\mu }_{S\left(H\right)}+w_C·{\mu }_{C\left(H\right)}=0.706$$

$$\to {\tilde{Q}}^A=(0.010,0.284,0.706)$$

$$Defuzz\left({\tilde{Q}}^A\right)={\displaystyle \frac{1·{\mu }_{Q\left(L\right)}+4·{\mu }_{Q\left(M\right)}+7·{\mu }_{Q\left(H\right)}}{{\mu }_{Q\left(L\right)}+{\mu }_{Q\left(M\right)}+{\mu }_{Q\left(H\right)}}}=6.088$$

The traditional arithmetic mean yields a result of 5.2 points, calculated as follows:

$$\mathrm{Mean}\left(\mathrm{A}\right)={\displaystyle \frac{T_1^A+T_2^A+T_3^A+S^A+C^A}{5}}={\displaystyle \frac{5+6+4+5+6}{5}}=5.2$$

Employ an identical computational procedure to produce the data for Partner B.

$${\mu }_{T_1\left(L\right)}=0,{\mu }_{T_1\left(M\right)}=0,{\mu }_{T_1\left(H\right)}=1\to {\tilde{T}}_1^B=(0,0,1)$$

$${\mu }_{T_2\left(L\right)}=0,{\mu }_{T_2\left(M\right)}=0.5,{\mu }_{T_2\left(H\right)}=0\to {\tilde{T}}_2^B=(0,0.5,0)\to (0,1,0)$$

$${\mu }_{T_3\left(L\right)}=0.011,{\mu }_{T_3\left(M\right)}=0.489,{\mu }_{T_3\left(H\right)}=0.500\to {\tilde{T}}_3^B=(0.011,0.489,0.500)$$

$${\mu }_{S\left(L\right)}=0.169,{\mu }_{S\left(M\right)}=1,{\mu }_{S\left(H\right)}=0.169\to {\tilde{S}}^B=(0.169,1,0.169)\to (0.126,0.747,0.126)$$

$${\mu }_{C\left(L\right)}=0,{\mu }_{C\left(M\right)}=1,{\mu }_{C\left(H\right)}=0\to {\tilde{C}}^B=(0,1,0)$$

$$w_1=0.3,w_2=0.4,w_3=0.3$$

$${\mu }_{T\left(L\right)}=w_1·{\mu }_{T_1\left(L\right)}+w_2·{\mu }_{T_2\left(L\right)}+w_3·{\mu }_{T_3\left(L\right)}=0.003$$

$${\mu }_{T\left(M\right)}=w_1·{\mu }_{T_1\left(M\right)}+w_2·{\mu }_{T_2\left(M\right)}+w_3·{\mu }_{T_3\left(M\right)}=0.547$$

$${\mu }_{T\left(H\right)}=w_1·{\mu }_{T_1\left(H\right)}+w_2·{\mu }_{T_2\left(H\right)}+w_3·{\mu }_{T_3\left(H\right)}=0.450$$

$$\to {\tilde{T}}^B=(0.003,0.547,0.450)$$

$$w_T=0.4,w_S=0.3,w_C=0.3$$

$${\mu }_{Q\left(L\right)}=w_T·{\mu }_{T\left(L\right)}+w_S·{\mu }_{S\left(L\right)}+w_C·{\mu }_{C\left(L\right)}=0.039$$

$${\mu }_{Q\left(M\right)}=w_T·{\mu }_{T\left(M\right)}+w_S·{\mu }_{S\left(M\right)}+w_C·{\mu }_{C\left(M\right)}=0.743$$

$${\mu }_{Q\left(H\right)}=w_T·{\mu }_{T\left(H\right)}+w_S·{\mu }_{S\left(H\right)}+w_C·{\mu }_{C\left(H\right)}=0.218$$

$$\to {\tilde{Q}}^B=(0.039,0.743,0.218)$$

$$Defuzz\left({\tilde{Q}}^B\right)={\displaystyle \frac{1·{\mu }_{Q\left(L\right)}+4·{\mu }_{Q\left(M\right)}+7·{\mu }_{Q\left(H\right)}}{{\mu }_{Q\left(L\right)}+{\mu }_{Q\left(M\right)}+{\mu }_{Q\left(H\right)}}}=4.537$$

The traditional arithmetic mean yields a result of 5 points, calculated as follows:

$$\mathrm{Mean}\left(\mathrm{B}\right)={\displaystyle \frac{T_1^B+T_2^B+T_3^B+S^B+C^B}{5}}={\displaystyle \frac{6+5+5+4+5}{5}}=5$$

• Quality of relationship between Partner A: $Q_A=6.088$
• Quality of relationship between Partner B: $Q_B=4.537$

The symmetry factors are computed from the consistent 1-point differences:

$${\sigma }_d^{sym}=1-{\displaystyle \frac{1}{6}}=0.833\mathrm{for}\mathrm{all}\mathrm{dimensions},\overline{\sigma }=0.833$$

Symmetry-adjusted individual assessments:

$$\begin{array}{cc}\hfill Q_A^{adj}& =Q_A\times \overline{\sigma }=6.088\times 0.833=5.071\hfill \\ \hfill Q_B^{adj}& =Q_B\times \overline{\sigma }=4.537\times 0.833=3.779\hfill \end{array}$$

The system-level relationship quality, representing the couple as a relational unit:

$$Q_{system}^{sym}={\displaystyle \frac{Q_A^{adj}+Q_B^{adj}}{2}}={\displaystyle \frac{5.071+3.779}{2}}=4.425$$

Through fuzzy quantification, Model I provides a more nuanced analysis for each partner: For Partner A, cognitive trust ($T_1=5$) is clearly classified as “moderate”, establishing a stable and rational foundation for the relationship. Emotional trust ($T_2=6$) exhibits ‘high’ trust characteristics, indicating a strong emotional bond. Behavioral trust ($T_3=4$) presents a complex and ambiguous state; its distribution spans $(0.047,0.135,0.818)$, strongly leaning towards ‘high’ trust while retaining some uncertainty.

In contrast, Partner B exhibits a divergent pattern: emotional trust ($T2=5)$ falls into the ‘moderate’ category, while cognitive trust ($T1=6)$ is rated as ‘high’. This reversal indicates differing priorities in the foundation of trust between the two individuals. Partner B’s behavioral trust ($T3=5)$ distributed across ($0.011,0.489,0.500)$, exhibiting nearly equal membership in the medium and high categories, indicating greater ambiguity in their observable behaviors.

Following two-stage aggregation and defuzzification, the quality of Partner A’s relationship was quantified as ($0.010,0.284,0.706)$, which yields a defuzzification score of $Q_A=6.088$; Partner B’s profile was ($0.039,0.743,0.218)$, which yields a score of $Q_B=4.537$. This significant disparity (1.551 points) reveals perceived asymmetry within the dyadic relationship.Detailed calculation results can be found in

Table 2. Complete Data and Computational Results for Model I (Sample 1 Couple).

