Few-Shot Community Detection in Graphs via Strong Triadic Closure and Prompt Learning

Abstract

Community detection is a fundamental task for understanding network structures, crucial for identifying groups of nodes with close connections. However, existing methods generally treat all connections in networks as equally important, overlooking the inherent inequality of connection strengths in social networks, and often require large quantities of labeled data. To address these challenges, we propose a few-shot community detection framework, Strong Triadic Closure Community Detection with Prompt (STC-CDP), which combines the Strong Triadic Closure (STC) principle, Graph Neural Networks, and prompt learning. The STC principle, derived from social network theory, states that if two nodes share strong connections with a third node, they are likely to be connected with each other. By incorporating STC constraints during the pre-training phase, STC-CDP can differentiate between strong and weak connections in networks, thereby more accurately capturing community structures. We design an innovative prompt learning mechanism that enables the model to extract key features from a small number of labeled communities and transfer them to the identification of unlabeled communities. Experiments on multiple real-world datasets demonstrate that STC-CDP significantly outperforms existing state-of-the-art methods under few-shot conditions, achieving higher F1 scores and Jaccard similarity particularly on Facebook, Amazon, and DBLP datasets. Our approach not only improves the precision of community detection but also provides new insights into understanding connection inequality in social networks.

1. Introduction

The community structure within social networks is a pivotal issue in the study of network science, referring to groups of nodes within the network that exhibit a significantly higher density of internal connections compared to connections with external nodes. Community structures are prevalent in real-world networks [1,2,3], and they hold substantial significance in uncovering network topology, predicting node behavior, and comprehending the dissemination of information. The identification of community structures has broad applications across various domains, ranging from market segmentation [4], recommendation systems [5], to fraud detection [6,7] and event organization [8]. Particularly in large-scale social networks, efficient and accurate community detection can aid platforms in optimizing content distribution, enhancing user engagement, and identifying potential malicious groups. With the rapid expansion of online social networks, traditional community detection methods face increasingly severe challenges. The exponential growth of network scale, dynamic changes in user behavior, and the complexity of community structures all demand continuous innovation in community detection techniques [9]. Especially under conditions where only limited labeled data are available, effectively identifying and extending specific types of communities has become a key research challenge, for instance, in security domains, identifying other potential suspicious groups in the network based on just a few known suspicious account clusters, or in marketing, discovering more potential customer groups with similar characteristics based on a small number of known high-value user communities.

Despite continuous advancements in community detection techniques, existing methods still face several key challenges. First, most algorithms treat all connections in networks as equally important, failing to consider the inherent inequality of connection strengths in social networks [10]. In real-world social networks, connection strengths vary significantly—some represent intimate relationships while others merely indicate superficial intersections, and these differences have a decisive impact on community structure formation [11,12]. Second, traditional algorithms typically require large quantities of labeled data for training, which are difficult to obtain in practical applications [13]. Although few-shot learning methods have been developed, they are highly sensitive to the quality of seed nodes and often suffer from insufficient flexibility and high computational overhead [14,15]. Third, existing methods generally lack adequate consideration of triangular structures that are common in social networks, despite their fundamental role in community formation [16]. To address these challenges, this paper introduces the Strong Triadic Closure (STC) principle to handle connection inequality in community detection. STC is an important concept in social network theory, stating that if two nodes share strong connections with a third node, they are likely to be connected themselves (either strongly or weakly) [17]. This principle has deep roots in sociology, with research by Granovetter [18] and Burt [19] demonstrating that triangular structures play a crucial role in social cohesion and information propagation. By combining STC with advanced deep learning techniques, we aim to improve the quality and accuracy of community detection, particularly under few-shot conditions.

This paper proposes a community detection framework based on Strong Triadic Closure Community Detection with Prompt—STC-CDP, which combines Graph Neural Networks, prompt learning, and the STC principle to achieve efficient few-shot community detection. Our main contributions include the following:

1.

Innovative STC Injection Mechanism: We propose a method to organically integrate STC properties into the Graph Neural Network training process, enabling the model to learn to distinguish between strong and weak connections, thereby more accurately characterizing community structures.

2.

Prompt-based Few-Shot Learning Framework: We design a parameter-efficient prompt learning framework that enables the model to extract key features from a small number of labeled communities and apply this knowledge to identify unlabeled similar communities.

3.

End-to-end Community Detection System: We implement a complete pipeline from pre-training and prompt learning to final community prediction, providing a practical solution for real-world applications.

Through experiments on multiple real-world datasets, we demonstrate the superiority of STC-CDP, particularly under few-shot conditions, where it achieves significantly higher F1 scores and Jaccard similarity compared to existing state-of-the-art methods. Our research not only provides a new technical path for community detection but also offers deep insights into understanding connection inequality in social networks.

The remainder of this paper is organized as follows: Section 2 reviews related work; Section 3 introduces problem definition and fundamental concepts; Section 4 elaborates on the STC-CDP method; Section 5 presents experimental settings and results analysis; and Section 6 concludes the paper and discusses future research directions.

2. Related Work

2.1. Community Detection Algorithms

The development of community detection algorithms has evolved from traditional partition-based methods to advanced deep learning approaches. Early methods such as Louvain [20] and Girvan–Newman [21] algorithms primarily relied on modularity metrics to identify communities. With the advancement of deep learning techniques, Graph Neural Network (GNN)-based community detection methods, such as [22,23], have demonstrated significant advantages in community partitioning tasks by learning low-dimensional embedding representations of nodes.

Community detection research can be broadly categorized into unsupervised and supervised methods. Unsupervised methods include optimization-based approaches, such as graph partitioning through optimizing modularity metrics [24], and matrix factorization methods that learn latent community representations by decomposing adjacency matrices [25]. In recent years, frameworks combining graph representation learning and community detection have made significant progress, as seen in [26,27,28,29,30]. These methods have substantially improved community detection accuracy through co-learning community membership and node representations. However, most of these methods lack precise identification capabilities for specific types of communities and typically require large quantities of labeled data.

Semi-supervised community detection has emerged as a recent research direction [31,32,33], aiming to discover similar communities in networks using a small number of labeled communities as training data. Zhang [34] proposed a seed expansion-based method that identifies communities by selecting seed nodes and gradually expanding them. However, this approach is highly sensitive to seed node quality, and inappropriate seed node selection may lead to inaccurate community partitioning. Wu et al. [13,35] improved this issue by introducing subgraph inference, further reducing dependence on seed node quality. Additionally, they incorporated prompt learning into community detection, significantly reducing the need for training data. Nevertheless, these methods fail to adequately consider the differences in connection strengths between nodes, assuming all connections have equal importance in community partitioning, which contradicts the inequality of connection strengths in real-world social networks [10]. This may result in insufficient accuracy in identifying certain communities.

2.2. Triadic Closure Principle and Its Applications in Network Analysis

The Triadic Closure principle is a core concept in social network analysis, describing the phenomenon of “friends of friends are friends.” Research by Bianconi et al. [16] and Granovetter [18] demonstrates that this principle not only helps understand social network structures but also predicts the formation of potential connections in networks. In graph theory, Triadic Closure is used to quantify the clustering coefficient of networks, where a high clustering coefficient typically indicates the presence of tight community structures in the network.

Kleinberg and Easley [17] further distinguished the different impacts of strong and weak ties in Triadic Closure, introducing the concept of Strong Triadic Closure (STC). As a significant work in this field, Sintos and Tsaparas [36] proposed the MINSTC problem, an optimization problem that minimizes the number of weak edges while satisfying STC properties. They proved that MINSTC was equivalent to solving the minimum vertex cover problem in the wedge graph Z(G). Tsourakakis et al. [37] extended that work, arguing that triangles (or other higher-order subgraph structures, i.e., motifs) in graphs were stronger signals of community structure, and thus these motifs could be leveraged to improve clustering effectiveness. Recent research, such as the work by Chakraborty et al. [38], combines graph mining with the Triadic Closure principle, applying STC to dense subgraph discovery. Shang et al. [39] proposed TriHetGCN, an extension of traditional Graph Convolutional Networks (GCNs) that incorporates explicit topological metrics—Triadic Closure and degree heterogeneity—to address the issue of GCNs ignoring node attributes and intrinsic structural relationships between node pairs.

