Hindawi Complexity Volume 2017, Article ID 3719428, 10 pages https://doi.org/10.1155/2017/3719428
Research Article Social Network Community Detection Using Agglomerative Spectral Clustering Ulzii-Utas Narantsatsralt and Sanggil Kang Department of Computer Engineering, Inha University, Incheon, Republic of Korea Correspondence should be addressed to Sanggil Kang;
[email protected] Received 18 April 2017; Revised 24 July 2017; Accepted 23 August 2017; Published 7 November 2017 Academic Editor: Katarzyna Musial Copyright © 2017 Ulzii-Utas Narantsatsralt and Sanggil Kang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Community detection has become an increasingly popular tool for analyzing and researching complex networks. Many methods have been proposed for accurate community detection, and one of them is spectral clustering. Most spectral clustering algorithms have been implemented on artificial networks, and accuracy of the community detection is still unsatisfactory. Therefore, this paper proposes an agglomerative spectral clustering method with conductance and edge weights. In this method, the most similar nodes are agglomerated based on eigenvector space and edge weights. In addition, the conductance is used to identify densely connected clusters while agglomerating. The proposed method shows improved performance in related works and proves to be efficient for real life complex networks from experiments.
1. Introduction In recent years, community detection in a network has become one of the main topics of fields, such as biology, computer science, physics, and applied mathematics [1–3]. In a network, 𝐺(𝑉, 𝐸), where 𝑉 is a set of nodes and 𝐸 is the edges (relation) between the nodes, a community is a group of nodes with tightly connected edges with each other and the nodes of community show similar characteristics. For example, in social network, people in a community show similar interest to a trend in a community, for example, buying the same products in online marketing. In a biology network, proteins in a community show similar specific functions, and, in the World Wide Web, sites clustered together show the same topic in their web page. Scientists in many fields made significant contributions to detecting communities by a number of different methods, such as graph partitioning [4, 5], hierarchical clustering [6, 7], and spectral clustering [8, 9]. In graph partitioning, a network is divided into clusters in such a way that the number of edges connecting the clusters is minimum, that is, the edges of a cluster are denser
inside than outside (also referred as conductance [10]). In addition, the number of lowest sized clusters needs to be specified. Girvan and Newman [3] introduced a popular graph partitioning algorithm. Girvan and Newman [3] use modularity (also referred as conductance) to cluster communities but the method is slower than other community detection algorithms [11, 12]. Later, Djidjev [13] proposed a computationally faster version of the algorithm. However, the definition of conductance is not always definite, and the definition can be false in some cases [5]. Therefore, graph partitioning still needs further inference. A number of methods have been proposed to solve this problem. One of the famous methods is introduced by Newman [5]. They use spectral clustering algorithm with modularity maximization, in which the modularity function is implemented for only possible clusters of network and the result proved that spectral clustering with conductance can efficiently cluster communities. Hierarchical clustering is used for complex networks because they often have a hierarchical structure [14]. Hierarchical clustering consists of a division [15] and agglomeration stage [16, 17]. In the division stage, a network is
2 deemed to be one cluster in the beginning and the network is then divided into clusters in each iteration, where the most dissimilar nodes are separated. In the agglomeration stage, similar nodes are agglomerated together until the termination criteria are met or the clusters agglomerate into a single community. However, hierarchical clustering needs a well-defined similarity function and the clustering can be inaccurate if all nodes are similar to each other. However, the problem of similar nodes and similarity function can be solved by projecting the nodes into high dimensional feature space using spectral clustering because the projected eigenvectors significantly distinguish the similar nodes into more distanced positions in feature space. The reason for using eigenvector space instead of using original point is that the properties of original clustering are made more distinct by the eigenvector space. In spectral clustering, original points are transformed into a set of points in eigenvector space and clustering is done by analyzing eigenvector space. One technique for clustering eigenvector space is to use 𝑘-means algorithm [18] where similar nodes are clustered together. However, traditional spectral clustering has a problem with model selection which depends on heuristics. The problem can be solved using weighted kernel spectral clustering (KSC) with primal and dual representations [19–21]. KSC [21] focuses on the principle that projections of similar nodes are clustered together in eigenvector space. In another work of KSC, an agglomeration technique is introduced to the KSC which is called agglomerative hierarchical kernel spectral clustering (AHKSC) [22]. AH-KSC uses eigenvector space to find distance between nodes and it agglomerates close distanced nodes. The main purpose of AH-KSC is to get hierarchical clustering but accuracy of AH-KSC does not improve significantly from KSC because AH-KSC allows indirectly connected nodes to be agglomerated together and also there are no termination criteria for satisfied community. The problems of KSC and AH-KSC are choosing eigenvector, improving accuracy of detected communities, and using only data generated by hand which usually do not show same characteristics as real life networks. The above-mentioned methods focus on decreasing the computation time or improving the accuracy of community detection. Methods for improving the computational time have been well studied and it can be solved by advances in technology and techniques [23–25], such as parallel computing and GPU programming. Improving the accuracy of community detection has been challenging task because networks are usually structured with great complexity with millions of nodes and edges. Hence, this paper proposes an agglomerative spectral clustering method with conductance and edge weights to improve the accuracy of community detection. The characteristics of the proposed method are well suited for accurate community detection in complex networks because the eigenvector space from spectral clustering provides well distinct points that are used for the similarity function in agglomeration. The conductance is used for the sensitive termination criteria of agglomeration and the edge weight is a major factor for evaluating a more accurate similarity. In addition, performance of the proposed
Complexity method was compared with that of AH-KSC and KSC using real life social network data with a ground-truth, which are the LiveJournal and Orkut network. This method can help improve the community detection performance from previous works [21, 22]. The remainder of this paper is as follows: Section 2 introduces the problem statement and background, which helps understand the proposed method. The core algorithm of the proposed method is explained in Section 3. The experiment is outlined in Section 4 and Section 5 reports the conclusions.