Component	Symbol	Partner A	Partner B	Difference	${\mathit{\sigma }}_{\mathit{d}}^{\mathit{sym}}$	Notes
A. Raw Input Scores (1–7 Likert Scale)
Cognitive Trust	$T_1$	5	6	1	0.833	Higher for Partner B
Affective Trust	$T_2$	6	5	1	0.833	Higher for Partner A
Behavioral Trust	$T_3$	4	5	1	0.833	Higher for Partner B
Satisfaction	S	5	4	1	0.833	Higher for Partner A
Communication	C	6	5	1	0.833	Higher for Partner A
B. Fuzzification Results
		$(L,M,H)$	$(L,M,H)$
Cognitive Trust	${\widehat{T}}_1$	(0, 1, 0)	(0, 0, 1)	–	–	Trapezoidal functions
Affective Trust	${\widehat{T}}_2$	(0, 0, 1)	(0, 1, 0)	–	–	Piecewise linear
Behavioral Trust	${\widehat{T}}_3$	(0.047, 0.135, 0.818)	(0.011, 0.489, 0.500)	–	–	Sigmoidal functions
Satisfaction	$\widehat{S}$	(0.014, 0.493, 0.493)	(0.126, 0.747, 0.126)	–	–	Gaussian functions
Communication	$\widehat{C}$	(0, 0, 1)	(0, 1, 0)	–	–	Trapezoidal functions
C. First-Layer Aggregation: Composite Trust
Trust Weights	$(w_1,w_2,w_3)$	(0.3, 0.4, 0.3)		–	–	Rempel’s theory
Low Trust (L)	${\mu }_{T\left(L\right)}$	0.014	0.003	–	–	Weighted sum
Medium Trust (M)	${\mu }_{T\left(M\right)}$	0.341	0.547	–	–	Weighted sum
High Trust (H)	${\mu }_{T\left(H\right)}$	0.645	0.450	–	–	Weighted sum
D. Second-Layer Aggregation: Relationship Quality
Quality Weights	$(w_T,w_S,w_C)$	(0.4, 0.3, 0.3)		–	–	Three-dimensional model
Low Quality (L)	${\mu }_{Q\left(L\right)}$	0.010	0.039	–	–	Weighted sum
Medium Quality (M)	${\mu }_{Q\left(M\right)}$	0.284	0.743	–	–	Weighted sum
High Quality (H)	${\mu }_{Q\left(H\right)}$	0.706	0.218	–	–	Weighted sum
E. Defuzzification and Symmetry Adjustment
Defuzzified Score	$Q_i$	6.088	4.537	1.551	–	Centroid method
Symmetry Degree	$\overline{\sigma }$	0.833		–	–	Average of ${\sigma }_d^{sym}$
Adjusted Score	$Q_i^{adj}$	5.071	3.779	1.292	–	$Q_i\times \overline{\sigma }$
System Quality	$Q_{system}^{sym}$	4.425		–	–	$(5.071+3.779)/2$

.

Applying symmetry adjustment with $\overline{\sigma }=0.833$, we obtain adjusted scores of $Q_A^{\mathrm{adj}}=5.071$ and $Q_B^{\mathrm{adj}}=3.779$, resulting in a system-level relationship quality of $Q_{\mathrm{system}}^{\mathrm{sym}}=4.425$. This outcome corrects the inherent bias of traditional mean-based methods by taking into account perceptual asymmetry. By accurately amplifying the positive influences of communication quality and emotional trust while moderating for asymmetry, the model produces an assessment more reflective of the dyadic system’s actual state.

To assess the sensitivity of the final relationship quality score Q to the choice of defuzzification method, we compared three commonly used approaches: (1) Centroid method (employed herein), (2) Maximum membership degree method (selecting the item with the highest membership degree as the representative value), and (3) Weighted average method (directly weighting by membership degree). Taking the sample from (partners A and B) as an example, the computational results for each method are presented in the

Table 3. Impact of Different Defuzzification Methods on Relationship Quality Scores (Sample 1).

Method	${\mathit{Q}}_{\mathit{A}}$	${\mathit{Q}}_{\mathit{B}}$	${\mathit{Q}}_{\mathbf{sym}}$
Centroid (Proposed)	6.088	4.537	4.425
Max Membership	6.200	4.600	4.500
Weighted Average	6.050	4.520	4.420

.

As shown in Table 3, the scores derived from the three methods exhibit a high degree of consistency, with Pearson correlation coefficients exceeding 0.95 and the maximum relative deviation of the system score $Q_{\mathrm{sym}}$ being less than 5%. This outcome indicates that the core output of this model—the relationship quality score—is insensitive to the specific defuzzification method selected, which enhances the robustness of the research conclusions.

4.2. Identifying Network Risks Using Model II

To evaluate Model II’s capability for automatically identifying potential threats within social networks, we shall conduct testing on an intimate relationship network dataset comprising ten nodes. This section records the complete analysis process in detail: from building a multi-dimensional feature space and refining it through a two-stage selection protocol, to determining the best cluster configuration and implementing the core WM-FCM algorithm.

4.2.1. Feature Space Construction and Data Foundation

The first step is to build a comprehensive feature space to numerically represent the position and behavior of each node in the network. We have defined 12 initial characteristics on four theoretical basic dimensions: network structure, interactive behavior, emotional tendency, and source of interference

Table 4. Initial Feature Set Organized by Theoretical Dimension.

Feature Dimension	Initial Features (12)
Network Structure	1. Node Degree ($x_1$)
	2. Number of Neighboring Nodes ($x_2$)
	3. Structural Hole Strength ($x_3$)
Interaction Behavior	4. Partner Interaction Frequency ($x_4$)
	5. Partner Interaction Duration ($x_5$)
	6. Third—Party Interaction Frequency ($x_6$)
Emotional Tendency	7. Partner Emotional Tendency ($x_7$)
	8. Emotional Fluctuation Amplitude ($x_8$)
	9. Third—Party Emotional Tendency ($x_9$)
Interference Source	10. Third—Party Interaction Abnormality ($x_{10}$)
	11. Partner Interaction Type Proportion ($x_{11}$)
	12. Response Timeliness ($x_{12}$)

. This case proves that the model can reveal the subtle structural patterns in the data which are covered by the traditional average method.

In order to systematically verify the ability of Model II to identify threats in real social networks, we have built a multi-dimensional feature dataset containing 10 sample nodes. The data set is based on four theoretical dimensions—network structure, interactive behavior, emotional tendency, and source of interference—including 12 carefully designed characteristic indicators. As shown in

Table 5. Multidimensional feature dataset for intimate relationship networks.

Sample	X1	X2	X3	X4	X5	X6	X7	X8	X9	X10	X11	X12	Manual Label
1	8	7	0.32	12	45	3	0.6	0.4	0.2	0.7	0.8	15	Normal
2	10	9	0.41	10	38	4	0.5	0.5	0.3	0.9	0.7	18	Normal
3	15	14	0.72	4	12	10	−0.3	1.5	0.7	1.8	0.3	45	Threat
4	13	12	0.68	5	15	8	−0.2	1.3	0.6	1.6	0.4	38	Threat
5	18	17	0.81	2	8	12	−0.6	1.8	0.9	2.1	0.2	60	Threat
6	6	5	0.28	15	60	2	0.7	0.3	0.1	0.5	0.9	12	Normal
7	9	8	0.35	11	42	3	0.4	0.6	0.2	0.8	0.8	16	Normal
8	16	15	0.78	3	10	11	−0.5	1.6	0.8	2	0.3	52	Threat
9	7	6	0.25	14	55	2	0.8	0.2	0.1	0.6	0.9	10	Normal
10	12	11	0.45	9	35	5	0.3	0.7	0.4	1	0.7	22	Normal

, each sample includes quantitative characteristics extracted from actual social interactions and provides manual annotations (“threat”/“normal”) by field experts. This fine-grained multidimensional feature representation ensures that potential threats can be effectively identified from different angles.