However, research on applying methods that combine graph representation learning with the Triadic Closure principle to targeted community detection tasks remains limited. First, existing graph representation learning methods often lack explicit modeling of community structures during the pre-training phase, making the generated node representations difficult to directly reflect community information. Second, most existing prompt learning methods are designed for tasks such as node classification or link prediction, relying on direct manipulation of node features while failing to adequately consider structural characteristics at the community level. Therefore, there exists a gap between these methods and the actual requirements when applied to community detection tasks.

3. Problem Definition and Preliminaries

3.1. Community Detection

Definition 1

(Community Detection). Given a graph $\mathcal{G}$, with m labeled communities $\dot{\mathcal{C}}=\{{\dot{\mathcal{C}}}^1,{\dot{\mathcal{C}}}^2,\dots ,{\dot{\mathcal{C}}}^m\}$ (where ${\forall }_{i=1}^m{\dot{\mathcal{C}}}^i\subset \mathcal{G}$) as training data, the goal is to find a set of other similar communities $\widehat{\mathcal{C}}$ such that $|\widehat{\mathcal{C}}| \gg |\dot{\mathcal{C}}|$ in $\mathcal{G}$.

This definition describes the few-shot community detection problem, where the objective is to identify other communities in the graph that share similar structural characteristics based on a small number of known community examples. This setting corresponds to real-world scenarios where manually labeling a large number of communities is costly, while automatically discovering communities with similar properties is highly valuable.

3.2. Strong Triadic Closure (STC)

Let $\mathcal{G}\equiv (\mathcal{V},\mathcal{E})$ be an undirected graph representing a social network, where the vertex set $\mathcal{V}$ corresponds to individuals and the edge set $\mathcal{E}$ corresponds to connections (relationships) between these individuals. Our goal is to label the relationships in the social network, classifying them as either strong or weak relationships.

We represent this labeling as a function $\mathsf{\ell }:\mathcal{E}\to \{W,S\}$, which maps each edge $e\in \mathcal{E}$ to a label W (weak relationship) or S (strong relationship). A pair of edges $e_1\equiv \{u,v\}\in \mathcal{E}$ and $e_2\equiv \{v,w\}\in \mathcal{E}$ is called a wedge if $\{u,w\}\notin \mathcal{E}$, denoted as $e_1\wedge e_2$ to represent the wedge between edges $e_1$ and $e_2$, where $u,v,w\in \mathcal{V}$.

Definition 2

(Strong Triadic Closure STC). Given a graph $\mathcal{G}$, if the labeling ℓ in the graph satisfies the Strong Triadic Closure (STC) property, then there does not exist a pair of edges $(u,v)$ and $(v,w)$ such that $\mathsf{\ell }(u,v)\equiv S$ and $\mathsf{\ell }(v,w)\equiv S$, but $(u,w)\notin \mathcal{E}$.

The Strong Triadic Closure (STC) property reveals an important phenomenon in social networks: if a vertex v has strong connections with vertices u and w, i.e., if $\mathsf{\ell }(u,v)\equiv S$ and $\mathsf{\ell }(v,w)\equiv S$, then u and w are more likely to form an edge in $\mathcal{E}$, which can be either a weak or strong connection [17], as shown in Figure 1. This property reflects the transitivity and cohesion of community structures in social networks.

Corollary 1

(Strong Triadic Closure Violation). Given a graph $\mathcal{G}$ and an edge labeling function ℓ, if $\mathsf{\ell }(u,v)\equiv S$, $\mathsf{\ell }(v,w)\equiv S$, and $(u,w)\notin \mathcal{E}$, then the vertex triplet $u,v,w\in \mathcal{V}$ constitutes an STC violation. Let $B(\mathsf{\ell },\mathcal{G})$ denote the total number of violations induced by labeling ℓ on graph $\mathcal{G}$.

This corollary defines the violation of STC constraints, providing a theoretical foundation for subsequent community detection using STC properties. The violation count $B(\mathsf{\ell },\mathcal{G})$ can be used as a metric to evaluate the quality of strong and weak relationship labeling in the graph and can also serve as an optimization objective to minimize STC violations to obtain edge labeling that better conforms to the structural characteristics of social networks.

4. STC-CDP: The Proposed Approach

The STC-CDP method consists of three main steps, edge labeling, pre-training, and fine-tuning, aiming to perform efficient community detection by combining the Strong Triadic Closure (STC) principle with Graph Neural Network (GNN) techniques. The framework consists of three main components: (1) STC-based edge labeling to distinguish strong and weak connections, (2) STC-enhanced contrastive learning pre-training to learn graph representations, and (3) prompt-based fine-tuning for few-shot community detection.

4.1. Edge Labeling Using STC

4.1.1. Graph-Theoretic Modeling of the STC Problem

We first define a wedge as a pair of edges sharing a common vertex, formally represented as a wedge triplet $W\equiv \left\{\right(u,v),(v,w\left)\right\}$, where v is the shared vertex. If $(u,w)\notin \mathcal{E}$, this wedge is called an open wedge (or open triangle). The Strong Triadic Closure (STC) property requires that at least one edge in each open wedge must be labeled as a weak relationship.

To handle the STC problem, we transform the original graph $\mathcal{G}$ into a dual graph ${\mathcal{G}}_W\equiv ({\mathcal{V}}_W,{\mathcal{E}}_W)$, called the wedge graph:

• ${\mathcal{V}}_W\equiv \left\{v_e\right|e\in \mathcal{E}\}$, where each edge e in the original graph is mapped to a vertex $v_e$ in the wedge graph;
• ${\mathcal{E}}_W\equiv \left\{(v_{e_1},v_{e_2})\right|\exists \mathrm{open}\mathrm{wedge}W,e_1,e_2\in W\}$.

Specifically, for each pair of edges $e_1\equiv (u,v)$ and $e_2\equiv (v,w)$ in the original graph $\mathcal{G}\equiv (\mathcal{V},\mathcal{E})$, if they form an open wedge (i.e., $(u,w)\notin \mathcal{E}$), we add an edge $(v_{e_1},v_{e_2})$ to the wedge graph ${\mathcal{G}}_W$. Through this transformation, we can convert the STC problem into finding a minimum edge set in the wedge graph ${\mathcal{G}}_W$ that satisfies the STC property constraints for all open wedges. This is closely related to the minimum vertex cover problem in graph theory.

In a graph $\mathcal{G}$, if the Strong Triadic Closure (STC) property is satisfied, there should be no open triangles that violate this property. Specifically, for any open triangle $\langle (u,v),(v,w)\rangle$, edges $(u,v)$ and $(v,w)$ cannot be simultaneously labeled as strong. This implies that in each open triangle, at least one edge must be labeled as weak to cover the triangle. We assume that the goal of social network construction is to establish strong relationships with others, therefore, our objective is to maximize the number of strong relationships while satisfying the STC property. This is equivalent to finding the minimum edge set to cover all open triangles in the graph and labeling these edges as weak relationships.

Following the approach in [36], we transform the problem into a Minimum Weak Edge Cover Problem, which aims to find the minimum edge set to cover all open triangles in the graph and label these edges as weak relationships. To solve the Minimum Weak Edge Cover Problem, we convert it into a minimum vertex cover problem. Specifically, for a graph $\mathcal{G}\equiv (\mathcal{V},\mathcal{E})$, a vertex set ${\mathcal{V}}_C\subseteq \mathcal{V}$ is a vertex cover of graph $\mathcal{G}$ if for each edge $(u,v)\in \mathcal{E}$, either vertex u or v belongs to the vertex set ${\mathcal{V}}_C$. The goal of the minimum vertex cover problem is to find the vertex set ${\mathcal{V}}_C$ with the minimum number of vertices. By selecting these vertices, we can cover all relevant edges, thereby indirectly covering all open triangles. This method effectively transforms the edge cover problem into a vertex cover problem, allowing us to utilize existing minimum vertex cover algorithms for solution.

4.1.2. STC Solution Based on Minimum Vertex Cover

After constructing the wedge graph ${\mathcal{G}}_W$, the STC labeling problem reduces to the minimum vertex cover (MVC) problem on ${\mathcal{G}}_W$. Specifically, any vertex cover $C\subseteq {\mathcal{V}}_W$ induces a weak-edge set in the original graph $\mathcal{G}$ that satisfies the STC constraint, and conversely, any STC-consistent labeling yields a vertex cover in ${\mathcal{G}}_W$.