2. Fundamental Concepts 2.1. Problem Statement. In KSC, the data are divided into training, validating, and test sets. In the training stage, the eigenvector space of the training data is signed, which is used for clustering the nodes in a network. The sign of the eigenvector points in the same cluster is identical. In the validating stage, model selection is performed to identify the clustering parameters. The eigenvector space of the test data is used to evaluate the clusters obtained from the training data using the hamming distance function. The problem with the KSC is that clustering depends on encoding/decoding eigenvectors space. The encoded values are all signed in KSC [21] and distinction between the two elements of the eigenvector is only “1” or “0” so that similar eigenvector points become noisy. For example, if in eigenvector space, 𝑒1(𝑙) = 0.001 and 𝑒2(𝑙) = −0.001, the two values can be binarized as “1” and “0,” respectively. Although, 𝑒1(𝑙) and 𝑒2(𝑙) are projected in similar feature space, the results of encoding show a different outcome. This problem can be solved by agglomerative hierarchical KSC (AH-KSC). In AH-KSC [22], instead of signing eigenvector space, the space is used as the data points to obtain the distance between nodes in a network and close distanced nodes are agglomerated together until there are only 𝑘 clusters or less. KSC and AH-KSC still have certain disadvantages. Both methods calculate the kernel matrix Ω by counting the number of edges connecting the common neighbors between two nodes, 𝑖 ∧ 𝑗: Ω𝑖𝑗 = ∑𝑘,𝑙∈𝑁𝑖𝑗 𝐴 𝑘𝑙 , where 𝑁𝑖𝑗 is a set of common neighbors of 𝑖 ∧ 𝑗, 𝑘 and 𝑙 are common neighbors, and 𝐴 𝑘𝑙 is adjacency matrix of the graph. However, the common neighbors between nodes can cause indirectly connected nodes to be clustered together so that the nodes in different clusters can be clustered. To solve this problem, the adjacency matrix is used as a kernel matrix so agglomerated nodes can be connected directly. In addition, KSC and AH-KSC use only the first 𝑘 − 1 eigenvectors for encoding/decoding but the remaining eigenvectors can still provide correlated information for clustering. To take this into consideration, in this study, all eigenvector space was used to evaluate the similarity between nodes. Furthermore, in AH-KSC, there were no termination criteria for agglomerating clusters. Therefore, the conductance was used as termination criterion during the agglomeration of satisfied clusters.
Complexity
3
2.2. Background. In general, KSC is described by a primaldual formulation. Given a network, 𝐺(𝑉, 𝐸), where 𝑉 denotes the vertices and 𝐸 the edges, and the training data 𝑉tr = 𝑁 {𝑥tr }𝑖=1tr , the primal problem [21] is 𝑘−1
min
𝑤𝑙 ,𝑒𝑙 ,𝑏𝑙
𝑘−1
1 1 −1 𝑙 𝑒 ∑ 𝑤(𝑙) 𝑤(𝑙) − ∑ 𝛾𝑙 𝑒(𝑙) 𝐷Ω 2 𝑙=1 2 𝑙=1 𝑡
𝑇
(1)
subject 𝑒(𝑙) = Φ𝑤(𝑙) + 𝑏𝑙 1𝑁tr , 𝑙 = 1, . . . , 𝑘 − 1, (𝑙) where 𝑒(𝑙) = [𝑒1(𝑙) , . . . , 𝑒𝑁 ] is the projection, which is the tr mapped points of training data in feature space with respect to the direction, 𝑤(𝑙) , 𝑙 indicates the number of score variables, −1 is the inverse which is needed to encode the 𝑘 clusters, 𝐷Ω matrix of the degree matrix of the kernel matrix, Ω, Φ is the 𝑁tr × 𝑑ℎ feature matrix, where Φ = [𝜑(𝑥1 )𝑇 ; . . . ; 𝜑(𝑥𝑁tr )𝑇 ], 𝛾𝑙 is the regularization constant, and 1𝑁tr is the 𝑁tr × 𝑁tr matrix of ones. The primal form of the data point is expressed as 𝑇
𝑒𝑖(𝑙) = 𝑤(𝑙) 𝜑 (𝑥𝑖 ) + 𝑏𝑙 , 𝑖 = 1, . . . , 𝑁tr ,
(2)
where 𝜑 : 𝑅𝑛 → 𝑅𝑑ℎ is the map to high dimensional feature space, where 𝑛 is the number of nodes in graph, 𝐺, 𝑑ℎ is the number of eigenvectors, and 𝑏𝑙 is the bias term. The dual problem related to the primal problem is −1 𝑀𝐷Ω𝛼𝑙 = 𝜆 𝑙 𝛼(𝑙) , 𝐷Ω
(3)
where Ω is the kernel matrix with 𝑖𝑗th entry, Ω𝑖𝑗 = 𝐾(𝑥𝑖 , 𝑥𝑗 ) = 𝜑(𝑥𝑖 )𝑇 𝜑(𝑥𝑖 ); 𝐷Ω is the diagonal matrix of Ω with elements of 𝑑𝑖𝑖 = ∑𝑗 Ω𝑖𝑗 ; 𝑀𝐷 is a centering matrix defined as −1 −1 𝑀𝐷 = 𝐼𝑁tr − (1/1𝑇𝑁tr 𝐷Ω 1𝑁tr )(1𝑁tr 1𝑇𝑁tr 𝐷Ω ), where 𝐼𝑁tr is the
𝑁tr × 𝑁tr identity matrix; 𝛼(𝑙) is the dual variable; and the kernel function 𝐾 is the similarity function of the graph. The parameters, such as the number of community 𝑘, are estimated using the training data, 𝑉tr , and validating data, 𝑉va . In addition, all the nodes are clustered in the training and validating stage. The eigenvector space is used to find unique code-word for all clusters 𝐴 𝑝 , 𝑝 = 1, . . . , 𝑘. The
codebook, 𝐶 = {𝑐𝑝 }𝑘𝑝=1 , can be obtained from rows of the binarized eigenvector matrix. Finally, the eigenvector space of the test data, 𝑉te , is decoded using the hamming distance [21] and the clustered result is evaluated. Therefore, eigenvector space is used to derive the similarity among nodes, which will be explained in detail in the following section.
3. Proposed Community Detection Algorithm This section presents details of agglomerative spectral clustering with the conductivity method. The eigenvector space is used to find the similarity among nodes and agglomerate the most similar nodes to make a new combined node in a network graph. The new combined node is added to the graph after agglomeration and the changed graph is iterated until the termination criteria are satisfied. To agglomerate two nodes, a similarity function is modified from the correlation distance function among the nodes as follows:
𝑁 𝑁 𝑖 𝑗 𝑖 𝑗 ∑𝑁 𝑛=1 (𝑥𝑛 ∗ 𝑥𝑛 ) − (1/𝑁) ∑𝑛=1 𝑥𝑛 ∑𝑛=1 𝑥𝑛
(𝑖, 𝑗) = CorDis = 1 − 𝑝 = 1 − √ ∑𝑁 𝑛=1
2 (𝑥𝑛𝑖 )
−
where (𝑖, 𝑗) is the similarity between nodes i and j in the range of [0,2] with 0 being perfect similarity and 2 being perfect dissimilarity. 𝑥𝑛𝑖 is the value of the eigenvector, 𝑒𝑖(𝑛) , in eigenvector space, 𝑖th row, and 𝑛th column. The eigenvector space is not enough to fully express the similarity among agglomerated nodes because the nodes connected to each other are projected into a similar place in feature space and it is very difficult to distinguish similar projections. On the other hand, these similar projections can be distinguished using the disparity of the edge connections between the nodes. Agglomerated nodes can have more than one connecting edge with each other. Therefore, more tightly connected nodes have more similarity. For example, in Figure 1, similar nodes are combined in the 1st iteration and node 𝑛6 has two connections to the agglomerated node of n4
𝑁 (1/𝑛) (∑𝑛=1
𝑥𝑛𝑖 )
2
√ ∑𝑁 𝑛=1
𝑗 2 (𝑥𝑛 )
−
𝑁 (1/𝑛) (∑𝑛=1
𝑗 2 𝑥𝑛 )
,
(4)
and n5 and has one connection to the agglomerated node of 𝑛7 and 𝑛8 , so that 𝑛6 is more likely to agglomerate with n5 and n4 . In the 2nd iteration, new agglomerated nodes are used to find new eigenvector space and agglomerate similar nodes. In doing so, the number of edges in the graph is unchanged and some nodes have more than one edge between them. As mentioned in the example of node n6 , the edges between two nodes are used as a mean to give a similarity score between nodes to improve the accuracy of the algorithm. On the other hand, the number of edges between nodes can be varied too much and the value of the similarity function in (4) is too different. Therefore, the similarity function will be overemphasized on a number of edges; that is, disregard the eigenvector space score. In the present study, a sigmoid function is used to normalize the edge values to solve the above-mentioned problem.