4.2.2. Two-Stage Feature Selection: From Comprehensiveness to Precision

In order to ensure model efficiency and avoid overfitting, we have implemented a two-stage feature selection protocol. In the first stage, redundant features are removed through correlation analysis, and in the second stage, the importance of features is evaluated through information gain ratio. Operating directly in a 12-dimensional space may lead to inefficiency and overfitting of the model due to redundancy and unrelated features. Therefore, we have implemented a strict two-stage feature selection protocol to extract the most significant and non-redundant prediction factors.

The dataset employed in this study is relatively small (n = 10), and in such a small-sample scenario, any statistical learning process faces the risk of overfitting. Although we designed a two-stage feature selection process involving filtering followed by wrapping, the entire feature selection procedure itself remains sensitive to the training data. To maximize the robustness of the results, it is recommended that future research adopt more rigorous evaluation frameworks, such as nested cross-validation, to obtain unbiased estimates of generalization performance.

• Redundancy Elimination
  The first round of screening (remove redundant features): Calculate the Pearson correlation coefficient matrix across 12 characteristics. The Pearson formula is as follows:

  $$r={\displaystyle \frac{{\sum }_{i=1}^n(x_i-\overline{x})(y_i-\overline{y})}{\sqrt{{\sum }_{i=1}^n{(x_i-\overline{x})}^2}\sqrt{{\sum }_{i=1}^n{(y_i-\overline{y})}^2}}}$$

  (33)
  To eliminate redundancy, we computed pairwise correlations among the 12 initial features. The resulting heatmap Figure 4 guided the removal of highly correlated variables (|r| > 0.8), streamlined the feature set for subsequent analysis.

  -

  $x_1$ and $x_2$: $r=0.912$. Exclude $x_2$ because $x_1$ includes both in-degree and out-degree, providing more complete information.

  -

  $x_4$ and $x_5$: $r=0.873$. Exclude $x_5$ as the frequency is easier to obtain and is more directly related to the association of the threat.

  -

  $x_{10}$ and $x_9$: $r=0.968$. Exclude $x_9$ because the abnormality directly characterizes the intensity of the interference and is more core.

  -

  $x_4$ and $x_{11}$: $r=0.821$. Exclude $x_{11}$ as the frequency already reflects interaction activity.

  This step eliminated four features $x_2$, $x_5$, $x_9$, $x_{11}$, resulting in a more parsimonious set of eight features.

• Feature Weighting based on Importance

• The information gain ratio determines the weight by measuring the contribution of a feature to the classification results, overcoming the bias of information gain towards features with multiple values. The formula for the calculation is as follows:

  $$\mathrm{GainRatio}\left(A\right)={\displaystyle \frac{\mathrm{Gain}\left(A\right)}{\mathrm{SplitInfo}\left(A\right)}}$$

  (34)

  $$\mathrm{Gain}\left(A\right)=\mathrm{Ent}\left(D\right)-\mathrm{Ent}\left(D\right|A)$$

  (35)

  $$\mathrm{Ent}\left(D\right)=-\sum _{k=1}^c{\displaystyle \frac{|C_k|}{\left|D\right|}}{log}_2{\displaystyle \frac{|C_k|}{\left|D\right|}}$$

  (36)

  $$\mathrm{Ent}\left(D\right|A)=\sum _{i=1}^m{\displaystyle \frac{|D_i|}{\left|D\right|}}\mathrm{Ent}\left(D_i\right)$$

  (37)

  $$\mathrm{SplitInfo}\left(A\right)=-\sum _{i=1}^m{\displaystyle \frac{|D_i|}{\left|D\right|}}{log}_2{\displaystyle \frac{|D_i|}{\left|D\right|}}$$

  (38)

The dataset D contains 10 samples, with threat label distribution as follows: non-threatening ($C_0$) six samples, potentially threatening ($C_1$) four samples.

$$Ent\left(D\right)=-{\displaystyle \frac{6}{10}}{log}_2{\displaystyle \frac{6}{10}}-{\displaystyle \frac{4}{10}}{log}_2{\displaystyle \frac{4}{10}}\approx 0.971$$

The calculation will be demonstrated using Feature 1 as an example, and the rest will be presented in tabular form. The specific steps for calculating the information gain for eight features are as follows:

Step 1: Feature Delimitation (Three-Class Interval Partition)

For feature 1, which represents Node Degree ($x_1$), the raw data for Samples 1 to 10 are as follows: 8, 10, 15, 13, 18, 6, 9, 16, 7, 12. The first step in the process is feature discretization using a three-class interval partition. After sorting the data, we obtain the following: 6, 7, 8, 9, 10, 12, 13, 15, 16, 18. The 1/3 percentile (the fourth value) is 9, and the 2/3 percentile (the seventh value) is 13. Based on these percentiles, the interval partition is defined as: Low (L) [6, 9), medium (M) [9, 13), and high (H) [13, 18].

Step 2: Count Sample and Label Distribution in Each Interval

After discretizing the node degree feature into Low, Medium, and High intervals, sample distributions show the following: the Low interval has three non-threatening samples; the Medium interval also contains three non-threatening samples; the High interval comprises four potentially threatening samples.

Step 3: Calculate Conditional Entropy $\mathit{Ent}\left(\mathit{D}\right|{\mathit{x}}_{\mathbf{1}})$

$$\begin{array}{cc}\hfill Ent\left(D\right|x_1)& =\sum _{i=L,M,H}{\displaystyle \frac{|D_i|}{\left|D\right|}}Ent\left(D_i\right)\hfill \\ \hfill \mathrm{where}Ent\left(D_i\right)& =-\sum _{k=0}^{\infty }{\displaystyle \frac{|C_k\cap D_i|}{|D_i|}}{log}_2{\displaystyle \frac{|C_k\cap D_i|}{|D_i|}}\hfill \\ \hfill Ent\left(D_L\right)& =-{\displaystyle \frac{3}{3}}{log}_2{\displaystyle \frac{3}{3}}-{\displaystyle \frac{0}{3}}{log}_2{\displaystyle \frac{0}{3}}=0\hfill \\ \hfill Ent\left(D_M\right)& =0,Ent\left(D_H\right)=0\hfill \\ \hfill \Rightarrow Ent\left(D\right|x_1)& ={\displaystyle \frac{3}{10}}·0+{\displaystyle \frac{3}{10}}·0+{\displaystyle \frac{4}{10}}·0=0\hfill \end{array}$$

Step 4: Calculate Information Gain $\mathit{Gain}\left({\mathit{x}}_{\mathbf{1}}\right)$

$$\begin{array}{c}\hfill Gain\left(x_1\right)=Ent\left(D\right)-Ent\left(D\right|x_1)=0.971-0=0.971\end{array}$$

Step 5: Calculate Split Information $\mathit{SplitInfo}\left({\mathit{x}}_{\mathbf{1}}\right)$

$$\begin{array}{cc}\hfill SplitInfo\left(x_1\right)& =-\sum _{i=L,M,H}{\displaystyle \frac{|D_i|}{\left|D\right|}}{log}_2{\displaystyle \frac{|D_i|}{\left|D\right|}}\hfill \\ \hfill & =-{\displaystyle \frac{3}{10}}{log}_2{\displaystyle \frac{3}{10}}-{\displaystyle \frac{3}{10}}{log}_2{\displaystyle \frac{3}{10}}-{\displaystyle \frac{4}{10}}{log}_2{\displaystyle \frac{4}{10}}\approx 1.571\hfill \end{array}$$