Theorem 1

(Minimum Vertex Cover). Given a graph $\mathcal{G}\equiv (\mathcal{V},\mathcal{E})$ and its corresponding wedge graph ${\mathcal{G}}_W\equiv ({\mathcal{V}}_W,{\mathcal{E}}_W)$, there exists a natural correspondence between the minimum vertex cover $C^{*}$ of ${\mathcal{G}}_W$ and the minimum weak edge set in $\mathcal{G}$ that satisfies the STC property.

Proof. 

We establish a bijective correspondence between STC-consistent labelings on $\mathcal{G}$ and vertex covers on the wedge graph ${\mathcal{G}}_W$.

(Necessity). Let $C\subseteq {\mathcal{V}}_W$ be a vertex cover of ${\mathcal{G}}_W$. Define an edge labeling $\mathsf{\ell }:\mathcal{E}\to \{\mathrm{weak},\mathrm{strong}\}$ by

$$\mathsf{\ell }\left(e\right)\equiv \left\{\begin{array}{cc}\mathrm{weak},\hfill & \mathrm{if}{v}_e\in C,\hfill \\ \mathrm{strong},\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(1)

Consider any open wedge in $\mathcal{G}$ formed by edges $e_1\equiv (u,v)$ and $e_2\equiv (v,w)$ with $(u,w)\notin \mathcal{E}$. By the wedge-graph construction, $(v_{e_1},v_{e_2})\in {\mathcal{E}}_W$. Because C is a vertex cover, at least one of $v_{e_1}$ or $v_{e_2}$ lies in C; hence, at least one of $e_1$ or $e_2$ is labeled weak by ℓ. Thus, no open wedge has both incident edges labeled strong, and ℓ satisfies the STC constraint.

(Sufficiency). Conversely, let $\mathsf{\ell }:\mathcal{E}\to \{\mathrm{weak},\mathrm{strong}\}$ be any labeling satisfying the STC constraint. Define $C\equiv \{v_e\in {\mathcal{V}}_W\mid \mathsf{\ell }\left(e\right)\equiv \mathrm{weak}\}$. For any edge $(v_{e_1},v_{e_2})\in {\mathcal{E}}_W$, the corresponding pair $e_1,e_2$ forms an open wedge in $\mathcal{G}$; hence, the STC constraint implies at least one of $e_1$ or $e_2$ is labeled weak. Therefore, at least one of $v_{e_1}$ or $v_{e_2}$ belongs to C, showing that C is a vertex cover of ${\mathcal{G}}_W$.

(Optimality). The correspondence preserves cardinality: the number of weak edges in ℓ equals $\left|C\right|$. Hence, a labeling with the minimum number of weak edges that satisfies STC corresponds exactly to a minimum vertex cover $C^{*}$ on ${\mathcal{G}}_W$.    □

By the established equivalence, taking $C^{*}$ (a minimum vertex cover of ${\mathcal{G}}_W$) and applying the above rule yields an STC-consistent labeling ℓ on $\mathcal{G}$ with the minimum number of weak edges.

Since the minimum vertex cover problem is NP-hard, we employ a greedy algorithm to find an approximate solution. The specific steps are shown in Algorithm 1. Algorithm 1 provides a greedy approximation. In addition, the classical maximal-matching-based greedy method yields a 2-approximation for minimum vertex cover (see Theorem 2).

Theorem 2

(Two-Approximation via Maximal Matching). The classical greedy algorithm that repeatedly selects an uncovered edge and adds both endpoints to the cover (equivalently, computes a maximal matching and returns all matched endpoints) is a 2-approximation for the minimum vertex cover problem.

Proof. 

Let M be any maximal matching of the graph, and let C be the set of all endpoints of edges in M. Then, C is a vertex cover: if an edge e had neither endpoint in C, it could be added to M, contradicting maximality. Moreover, any vertex cover must contain at least one endpoint of every edge of M; hence,

$$|C^{*}|\ge |M|,$$

(2)

where $C^{*}$ is an optimal cover. By construction, $\left|C\right|=2\left|M\right|$; thus, combining with (2) yields

$$\left|C\right|=2\left|M\right|\le 2|C^{*}|.$$

(3)

Therefore, the maximal-matching-based greedy algorithm achieves a 2-approximation.    □

Theorem 3

(Propagation of MVC Approximation Error to STC-CDP). Suppose the MVC solution used to construct the STC labeling has relative error ϵ (with respect to the number of selected vertices). Then, the induced STC labeling error has a bounded downstream impact on community detection.

Proof. 

Let ${\mathsf{\ell }}^{*}$ be the STC labeling induced by an optimal MVC solution and let ℓ be the labeling induced by the approximate solution, both on $\mathcal{G}\equiv (\mathcal{V},\mathcal{E})$. Define the error set ${\mathcal{E}}_{\mathrm{err}}\equiv \{e\in \mathcal{E}:\mathsf{\ell }\left(e\right)\ne {\mathsf{\ell }}^{*}\left(e\right)\}$. By assumption, $|{\mathcal{E}}_{\mathrm{err}}| \le ϵ|\mathcal{E}|$.

Define strong-edge subgraphs ${\mathcal{G}}_s^{*}\equiv (\mathcal{V},{\mathcal{E}}_s^{*})$ and ${\mathcal{G}}_s\equiv (\mathcal{V},{\mathcal{E}}_s)$ with

$$\begin{array}{cc}\hfill {\mathcal{E}}_s^{*}& \equiv \{e\in \mathcal{E}:{\mathsf{\ell }}^{*}\left(e\right)=\mathrm{strong}\},\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill {\mathcal{E}}_s& \equiv \{e\in \mathcal{E}:\mathsf{\ell }(e)=\mathrm{strong}\}.\hfill \end{array}$$

(5)

Then, the symmetric difference is bounded by

$$|{\mathcal{E}}_s^{*}▵{\mathcal{E}}_s|\le |{\mathcal{E}}_{\mathrm{err}}|\le ϵ|\mathcal{E}|.$$

(6)

Consider a GNN encoder with normalized propagation matrix $\tilde{A}\equiv D^{-1/2}A{D}^{-1/2}$, where A is the adjacency matrix (determined by the strong-edge subgraph) and D the degree matrix. Let $h_v$ and $h_v^{*}$ be the node embeddings under ℓ and ${\mathsf{\ell }}^{*}$, respectively. For any node v, we have

$$\parallel h_v-h_v^{*}{\parallel }_2=\parallel \sum _{u\in \mathcal{N}\left(v\right)}{\tilde{A}}_{vu}(f_u-f_u^{*}){\parallel }_2\le \sum _{u\in \mathcal{N}\left(v\right)}|{\tilde{A}}_{vu}| · \parallel f_u-f_u^{*}{\parallel }_2,$$

(7)

where $f_u$ and $f_u^{*}$ are the intermediate representations depending on the edge labels. Since differences are localized to edges in ${\mathcal{E}}_{\mathrm{err}}$, there exists a collection of edge feature perturbations ${\left\{{\varphi }_e\right\}}_{e\in {\mathcal{E}}_{\mathrm{err}}}$ such that $\parallel f_u-f_u^{*}{\parallel }_2 \le {\sum }_{e\in {\mathcal{E}}_{\mathrm{err}}}{\parallel {\varphi }_e\parallel }_2$ with coefficients bounded by the network’s Lipschitz constants. Hence,

$$\parallel h_v-h_v^{*}{\parallel }_2\le C_1 · |{\mathcal{E}}_{\mathrm{err}}|\le C_1ϵ\left|\mathcal{E}\right|,$$

(8)

for a constant $C_1$ depending on the architecture and normalization.

For similarity-based community assignment with $\mathrm{sim}(h_i,h_j)\equiv h_i^{\top }{h}_j$, we obtain

$$|\mathrm{sim}(h_i,h_j)-\mathrm{sim}(h_i^{*},h_j^{*})|\le \parallel h_i-h_i^{*}{\parallel }_2 · \parallel h_j{\parallel }_2+\parallel h_i^{*}{\parallel }_2 · {\parallel h_j-h_j^{*}\parallel }_2.$$

(9)

Thus, similarity perturbations are bounded by $O\left(ϵ\right|\mathcal{E}\left|\right)$, which in turn induces a bounded change in the community matching used to compute the F1 score. Let ${\mathcal{C}}^{*}$ and $\mathcal{C}$ be the ground-truth and predicted community sets, respectively. Then,

$$|\mathrm{F}1\left(\mathcal{C}\right)-\mathrm{F}1\left({\mathcal{C}}^{*}\right)|\le C_2ϵ\left|\mathcal{E}\right|{\displaystyle \frac{\sqrt{d}}{\left|\mathcal{V}\right|}},$$

(10)

where d is the embedding dimension, and $C_2$ is a constant depending on the matching rule and normalization.    □

Algorithm 1 Wedge graph-based STC edge labeling algorithm.

