Detecting local communities in complex network via the optimization of interaction relationship between node and community

Seed selection

The goal of the seed selection method is to identify the core node of the community where the target node is located, which can improve the quality of the initial community (Wang et al., 2016). To obtain high-quality seeds as the initial community for expansion, a variety of seed selection methods, had been proposed by scholars. Lancichinetti, Fortunato & Kertész (2009) used a random selection method which is the simplest and most time-saving method to select nodes as seeds. However, the random selection method will make the algorithm unstable, which results in uncontrollable results. Similarly, Baumes et al. (2005) also used a random selection method, but instead of selecting random nodes, they replaced random edges as seeds. However, searching for random edges as seeds will generate many duplicate communities, which will lead to an increase in the algortithm’s time complexity of the algorithm and thus require a lot of computation time. Lee et al. (2010) explored k-clique, which is a complete subgraph with k vertices, of the target node as seeds. Based on the seed selection method, Lee et al. (2010) proposed a Greed Clique Expansion (GCE) algorithm. Furthermore, Li, Wang & Cui (2014) took maximum cliques as the seed, searched using depth and breadth search methods, and merged different communities into a larger sub-graph according to the given rules. To eliminate the influence of the seed quality on the local community detection algorithm, Ding, Zhang & Yang (2018) proposed a core member searching method that iteratively replaces the initial node with the candidate seed that has greater local influence and is most similar to the given node. Cheng et al. (2020) ranked nodes of networks according to the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), and the node with the highest score was used as the seed. Ni et al. (2020) took with the NGC node (Luo et al., 2017), the nearest node with the greater centrality, and selected nodes with greater fuzzy relationships among their NGC nodes are considered to be the seeds.

Community expansion

The goal of the community expansion is to expand the initial community into the resulting community through an expansion mechanism. The commonly used expansion mechanisms are the quality function (Kanawati, 2015; Zhang et al., 2015; Peng & Jing, 2016; Liakos, Ntoulas & Delis, 2016; Zhu, Chen & Zeng, 2020) and influence spreading (Kloster & Gleich, 2014; Hu, Yang & Wong, 2016; He et al., 2015; Yao et al., 2016; You, Ma & Liu, 2020). The quality function is a measure of the quality of community division results derived from the definition of community structure. Newman & Girvan (2004) proposed modularity as the quality of the community for measuring the community quality. According to the definition of modularity, high-quality communities should have a tight internal structure and loose external links between communities. Guo et al. (2022) proposed an improved algorithm that takes the which take local modularity density as the quality function.

The influence spreading method expands the community by calculating the influence of nodes and spreading these influences throughout the network. Raghavan, Albert & Kumara (2007) proposed the Label Propagation algorithm (LPA) based on an epidemic spreading model. LPA assigns each node of the network a unique label and spreads these labels over the entire network. Xu, Guo & Yang (2020) proposed a novel similarity measure based on a two-level neighborhood (TNS). Using TNS as a basis, they also proposed an improved LPA algorithm.

Similarity index

Nodes within the same community exhibit high similarity, whereas those between communities are not typically similar (Malliaros & Vazirgiannis, 2013). Therefore, similarity index can also measure the memberships between nodes and communities. Ding, Zhang & Yang (2020) proposed the local expansion and boundary rechecking (LEBR) algorithm, which optimized the membership between nodes and communities to expand communities, rather than optimizing the quality function. Ding, Zhang & Yang (2020) demonstrated that LEBR is highly effective at detect communities with diverse structures; thus, the limitation problem caused by the quality function is avoided. Table 1 displays commonly used node similarity indices.

In recent years, scholars have proposed various node similarity indices, leading to progress in node similarity calculation accuracy. The similarity indices related to this article are as follows.

Zhang, Ding & Yang (2019) reported that the similarity between two adjacent nodes increases as their k-core value grows larger. To distinguish between external and internal nodes of the community, they introduced the concept of local k-core value in their algorithm. Furthermore, the contribution of two adjacent nodes to their similarity should be different. The core similarity (CS) between two nodes is defined as follows.