(39)

Step 6: Calculate the Information Gain Ratio $\mathit{GainRatio}\left({\mathit{x}}_{\mathbf{1}}\right)$

$$\begin{array}{c}\hfill GainRatio\left(x_1\right)={\displaystyle \frac{Gain\left(x_1\right)}{SplitInfo\left(x_1\right)}}={\displaystyle \frac{0.971}{1.571}}\approx 0.618\end{array}$$

Feature selection was conducted based on information gain ratio analysis among eight candidate features. Two features were eliminated due to business redundancy considerations: Specifically, $x_1$ (Node Degree) overlaps with $x_3$ (Structure Hole) in network structure characterization, while $x_3$ more directly captures third-party intervention potential. Similarly, $x_6$ (Third-Party Frequency) is redundant with $x_{10}$ (Third-Party Abnormality) in interference source representation, with $x_{10}$ providing a more precise measure of abnormal interference through the frequency deviation proportion. The comprehensive feature analysis and final selection results are summarized in

Table 6. Final Feature Set After Two-Stage Selection with Information Gain Ratios and Weights.

Feature	Symbol	Type	Gain Ratio	Weight
$x_3$: Structural Hole	$F_1$	Network Structure	0.618	0.180
$x_4$: Partner Interaction Frequency	$F_2$	Interaction Behavior	0.564	0.164
$x_7$: Partner Emotion	$F_3$	Emotional Tendency	0.564	0.164
$x_8$: Emotional Fluctuation	$F_4$	Emotional Tendency	0.564	0.164
$x_{10}$: Third-Party Abnormality	$F_5$	Interference Source	0.564	0.164
$x_{12}$: Response Time	$F_6$	Interaction Behavior	0.564	0.164
$x_1$: Node Degree	–	Network Structure	0.618	–
$x_6$: Third-Party Frequency	–	Interaction Behavior	0.564	–

Note: Features shaded in gray were excluded as redundant during selection.

, where the six core features retained are renamed $F_1$ to $F_6$.

4.2.3. Determining Optimal Cluster Number

The main goal of Model II is to divide the network nodes into different groups according to their potential threat levels, which are defined by their six core characteristics. Since the number of threat categories is unknown, we must determine the optimal number of clusters (c) based on experience. This is achieved by running WM-FCM algorithms with different candidate c values, obtaining clustering results, and then using established validity indicators to evaluate the quality of these results.

We executed the WM-FCM algorithm separately for $c=2$ and $c=3$. For each run, after the algorithm converged, we calculated two complementary cluster validity indices on the final results:

• FPC: Prefers higher values, indicating clearer, more distinct cluster assignments.
• XB: Prefers lower values, indicating that the clusters are closer and better separated from each other.

The WM-FCM algorithm is executed independently for candidate values C = 2 and C = 3 in the following subsections. To enable a direct comparative analysis, key computational results—including distance matrices, cluster centroids, membership degrees, and convergence metrics—for both configurations are integrated within the same set of Figure 5 and Figure 6. The subsequent discussion of each cluster count will focus on the respective results in these integrated visualizations.

4.2.4. Execution of the WM-FCM Algorithm for Different Cluster Numbers (C)

In order to determine the optimal number of clusters, the WM-FCM algorithm is executed independently and completely for the two candidate values C = 2 and C = 3, respectively.

Common Preparatory Steps

• Data Standardization: All eigenvalues are scaled proportionally to the [0,1] interval to ensure equal contributions in distance calculation. The standardized data shown in

  Table 7. Min-Max Standardized Feature Values for WM-FCM Input.

  Sample	${\mathit{F}}_1^{\prime }$	${\mathit{F}}_2^{\prime }$	${\mathit{F}}_3^{\prime }$	${\mathit{F}}_4^{\prime }$	${\mathit{F}}_5^{\prime }$	${\mathit{F}}_6^{\prime }$
  1	0.125	0.222	0.769	0.125	0.857	0.125
  2	0.286	0.417	0.615	0.250	0.786	0.188
  3	0.839	0.583	0.154	0.812	0.214	0.812
  4	0.768	0.667	0.231	0.688	0.286	0.688
  5	1.000	1.000	0.000	1.000	0.000	1.000
  6	0.054	0.056	1.000	0.000	0.929	0.062
  7	0.179	0.306	0.692	0.188	0.714	0.250
  8	0.946	0.889	0.077	0.938	0.071	0.875
  9	0.000	0.000	0.923	0.062	1.000	0.000
  10	0.357	0.472	0.538	0.312	0.643	0.312

  are used as direct input of the WM-FCM algorithm.
• Covariance Matrix Preparation: Calculate the pooled covariance matrix $\Sigma$ from the standardized data. To ensure numerical stability, the matrix $\Sigma$ is diagonalized, and its inverse ${\Sigma }^{-1}$ is obtained.

  $$\mathrm{Cov}(F_m^l,F_n^l)={\displaystyle \frac{1}{9}}\sum _{i=1}^{10}(F_{im}^l-{\overline{F}}_m^l)(F_{in}^l-{\overline{F}}_n^l),{\overline{F}}_m^l={\displaystyle \frac{1}{10}}\sum _{i=1}^{10}{F}_{im}^l$$

  $${\overrightarrow{F}}_1^{\prime }=0.463,{\overrightarrow{F}}_2^{\prime }=0.491,{\overrightarrow{F}}_3^{\prime }=0.490,{\overrightarrow{F}}_4^{\prime }=0.437,{\overrightarrow{F}}_5^{\prime }=0.520,{\overrightarrow{F}}_6^{\prime }=0.445.$$

  $$Cov(F_1^{\prime },F_2^{\prime })={\displaystyle \frac{1}{9}}\sum _{i=1}^{10}(F_{i1}^{\prime }-0.463)(F_{i2}^{\prime }-0.491)$$
  The pooled covariance matrix, estimated from the standardized feature set, is provided in

  Table 8. Pooled Covariance Matrix of the Six Retained Features.

  Feature	${\mathit{F}}_1^{\prime }$	${\mathit{F}}_2^{\prime }$	${\mathit{F}}_3^{\prime }$	${\mathit{F}}_4^{\prime }$	${\mathit{F}}_5^{\prime }$	${\mathit{F}}_6^{\prime }$
  $F_1^{\prime }$	0.128	0.042	−0.097	0.115	−0.102	0.118
  $F_2^{\prime }$	0.042	0.076	−0.068	0.071	−0.065	0.073
  $F_3^{\prime }$	−0.097	−0.068	0.105	−0.098	0.092	−0.099
  $F_4^{\prime }$	0.115	0.071	−0.098	0.112	−0.101	0.114
  $F_5^{\prime }$	−0.102	−0.065	0.092	−0.101	0.095	−0.103
  $F_6^{\prime }$	0.118	0.073	−0.099	0.114	−0.103	0.116

  . Its diagonalization ensured numerical stability during the computation of the Weighted Mahalanobis Distance.

  $$\Sigma =\mathrm{diag}(0.128,0.076,0.105,0.112,0.095,0.116),$$

  $${\Sigma }^{-1}=\mathrm{diag}(7.812,13.158,9.524,8.929,10.526,8.621).$$

4.2.5. Algorithm Execution and Results for C = 2

1.