Require:  Graph $\mathcal{G}\equiv (\mathcal{V},\mathcal{E})$ Ensure:  Edge labeling function ℓ satisfying STC property   1: Construct wedge graph ${\mathcal{G}}_W\equiv ({\mathcal{V}}_W,{\mathcal{E}}_W)$:   2:    ${\mathcal{V}}_W\leftarrow \left\{v_e\right|e\in \mathcal{E}\}$   3:    ${\mathcal{E}}_W\leftarrow \varnothing$   4: for all $v\in \mathcal{V}$ and all pairs $u,w\in N\left(v\right)$ do   5:    if $(u,w)\notin \mathcal{E}$ then   6:      ${\mathcal{E}}_W\leftarrow {\mathcal{E}}_W\cup \left\{(v_{(u,v)},v_{(v,w)})\right\}$   7:    end if   8: end for   9: Compute approximate minimum vertex cover C of ${\mathcal{G}}_W$:   10:    $C\leftarrow \varnothing$   11: while  ${\mathcal{E}}_W\ne \varnothing$  do   12:    Select vertex $v_{max}\in {\mathcal{V}}_W$ with maximum degree   13:    $C\leftarrow C\cup \left\{v_{max}\right\}$   14:    Remove $v_{max}$ and all its adjacent edges   15: end while   16: for all $e\in \mathcal{E}$ do   17:    if $v_e\in C$ then   18:        $\mathsf{\ell }\left(e\right)\leftarrow \mathrm{weak}$   19:    else   20:        $\mathsf{\ell }\left(e\right)\leftarrow \mathrm{strong}$   21:    end if   22: end for   23: return labeling function ℓ

Corollary 2.

1. If $ϵ\le 1/\left|\mathcal{E}\right|$, then $|\mathrm{F}1\left(\mathcal{C}\right)-\mathrm{F}1\left({\mathcal{C}}^{*}\right)| \le O\left(\frac{\sqrt{d}}{\left|\mathcal{V}\right|}\right)$, showing a vanishing error as the graph grows.

2. For dense graphs with $\left|\mathcal{E}\right|=O\left(\right|\mathcal{V}{|}^2)$, the bound becomes $O\left(ϵ\right|\mathcal{V}\left|\sqrt{d}\right)$; for sparse graphs with $\left|\mathcal{E}\right|=O\left(\right|\mathcal{V}\left|\right)$, it simplifies to $O\left(ϵ\sqrt{d}\right)$.

4.2. STC-Enhanced Contrastive Learning Pre-Training

After completing the STC edge labeling algorithm described in Section 4.1.2, we input the labeling results as edge attributes into the Graph Neural Network (GNN) for pre-training. The pre-training framework employs a multi-level contrastive learning strategy that fully leverages edge label information to enhance the model’s ability to understand graph structures. The pre-training phase is designed with three key objectives: node-level contrastive loss, subgraph-level contrastive loss, and edge prediction loss. These objectives work together to enable the model to learn structural features in graphs and identify potential community patterns.

Specifically, the Graph Neural Network encoder ${\mathrm{GNN}}_{\Theta }(·)$ receives the STC-labeled graph as input, where edge labels (strong/weak) are converted into edge attributes. By learning this special structural information, the model can more accurately capture community structures in networks. The goal of the pre-training phase is to make the GNN model learn core features of graph structures, particularly considering the impact of edges with different strengths on community structures. To this end, we design a contrastive learning framework specifically for STC characteristics, enabling the model to understand the different roles of weak and strong edges in community formation.

4.2.1. STC-Based Representation Learning Framework

Given a graph $\mathcal{G}\equiv (\mathcal{V},\mathcal{E})$ with edges $\mathcal{E}$ labeled by the STC algorithm, where the edge labeling is $\mathsf{\ell }:\mathcal{E}\to \{\mathrm{weak},\mathrm{strong}\}$, our goal is to learn a graph encoder ${\mathrm{GNN}}_{\Theta }(·)$ that can capture community structures.

The representation learning framework consists of two key components:

• Node-level Contrastive Learning: Learns the consistency between node representations and their corresponding community representations.
• Community-level Contrastive Learning: Learns the consistency between the original community structure and the perturbed community structure, where the perturbation retains strong edges and preferentially removes weak edges.

Formally, for a node $v\in \mathcal{V}$, its representation is defined as:

$$z\left(v\right)\equiv {\mathrm{GNN}}_{\Theta }(X,\mathcal{E})\left[v\right]$$

(11)

where $X\in {\mathbb{R}}^{\left|\mathcal{V}\right|\times d}$ is the node feature matrix, $\mathcal{E}\in {\mathbb{R}}^{2\times \left|\mathcal{E}\right|}$ is the edge index matrix, $\Theta$ denotes the parameters of the GNN, and d is the dimension of node features. ${\mathrm{GNN}}_{\Theta }(X,\mathcal{E})\left[v\right]$ denotes the embedding of node v produced by the GNN, with dimension ${\mathbb{R}}^h$, where h is the hidden-layer dimension.

For a subgraph ${\mathcal{G}}_S\equiv ({\mathcal{V}}_S,{\mathcal{E}}_S)$, its representation is defined as:

$$z\left({\mathcal{G}}_S\right)\equiv \mathrm{READOUT}\left(\left\{z\left(v\right)\right|v\in {\mathcal{V}}_S\}\right)$$

(12)

where ${\mathcal{V}}_S\subseteq \mathcal{V}$ is the set of nodes in the subgraph, ${\mathcal{E}}_S\subseteq \mathcal{E}$ is the set of edges in the subgraph, and $\mathrm{READOUT}:{\mathbb{R}}^{|{\mathcal{V}}_S|\times h}\to {\mathbb{R}}^h$ is a pooling function that aggregates the set of node representations $\left\{z\left(v\right)\right|v\in {\mathcal{V}}_S\}$ into a subgraph representation vector. Common pooling functions include mean pooling, max pooling, or attention pooling.

4.2.2. STC-Guided Contrastive Learning

Our contrastive learning framework leverages the STC property and learns representations through the following two key loss functions:

Node-Community Contrastive Loss: Encourages the alignment between node representations and their corresponding community representations:

$${\mathcal{L}}_{\mathrm{node}}(\Theta )\equiv -\sum _{v\in {\mathcal{V}}_B}log{\displaystyle \frac{exp(z\left(v\right)·z\left({\mathcal{G}}_v\right)/\tau )}{{\sum }_{{\mathcal{G}}^{\prime }\in \mathcal{B}}exp(z\left(v\right)·z\left({\mathcal{G}}^{\prime }\right)/\tau )}}$$

(13)

where ${\mathcal{V}}_B$ is the set of nodes in the batch, ${\mathcal{G}}_v$ is the subgraph containing node v, $\mathcal{B}$ is the set of subgraphs in the batch, and $\tau$ is the temperature parameter.

STC-Guided Community Contrastive Loss: Encourages the alignment between the original community representation and the perturbed community representation that retains strong edges:

$${\mathcal{L}}_{\mathrm{subg}}(\Theta )\equiv -\sum _{{\mathcal{G}}_S\in \mathcal{B}}log{\displaystyle \frac{exp(z\left({\mathcal{G}}_S\right)·z\left({\tilde{\mathcal{G}}}_S\right)/\tau )}{{\sum }_{{\mathcal{G}}^{\prime }\in \mathcal{B}}exp(z\left({\mathcal{G}}_S\right)·z\left({\mathcal{G}}^{\prime }\right)/\tau )}}$$

(14)

where ${\tilde{\mathcal{G}}}_S$ is the perturbed version of ${\mathcal{G}}_S$, and the perturbation strategy is based on STC labeling, retaining edges with higher strength.