(1) $$\mathbf{S}({\mathit{v}}_{\mathit{i}},{\mathit{v}}_{\mathit{j}})={\displaystyle \frac{{\mathbf{K}}_{\mathit{N}({\mathit{v}}_{\mathit{i}})\cap \mathit{N}({\mathit{v}}_{\mathit{j}})}({\mathit{v}}_{\mathit{i}})}{{\mathbf{K}}_{\mathbf{V}}({\mathit{v}}_{\mathit{i}})}\sqrt{{\mathbf{K}}_{\mathbf{V}}({\mathit{v}}_{\mathit{i}})\ast {\mathbf{K}}_{\mathbf{V}}({\mathit{v}}_{\mathit{j}})}}$$where ${\mathrm{K}}_{\mathrm{N}({\mathrm{v}}_{\mathrm{i}})\cap \mathrm{N}({\mathrm{v}}_{\mathrm{j}})}({\mathrm{v}}_{\mathrm{i}})$ is the local k-core value of node ${\mathrm{v}}_{\mathrm{i}}$ in the interaction of neighborhood of ${\mathrm{v}}_{\mathrm{i}}$ and ${\mathrm{v}}_{\mathrm{j}}$, ${\mathrm{K}}_{\mathrm{V}}(v_i)$ is the k-core value of node ${\mathrm{v}}_{\mathrm{i}}$ in the whole network.

Inspired by the RA index and local path (LP) similarity index (Zhou, Lü & Zhang, 2009), Xu, Guo & Yang (2020) proposed a novel similarity index based on a two-level neighborhood of nodes. RA makes full use of the topological information of nodes to improve the accuracy of similarity between nodes. LP similarity index and the two-level neighborhood similarity (TNS) index are defined as follows.

(2) $$\mathbf{S}={\mathbf{A}}^2+\text{α}{\mathbf{A}}^3$$where S denotes the similarity matrix, A denotes the node adjacent matrix and $\mathrm{\alpha }$ denotes the free parameter.

(3) $$\mathbf{S}({\mathit{v}}_{\mathit{i}},{\mathit{v}}_{\mathit{j}})={\sum }_{{\mathit{v}}_{\mathit{l}}\in \mathit{N}({\mathit{v}}_{\mathit{i}})\cap \mathit{N}({\mathit{v}}_{\mathit{j}})}{\displaystyle \frac{1}{{\mathbf{d}}_{{\mathit{v}}_{\mathit{l}}}}+{\sum }_{{\mathit{v}}_{\mathit{m}}\in \mathit{N}({\mathit{v}}_{\mathit{i}}),{\mathit{v}}_{\mathit{n}}\in \mathit{N}({\mathit{v}}_{\mathit{j}})}{\displaystyle \frac{{\mathbf{A}}_{\mathbf{m}\mathbf{n}}}{\sqrt{{\mathbf{d}}_{{\mathit{v}}_{\mathit{m}}}{\mathbf{d}}_{{\mathit{v}}_{\mathit{n}}}}}}}$$

Liu et al. (2022) introduced a node similarity index named CN, which combines the common neighbors and degree of node. CN is defined as follows.

(4) $$\mathbf{S}({\mathit{v}}_{\mathit{i}},{\mathit{v}}_{\mathit{j}})=|\mathit{N}({\mathit{v}}_{\mathit{i}})\cap \mathit{N}({\mathit{v}}_{\mathit{j}})|+{\displaystyle \frac{1}{{\mathbf{d}}_{{\mathit{v}}_{\mathit{i}}}{\mathbf{d}}_{{\mathit{v}}_{\mathit{j}}}}}$$

Motivation

As described in “Related Works”, researches in the field of seed selection methods, community expansion methods and similarity indices have made a lot of progress. However, there are still problems with the implementation of local community detection algorithms, which prevent accurate results from being obtained.

In the realm of community detection algorithms, one of the most significant challenges is the seed dependence problem. Essentially, the quality of the given node determines the accuracy of the resulting community partition. To address this issue, Ding, Zhang & Yang (2018) proposed a seed selection method called SSSC that effectively solves the seed dependence problem. This problem arises when the accuracy of the community detection algorithm depending heavily on the quality of the given seed. Specifically, the method involves comparing the centrality between the alternative seed and the given node, and then comparing the similarity between these two nodes with the maximum similarity obtained before. However, this method is correct only when the alternative seed and the given node are in the same community. In cases where the alternative seed and the given node are not in the same community, but the alternative seed has the greatest centrality and greater similarity with the given node, SSSC will still consider this node as the alternative seed of the given node. This leads to incorrect results, which we refer to as the seed deviation problem. As such, further research is required to address this issue and improve the accuracy of community detection algorithms.