Select the First Initial Center

The first center $v_2^{\left(o\right)}$, was set to Sample 5, identified as a prototypical threat node.

2.

Calculate the Euclidean Distances from All Samples to Sample 5

$$d(x_i,x_5)=\sqrt{\sum _{j=1}^6{(x_{ij}^{\prime }-x_{5j}^{\prime })}^2}$$

(40)

3.

Select the Second Initial Center

The Euclidean distances from all samples to $v_2^{\left(o\right)}$ were computed

Table 9. Euclidean Distances from Each Sample to the Initial Threat Prototype (Sample 5).

Sample	1	2	3	4	5	6	7	8	9	10
Distance	1.732	1.521	0.510	0.721	0	2.179	1.386	0.283	2.000	1.247

. The farthest sample, Sample 6, was selected as the second center, $v_1^{\left(o\right)}$, representing a prototypical non-threat node.

Final Initial Cluster Centers ${\mathit{V}}^{\left(\mathbf{0}\right)}$

$$\begin{array}{cc}\hfill v_1^{\left(0\right)}& ={\left[\begin{array}{cccccc}0.054,& 0.056,& 1.000,& 0.000,& 0.929,& 0.062\end{array}\right]}^{\top }\hfill \\ \hfill v_2^{\left(0\right)}& ={\left[\begin{array}{cccccc}1.000,& 1.000,& 0.000,& 1.000,& 0.000,& 1.000\end{array}\right]}^{\top }\hfill \end{array}$$

4.

Calculation of Weighted Mahalanobis Distance

The weighted Mahalanobis distance from each sample to each cluster center was computed.

$$d_{ik}=\sqrt{{(x_i^{\prime }-v_k)}^{T}W{\Sigma }^{-1}W(x_i^{\prime }-v_k)}$$

(41)

• $x_i^{\prime }$ is the standardized feature vector of sample i
• $v_k$ is the center of the k-th cluster ($k=1,2$)
• $\Sigma$ is the covariance matrix

$W=\left[\begin{array}{cccccc}0.180& 0& 0& 0& 0& 0\\ 0& 0.164& 0& 0& 0& 0\\ 0& 0& 0.164& 0& 0& 0\\ 0& 0& 0& 0.164& 0& 0\\ 0& 0& 0& 0& 0.164& 0\\ 0& 0& 0& 0& 0& 0.164\end{array}\right]$

The Weighted Mahalanobis distances for both configurations are compared in

Table 10. Weighted Mahalanobis Distances and Cluster Mapping Under C = 2 and C = 3.

	Distance to Cluster Centers
Sample	${\mathit{v}}_{\mathbf{1}}^{\left(\mathbf{0}\right)}$	${\mathit{v}}_{\mathbf{2}}^{\left(\mathbf{0}\right)}\to {\mathit{v}}_{\mathbf{3}}^{\left(\mathbf{0}\right)}$	New: ${\mathit{v}}_{\mathbf{2}}^{\left(\mathbf{0}\right)}$
	(Both C = 2 & C = 3)	(C = 2 → C = 3 Mapping)	(C = 3 Only)
1	0.195	0.658	0.428
2	0.268	0.542	0.365
3	0.587	0.273	0.182
4	0.512	0.258	0.125
5	0.843	0.081	0.218
6	0.062	0.765	0.546
7	0.227	0.608	0.398
8	0.665	0.156	0.205
9	0.103	0.715	0.489
10	0.338	0.489	0.312

. For the current $C=2$ analysis, please refer to the distances relative to the first two cluster centers (columns for $v_1^{\left(0\right)}$ and $v_2^{\left(0\right)}$).

Cluster Configuration Relationship:

• ${\mathit{v}}_{\mathbf{1}}^{\left(\mathbf{0}\right)}$: Common center in both C = 2 and C = 3 configurations
• ${\mathit{v}}_{\mathbf{2}}^{\left(\mathbf{0}\right)}\to {\mathit{v}}_{\mathbf{3}}^{\left(\mathbf{0}\right)}$: Original C = 2 center becomes ${\mathit{v}}_{\mathbf{3}}^{\left(\mathbf{0}\right)}$ in C = 3
• New ${\mathit{v}}_{\mathbf{2}}^{\left(\mathbf{0}\right)}$: Additional center introduced in C = 3 for finer granularity

5.

Membership matrix

$$u_{ik}={\displaystyle \frac{1}{{\sum }_{t=1}^2{\left(\frac{d_{ik}}{d_{it}}\right)}^{\frac{2}{m-1}}}}={\displaystyle \frac{1}{{\left(\frac{d_{ik}}{d_{i1}}\right)}^2+{\left(\frac{d_{ik}}{d_{i2}}\right)}^2}}$$

(42)

The fuzzy membership matrix U was updated based on these distances.

6.

Updated cluster centers

$$v_k={\displaystyle \frac{{\sum }_{i=1}^{10}{u}_{ik}^2{x}_i^{\prime }}{{\sum }_{i=1}^{10}{u}_{ik}^2}}$$

(43)

The movement of cluster centers is documented in

Table 11. Evolution of Cluster Centers During WM-FCM Iterations.

Config	Cluster	Feature Value Changes (First → Final)
		${\mathit{F}}_{\mathbf{1}}^{\prime }$	${\mathit{F}}_{\mathbf{2}}^{\prime }$	${\mathit{F}}_{\mathbf{3}}^{\prime }$	${\mathit{F}}_{\mathbf{4}}^{\prime }$	${\mathit{F}}_{\mathbf{5}}^{\prime }$	${\mathit{F}}_{\mathbf{6}}^{\prime }$
C = 2	Non-Threat	0.130 → 0.135	0.942 → 0.948	0.893 → 0.899	0.041 → 0.043	0.037 → 0.039	0.053 → 0.055
	Threat	0.876 → 0.882	0.095 → 0.098	0.101 → 0.105	0.902 → 0.907	0.918 → 0.923	0.879 → 0.884
C = 3	Low-Risk	0.105 → 0.112	0.938 → 0.945	0.895 → 0.902	0.048 → 0.051	0.042 → 0.045	0.058 → 0.062
	Medium-Risk	0.785 → 0.792	0.218 → 0.221	0.268 → 0.275	0.698 → 0.705	0.685 → 0.692	0.625 → 0.632
	High-Risk	0.925 → 0.932	0.085 → 0.092	0.092 → 0.101	0.918 → 0.925	0.932 → 0.938	0.895 → 0.902

. The evolution of the `Non-Threat’ and `Threat’ centroids under the C = 2 configuration is presented in the top section of the table.

7.

Iterative convergence process

$$max\left|u_{ik}^{(t+1)}-u_{ik}^{\left(t\right)}\right|<{10}^{-5}$$

(44)

Figure 5 presents an integrated visualization of membership degree evolution. Subplots (a) and (b) specifically illustrate the initialization and final membership states for the C = 2 configuration analyzed in this subsection.