Formal Definition of the Perturbation Strategy: Given a subgraph ${\mathcal{G}}_S\equiv ({\mathcal{V}}_S,{\mathcal{E}}_S)$ and its STC edge labeling, the perturbation operation $\mathcal{P}$ is defined as:

$$\mathcal{P}\left({\mathcal{G}}_S\right)\equiv ({\mathcal{V}}_S,{\mathcal{E}}_S^{\prime })$$

(15)

where ${\mathcal{E}}_S^{\prime }\subseteq {\mathcal{E}}_S$, and 

$$P(e\in {\mathcal{E}}_S^{\prime }|e\in {\mathcal{E}}_S)\equiv \left\{\begin{array}{cc}{p}_{\mathrm{strong}},\hfill & \mathrm{if}\mathsf{\ell }\left(e\right)\equiv \mathrm{strong}\hfill \\ p_{\mathrm{weak}},\hfill & \mathrm{if}\mathsf{\ell }\left(e\right)\equiv \mathrm{weak}\hfill \end{array}\right.$$

(16)

Typically, $p_{\mathrm{strong}}>p_{\mathrm{weak}}$ is set to ensure that strong edges are preferentially retained, which is consistent with the STC assumption that strong edges are more important for community structure.

4.2.3. Edge Prediction Auxiliary Task

To further leverage the edge labeling information provided by STC, we introduce edge prediction as an auxiliary task. This task requires the model to predict the type of each edge (no edge, weak edge, or strong edge), which is formalized as:

$${\mathcal{L}}_{\mathrm{edge}}(\Theta )\equiv -\sum _{(u,v)\in {\mathcal{E}}_B}log{P}_{\Theta }(\mathsf{\ell }(u,v)|z\left(u\right),z\left(v\right))$$

(17)

where ${\mathcal{E}}_B$ is the set of edges in the batch, $P_{\Theta }$ denotes the probability distribution for predicting the edge label based on the node representations, and $\mathsf{\ell }(u,v)$ is the ground-truth label of edge $(u,v)$ (0 for weak edge, 1 for strong edge), while $z\left(u\right)$ and $z\left(v\right)$ are the representation vectors of nodes u and v, respectively.

The final training objective is a weighted combination of these losses:

$$\mathcal{L}(\Theta )\equiv {\lambda }_{\mathrm{node}}{\mathcal{L}}_{\mathrm{node}}(\Theta )+{\lambda }_{\mathrm{subg}}{\mathcal{L}}_{\mathrm{subg}}(\Theta )+{\lambda }_{\mathrm{edge}}{\mathcal{L}}_{\mathrm{edge}}(\Theta )$$

(18)

where ${\lambda }_{\mathrm{node}}$, ${\lambda }_{\mathrm{subg}}$, and ${\lambda }_{\mathrm{edge}}$ are hyperparameters that balance the contributions of each loss term.

In this way, the learned representations not only capture the relationships between nodes and their communities but also encode the edge strength information provided by STC, thereby enabling a better understanding of community structures.

The detailed workflow of the pre-training process is summarized in Algorithm 2.

Algorithm 2 STC-based graph pre-training algorithm.

Require:  Graph $\mathcal{G}\equiv (\mathcal{V},\mathcal{E},\mathcal{A})$ with STC edge labels, where $\mathcal{A}$ denotes edge attributes Ensure:  Pre-trained GNN model   1: Initialize GNN parameters $\Theta$   2: for epoch = 1 to epochs do   3:    Randomly sample a batch of nodes $\mathcal{B}\subset \mathcal{V}$   4:    for each node $v\in \mathcal{B}$ do   5:      Extract the k-hop subgraph ${\mathcal{N}}_v$   6:    end for   7:    for each subgraph ${\mathcal{N}}_v$ do   8:      Create a perturbed version ${\tilde{\mathcal{N}}}_v$   9:    end for   10:  Use the GNN to process subgraphs and obtain node embeddings ${\mathbf{z}}_i$ and summary vectors ${\mathbf{s}}_i$   11:    Compute ${\mathcal{L}}_{\mathrm{node}}$, ${\mathcal{L}}_{\mathrm{subg}}$, and ${\mathcal{L}}_{\mathrm{edge}}$   12:    Calculate the total loss ${\mathcal{L}}_{\mathrm{total}}\equiv \alpha {\mathcal{L}}_{\mathrm{node}}+\beta {\mathcal{L}}_{\mathrm{subg}}+\gamma {\mathcal{L}}_{\mathrm{edge}}$   13:    Update parameters $\Theta$ to minimize ${\mathcal{L}}_{\mathrm{total}}$   14: end for   15: return the trained GNN model

4.3. Prompt Learning and Knowledge Transfer

After the pre-training phase, the GNN encoder has acquired an understanding of the underlying community structures in the network. To apply this knowledge to downstream tasks, we adopt a prompt learning framework, using a small number of labeled samples to guide the model in community discovery.

4.3.1. Prompt Function Design

We design a simple yet efficient prompt function ${\mathrm{PT}}_{\Phi }(·)$, which takes node embeddings as input and predicts, via a multi-layer perceptron (MLP), whether nodes belong to the same community:

$${\widehat{\mathcal{C}}}_v\equiv {\mathrm{PT}}_{\Phi }\left({\mathcal{N}}_v\right)$$

(19)

where ${\widehat{\mathcal{C}}}_v$ denotes the candidate community centered at node v, and ${\mathcal{N}}_v$ represents its k-hop neighborhood (K-EGO network).

The prompt function is implemented by comparing the embedding relationships between the central node and each node in its K-EGO network, performing a binary classification prediction:

$${\widehat{\mathcal{C}}}_v\equiv \{u\in {\mathcal{N}}_v\mid \sigma \left({\mathrm{PT}}_{\Phi }(z\left(u\right),z\left(v\right))\right)\ge \tau \}$$

(20)

where $z\left(u\right)$ and $z\left(v\right)$ are node embeddings provided by the pre-trained GNN, ${\mathrm{PT}}_{\Phi }$ is the parameterized prompt function, $\sigma$ is the sigmoid function that converts the output to a probability between 0 and 1, and $\tau$ is the threshold parameter (default value is 0.2).

4.3.2. Edge Weight-Aware K-EGO Network Construction

During the prompt learning phase, we introduce an edge weight-based K-EGO network construction strategy. Given a graph $\mathcal{G}\equiv (\mathcal{V},\mathcal{E},\mathcal{A})$, where $\mathcal{A}$ is the set of edge attributes, each edge $e_{i,j}\in \mathcal{E}$ is assigned a weight according to its attribute $a_{i,j}\in \mathcal{A}$:

$$w\left(e_{i,j}\right)\equiv \left\{\begin{array}{cc}{w}_{\mathrm{strong}},\hfill & \mathrm{if}{a}_{i,j}\equiv 1(\mathrm{strong}\mathrm{edge})\hfill \\ w_{\mathrm{weak}},\hfill & \mathrm{if}{a}_{i,j}\equiv 0(\mathrm{weak}\mathrm{edge})\hfill \end{array}\right.$$

(21)

where $w_{\mathrm{strong}}>w_{\mathrm{weak}}$ (typically, $w_{\mathrm{strong}}\equiv 5.0$, $w_{\mathrm{weak}}\equiv 1.0$).

When constructing the K-EGO network for node v, the probability of selecting an edge is proportional to its weight:

$$P(e_{i,j}\in {\mathcal{N}}_v)\propto w\left(e_{i,j}\right)$$

(22)

In this way, strong edges are more likely to be retained, thereby better preserving community structure information in the K-EGO network.

4.3.3. Training Strategy with Positive–Negative Sample Balancing

During the training of the prompt function, the selection of positive and negative samples is crucial for model performance. Given a node v in community ${\mathcal{C}}_i$, the nodes in its K-EGO network ${\mathcal{N}}_v$ can be divided into two categories:

• Positive samples: ${\mathcal{P}}_v\equiv \{u\in {\mathcal{N}}_v|u\in {\mathcal{C}}_i\}$;
• Negative samples: ${\mathcal{N}}_v\setminus {\mathcal{P}}_v$.

Since negative samples usually far outnumber positive samples, we adopt a weighted sampling strategy to balance the ratio of positive and negative samples. For each negative sample u, its probability of being selected is:

$$P(u\in {\mathcal{N}}_v^{\mathrm{sampled}})\propto {\displaystyle \frac{1}{w\left(e_{v,u}\right)}}$$

(23)

This means that negative samples connected by weak edges are more likely to be selected, while those connected by strong edges are less likely to be chosen. The intuition behind this strategy is that nodes connected by strong edges are more likely to belong to the same community and thus are less reliable as negative samples, while nodes connected by weak edges are more likely to belong to different communities and thus are more reliable as negative samples.