Secondly, a local community detection algorithm typically optimize only one type of quality function during the process of community expansion. While this approach my yield satisfactory results for certain types of networks, it can lead to less efficient performance when dealing with other types of networks. In particular, a quality function that describes a community with only one structural feature may not be sufficient for more complex networks (Ding, Zhang & Yang, 2020). As a result, community detection algorithms may face the quality function limitation problem. As such, further research is required to address this issue and improve the accuracy of community detection.

To solve the problems of seed dependence and seed deviation, we propose an improve seed selection method that first considers similarity first. This ensures that the alternative seed and the given node are in the same community. We then calculate the node centrality. We consider the problem with a simple example in Fig. 1. As shown in Fig. 1, the similarity between v₃ and v₁ is lower than that between v₂ and v₁, but v₃ has a greater node centrality (9) than v₂, which has a centrality of (6). In this condition, SSSC considers v₃ as the alternative seed of v₁ under this condition. However, since v₂ is more similar to v₁ than v₃ is. Additionally, v_3, and v₁ are in the same community. Therefore, v₂ is actually the alternative seed of v₁.

To mitigate the quality function limitations problem, a novel local community detection algorithm based on the optimization of the interaction relationships is proposed, which deprecates the quality function. The proposed motivation is to develop with precision similarity indices that can accurately calculate the interaction relationship. To achieve higher precision in measuring node similarity than existing measures, this study gradually obtains more neighbourhood information gradually.

Problem definition

This study focuses on a graph called $G=(V,E)$. The node set composed of all nodes in the graph is represented by V. The link set composed of all links between these nodes is represented by E, and A is a two-dimensional array called adjacent matrix that records whether two nodes are connected. $A_{ij}=1$ denotes that there is a link between node i and node j that is connected; otherwise, $A_{ij}=0$.

The given node denotes an initial node given in local community detection algorithms. A community C denotes a collection of nodes and their connected links, where C = {v₁, v₂,…, v_j} (C ∈ C, v_i ∈ V). The initial community denotes the community composed of seed and its part of neighbors. The expending community denotes the community in expanding. The result community denotes the community detected by algorithms. This study aims to detect a community C where the given node really locates.

Basic definitions

Definition 1 (Node neighbors). The node neighbors of node v are defined as follows:

(5) $$\mathit{N}(\mathit{v})=\{\mathit{u}|\mathit{u}\in \mathit{V},{\mathit{A}}_{\mathit{u}\mathit{v}}=1\},\mathit{v}\in \mathit{V}$$where A is the adjacent matrix of graph G, and if A_uv = 1, it means that there is a link between node v and node u. The definition of node neighbors is a set of nodes with links connected to the node.

Definition 2 (Node influential scope). The node influential scope of node v is defined as follows:

(6) $$\mathit{N}\mathit{I}(\mathit{v})=\{\mathit{u}|\mathit{u}\in \mathit{N}(\mathit{v}),{\mathit{A}}_{\mathit{u}\mathit{v}}=1\},\mathit{v}\in \mathit{V}$$where N(v) denotes node neighbors defined in Definition 1. The definition of node influential scope is a set of nodes consists of node neighbors and node itself.

Definition 3 (Community neighbors). The community neighbors of community C is defined as follows:

(7) $$\mathit{N}(\mathit{C})=\{\mathit{u}|\mathit{u}\notin \mathit{C},\mathit{N}(\mathit{v}),\mathrm{\exists }\mathit{v}\in \mathit{C},(\mathit{u},\mathit{v})\in \mathbf{E}\},\mathit{v}\in \mathit{V},\{u|u\in |C,\mathrm{\exists }v\in C,(u,v)\in E\}$$where E denotes the set of links of network G. The definlition of community neighbors is a set of external nodes that have links connected to the members of the community.

Definition 4 (Node degree). The node degree of node v is defined as follows:

(8) $$\mathit{d}(\mathit{v})=|\mathit{N}(\mathit{v})|,\mathit{v}\in \mathit{V}$$

The definition of node degree is the number of node links.