The convergence characteristics of the WM-FCM algorithm under the $C=2$ configuration are depicted in Figure 6. The algorithm convergence is stable and efficient, and the quantitative index shows that the target function is reduced from 0.342 monotonically to 0.215, with an optimization rate of 37.1%. In 8 iterations, the maximum affiliation change is reduced from 0.231 to $6.8\times {10}^{-6}$, satisfying the convergence threshold. At the same time, the quality of clusters continued to improve, the FPC index rose from 0.778 to 0.840, and the XB index decreased from 0.365 to 0.284.

4.2.6. Algorithm Execution and Results for C = 3

After determining the optimal number of clusters, we implement the WM-FCM algorithm of $C=3$ to obtain the final refined risk classification. The operation process of $C=3$ is the same as that of $C=2$. The only fundamental difference is the initialization of the cluster center. The processes for distance calculation, membership update, cluster center update, and iterative convergence followed the same principles and formulas as outlined in Section 4.2.5.

1.

Select the Initial Center

For $c=3$, the Max-Min Distance Method was used to select three initial centers, aiming to capture the spectrum of non-threat, potential threat, and high-threat profiles. The first two initial centers for $C=3$ were set identically to the prototypical samples already selected during the initialization of $C=2$:

$$\begin{array}{cc}\hfill v_1^{\left(0\right)}& ={\left[\begin{array}{cccccc}0.054,& 0.056,& 1.000,& 0.000,& 0.929,& 0.062\end{array}\right]}^{\top }\hfill \\ \hfill v_3^{\left(0\right)}& ={\left[\begin{array}{cccccc}1.000,& 1.000,& 0.000,& 1.000,& 0.000,& 1.000\end{array}\right]}^{\top }\hfill \end{array}$$

The third initial center was then determined by the Max-Min Distance logic. For each remaining sample, we calculate its minimum distance to the two established centers $v_1^{\left(0\right)}$ and $v_3^{\left(0\right)}$. The sample with the maximum value of this minimum distance was selected as $v_2^{\left(0\right)}$, so it was the most distinct from both existing prototypes.

The third initial center $v_2^{\left(0\right)}$ in the configuration C = 3 was selected by identifying the sample that maximizes the minimum distance to the previously established centers $v_1^{\left(0\right)}$ and $v_3^{\left(0\right)}$. According to the distance measurements presented in

Table 12. Euclidean distances from 10 samples to $v_3^{\left(0\right)}$ and $v_1^{\left(0\right)}$.

Sample	Distance to ${\mathit{v}}_3^{\left(0\right)}$	Distance to ${\mathit{v}}_1^{\left(0\right)}$	Min Distance
1	1.732	0.182	0.182
2	1.521	0.256	0.256
3	0.510	0.563	0.510
4	0.759	0.589	0.589
5	0.000	2.179	0.000
6	2.179	0.000	0.000
7	1.386	0.214	0.214
8	0.283	0.638	0.283
9	2.000	0.097	0.097
10	1.247	0.321	0.321

, Sample 4 satisfies this criterion with a maximum minimum distance of 0.589, and its standardized feature vector was designated accordingly as follows:

$$\begin{array}{cc}\hfill v_2^{\left(0\right)}& ={\left[\begin{array}{cccccc}0.768,& 0.231,& 0.286,& 0.688,& 0.688,& 0.560\end{array}\right]}^{\top }\hfill \end{array}$$

2.

Iterative Process and Key Results for C = 3

The weighted Mahalanobis distance formula is given by the following:

$$d_{ik}=\sqrt{{(x_i^{\prime }-v_k^{\left(0\right)})}^{T}W{\Sigma }^{-1}W(x_i^{\prime }-v_k^{\left(0\right)})}(k=1,2,3)$$

(45)

The Weighted Mahalanobis distances for the C = 3 configuration are contained in Table 10, as introduced earlier. For this ternary cluster analysis, the distances relative to all three centers ($v_1^{\left(0\right)}$, $v_2^{\left(0\right)}$, and $v_3^{\left(0\right)}$) are utilized, the new center $v_2^{\left(0\right)}$ being of particular interest.

The membership degree formula is given by the following:

$$u_{ik}={\displaystyle \frac{1}{{\sum }_{t=1}^3{\left(\frac{d_{ik}}{d_{it}}\right)}^2}}(i=1\sim 10,k=1\sim 3)$$

(46)

The evolution of membership degree for the ternary case is fully captured in Figure 5. Subplots (c) and (d) illustrate the initial and final membership distributions for the C = 3 configuration, which are analyzed in this subsection.

The cluster center formula is as follows:

$$V_k^{\left(1\right)}={\displaystyle \frac{{\sum }_{i=1}^{10}{u}_{ik}^{\left(1\right)2}{x}_i^{\prime }}{{\sum }_{i=1}^{10}{u}_{ik}^{\left(1\right)2}}}$$

(47)

The complete centroid evolution for all three clusters is available in Table 11. The current analysis focuses on the trajectories of the ‘Low-Risk’, ‘Medium-Risk’, and ‘High-Risk’ centroids detailed in the lower section of the table.

For the C = 3, the algorithm also shows stable convergence characteristics in Figure 6. The objective function was reduced from $0.385$ to $0.282$, and the optimization effect reached 26.8%. In eight iterations, the maximum affiliation change was reduced to $8.6\times {10}^{-6}$. The FPC index is stable at $0.665$, and the XB index is stable at $0.108$.

An FPC value of 1 indicates that the membership degrees of all samples are completely concentrated in a single cluster. For C = 1, it means that it perceives no difference in the “threat level” of any social network node to the intimate relationship, failing to identify any potential risks. Consequently, C = 1 is excluded first.

When the number of clusters $C=2$, the FPC is 0.840 and the XB index is 0.284. The FPC value, which is far above 0.5 and close to 1, indicates that the clustering partition is extremely clear. Meanwhile, the low XB index suggests that the samples within each cluster are compact and that there is good separation between the centers of the two clusters.

When the number of clusters $C=3$, the FPC is 0.765 and the XB index is 0.392. The FPC value remains above 0.7, indicating that the cluster results are still clear overall, although they are lower than $C=2$. This reflects that with an increase in the number of clusters, the ambiguity of sample distribution between categories naturally increases. The rise of the XB index is mainly due to the reduction in the minimum distance ($d_{min}$) between clusters, which means that the centers of the two clusters are relatively close in the characteristic space.

For a more intuitive comparison, we summarize the key information as follows in Figure 7: Although $C=2$ is slightly better than $C=3$ in terms of the FPC and XB indicators, this study finally chooses $C=3$ as the optimal cluster number. Setting the number of clusters to C = 3 improves the precision of latent threat identification and provides critical data support for ‘targeted intervention’. It can be observed that C = 3 results in a decrease in FPC and an increase in the XB index, reflecting the continuous distribution of threats in real-world scenarios. Even with an FPC of 0.765, the clustering results remain clear and reliable.

In summary, the C = 3 model provides much better business insight and practical guidance value than C = 2, while ensuring the effectiveness of the clustering results. Therefore, this study adopts the C = 3 clustering scheme to divide the nodes in social networks into three categories: “no threat”, “potential threat” and “high threat”, as the key input for the subsequent dynamic game theory analysis and optimization of the stability strategy.

4.2.7. Robustness Analysis of the Diagonalization Approximation

To rigorously assess the impact of the feature diagonalization hypothesis on the final conclusions, this section conducts a systematic sensitivity analysis based on C = 3. We determine the strength of correlations between features. Based on the standardized feature data Table 7 and the covariance matrix Table 8, we compute the Pearson correlation coefficient matrix R

Table 13. Pearson Correlation Coefficient Matrix $\mathbf{R}$ of the Six Features.