4
Complexity
n5
Find the most
Combine the nodes
similar nodes using
with the most
similarity function
similar node
n5 n6
n7
n8
n5
n6
n7
n6
n7
n8
n4
n4
C n8
n4 C
n1
n10
n9
n1
n3
n2
Inside edge Outside edge (a) Initial graph
n10
n9
n1
n3
n10
C n3
n2
n2
Inside edge Outside edge
Inside edge Outside edge
(b) 1st iteration
n9
(c) 2nd iteration
Figure 1: Example of agglomerating nodes.
Equation (4) is modified, as expressed in (𝑖, 𝑗) 2
=
2
2
𝑁 𝑁 𝑁 𝑖 𝑗 𝑖 𝑗 𝑖 2 𝑖 √ ∑𝑁 (𝑥𝑗 ) − (1/𝑛) (∑𝑁 𝑥𝑗 ) )) √ 𝑁 (1 − ((∑𝑁 𝑛 𝑛=1 (𝑥𝑛 ∗ 𝑥𝑛 ) − (1/𝑁) ∑𝑛=1 𝑥𝑛 ∑𝑛=1 𝑥𝑛 ) / ∑𝑛=1 (𝑥𝑛 ) − (1/𝑛) (∑𝑛=1 𝑥𝑛 ) 𝑛=1 𝑛=1 𝑛
0.5/ (1 + 𝑒((−5/𝐸max )∗𝐸(𝑖,𝑗)) )
where 𝐸max is the maximum number of edges and 𝐸(𝑖, 𝑗) is the number of edges between nodes 𝑖 ∧ 𝑗. The vertical value of the sigmoid graph is deemed to be the edge similarity score and starts from 0.5 to 1 while the horizontal value, which is the number of edges, ranges from 0 to the maximum number of edges. Equation (5) is used to find the most similar node of node 𝑖 from the other nodes in graph 𝐺. At the first iteration, the first node becomes a candidate and if there is a more similar node than the candidate, the candidate is then replaced with it. The process continues until the similarity of all nodes is evaluated. Thus, the most similar node to node 𝑖 is determined by 𝑖 𝑛ms = min (𝑖, 𝑘) , 𝑘
(6)
𝑖 where 𝑘 ∈ 𝑁, 𝑁 is all the nodes in graph 𝐺, and 𝑛ms is the most similar node of node 𝑖. Furthermore, to obtain a more accurate clustering result, this study considers the definition of a good community,
(5) ,
which is “density of the edge connection should be higher inside than outside” [10]. The similar nodes are agglomerated together in every iteration and the agglomerated nodes become a clustered community after a few iterations. If the cluster is connected tightly inside and sparsely connected to outside, there is no need for further agglomeration because the cluster is sufficiently satisfied to be a good community and agglomeration for this community is terminated. In addition, two communities are agglomerated when they are tightly connected to each other. For example, in Figure 1(c), where the inside edges are straight lines and the outside edges are dotted lines, the graph is clustered into three agglomerated communities, such as 𝐶1 , 𝐶2 , and 𝐶3 . In the case of community 𝐶1 , it has three inside edges and two outside edges connected to both 𝐶2 and 𝐶3 so that 𝐶1 has a denser connection inside than outside. Consequently, no further agglomeration is needed. To consider the ratio of the inside and outside edges into (5), the two possible cases are
Complexity
5
divided when node 𝑗 is a candidate as the most similar node for node 𝑖: (1) 𝑁𝑖 < 𝑁𝑗 (2) 𝑁𝑖 > 𝑁𝑗 where 𝑁𝑖 is the number of nodes inside node 𝑖 and 𝑁𝑗 is the number of nodes inside node 𝑗. In the first case, the number of inside edges of node 𝑖 is at most equal to the number of outside edges connecting to node 𝑗. However in the next case, the number of inside edges of node 𝑖 is more than the number of outside edges connecting to node 𝑗. Therefore, to agglomerate only tightly connected nodes together, (5) can be modified using the inside and outside edges: 𝐸𝑖 < 𝐸𝑖𝑗 ∗ 𝜇, 𝑁𝑖 < 𝑁𝑗 𝑖 = min (𝑖, 𝑘) → { }, 𝑛ms 𝑘 𝐸𝑗 < 𝐸𝑖𝑗 ∗ 𝜇, 𝑁𝑖 > 𝑁𝑗
(7)
where 𝐸𝑖 is the inside edges of node 𝑖, 𝐸𝑗 is the inside edges of node 𝑗, 𝐸𝑖𝑗 is the edges connecting node 𝑖 ∧ 𝑗, and 𝜇 is the community density parameter. After finding the similarity between nodes using (7), agglomeration of the most similar nodes starts. In every iteration, the most similar node for each node was found and if the most similar node of 𝑛𝑖 is 𝑛𝑗 , the opposite is not definite for 𝑛𝑗 . Therefore, only the case in which both nodes choose each other is agglomerated as the most similar node. Thus, the agglomerated node 𝑛ag is 𝑖 =𝑗 𝑛ms 𝑛ag = 𝑛𝑖 𝑛𝑗 → { 𝑗 }, 𝑛ms = 𝑖
(8)
Input: Graph 𝐺, Nodes 𝑉, Edges 𝐸, density parameter 𝜇 Output: Hierarchically clustered communities (1) Find the eigenvectors 𝛼𝑙 = (𝑙) −1 ] of 𝐷Ω 𝑀𝐷 Ω [𝛼1(𝑙) , . . . , 𝛼𝑁 tr (2) Find the similarities of each node i and j using eigenvectors with Eq. (5) (3) Compute the most similar node i using Eq. (7) (4) Agglomerate the node i and j if the two nodes chose each other as the most similar node (5) Re-initialize the graph with the agglomerated nodes and start the next iteration (6) Agglomerate the nodes into hierarchical clusters when the iteration is finished Pseudocode 1
The evaluation is done using the measurement metrics, such as precision (𝑃), recall (𝑅), and 𝐹-score. 𝑃=
𝑇𝑝 𝑇𝑝 + 𝐹𝑝
,
(9)
where 𝑇𝑝 is the number of nodes that are correctly clustered and 𝐹𝑝 is the number of nodes that are falsely clustered. 𝑅=
𝑇𝑝 𝑇𝑝 + 𝐹𝑛
,
(10)
where 𝑘 ∈ 𝑁 and 𝑁 is all the nodes in graph 𝐺. The termination condition is met when all agglomerated nodes are connected more tightly inside than outside as seen in Pseudocode 1.
where 𝐹𝑛 is the number of nodes that are supposed to be clustered but failed to do so.