Definition 5 (Local centrality). The local centrality of node v is defined as follows:

(9) $$\mathit{L}\mathit{C}(\mathit{v})=\{{\mathit{v}}_{\mathit{i}},{\mathit{v}}_{\mathit{j}}|{\mathit{v}}_{\mathit{i}},{\mathit{v}}_{\mathit{j}}\in \mathit{N}\mathit{I}(\mathit{v}),{\mathit{A}}_{\mathit{i}\mathit{j}}=1\},\mathit{v}\in \mathit{V}$$

We measure node centrality by examining links within the node’s influential scope defined in Definition 2. The more links there are within the scope, the greater the node’s centrality.

Definition 6 (Node similarity 1). The first similarity index proposed in this article between node v_i and v_j is defined as follows:

(10) $$\mathit{S}1({\mathit{v}}_{\mathit{m}},{\mathit{v}}_{\mathit{n}})=\{{\mathit{v}}_{\mathit{i}},{\mathit{v}}_{\mathit{j}}|{\mathit{v}}_{\mathit{i}},{\mathit{v}}_{\mathit{j}}\in \mathit{N}({\mathit{v}}_{\mathit{m}})\cap \mathit{N}({\mathit{v}}_{\mathit{n}}),{\mathit{A}}_{\mathit{i}\mathit{j}}=1\},\mathit{v}\in \mathit{V}$$

We measure the similarity between the neighborhood of $v_m$ and $v_n$ by analyzing the links between them. The more links there are within the two nodes’ influential scope, the greater their similarity. We can describe this similarity index with the simple example shown in Fig. 2, where $S1(v_1,v_2)=14$.

Definition 7 (Node similarity 2). The second similarity index proposed in this article between node v_i and v_j is defined as follows:

(11) $$S2(v_m,v_n)={\sum }_{v_i,v_j\in N(v_m)\cap N(v_n),A_{ij}=1}|d(v_i)+d(v_j)|,v\in V$$

The contribution of each link within the node influential scope to the similarity of two nodes is likely not the same. Therefore, we assign weights to the links based on the degree of nodes on both sides of the link. The similarity between nodes is then calculated as the degree sum of the nodes at both ends of the link within the common influence scope.

Definition 8 (Node similarity 3). The third similarity index proposed in this article between node v_i and v_j is defined as follows:

(12) $$\mathit{S}3({\mathit{v}}_{\mathit{m}},{\mathit{v}}_{\mathit{n}})={\sum }_{{\mathit{v}}_{\mathit{l}}\in \mathit{N}({\mathit{v}}_{\mathit{m}})\cap \mathit{N}({\mathit{v}}_{\mathit{n}})}|\mathit{N}\mathit{I}({\mathit{v}}_{\mathit{l}})|\mathbf{\ast }\mathit{S}2({\mathit{v}}_{\mathit{m}},{\mathit{v}}_{\mathit{n}}),\mathit{v}\in \mathit{V}$$

The contribution of each node within the node influential scope compared to the similarity of two nodes is likely not the same. Thus, Definition 7 describes the similarity index within the influence scope of these two adjacent nodes. Based on Definition 7, we multiply the number of nodes within the influence scope of each node in the common influential scope of two adjacent nodes by the similarity index, and sum all that of nodes in the scope. We can show this similarity index with the simple example in Fig. 2, where $S1(v_1,v_2)=(3+5+3+7+5+9+3+1+5+1)\ast 102=\mathrm{4,284}$.

Definition 9 (Node community similarity). The node community similarity between node v and community C is defined as follows:

(13) $$\mathit{N}\mathit{C}\mathit{S}(\mathit{v},\mathit{C})={\sum }_{\mathit{u}\in (\mathit{N}(\mathit{v})\cap \mathit{C})}\mathit{N}\mathit{S}(\mathit{u},\mathit{v}),\mathit{u},\mathit{v}\in \mathit{V}$$where $\mathit{N}\mathit{S}(\mathit{u},\mathit{v})$ represents a method of node similarity calculation.

We calculate the similarity between node v and community C by sum the similarity between node v and each node in community which has a link with node v.

Proposed algorithm

Algorithm 1 shows the pseudocode of OIRLCD. To facilitate readers’ understanding of the proposed algorithm, we provide flowcharts of the seed selection process and community expansion process in Figs. 3 and 4, respectively. This section provides a detailed description of the proposed algorithm.

Initialization (Lines 1–4). Line 2 initializes the empty community C, which will store the final result. Based on Definition 4, Line 3 calculates the node degree of each node in node set V. Based on Definition 5, Line 4 calculates the local centrality of each node in node set V.