Finally, the prompt function is trained by optimizing the following loss function:

$$\begin{array}{cc}\hfill {\mathcal{L}}_{\mathrm{PT}}(\Phi )& \equiv \sum _{{\mathcal{C}}_i\in {\mathcal{C}}_{\mathrm{train}}}\sum _{v\in {\mathcal{C}}_i}\left(\sum _{u\in {\mathcal{P}}_v}{\mathcal{L}}_{\mathrm{BCE}}(\sigma \left({\mathrm{PT}}_{\Phi }(z\left(u\right),z\left(v\right))\right),1)\right.\hfill \\ \hfill & \left.+\sum _{u\in {\mathcal{N}}_v^{\mathrm{sampled}}}{\mathcal{L}}_{\mathrm{BCE}}(\sigma \left({\mathrm{PT}}_{\Phi }(z\left(u\right),z\left(v\right))\right),0)\right)\hfill \end{array}$$

(24)

4.3.4. Community Prediction Process

Based on the pre-trained GNN and prompt function, we achieve large-scale community discovery from a small number of labeled communities through knowledge transfer. Specifically, for each node $v\in \mathcal{V}$, we generate a candidate community ${\widehat{\mathcal{C}}}_v\equiv \{u\in {\mathcal{N}}_v^w|\sigma \left({\mathrm{PT}}_{\Phi }(z\left(u\right),z\left(v\right))\right)\ge \tau \}$ based on its weighted K-EGO network ${\mathcal{N}}_v^w$. By computing the community embedding $z\left({\widehat{\mathcal{C}}}_v\right)\equiv \mathrm{READOUT}\left(\left\{z\left(u\right)\right|u\in {\widehat{\mathcal{C}}}_v\}\right)$ and $z\left({\mathcal{C}}_i\right)\equiv \mathrm{READOUT}\left(\left\{z\left(u\right)\right|u\in {\mathcal{C}}_i\}\right)$, and using the Euclidean distance $d({\widehat{\mathcal{C}}}_v,{\mathcal{C}}_i)\equiv \left|\right|z\left({\widehat{\mathcal{C}}}_v\right)-z\left({\mathcal{C}}_i\right){\left|\right|}_2$ to evaluate similarity, we select the k most similar candidate communities for each training community. This enables knowledge transfer from training communities to target communities, ultimately yielding the predicted community set ${\mathcal{C}}_{\mathrm{pred}}\equiv {\bigcup }_{i=1}^{|{\mathcal{C}}_{\mathrm{train}}|}{\mathcal{S}}_i$.

4.4. Complexity Analysis of STC

The traditional STC algorithm exhibits significant computational inefficiency in wedge detection and minimum vertex cover computation, rendering it unsuitable for practical applications. We optimized these components to enhance computational efficiency. This section analyzes the computational complexity of our improved methods for wedge structure detection and minimum vertex cover calculation.

4.4.1. Complexity of Wedge Detection

The basic triplet search method identifies all potential triads through three nested loops, considering triplets $(u,v,w)$, where u and w are not directly connected but both are connected to v. This method examines each node pair and their neighbors to identify potential wedge structures. The time complexity is primarily determined by the three nested loops: the outer loop iterates over all nodes, the middle loop iterates over each node’s neighbors, and the inner loop iterates over the neighbors of the neighbors. Thus, the overall time complexity can be expressed as:

$$T_1=\sum _{u\in \mathcal{V}}\sum _{v\in \mathcal{N}\left(u\right)}\sum _{w\in \mathcal{N}\left(v\right)}1=\sum _{u\in \mathcal{V}}\sum _{v\in \mathcal{N}\left(u\right)}{d}_v=\sum _{v\in \mathcal{V}}{d}_v^2$$

(25)

where $\mathcal{V}$ is the set of nodes, $\mathcal{N}\left(u\right)$ is the set of neighbors of node u, and $d_v$ is the degree of node v. In the worst case, if every node is connected to every other node (a complete graph), then $d_v=n-1$ for all v, simplifying the time complexity to $O\left(n^3\right)$.

We propose the common neighbor search method, which identifies wedge structures by calculating the common neighbors of node pairs. The core idea is to examine these neighbors only if two nodes share a common neighbor. The time complexity of this method is primarily determined by two loops and set intersection operations. The outer loop iterates over all nodes, and the inner loop iterates over each node’s neighbors and calculates other nodes with common neighbors. Thus, the overall time complexity can be expressed as:

$$T_2=\sum _{u\in \mathcal{V}}\sum _{v\in \mathcal{N}\left(u\right)}min(d_u,d_v)=\sum _{(u,v)\in \mathcal{E}}min(d_u,d_v)$$

(26)

where $\mathcal{E}$ is the set of edges. This method significantly reduces computational overhead by avoiding unnecessary nested loops through efficient set intersection operations.

The complexity ratio of the two methods is:

$$R={\displaystyle \frac{T_1}{T_2}}={\displaystyle \frac{{\sum }_{v\in \mathcal{V}}{d}_v^2}{{\sum }_{(u,v)\in \mathcal{E}}min(d_u,d_v)}}$$

(27)

To quantify the performance improvement, we consider the characteristics of different graph structures. For dense graphs, we can estimate the number of edges $m\approx {\displaystyle \frac{n·d_{\mathrm{avg}}}{2}}$, and when the degree distribution is uniform, $min(d_u,d_v)\approx d_{\mathrm{avg}}$, thus:

$$R\approx {\displaystyle \frac{n·d_{\mathrm{avg}}^2}{m·d_{\mathrm{avg}}}}={\displaystyle \frac{n·d_{\mathrm{avg}}}{m}}=2$$

(28)

For sparse graphs with average degree $d_{\mathrm{avg}}\ll n$, we have:

$$R\approx {\displaystyle \frac{n·\mathbb{E}\left[d_v^2\right]}{m·\mathbb{E}[min(d_u,d_v)]}}$$

(29)

where $\mathbb{E}$ represents expectation, $\mathbb{E}\left[d_v^2\right]\approx d_{\mathrm{avg}}^2+\mathrm{Var}\left(d_v\right)$, and $\mathbb{E}[min(d_u,d_v)]\approx {\displaystyle \frac{d_{\mathrm{avg}}}{2}}$.

In social networks or other networks with power-law degree distributions, most nodes have degrees close to the average degree, while only a few nodes have degrees significantly higher than the average. These extreme values contribute substantially to the variance but represent a small proportion of the entire network. Therefore, for simplification, we can assume that the degree variance $\mathrm{Var}\left(d_v\right)$ is relatively small compared to $d_{\mathrm{avg}}^2$ and can be neglected. Consequently, the performance improvement ratio R can be simplified as:

$$R\approx {\displaystyle \frac{n·d_{\mathrm{avg}}^2}{m·\frac{d_{\mathrm{avg}}}{2}}}={\displaystyle \frac{2·n·d_{\mathrm{avg}}}{m}}$$

(30)

Since in sparse graphs, the number of edges m is much less than ${\displaystyle \frac{n(n-1)}{2}}$, this ratio is typically greater than two, depending on the sparsity of the graph.

4.4.2. Complexity of Minimum Vertex Cover

After constructing the wedge graph, we need to solve the minimum vertex cover problem on the wedge graph. Since the size of the wedge graph may be substantially larger than the original graph, traditional vertex cover algorithms face efficiency bottlenecks when dealing with large-scale wedge graphs. To address this challenge, we designed three vertex cover algorithms with different complexity characteristics to accommodate wedge graphs of varying sizes. Given the wedge graph $\mathcal{W}=({\mathcal{V}}_W,{\mathcal{E}}_W)$, where $|{\mathcal{V}}_W|=m_w$ (number of nodes in the wedge graph) and $|{\mathcal{E}}_W|=e_w$ (number of edges in the wedge graph), the minimum vertex cover problem requires finding the smallest node set $\mathcal{C}\subseteq {\mathcal{V}}_W$ such that every edge $e\in {\mathcal{E}}_W$ has at least one endpoint in $\mathcal{C}$.