4. Experiment
where 𝐹score is the harmonic mean of precision and recall. The results of clustering are shown by varying the value of 𝜇, which is the community density parameter in Figure 2, where there are 3 communities. The right side community is the smallest with 61 nodes, the middle community has 109 nodes, and the left side community is the largest community with 331 communities. An optimal value was evaluated from the training data by a trial-and-error method. Beginning with 𝜇 = 1, as shown in Figure 2(a), each community is clustered into small sized clusters internally. The clustering performance is improved by increasing the value of 𝜇 up to 𝜇 = 3. The middle and right side communities are clustered successfully but the largest community on the left side failed to cluster because the left side community is very complex with many edges. In this case, the clustering performance can be improved by relaxing 𝜇. The three communities are successfully clustered when 𝜇 = 4, as shown in Figure 2(c). If the value of 𝜇 continues to increase, the clustering performance becomes worse than 𝜇 = 4 because the clustering criteria are too relaxed as 𝜇 increases. With 𝜇 =
This section presents the results of the proposed method and compares the data with that of conventional community detection works [21, 22] by varying the value of parameters. LiveJournal and Orkut are used for evaluation as the groundtruth social network. LiveJournal is a blogging and social networking site that has been around since 1999. LiveJournal data has 4 million nodes and 35 million edges. The LiveJournal ground-truth data has 287,512 communities. In order to show the change of detected community by varying the density parameter for different networks, we also use Orkut network because the density difference of two networks can clearly emphasize the importance of choosing optimal density parameter. Orkut is a free online social network where users form friendship with each other. Orkut data has 3 million nodes and 117 million edges. The Orkut ground-truth data has 6,288,363 communities. The network is massive and complex, which makes more difficult clustering task. The dataset is available at https://snap.stanford.edu/data/.
𝐹score =
2∗𝑃∗𝑅 , 𝑃+𝑅
(11)
6
Complexity 2167209
1909241 1726372
2167209
1909241 1726372
867756 453835 1930314
2167550
1915102 2683105
480565 700570
46340
707769 348839 830910 1921843 2257729 347928
1937967 46454 867566
1915579
749144
547590
977046 1953312 578515
2821219
839123 350777
370438 2899547 3019969
2624671
2154232 806486 2664372 2568937 2821041 2568915 776589 728599 833496 2568900 2206381 1962422 1915048 2164094 2568936 2596049 1930860 1883999 1962235 1957581 1937799 349484 866836 806382 424606 1937557 1867785 466957 2174079350087 9694712568895 867255 3679663693882568903 1902498 728523 867164 224752 2684588 1951132 41058 2568869 2568889 2568896 2568925 1959839 834739 23304 372360 2206383 2568876 2684571 2266888 367928 349176 1915849 2821237 2681959
353546226374 369487 347499 1930588 372390 369430 368612 830428 865042
351406
867756
1915102 2683105
480565 700570
2337498 1899299 705277 370411 1965576 364807
867862 32769 350542 869979 1951893 806358 2183234806458 369314 1909345 806474 867601 2435272 3069776 1937955 806545 806408 454083 2627791 1846252291509 8123 2684521 806547 2504992 806689 347573 3069785 347368 728517 2206391 806384 3003140 2269305 835799 2730269 368521 3069777 454096 2568873779952 350407 367229 1929277 1949828 2160570 453913 967944 1946733 2633614 526275 350454 2899529 729780 368814 1948721 3069787 348320 3069781 368049 350303 3069760 2266811 2200468 371308 2865827 349517 348472 1941756 349594 353245 3069767 1946504 2031030 733715 2009329 1949269 2205710 1942228 2009330
453835 1930314
2167550
2627886
347252
2159180
1974004 578522 3512802684597 699079 350116 1885100 1874568 364847 18840452164103 967935 1883166 2164336
2568885 1953847 728585 728522 2568898 2691438 806466 1915049 2568913 3532171948019 2411838347487 1939142 2684603 377143 8691612441153 1961896 2758790 350548 2684642 2041061 2152870 336318 348645 2684538 833412 728567 2568917 728549 2684618 1956874 2155795 2568919 88156 2568926 351590 975385349711 2629492 2568907 2568941 695345 149684 226702 1937545 83778 2164102 728583 2568892 349074 695472 700985 1883982 367687 352382 127952 806501 806364 728516 919975 2684644 2352234 372508 352823967442 116298 909485 369469 349509 352978 337929 608477 2498495 351154 1937296 1880990 1948481 296082 8195 1921398 683365 1949007 349585 347633 357619 695397 368794 866844 832181 306352037147 226920 71780 371142 1908108 37667794807 366751 2154219967848 577275 369574 436887 901020 149617 295702 2150922 370642 367422353681 1913475 797865 641645 728541 247155 353447 563626 2630493 367067 46277 2024676 446922194697 1929805 834058 245239 83724 695443 2568868 2524148 866810 347603 698926 10941 1954771 352750 346867 695704 2568884 1915088370514 867445 697295 1915878697535 350340 577274 806506 361256 793687 1957123 83596 648159 801903 368736 1910956 1927286 342895 2174407 834191 348431 1940756 370128 2314177 226808 420386 80148 95031 367388 377661 698914 837964 480345 1896257 698913 2247234 94780 349968968494 777196 986764 2174473 475867 727917 83684 349683 369921 366628 477696 456859 2174837 348258 347211 10909 727922 351573 771463 147061 697520 1037407 85745 775025 2188124 1929140 352224 83534 370626 2174340 699703 695654 784004 617618 2886476 1916792 353282 2995740 2710857 2438849 446626 367562 1298220
1909267724582 370284
46340
707769 348839 830910 1921843 2257729 347928
1937967
2655654 2671725 352529 969777 95311 969826 2633171 970220 970044
350777
446626 367562 1298220
2821219
970477
25705202190207 970172 970182 970311 969773 348045 967544 1968227 970619 969790 969799 2887917 2671745 970280 969814 2581808 2655508 350968 2671776 970246 2159233 970223 2953795 969791 41085 2671772 969786 1883995 182396
1867789
1937665
547590
839123
578515
926394
970123
970170
1883406