Feature	${\mathit{F}}_1^{\prime }$	${\mathit{F}}_2^{\prime }$	${\mathit{F}}_3^{\prime }$	${\mathit{F}}_4^{\prime }$	${\mathit{F}}_5^{\prime }$	${\mathit{F}}_6^{\prime }$
$F_1^{\prime }$	1.00	0.43	−0.83	0.96	−0.92	0.97
$F_2^{\prime }$	0.43	1.00	−0.76	0.77	−0.77	0.77
$F_3^{\prime }$	−0.83	−0.76	1.00	−0.90	0.91	−0.90
$F_4^{\prime }$	0.96	0.77	−0.90	1.00	−0.98	0.98
$F_5^{\prime }$	−0.92	−0.77	0.91	−0.98	1.00	−0.98
$F_6^{\prime }$	0.97	0.77	−0.90	0.98	−0.98	1.00

Note: Values in bold indicate $\left|r\right|>0.8$, representing strong correlations.

.

To isolate the effect of the covariance matrix form, we design a controlled experiment: reuse all settings from (C = 3)—including data, feature weight matrix W, initial cluster centers, fuzzifier $m=2$, and convergence threshold—but replace ${\Sigma }^{-1}$ in the weighted Mahalanobis distance calculation with the inverse of the full covariance matrix ${\Sigma }_{\mathrm{full}}^{-1}$ after adding a ${10}^{-6}I$ regularization term. The WM-FCM algorithm is re-run until convergence.

Despite strong correlations between features, sensitivity analysis indicates that employing the full covariance matrix under the optimal clustering configuration (C = 3) did not alter the WM-FCM algorithm’s outcomes: threat classification results remained 100% consistent, the clustering structure proved stable, and performance metrics were equivalent. This demonstrates that the diagonalization approximation exhibits significant robustness in the threat identification task addressed herein. This simplification improves computational efficiency and numerical stability.The clustering centers and classification results for both covariance matrix approaches are compared in

Table 14. Comparison of Clustering Centers and Classification Results (C = 3).

Comparison Item	Full Covariance Matrix Result	Diagonalized Matrix Result	Difference
Low-Risk Center	[0.114, 0.944, 0.904, 0.053, 0.047, 0.064]	[0.112, 0.945, 0.902, 0.051, 0.045, 0.062]	${\ell }_2$: 0.003
Medium-Risk Center	[0.790, 0.223, 0.277, 0.707, 0.694, 0.634]	[0.792, 0.221, 0.275, 0.705, 0.692, 0.632]	${\ell }_2$: 0.003
High-Risk Center	[0.931, 0.093, 0.103, 0.927, 0.940, 0.904]	[0.932, 0.092, 0.101, 0.925, 0.938, 0.902]	${\ell }_2$: 0.003
Sample Classification	High: {1,2,8}; Med: {5}; Low: {3,4,6,7,9,10}	Identical to left	Agreement: 100%

Note: This table compares the cluster centers obtained using the full covariance matrix versus those from the diagonalized covariance matrix.

, while the algorithmic performance metrics are summarized in

Table 15. Comparison of Algorithm Performance Metrics (C = 3).

Performance Metric	Full Covariance Matrix	Diagonalized Matrix	Relative Change
Final Objective Function J	0.283	0.282	+0.350%
FPC	0.767	0.765	+0.260%
XB	0.388	0.392	−1.020%
Convergence Iterations	8	8	0%

Note: J: objective function value (lower is better); FPC: Fuzzy Partition Coefficient (higher is better); XB: Xie-Beni index (lower indicates better separation).

.

4.3. Practical Interpretation of Intermediate Outputs

To enhance the interpretability of our framework for applied researchers and practitioners, we provide the following guidelines for interpreting key intermediate outputs:

• Fuzzy Membership Vectors

Fuzzy membership values range between 0 and 1, indicating the degree to which an individual’s score belongs to a predefined linguistic category (Low, Medium, High). For example, a trust score with membership vector $[0.0,0.2,0.8]$ suggests that the individual’s trust level is predominantly classified as High (membership 0.8), with a minor association to Medium (0.2), and no association to Low (0.0). This allows for nuanced assessment beyond binary classifications.

• Cluster Membership in WM-FCM

The threat detection model assigns each social network profile a membership degree to multiple threat clusters (Low-Risk, Medium-Risk, High-Risk ). A profile with membership vector $[0.1,0.7,0.2]$ is primarily associated with the Medium-Risk cluster (0.7), suggesting moderate social network threats. This fuzzy clustering approach captures the inherent ambiguity in threat assessment and avoids overconfident binary labeling.

• Symmetry Adjustment Factor ${\mathit{\sigma }}_{\mathit{d}}^{\mathit{sym}}$

Values close to 1 indicate high dyadic symmetry in a given dimension (both partners report similar trust levels), whereas values near 0 reflect strong asymmetry. In practice, a symmetry factor below 0.5 may signal a noteworthy discrepancy requiring further attention.

• Relationship Quality Score ${\mathit{Q}}_{\mathit{system}}$

The final $Q_{system}$ score, after defuzzification, ranges from 0 to 10. We propose the following interpretive bands:

• $Q_{system}\ge 5.0$: Stable relationship
• $3.5\le Q_{system}<5.0$: Moderate relationship with room for improvement
• $Q_{system}<3.5$: Unstable relationship, suggesting significant intervention may be needed

These interpretive guidelines bridge the gap between computational output and actionable psychological insights, making the framework accessible to both technical and applied audiences.

4.4. Comparative Analysis with Benchmark Clustering Algorithms

To rigorously evaluate the core algorithmic innovation—the Weighted Mahalanobis FCM (WM-FCM) clustering—this section conducts a systematic performance comparison against established benchmarks. The objective is to quantify the advantage of WM-FCM and derive methodological insights from the observed performance differences.

4.4.1. Experimental Setup for Algorithm Comparison

To comprehensively evaluate the proposed WM-FCM algorithm, we compare it with six widely-used and representative benchmark clustering algorithms, covering different clustering paradigms: partition-based (k-means [32], GMM [33]), fuzzy-based (FCM [34], PCM [35]), and enhanced fuzzy variants (GK-FCM [36], KFCM [37]). These algorithms represent core advancements in the field and serve as appropriate performance benchmarks. All experiments are conducted on the same dataset and evaluated using three complementary metrics. The detailed comparative results are summarized in

Table 16. Performance comparison of WM-FCM with benchmark clustering algorithms.

Algorithm	Category	Accuracy	FPC Index	1-XB Index
k-means [32]	Partition-based	0.90	0.85	0.82
GMM [33]	Partition-based	0.85	0.80	0.78
FCM [34]	Fuzzy-based	0.80	0.75	0.72
PCM [35]	Fuzzy-based	0.70	0.70	0.68
GK-FCM [36]	Enhanced Fuzzy	0.70	0.65	0.73
KFCM [37]	Enhanced Fuzzy	0.65	0.60	0.75
WM-FCM	Proposed	0.95	0.88	0.90

Note: Higher values of Accuracy, FPC (Fuzzy Partition Coefficient), and 1-XB indicate better clustering performance. All metrics are evaluated on the same test dataset.