• Standard Greedy Algorithm: Uses a max-heap to maintain node priorities, iteratively adding the node with the highest degree to the cover set. The time complexity of this algorithm is $O(e_{w}log{m}_w)$, where the heap operation overhead is $O(log{m}_w)$, executed $O\left(e_w\right)$ times.
• Simplified Greedy Algorithm:To avoid the overhead of heap maintenance, this algorithm directly iterates over the nodes to find the one with the highest degree. The time complexity of this algorithm is $O(e_w·m_w)$, where each search for the node with the highest degree requires $O\left(m_w\right)$ time, executed $O\left(e_w\right)$ times.
• Fast Batch Algorithm: Employs a batch processing strategy, selecting the top $\lceil \alpha m_w\rceil$ nodes with the highest degrees at a time (where $\alpha =0.1$), reducing the number of iterations to enhance efficiency. The time complexity of this algorithm is $O(e_{w}log{m}_w)$, where each batch processing requires $O(m_{w}log{m}_w)$ time for sorting, with approximately ${log}_{1/\alpha }{m}_w={log}_{10}{m}_w$ iterations.

The adaptive algorithm selection strategy based on the size of the wedge graph is as follows:

$$\mathrm{Algorithm}=\left\{\begin{array}{cc}\mathrm{Fast}\left(\mathcal{W}\right)\hfill & \mathrm{if}{m}_w>\mathrm{50,000}\hfill \\ \mathrm{Simple}\left(\mathcal{W}\right)\hfill & \mathrm{if}\mathrm{10,000}<m_w\le \mathrm{50,000}\hfill \\ \mathrm{Standard}\left(\mathcal{W}\right)\hfill & \mathrm{if}{m}_w\le \mathrm{10,000}\hfill \end{array}\right.$$

(31)

This tiered strategy is theoretically justified by the performance characteristics of different algorithms at various scales. For small-scale wedge graphs ($m_w\le \mathrm{10,000}$), the standard algorithm’s precision and heap operation efficiency make it the optimal choice. For medium-scale wedge graphs ($\mathrm{10,000}<m_w\le \mathrm{50,000}$), the simplified algorithm avoids the overhead of heap maintenance, achieving a good balance between implementation complexity and performance. For large-scale wedge graphs ($m_w>\mathrm{50,000}$), the fast batch algorithm significantly improves processing efficiency by reducing the number of iterations.

The performance improvement of the fast batch algorithm over the simplified algorithm can be analyzed as follows, given the number of wedge graph edges $e_w$ and nodes $m_w$:

$$R={\displaystyle \frac{T_{\mathrm{Simple}}}{T_{\mathrm{Fast}}}}={\displaystyle \frac{e_w·m_w}{e_{w}log{m}_w}}={\displaystyle \frac{m_w}{log{m}_w}}$$

(32)

For typical large-scale wedge graphs, this improvement is quite significant. By adopting this tiered optimization strategy, the STC algorithm can achieve good performance on wedge graphs of varying sizes, providing a feasible solution for community detection in large-scale social networks.

5. Experiments

In this section, we present the results of a comprehensive evaluation of the STC-CDP method to verify the effectiveness of combining the Strong Triadic Closure (STC) principle with prompt learning for few-shot community detection. All experiments were conducted on an NVIDIA 3090 GPU, implemented using PyTorch 1.13.1 and PyTorch-Geometric 2.6.1 frameworks. We adopted widely recognized community detection evaluation metrics and report the average results and standard deviations over multiple runs to ensure the reliability and stability of the experimental results. The code is published at https://github.com/zhouyeqin/STC-CDP.git (accessed on 8 September 2025).

Our experimental design aimed to answer the following key research questions:

RQ1 (Overall Performance): How does STC-CDP perform on few-shot community detection tasks compared to existing state-of-the-art methods?

RQ2 (Ablation Study): What are the specific contributions of the STC module and the prompt learning mechanism to the performance of STC-CDP?

RQ3 (Parameter Sensitivity): How do key hyperparameters (such as STC weight and the number of labeled communities) affect the performance of STC-CDP?

RQ4 (Computational Efficiency): How does the computational efficiency and resource consumption of STC-CDP compare to baseline methods?

5.1. Experimental Setup

5.1.1. Datasets

We used five widely adopted real-world social network datasets, each containing overlapping communities of different scales and characteristics: Facebook, Amazon, DBLP, Livejournal, and Twitter, accessed on 1 July 2025 (http://snap.stanford.edu/data/).

Table 1. Dataset statistics.

Datasets	Nodes	Edges	#$\mathcal{C}$	$|\overline{\mathcal{C}}|$
Amazon	13,178	33,767	4517	9.3
DBLP	114,095	466,761	4559	8.4
Twitter	87,760	1,293,985	2838	10.9
Facebook	3622	72,964	130	15.6
Livejournal	69,860	911,179	1000	13.0

presents the basic statistics of these datasets.

5.1.2. Baseline Methods

We compared STC-CDP with the following representative methods:

• SEAL [34]: A method that learns heuristic rules for target communities based on generative adversarial networks.
• CLARE [35]: Proposes a subgraph-based inference framework, including a locator and a rewriter.
• ProCom [13]: A few-shot community detection method that adopts a prompt learning strategy.

5.1.3. Evaluation Metrics

We used bidirectional matching F1 score and Jaccard similarity as the main evaluation metrics, which are widely recognized standard measures in the field of community detection. Given M ground-truth communities ${\dot{\mathcal{C}}}^{\left(i\right)}$ and N predicted communities ${\widehat{\mathcal{C}}}^{\left(j\right)}$, the score was calculated as follows:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1}{N}}\sum _j\underset{i}{max}\delta \left({\widehat{\mathcal{C}}}^{\left(j\right)},{\dot{\mathcal{C}}}^{\left(i\right)}\right)+{\displaystyle \frac{1}{M}}\sum _i\underset{j}{max}\delta \left({\widehat{\mathcal{C}}}^{\left(j\right)},{\dot{\mathcal{C}}}^{\left(i\right)}\right)\right)\hfill \end{array}$$

(33)

where $\delta$ can be either the F1 function or the Jaccard function.

5.1.4. Implementation Details

The hyperparameter settings for STC-CDP are shown in

Table 2. Hyperparameters in STC-CDP.

Component	Hyperparameter	Value
Encoding	Batch size	256
	Number of epochs	30
	Learning rate	1 $\times {10}^{-3}$
	Implementation of ${\mathrm{GNN}}_{\Theta }(·)$	2-layer GCN
	K-EGO subgraph	2
	Embedding dimension	128
	Temperature $\tau$	0.1
	Ratio $\rho$ for corruption	0.85
	Loss weight $\lambda$	1
Sampling	Batch size	32
	Number of epochs	100
	Embedding dimension	64
	MLP layers	3
	LGPNs layers	3
	Learning rate	1 $\times {10}^{-2}$
	Discount factor $\gamma$	1
Fine-tuning	Implementation of ${\mathrm{PF}}_{\Phi }(·)$	2-layer MLP
	Number of epochs	30
	Learning rate	1 $\times {10}^{-3}$
	K-EGO subgraph	3
	Number of prompts m	20
	Threshold value $\alpha$	0.2

.

5.2. Experimental Results and Analysis

5.2.1. Overall Performance Comparison (RQ1)

Table 3. Performance comparison of different methods on five datasets.The best results are highlighted in bold.

Method	Facebook		Amazon		DBLP		LiveJournal		Twitter
	F1	Jaccard	F1	Jaccard	F1	Jaccard	F1	Jaccard	F1	Jaccard
SEAL	31.02	20.33	84.43	75.95	39.47	31.75	38.30	30.27	22.09	14.32
CLARE	26.71	18.36	79.19	73.16	45.80	36.18	49.57	40.61	17.10	10.76
ProCom	37.22	26.87	84.44	75.96	50.87	39.56	53.93	44.45	29.38	19.6
STC-CDP (Ours)	39.51	29.15	85.05	76.54	57.35	46.33	55.01	44.66	31.87	21.45

presents the overall performance comparison of STC-CDP and baseline methods across five datasets. The results shown are the averages of ten runs under identical hardware conditions for each model, facilitating a comprehensive comparison. It is evident that STC-CDP achieved the best performance across all datasets. This demonstrates the effectiveness of integrating the STC principle with prompt learning to capture community structures. The results clearly show the consistent superiority of our proposed method across different evaluation metrics and datasets.

The performance of STC-CDP is notably enhanced on the DBLP dataset, as indicated by its superior F1 scores and Jaccard Similarity in Table 3. This improvement is attributed to DBLP’s lower edge density, as shown in Figure 2, which makes the addition of edge attributes by STC-CDP more impactful. In contrast, the moderate improvements observed on denser datasets like Facebook and Twitter are likely due to their higher edge densities, which diminish the relative benefit of added edge attributes. The edge density, calculated as the ratio of actual to maximum possible edges, thus plays a pivotal role in determining the effectiveness of STC-CDP. This analysis underscores STC-CDP’s particular efficacy in sparse graph environments.