2632477
347199 1931806
453819
347441
1635578 95257
2468261 2986851 24535322468756
2151401
2153515
1898695
350133
351406
2337498 1899299 705277 370411 1965576 364807
867862 32769 350542 869979 1951893 806358 2183234806458 369314 1909345 806474 867601 2435272 3069776 1937955 806545 806408 454083 2627791 1846252291509 8123 2684521 806547 2504992 806689 347573 3069785 347368 728517 2206391 806384 3003140 2269305 835799 2730269 368521 3069777 454096 2568873779952 350407 367229 1929277 1949828 2160570 453913 967944 1946733 2633614 526275 350454 2899529 729780 368814 1948721 3069787 348320 3069781 368049 350303 3069760 2266811 2200468 371308 2865827 349517 348472 1941756 353245 349594 3069767 1946504 2031030 733715 2009329
2624671
977046 1953312
370874
2570417 1942141
370438 2899547 3019969
1949269 2205710 1942228 2009330
2167209
700570
1937967
1867789
707769 348839 830910 1921843 2257729 347928
2821219
578522 3512802684597 699079 350116 1885100 1874568 364847 18840452164103 967935 1883166 2164336
1930860 1883999 424606 466957 2174079350087 728523
547590
977046 1953312
970123
970170
1883406
2632477
347199 1931806
453819
347441
1635578 95257
2468261 2986851 24535322468756
2151401 1898695
2153515
(b) 𝜇 = 3 2167209
453835 1930314 1909267724582 370284
1909241 1726372
839123
350777
867756
700570
2337498 1899299 705277 351406 370411 867862 1965576 32769 350542 364807 869979 1951893 806358 2183234806458 369314 1909345 806474 867601 2435272 3069776 1937955 806545 806408 454083 2627791 1846252291509 8123 2684521 806547 2504992 806689 347573 3069785 347368 728517 2206391 806384 3003140 2269305 835799 2730269 368521 3069777 454096 2568873779952 350407 367229 1929277 1949828 2160570 453913 967944 1946733 2633614 526275 350454 2899529 729780 368814 1948721 3069787 348320 3069781 368049 350303 3069760 2266811 2200468 371308 2865827 349517 348472 1941756 349594 353245 3069767 1946504 2031030 733715 2009329 1949269 2205710 1942228 2009330
1915102 2683105
480565
370438 2899547 3019969
2624671
453835 1930314
2167550
2627886
347252
2159180
1974004
2154232 806486 2664372 2568937 2821041 2568915 776589 728599 833496 2568900 2206381 1962422 1915048 2164094 2568936 2596049 1962235 1957581 1937799 349484 866836 806382 1937557 1867785 9694712568895 867255 3679663693882568903 1902498 867164 224752 2684588 1951132 41058 2568869 2568889 2568896 2568925 1959839 834739 23304 372360 2206383 2568876 2684571 2266888 367928 349176 1915849 28212372568885 2681959 1953847 728585 2691438 728522 2568898 806466 1915049 353546226374 2568913 3532171948019 2411838347487 1939142 2684603 377143 446626 367562 369487 8691612441153 347499 1930588 1961896 2758790 350548 372390 369430 2041061 2684642 2152870 1298220 368612 336318 348645 830428 2684538 865042 833412 1956874 728567 2568917 728549 2684618 2155795 2568919 88156 2568926 351590 975385349711 2629492 2568907 2568941 695345 149684 226702 1937545 83778 2164102 728583 2568892 349074 695472 700985 1883982 367687 352382 127952 806501 806364 728516 919975 2684644 2352234 372508 352823967442 116298 909485 369469 349509 352978 337929 608477 2498495 351154 1937296 1880990 1948481 296082 8195 1921398 683365 1949007 349585 347633 357619 866844 695397 368794 832181 306352037147 226920 71780 371142 1908108 37667794807 366751 2154219967848 577275 369574 436887 901020 149617 295702 2150922 370642 367422353681 1913475 797865 641645 728541 247155 353447 563626 2630493 367067 46277 2024676 446922194697 1929805 834058 245239 83724 695443 2568868 2524148 866810 347603 698926 10941 1954771 352750 346867 695704 2568884 1915088370514 867445 697295 1915878697535 350340 577274 806506 361256 793687 1957123 83596 648159 801903 368736 1910956 1927286 342895 2174407 834191 348431 1940756 370128 2314177 226808 420386 80148 95031 367388 377661 698914 837964 480345 1896257 698913 2247234 94780 349968968494 777196 986764 2174473 475867 727917 83684 349683 369921 366628 477696 456859 2174837 348258 347211 10909 727922 351573 771463 147061 697520 1037407 85745 775025 2188124 1929140 352224 83534 370626 2174340 699703 695654 617618 784004 2886476 1916792 353282 2995740 2710857 2438849 46454
867566
25705202190207 970172 970182 970311 969773 348045 967544 1968227 970619 969790 969799 2887917 2671745 970280 969814 2581808 2655508 350968 2671776 970246 2159233 970223 2953795 969791 41085 2671772 969786 1883995 182396
1937665
1909241 1726372
2167550
1915102 2683105
480565
578515
926394
970477
2570417 1942141
867756
1915579
370874
2655654 2671725 352529 969777 95311 969826 2633171 970220 970044
2305396
(a) 𝜇 = 1
749144
2671767
2671706
350133
2305396
46340
2627886
347252
2159180
1974004 578522 3512802684597 699079 350116 1885100 1874568 364847 18840452164103 967935 1883166 2164336
2154232 806486 2664372 2568937 2821041 2568915 776589 728599 833496 2568900 2206381 1962422 1915048 2164094 2568936 2596049 1930860 1883999 1962235 1957581 1937799 349484 866836 806382 424606 1937557 1867785 466957 2174079350087 9694712568895 867255 3679663693882568903 1902498 728523 867164 224752 2684588 1951132 41058 2568869 2568889 2568896 2568925 1959839 834739 23304 372360 2206383 2568876 2684571 2266888 367928 349176 1915849 28212372568885 2681959 1953847 728585 2691438 728522 2568898 806466 1915049 353546226374 2568913 3532171948019 2411838347487 1939142 2684603 377143 369487 869161 347499 2758790 1930588 1961896 2441153 350548 372390 369430 2152870 2684642 2041061 368612 336318 348645 830428 2684538 865042 833412 1956874 728567 2568917 728549 2684618 2155795 2568919 88156 351590 2568926 975385349711 2629492 2568907 2568941 695345 149684 226702 1937545 83778 2164102 728583 2568892 349074 695472 700985 1883982 367687 352382 127952 806501 806364 728516 919975 2684644 2352234 372508 352823967442 116298 909485 369469 349509 352978 337929 608477 2498495 351154 1937296 1880990 1948481 296082 8195 1921398 683365 1949007 349585 347633 357619 866844 695397 368794 832181 306352037147 226920 71780 371142 1908108 37667794807 366751 2154219967848 577275 369574 436887 901020 149617 295702 2150922 367422 370642 353681 1913475 797865 641645 728541 247155 353447 563626 2630493 367067 46277 194697 2024676 446922 1929805 834058 245239 83724 695443 2568868 2524148 866810 347603 698926 10941 1954771 352750 695704 2568884 346867 1915088370514 867445 697295 1915878697535 350340 577274 806506 361256 793687 1957123 83596 648159 801903 368736 1910956 1927286 342895 2174407 834191 348431 1940756 370128 2314177 226808 420386 80148 95031 367388 377661 698914 837964 480345 1896257 698913 2247234 94780 349968968494 777196 986764 2174473 475867 727917 83684 349683 369921 366628 477696 456859 2174837 348258 347211 10909 727922 351573 771463 147061 697520 1037407 85745 775025 2188124 1929140 352224 83534 370626 2174340 699703 695654 784004 617618 2886476 1916792 353282 2995740 2710857 2438849 46454