Seed selection (Lines 4–21). The seed selection process searches for the core node of the community where the given node is located as the alternative seed. To find the alternative seed for a given node, two requirements must be met. First, the alternative seed must have the maximum similarity to the given node to ensure that they are in the same community. Second, the alternative seed must have the greater local centrality than the given node to ensure that the alternative seed is closer to the center of the community than the given node. As shown in Algorithm 1, Line 7 sets max_similarity as zero to store the greatest similarity value. Line 10 obtains all neighboring nodes $N(v_{temp})$ of $v_{temp}$ based on Definition 1. Line 12 calculates node similarity based on Definition 8 to ensure that all the comparison algorithms with different similarity indices should have the same seed node for community expansion. For each node in $N(v_{temp})$, the process will replace the previous alternative seed (Line 16) if it satisfies the two conditions mentioned above. Executing the program until all nodes in $N(v_{temp})$ are calculated (Lines 11–19). Lines 8–20 search for the alternative seed until no neighboring nodes of the current alternative seed meet the conditions. Line 20 sets the current alternative seed as the seed and sends this seed to the community expansion process.

Community expansion (Lines 21–39). For community expansion, the proposed algorithm gradually adds the community neighbors that meet specific conditions. An eligible community neighbor is one whose similarity to the community is greater than its similarity to the rest of the nodes in the network. As shown in Algorithm 1, Line 23 initializes the initial community $C$ as the influential scope of the seed. Line 24 excluded nodes which is not meeting the conditions above. Line 29 obtains all community neighbors $N(C_{temp})$ of $C_{temp}$ based on Definition 3. Line 33 calculates the node community similarity $NCS(v_i,C)$ based on Definition 9 between node $v_i$ and the community $C$. The remaining nodes of the network G are regarded as community $\overline{C}$. Line 34 calculates the node community similarity $NCS(v_i,\overline{C})$ between node $v_i$ and the community $\overline{C}$. When $NCS(v_i,C)>NCS(v_i,\overline{C})$, $v_i$ should be added to the community $C$. Executing the program until all nodes in $N(C_{temp})$ are calculated (Lines 31–38). Lines 27–39 execute community expansion until no community neighboring nodes of the current community $C$ meet the conditions. Line 20 returns the current community $C$.

Time complexity analysis

This section analyzes the time complexity of OIRLCD. All the comparison algorithms are executed on the network $G=(V,E)$ where the average degree $\overline{d}$.

First, the initialization process includes Definition 4 and Definition 5, with the time complexity of $O(\overline{d})$ and $O({\overline{d}}^2)$. So the first process needs $O({\overline{d}}^2)$. Secondly, the seed selection process use the Definition 8 to calculate the node similarity, and its time complexity is $O({\overline{d}}^2)$. So the seed selection process needs $O({\overline{d}}^3)$. Thirdly, the community expansion process first calculates the community neighbors based on Definition 3 with time complexity $O({\overline{d}}^2)$. The mean distance from the community edge to its core node is set as $\overline{r}$. The proposed similarity index defined as Definition 6, Definition 7 and Definition 8 with time complexity $O({\overline{d}}^2)$, $O({\overline{d}}^2)$, $O({\overline{d}}^2)$, respectively. Finally, the overall time complexity of our three proposed algorithm need $O(\overline{r}{\overline{d}}^2)$.

Calculating Jaccard similarity index needs $O(1)$, calculating Salton similarity index needs $O(1)$, calculating RA similarity index needs $O(\overline{d})$, calculating the CS similarity index needs $\mathrm{O}(n{\overline{d}}^3)$, calculating the TNS similarity index needs $\mathrm{O}(n{\overline{d}}^2)$ and calculating the CN similarity index needs $\mathrm{O}({\overline{d}}^2)$. Finally, the overall time complexity of the proposed algorithms and the comparison algorithms are listed in Table 2. The symbols in the table that C denotes the size detected community; $\overline{|C|}$ denotes the size of detected community; $|S|$ denotes the size of the shell sub-network of C; $|N|$ denotes the size of the neighbor sub-network of C.

Evaluation criteria

Normalized mutual information (Danon et al., 2005) (NMI) and F-score (Li, Wang & Wu, 2015) is two widely used methods for evluating community quality. This study verified the resulting communities of OIRLCD and comparison algorithms on these two indicators.