5.2.2. Ablation Study (RQ2)

To verify the contribution of each component in STC-CDP, we conducted a detailed ablation study, and the results are shown in

Table 4. Ablation study results of STC-CDP.The best results are highlighted in bold.

Variant	Facebook		Amazon		DBLP		Livejournal		Twitter
	F1	Jaccard	F1	Jaccard	F1	Jaccard	F1	Jaccard	F1	Jaccard
Basic GNN	33.8	24.1	83.3	74.8	45.7	35.3	41.5	33.6	27.0	18.0
GNN+STC	34.3	24.7	83.5	74.9	47.3	36.8	47.0	38.6	29.5	19.7
GNN + Prompt Learning	38.8	28.2	84.3	75.9	51.4	40.2	54.0	44.6	31.1	20.8
STC-CDP (Full Model)	39.5	29.1	85.1	76.5	57.4	46.3	55.0	44.7	31.9	21.5

. Research indicates that both the STC principle and the prompt learning mechanism significantly enhance model performance. Specifically, adding only the STC principle increased the average F1 score by 1.7 percentage points across the five datasets. The addition of the prompt learning mechanism yielded an even greater performance enhancement. The complete STC-CDP model, which integrates the benefits of both mechanisms, achieved the best performance across all datasets, particularly achieving a 10-percentage-point improvement over the base GNN on the DBLP and Livejournal dataset. These findings underscore a synergistic effect between the STC principle and prompt learning: STC strengthens the model’s comprehension of connection strength, while prompt learning enhances the model’s capability to learn from a limited number of samples. Together, they complement each other and collectively enhance the model’s performance in community detection tasks.

5.2.3. Parameter Sensitivity Analysis (RQ3)

To gain deeper insight into the robustness and performance characteristics of the STC-CDP model, we conducted a sensitivity analysis of key hyperparameters, focusing on the STC loss weight and the number of labeled communities. Using the Facebook dataset as a benchmark, we systematically tuned these parameters to investigate their impact on model performance. The results are shown in Figure 3 and Figure 4. The F1 score initially increases with higher weight values, reaching peak performance at 0.3, then gradually decreases when the weight exceeds this optimal value. The model achieves optimal performance with two prompts, with diminishing returns when using more prompts, demonstrating the efficiency of few-shot learning.

Impact of Edge Prediction Loss Weight: As shown in Figure 3, the edge prediction loss weight has a significant impact on model performance. As the weight increases from 0 to 0.3, the model performance exhibits a clear upward trend, which fully validates the effectiveness of the STC principle in community detection tasks. However, when the weight exceeds 0.3, model performance begins to decline. This phenomenon may be attributed to two factors: first, an excessively high STC constraint may cause the model to focus too much on Triadic Closure structures, thereby neglecting other important community features; second, overly strong constraints may introduce additional noise, affecting the model’s generalization ability. This finding provides important guidance for model tuning, suggesting that the edge prediction loss weight should be set at around 0.3 to achieve optimal performance.

Impact of the Number of Prompts: As demonstrated in Figure 4, the number of prompts significantly affects model performance. We observed that model performance initially increases and then plateaus as the number of prompts changes. Specifically, when the number of prompts increases from one to two, the model performance reaches its peak at approximately 0.3951, and as the number continues to increase (up to eight prompts), the performance improvement gradually levels off. This phenomenon has important practical implications: first, it confirms that STC-CDP can effectively learn from a small number of labeled samples, which is highly consistent with our original intention in designing a few-shot learning framework; second, the results indicate that only two prompts are required to achieve near-optimal performance, greatly reducing the annotation cost in real-world applications. This finding not only validates the efficiency of the model but also provides important guidance for parameter configuration in practical deployment.

Overall, the parameter sensitivity analysis reveals the response characteristics of the STC-CDP model to key hyperparameters, providing reliable recommendations for parameter configuration in real-world applications. At the same time, these findings further confirm the rationality and effectiveness of the model design.

5.2.4. Computational Efficiency Analysis (RQ4)

Computational Efficiency Analysis of STC-CDP: All baseline models were rerun on identical hardware, and our algorithm exhibited a marginal advantage.

Table 5. Efficiency study in terms of total running time.The best results are highlighted in bold.

Method	Facebook	Amazon	DBLP	Livejournal	Twitter
SEAL	35 m	35 m	30 m	88 m	46 m
CLARE	2 m	2 m	21 m	26 m	37 m
ProCom	7 m	9 m	29 m	29 m	25 m
STC-CDP (Ours)	2 m	5 m	12 m	14 m	11 m

offers a comparative analysis of the computational efficiency of various methods across five datasets. The results indicate that STC-CDP significantly outperformed mainstream methods such as SEAL, CLARE, and ProCom in terms of training time on most datasets. The efficiency of STC-CDP was primarily attributed to the incorporation of additional edge information, which facilitated faster convergence during training. Although STC-CDP had a slightly longer training time on the Amazon datasets compared to CLARE, it offered advantages in model expressiveness and scalability, enabling efficient training while maintaining accuracy. In summary, STC-CDP surpasses most mainstream methods in computational efficiency and can effectively handle large-scale social networks under limited resource conditions, making it suitable for real-world applications.

Further analysis of the training loss evolution between the ProCom and our STC-CDP models on the DBLP dataset revealed that the STC-CDP model demonstrated a marginally superior performance over ProCom. As depicted in Figure 5, the STC-CDP model exhibited a more rapid decline in training loss throughout the training epochs, indicating a more efficient learning process. This observation further substantiates the hypothesis that the incorporation of additional edge attributes is advantageous for the training process.

Efficiency Analysis of Edge Labeling with STC: We conducted a comparative analysis of the efficiency of labeling edge labels using different datasets with the STC method. The time spent by two STC algorithms to label edge labels across various datasets is recorded in

Table 6. Efficiency analysis of edge labeling with STC.

Method	DBLP	Amazon	Twitter	LiveJournal	Facebook
Traditional STC Time	Over 24 h	20 m	Over 24 h	Over 24 h	24 h
Improved STC Time	55 s	4 m	7.2 m	8.2 m	40 m

. To elucidate the correlation between labeling efficiency and dataset attributes, Figure 6 presents the runtime of the enhanced STC method across five datasets, detailing the number of nodes, edges, average node degree, and runtime. The analysis reveals that labeling efficiency is not highly correlated with the quantity of edges and nodes but shows a fundamental positive correlation with the average node degree. Subsequently, we present the node degree distribution plots for the five datasets in Figure 7. Our findings conclude that, as demonstrated by the DBLP dataset, labeling algorithms operate more swiftly under conditions where node degrees are generally low, irrespective of the magnitude of nodes and edges, as evidenced by the Facebook and LiveJournal datasets. Conversely, a higher prevalence of nodes with high degrees leads to increased algorithmic duration. This further illustrates the algorithm’s particular suitability for extremely sparse graph data, and of course, the time expenditure for generally sparse datasets falls within acceptable limits.

6. Conclusions

This study proposed STC-CDP, a few-shot community detection framework integrating Strong Triadic Closure (STC) with prompt learning. The main contributions are as follows: First, we introduced the STC principle to community detection, addressing limitations in handling connection strength inequality. STC provides a theoretical foundation for distinguishing strong and weak ties, enabling more accurate identification of community structures. Experiments showed STC-based modeling significantly improved detection accuracy across real-world datasets.

Second, we designed a parameter-efficient prompt learning framework that alleviated few-shot detection challenges. By combining pre-training with prompt adaptation, STC-CDP extracts key features from limited labeled communities and transfers knowledge to unlabeled ones, reducing data dependency and computational costs.

Third, ablation studies verified the synergistic effect between STC and prompt learning. STC enhanced network structure understanding while prompt learning improved few-shot generalization. Their combination outperformed individual methods, demonstrating framework effectiveness.

Despite these advances, limitations remain. Future directions include extending our method to dynamic networks, exploring continuous connection strength representations, and improving theoretical frameworks. STC-CDP provides new theoretical perspectives for community detection, advancing social network analysis with broad applications in social media analysis, market segmentation, public health, and information dissemination.