867566
2671767
2671706
1915579
749144
1909267724582 370284
46340
749144
1937967
970182
348045 970619 969790 969799 2671745 970280 969814 2655508 970246 969791
1867789
1937665 2570417 1942141
970170
350968
2671776 2581808 2159233 970223
41085 2671772 969786
2887917
2953795
1883995
970123
182396 1883406
2632477
347199 1931806
453819
347441
1635578 95257
2468261 2986851 2151401 1898695
2337498 1899299 705277 351406 370411 867862 1965576 32769 350542 364807 869979 1951893 806358 2183234806458 369314 1909345 806474 867601 2435272 3069776 1937955 806545 806408 454083 2627791 1846252291509 8123 2684521 806547 2504992 806689 347573 3069785 347368 728517 2206391 806384 3003140 2269305 835799 2730269 368521 3069777 454096 2568873779952 350407 367229 1929277 1949828 2160570 453913 967944 1946733 2633614 526275 350454 2899529 729780 368814 1948721 3069787 348320 3069781 368049 350303 3069760 2266811 2200468 371308 2865827 349517 348472 1941756 349594 353245 3069767 1946504 2031030 733715 2009329
1930860 1883999 424606 466957 2174079350087 728523
350777
2821219
25705202190207 970172 970311 969773 967544 1968227
24535322468756 2153515
350133
2627886
370438 2899547 3019969
2624671
839123
578515
926394
2655654 2671725 352529 969777 95311 969826 2633171 970220 970044 970477
707769 348839 830910 1921843 2257729 347928
547590
977046 1953312
370874
578522 3512802684597 699079 350116 1885100 1874568 364847 18840452164103 967935 1883166 2164336
347252
2159180
2154232 806486 2664372 2568937 2821041 2568915 776589 728599 833496 2568900 2206381 1962422 1915048 2164094 2568936 2596049 1962235 1957581 1937799 349484 866836 806382 1937557 1867785 9694712568895 867255 3679663693882568903 1902498 867164 224752 2684588 1951132 41058 2568869 2568889 2568896 2568925 1959839 834739 23304 372360 2206383 2568876 2684571 2266888 367928 349176 1915849 28212372568885 2681959 1953847 728585 2691438 728522 2568898 806466 1915049 353546226374 2568913 3532171948019 2411838347487 1939142 2684603 377143 446626 367562 369487 8691612441153 347499 1930588 1961896 2758790 350548 372390 2041061 369430 2684642 2152870 1298220 368612 336318 348645 830428 2684538 865042 833412 1956874 728567 2568917 728549 2684618 2155795 2568919 88156 2568926 351590 975385349711 2629492 2568907 2568941 695345 149684 226702 1937545 83778 2164102 728583 2568892 349074 695472 700985 1883982 367687 352382 127952 806501 806364 728516 919975 2684644 2352234 372508 352823967442 116298 909485 369469 349509 352978 337929 608477 2498495 351154 1937296 1880990 1948481 296082 8195 1921398 683365 1949007 349585 347633 357619 866844 695397 368794 832181 306352037147 226920 71780 371142 1908108 37667794807 366751 2154219967848 577275 369574 436887 901020 149617 295702 2150922 367422353681 370642 1913475 797865 641645 728541 247155 353447 563626 2630493 367067 46277 194697 2024676 446922 1929805 834058 245239 83724 695443 2568868 2524148 866810 347603 698926 10941 1954771 352750 695704 2568884 346867 1915088370514 867445 697295 1915878697535 350340 577274 806506 361256 793687 1957123 83596 648159 801903 368736 1910956 1927286 342895 2174407 834191 348431 1940756 370128 2314177 226808 420386 80148 95031 367388 377661 698914 837964 480345 1896257 698913 2247234 94780 349968968494 777196 986764 2174473 349683 727917 83684 475867 369921 366628 477696 456859 2174837 348258 347211 10909 727922 351573 771463 147061 697520 1037407 85745 775025 2188124 1929140 352224 83534 370626 2174340 699703 695654 617618 784004 2886476 1916792 353282 2995740 2710857 2438849 46454
867566
2671767
2671706
1915579
1909267724582 370284
1974004
19492692205710 1942228 2009330
1867789
1937665
2671767 370874
2671706
926394
2655654 2671725 352529 969777 95311 969826 2633171 970220 970044 970477
25705202190207 970172 970182 970311 969773 348045 967544 1968227 970619 969790 969799 2887917 2671745 970280 969814 2581808 2655508 350968 2671776 970246 2159233 970223 2953795 969791 41085 2671772 969786 1883995 182396
2570417 1942141
970170
970123
1883406
2632477
347199 1931806
453819
347441
1635578 95257
2468261 2986851 2151401 1898695
24535322468756 2153515
350133
2305396
2305396
(c) 𝜇 = 4
(d) 𝜇 = 5 2167209
1909241 1726372 867756 453835 1930314
2167550
1915102 2683105
480565 700570
578515
2821219
749144
1915579
2627886
347252
2159180