Datasets

Experimental settings

In this experiment, we refer to the proposed algorithms based on Definition 6, Definition 7 and Definition 8 as OIRLCDF, OIRLCDS and OIRLCDT respectively. Additionally, each algorithm has a version that uses SSCS called Algorithm1 and a version that uses SSSC called Algorithm 2. For example, OIRLCDF1 and OIRLCDF2 represent the algorithms that use SSCS and those that use SSSC, respectively. During the comparative experiments, we only replace the similarity indices of node similarity in the proposed algorithms. We name the corresponding algorithms based on the similarity index used. Three commonly used similarity indices involved in this experiment are the Jaccard similarity index (Jaccard, 1901), Salton similarity index (Salton & McGill, 1986) and RA similarity index (Zhou, Lü & Zhang, 2009). Three novel complex similarity indices involved in this experiment are CS (Zhou, Lü & Zhang, 2009), TNS (Xu, Guo & Yang, 2020) and CN (Liu et al., 2022).

In addition, we compare OIRLCD to five existing local community detection algorithms: LWP (Luo, Wang and Promislow) (Luo, Wang & Promislow, 2008), Chen (Chen, Zaï & Goebel, 2009), LS (link similarity) (Chen, Zaï & Goebel, 2009), LCD (local community detection based on maximum cliques) (Wu et al., 2012) and RTLCD (Zhang, Ding & Yang, 2019).

Luo, Wang & Promislow (2008) proposed an improved quality function M based on the Clauset algorithm (Clauset, 2005). M is calculated by dividing the inner links of the community by the links between communities. Similar to the Clauset algorithm, the LWP algorithm expands the community by optimizing the quality function M. To manage outliers, Chen, Zaï & Goebel (2009) proposed a local community detection algorithm based on quality function L. The Chen algorithm rechecks the removed nodes to identify whether they optimize the quality function; this operation can reduce the effects of outliers. Wu et al. (2012) proposed a local community detection algorithm based on link similarity (LS) that calculates similarity based on the intersection of adjacent nodes of the node and adjacent nodes of the community. The LS algorithm expands communities by optimizing node similarity. Fanrong et al. (2014) proposed a local community detection algorithm (LCD) based on maximum network cliques. The LCD algorithm searches maximum cliques as seeds and expands from these nodes by optimizing the distribution of maximum cliques. Zhang, Ding & Yang (2019) proposed a robust community detection algorithm RTLCD which consists of two stages: seed selection stage and community expansion stage. During seed selection stage, RTLCD searches the core node as the alternative seed of the given node, which solves the seed-dependent problem. In the community expansion stage, RTLCD expands the community by node relation strength, which maintains seed validity.

Additionally, the proposed and other algorithms are applied to the dataset mentioned above, with the exception that any algorithm running for more than 24 h is stopped.

Experimental results on real-world networks

Table 6 describes the performance of the proposed algorithms and five existing local community detection algorithms based on NMI, Recall, Precision, F-score and time metrics in six real-world networks. The best and the second-best values are marked in bold. Table 7 lists the percentage gains in terms of NMI and F-Measure for algorithms using SSCS compared to algorithms using SSSC.

From Table 6, we can observe that OIRLCDT outperforms OIRLCDS in the NMI, Recall, Precision, F-score metrics; and OIRLCDS outperforms OIRLCDT in these metrics. Improving the precision of the similarity index improves the performance of algorithms on each metric. This phenomenon demonstrates that enhancing the precision of the similarity index can increase the accuracy of detecting local communities. Notably, OIRLCDT outperforms all the other comparison algorithms, except LCD, on each metric of all six real-world networks. Therefore, OIRLCDT can effectively detect local communities and exhibit better performance than the existing algorithms tested in this study. However, LCD can achieve good performance, but it lacks scalability in three large real-world networks; these results indicate that LCD is not competitive in large real-world networks. It is further observed that OIRLCDF, OIRLCDS and OIRLCDT show gradual improvement in the time metric, suggesting that it takes more time to enhance the accuracy of similarity indicators in detecting communities.

Table 7 shows that, when using the same similarity index, the algorithm using SSCS outperforms the noe using SSSC. This result suggests that SSCS is more effective than SSSC in finding the core node. Table 6 shows that the algorithm using SSCS takes more time than the one using SSSC. For the Karate network, SSCS performs the same as SSSC because the Karate network is so small for different seed selection strategies to make a significant difference.

Experimental results on artificial networks