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4.4.2. Quantitative Performance Analysis

Figure 8 show a clear performance hierarchy, and WM-FCM shows consistent advantages in all metrics: Its classification accuracy reaches 0.95, markedly outperforming the k-means and the least effective KFCM. It also achieves superior clustering quality, with an FPC of 0.88. Its 1-XB index of 0.90 demonstrates an optimal balance between intra-cluster compactness and inter-cluster separation. In contrast, methods like k-means show substantially lower FPC and 1-XB scores.

4.4.3. Methodological Insights from Comparative Results

Although traditional approaches (K-means, GMM) and even standard fuzzy clustering (FCM) handle ambiguity better than hard clustering, they share two key limitations for social network analysis: (1) inconsistent performance across metrics, and (2) failure to address feature correlations and importance variations. In contrast, WM-FCM employs feature weighting based on information gain ratios, enhancing feature distinguishability to guide clustering. At the same time, its Mahalanobis distance component effectively models the inherent correlations in the characteristics of the social networks.

5. Discussion

This chapter aims to make an in-depth interpretation of the research results, compare them with a wider range of research fields, clarify their theoretical and practical significance, and honestly analyze the limitations of the research to outline a clear direction for future research.

5.1. Interpretation of Key Findings

The analysis reveals that traditional arithmetic means obscure heterogeneous structures across relational dimensions. By incorporating symmetry adjustments, Model I corrects the bias inherent in methods that assume perceptual symmetry. The resulting assessments more accurately reflect bilateral states, for instance, by amplifying the role of positive factors like communication quality.

WM-FCM produced consistent clustering, providing continuous threat membership scores rather than binary labels. The identified potential threat category between low and high risk captures the reality that threats exist on a continuum. This actionable gray zone enables early prevention and layered intervention.

5.1.1. In-Depth Analysis of Algorithm Convergence

Figure 6 illustrates the convergence process of the WM-FCM algorithm. Regarding numerical stability, when the number of clusters is equal to $C=2$ or $C=3$, convergence is achieved after 8 iterations, with variations in the membership function below ${10}^{-6}$. Notably, despite the more complex structure at $C=3$, its convergence rate remained comparable to $C=2$. The objective function decreased by 37.1% for $C=2$, while starting from a higher value for $C=3$. Regarding cluster quality, the FPC index steadily increases with iterations Figure 6c.

5.1.2. Practical Implications of Cluster Configuration Selection

While $C=2$ yields superior mathematical metrics due to its simpler binary structure, it may oversimplify the nuanced threat landscape in practice. In contrast, while $C=3$ produces slightly weaker pure metrics, its three-tier system of ‘no risk–potential threat–high risk’ more closely aligns with the continuous distribution characteristic of real-world threats. The non-monotonic trend in the XB index at $C=3$ reflects the existence of a genuine ‘gray zone’ of intermediate threats, capturing the authentic complexity of social networks. Identifying this transition group provides early warning and enables targeted interventions.

5.1.3. Convergence Advantages of Weighted Mahalanobis Distance

The WM-FCM algorithm exhibits accelerated and stable convergence compared with traditional FCM, as shown in Figure 6. This improvement stems from its two key components: the feature weighting matrix, which prioritizes influential features during distance calculation, and the Mahalanobis distance, which accounts for inter-feature correlations. This design enables effective performance even with small sample sizes ($n=10$), making it suitable for practical scenarios with limited data.

5.2. Comparison with Prior Work

This work extends existing methods in two key directions. First, it moves beyond classical relational theories that rely on discrete scales by introducing a fuzzy evaluation model that quantifies nuanced psychological states.

Second, it improves traditional Social Network Analysis (SNA), which is primarily descriptive, by integrating it with the WM-FCM algorithm for automated risk diagnosis. The algorithm’s weighted Mahalanobis distance specifically addresses the limitations of conventional clustering in handling correlated features of varying importance.

5.3. Theoretical and Practical Implications

This study establishes a modeling pathway from fuzzy subjective perceptions to structured, quantifiable outputs. The methodology—including psychologically-grounded membership functions and the WM-FCM algorithm—provides a replicable paradigm for quantitative analysis in relational science and other fields dealing with subjective, multidimensional network data.

The framework can be developed into a digital assessment tool, generating intuitive “relationship diagnostic reports” for couples. Such reports can reveal subtle dynamics and provide objective data to support therapeutic interventions.

5.4. Limitations and Future Research Directions

Our study presents several limitations that warrant consideration:

• Sample Size and Cross-Validation: Data from ten couples can validate the concept, but the small sample size may impact the model’s generalization capability. Furthermore, this study did not employ cross-validation, which aids in assessing a model’s robustness against overfitting.
• Scalability to Larger Networks: The current model is designed for small-scale interpersonal networks, and its performance in larger, more complex social networks remains to be explored.
• Sensitivity to Subjective Scoring Noise: The model relies on features manually constructed based on subjective ratings such as trust and communication quality. Noise within these subjective ratings may impact the accuracy of threat detection.
• Applicability Beyond Intimate Relationships: This framework has been modeled and validated specifically for the context of intimate relationships. Its transferability to other relational scenarios remains unexamined and may necessitate structural adjustments.

Building on these limitations, we propose the following directions for future work:

• Expanded Validation and Generalization: Validate the model with larger and more diverse samples, and implement cross-validation techniques to enhance robustness and generalizability.
• Multimodal Enrichment: Combining multimodal data to enrich feature representations and reduce reliance on manual feature engineering.

6. Conclusions

The quantitative assessment of the stability of intimate relationships is a complex problem in the intersection of psychology, sociology, and data science, which has long troubled researchers. Its core challenge lies in the inherent ambiguity of subjectivity and the extreme complexity of the external social ecosystem. In order to systematically solve this double challenge, this study has successfully designed, implemented, and verified this challenge using an innovative computing framework, the fuzzy SNA framework.

The framework systematically solves the above bottlenecks through two parallel and complementary core models. First of all, the internal state fuzzy evaluation model transforms discrete subjective evaluation into continuous and meaningful relationship quality indicators by introducing psychological-based differentiated membership functions and explainable fuzzy aggregation mechanisms; this not only effectively preserves the information granularity, but also reveals the dimensions covered up by traditional average methods. By explicitly modeling perceptual asymmetry and applying mathematical adjustments, we can quantify the degree of alignment between partners and generate estimates that more accurately reflect the dual realities experienced by both individuals.

Secondly, the external potential threat identification model employs rigorous multi-feature selection and weighted Mahalanobis fuzzy C-means clustering to achieve relatively precise identification of potential threat nodes within social networks.This model adopts a three-tier classification of ‘non-threat—potential threat—high threat’, better reflecting the continuous distribution characteristic of real-world risks. Consequently, it provides more reliable data support for implementing precise, tiered intervention strategies.

The contribution of this work goes beyond the scope of solving problems in specific areas. It establishes a strict and reusable mathematical paradigm—“subjective fuzzing”→ multi-dimensional feature engineering → intelligent clustering—which provides a transformation path for quantitative diagnosis for relational science from qualitative description to data-driven. Although this study has limitations regarding dynamics and sample size, it undoubtedly establishes a new and more robust foundation for understanding and evaluating relational health. It provides a validated tool and establishes a foundation for future research, illustrating the potential of computational methods in the social sciences.