578522 3512802684597 699079 350116 1885100 1874568 2624671 364847 18840452164103 967935 707769 1937967 1883166 2164336 348839 2154232 806486 2664372 2568937 2821041 2568915 776589 830910 1921843 728599 46454 833496 2257729 347928 2568900 2206381 1962422 1915048 2164094 2568936 2596049 1930860 1883999 867566 1962235 547590 1957581 1937799 349484 866836 806382 424606 1937557 1867785 466957 2174079350087 9694712568895 977046 1953312 867255 3679663693882568903 1902498 728523 867164 224752 2684588 1951132 41058 839123 2568869 2568889 2568896 2568925 1959839 834739 23304 372360 2206383 2568876 2684571 2266888 367928 349176 1915849 28212372568885 2681959 350777 1953847 728585 728522 2568898 2691438 806466 1915049 353546226374 2568913 3532171948019 2411838347487 1939142 2684603 377143 446626 367562 369487 8691612441153 347499 1930588 1961896 2758790 350548 372390 369430 2152870 2684642 2041061 1298220 368612 336318 348645 830428 2684538 865042 833412 728567 2568917 728549 2684618 1956874 2155795 2568919 88156 2568926 351590 975385349711 2629492 2568907 2568941 695345 149684 226702 1937545 83778 2164102 728583 2568892 349074 695472 700985 1883982 367687 352382 127952 806501 806364 728516 919975 2684644 2352234 372508 352823967442 116298 909485 369469 349509 352978 337929 608477 2498495 351154 1937296 1880990 1948481 296082 8195 1921398 683365 1949007 349585 347633 357619 866844 695397 368794 832181 306352037147 226920 71780 371142 1908108 37667794807 366751 2154219967848 577275 369574 436887 901020 149617 295702 2150922 367422353681 370642 1913475 797865 641645 728541 247155 353447 563626 2630493 367067 46277 194697 2024676 446922 1929805 834058 245239 83724 695443 2568868 2524148 866810 347603 698926 10941 1954771 352750 695704 2568884 346867 1915088370514 867445 697295 1915878697535 350340 577274 806506 361256 793687 1957123 83596 648159 801903 368736 1910956 1927286 342895 2174407 834191 348431 1940756 370128 2314177 226808 420386 80148 95031 367388 377661 698914 837964 480345 1896257 698913 2247234 94780 349968968494 777196 986764 2174473 349683 727917 83684 475867 369921 366628 477696 456859 2174837 348258 347211 10909 727922 351573 771463 147061 697520 1037407 85745 775025 2188124 1929140 352224 83534 370626 2174340 699703 695654 784004 617618 2886476 1916792 353282 2995740 2710857 2438849 46340
1909267724582 370284
1974004 370438 2899547 3019969
2337498 1899299 705277 351406 370411 867862 1965576 32769 350542 364807 869979 1951893 806358 2183234806458 1909345 806474 867601 2435272 3069776 1937955 806545 806408 454083 2627791 1846252291509 8123 2684521 806547 2504992 806689 347573 3069785 347368 728517 2206391 806384 3003140 2269305 835799 2730269 368521 3069777 454096 2568873779952 350407 367229 1929277 1949828 2160570 453913 967944 1946733 2633614 526275 350454 2899529 729780 368814 1948721 3069787 348320 3069781 368049 350303 3069760 2266811 2200468 371308 2865827 349517 348472 1941756 349594 353245 3069767 1946504 2031030 733715 2009329 369314
1949269 2205710 1942228 2009330
1867789
2671767 370874
2671706
926394
2655654 2671725 352529 969777 95311 969826 2633171 970220 970044 970477 2671745
25705202190207 970172 970311 969773 348045 967544 1968227 970619 969790 969799 2887917 970280 969814 970182
2655508 970246 969791
1937665 2570417 1942141
970170
350968
2671776 2581808 2159233 970223
41085 2671772 969786
2953795
1883995
970123
182396 1883406
2632477
347199 1931806
453819
347441
1635578 95257
2468261 2986851 2151401 1898695
24535322468756 2153515
350133
2305396
(e) 𝜇 = 7
Figure 2: Varying the results of the density parameter 𝜇 to obtain the optimal value.
5, the left side community is clustered into 4 communities, as shown in Figure 2(d), and when the value of 𝜇 reaches 7, the left side community is separated into smaller communities, as shown in Figure 2(e). Figure 3 shows comparison by varying values of density parameter 𝜇 of Orkut network. Unlike LiveJournal network, Orkut is a more densely clustered network where the ratio of node and edges of LiveJournal is 1 : 8.6 whereas Orkut is 1 : 38.1. Therefore, the density parameter of Orkut requires being more strict compared to LiveJournal because the clusters are all densely clustered with each other. If the density parameter is not strict, it will allow the densely clustered nodes to be agglomerated together. First column of Figure 3 shows the ground-truth community which is colored in yellow and the following columns are detected networks by varying density parameters ranging from 0.1 to 4. As shown in the first row of Figure 3, the detected community with density parameter 0.1 shows well desired result but the accuracy has sharply dropped when we increased the density parameter because the detected community’s size has
continuously increased. The second row of Figure 3 shows different characteristics compared to the first row where the seed node has agglomerated into different cluster due to the relaxed density parameter. The third row network has similar characteristics to the second row network which shows that relaxed density parameter could lead to less densely clustered community. Fourth row network has similar results to the first row which shows that if we allow relaxed density parameter, the network will continue to expand. The optimal result of community detection has been obtained with 0.1 in Orkut network while the optimal value is 4 in LiveJournal network. Therefore, the experiment result shows that the density parameter is closely related to density of the network where the denser network requires stricter density parameter. It means that when evaluating the optimal value of density parameter, the density of the network should be considered. Figure 4 shows the results of an analysis of the agglomeration process of the proposed method with 𝜇 = 4 and AHKSC as the number of iterations increase. Figure 4(a) shows the early stage agglomeration result of middle community
Complexity
7
= 0.1
Ground-Truth
=1
2658126 1422221 899761 2412062 8013961612762 2101633 1256304 2101508 1420867 2782124 2071915 2501534 6108211481117 1534112 1904930 2308819 1795757 1302830 2071702 2121579 14809751668863 2115323 1562603 1147991 1072646 1481265
2658126 1422221 899761 2412062 801396 1612762 2101633 1256304 2101508 1420867 2782124 2071915 2501534 6108211481117 1534112 1904930 2308819 1795757 1302830 2071702 2121579 1480975 1668863 2115323 1562603 1147991 1072646 1481265 1883625 723995 2071918 2120143 1691870 1492536 1905536 1034233 1987391 2625674 1602491 1408975 13759651408884 466613 1810619 1264358 1379264 1775464 1905912 1297432 2091231 734826 1459160 1558007 2354551 1420784 2134243 2248669 1833619 2120126 1368311 1557999 1029079 1396795 1686582 1987521 1177808 1221679 211260 1987240 2335153 2601388 2137044 1987531 2069988 1987357 1987503 1987510 2137239 1987439 19872241987189 1310012 1987366 1987266 1987204 1987554 1860214 1987450 1987484 1987463 1987499 2188016 1987408 1987471 1987250 1987430 1987406 1987393 890186 999432 890340 1987280 1987533 1987535 1987465 19874591987187 1987539 1276075 1987428 1009257 1987416 1518390 1303966 19875751987519 2111281 1987494 19875561987311 1987529 1987382 1987454 1987318 164222 1987256 1987361 1987198 1285713 1987402 1284174 1987385 1987307 1987461 1987452 1437756 1987234 1987542 1987229 1987338 1987288 1987315 1987325 1840963 890116 1987417 2114742 1987558 1987502 1987242 1987188 1987328 1987399 1987545 1351473 1987409 478882 1987232 1987482 1987210 1987253 1987526 1987438 1987202 1987351 1987551 18257291832949 1987467 1987304 1987568 1987314 1987515 1987203 2120776 1987490 1987421 1987447 1987572 19873351987413 1987507 1280211 1987341 48126 18000261563534 1862284 1987260 1987237 1987433 1987573 1987443 1987448 1987387 1987247 1092732 1451970 1987369 1987258 1437258 1987215 1122196 1050829 1987511 1987194 1987372 1987218 1840841 1987321 1987208 1987396 1987498 1987478 1987517 1987426 1165635 1987294 1860586 19872691987196 1987522 618 1987560 1842201 1987332 1520863 1987566 1987434 1987344 1987301 1987513 1987492 1987557 1987276 1987540 1987543 1987271 1987570 1987355 1987263 1987424 1987501 1987291 1987306 1437223 1987547 1987326 1987200 1987486 1987296 1987477 1987469 1987456 1987505 1987279 1987376 1987562 1987473 1987226 1987488 1987524 1987564 1987441 1987216 1987348 1987480 1987508 1987496 1987285 1987274 1987475 1987537 1987436 1987549 1987363 1987282 1987445 1987223 1987244 1987374 1987212 1201462 1987227 1205996 1987190 1987389 1987451
723995
1883625
2071918
1691870 1034233
1905536 1987391 1602491 1408975
2354551
1492536
2120143
2625674 1408884 1375965 466613 1810619 1264358 1905912 1297432 2091231 1459160
1775464 734826
1029079
1987521
1379264
1558007 2248669 1833619 2120126 1368311 1557999 1396795 1686582 1177808 1221679 211260
1420784
2134243
1987240 1987531 1987357 1987503 1987510 19872241987189 1987366 1987266 1987554 1987450 19874841987463 1987499 1987408 1987471 1987250 1987430 1987393 1987280 1987533 1987406 198753 5 1987465 19874591987187 1987539 1987428
2335153 2601388
2137044 2069988 2137239 1310012
1987439
1987204 1860214 2188016
1276075 1009257 1987416 1518390 1303966 19875751987519 1987494 19875561987311 1987529 1987382 1987454 1987318 164222 1987256 1987361 1987198 1987402 1284174 1987385 1987307 1987234 1987461 1987452 1437756 1987542 1987229 1987338 1987288 1987315 1987325 1840963 1987417 2114742 1987558 1987502 19872421987188 1987328 1987399 1987545 1351473 1987409 478882 1987232 1987482 1987210 1987253 1987526 1987438 1832949 1987202 1987351 1987551 1825729 1987467 1987568 1987304 1987314 1987515 1987203 2120776 1987490 1987421 1987447 1987572 19873351987413 1987507 1280211 1987341 48126 18000261563534 1862284 1987260 1987237 1987433 1987573 1987443 1987448 1987387 1987247 14519701092732 1987369 1987258 14372581987215 1122196 1050829 1987511 1987194 1987372 1987218 1840841 1987321 1987208 1987396 1987498 1987478 1987517 1987426 1165635 1987294 1860586 19872691987196 1987522 1842201 1987560 1987332 618 1520863 1987566 1987434 1987344 1987301 1987513 1987492 1987557 1987276 1987540 1987543 1987271 1987570 1987355 1987263 1987424 1987501 1987291 1987306 1437223 1987547 1987200 1987486 1987296 1987477 1987326 1987469 1987456 1987505 1987279 1987376 1987562 1987473 1987226 1987488 1987524 1987564 1987441 1987216 1987348 1987480 1987508 1987496 1987285 1987274 1987475 1987537 1987436 1987549 1987363 1987282 1987445 1987223 1987244 1987374 1987212 1201462 1987227 1987190 1205996 1987389 1987451
890186 999432
=4
2658126 1422221 899761 2412062 801396 1612762 2101633 1256304 2101508 1420867 2782124 2071915 2501534 6108211481117 1534112 1904930 2308819 1795757 1302830 2071702 2121579 1480975 1668863 2115323 1562603 1147991 1072646 1481265 1883625 723995 2071918 2120143 1691870 1492536 1905536 1034233 1987391 2625674 1602491 1408975 13759651408884 466613 1810619 1264358 1379264 1775464 1905912 1297432 2091231 734826 1459160 1558007 2354551 1420784 2134243 2248669 1833619 2120126 1368311 1557999 1029079 1396795 1686582 1987521 1177808 1221679 211260 1987240 2335153 2601388 2137044 1987531 2069988 1987357 1987503 1987510 2137239 1987439 19872241987189 1310012 1987366 1987266 1987204 1987554 1860214 1987450 1987484 1987463 1987499 2188016 1987408 1987471 1987250 1987430 1987406 1987393 890186 999432 890340 1987280 1987533 1987535 1987465 19874591987187 1987539 1276075 1987428 1009257 1987416 1518390 1303966 19875751987519 2111281 1987494 19875561987311 1987529 1987382 1987454 1987318 164222 1987256 1987361 1987198 1285713 1987402 1284174 1987385 1987307 1987461 1987452 1437756 1987234 1987542 1987229 1987338 1987288 1987315 1987325 1840963 890116 1987417 2114742 1987558 1987502 1987242 1987188 1987328 1987399 1987545 1351473 1987409 478882 1987232 1987482 1987210 1987253 1987526 1987438 1987202 1987351 1987551 18257291832949 1987467 1987304 1987568 1987314 1987515 1987203 2120776 1987490 1987421 1987447 1987572 19873351987413 1987507 1280211 1987341 48126 18000261563534 1862284 1987260 1987237 1987433 1987573 1987443 1987448 1987387 1987247 1092732 1451970 1987369 1987258 1437258 1987215 1122196 1050829 1987511 1987194 1987372 1987218 1840841 1987321 1987208 1987396 1987498 1987478 1987517 1987426 1165635 1987294 1860586 19872691987196 1987522 618 1987560 1842201 1987332 1520863 1987566 1987434 1987344 1987301 1987513 1987492 1987557 1987276 1987540 1987543 1987271 1987570 1987355 1987263 1987424 1987501 1987291 1987306 1437223 1987547 1987326 1987200 1987486 1987296 1987477 1987469 1987456 1987505 1987279 1987376 1987562 1987473 1987226 1987488 1987524 1987564 1987441 1987216 1987348 1987480 1987508 1987496 1987285 1987274 1987475 1987537 1987436 1987549 1987363 1987282 1987445 1987223 1987244 1987374 1987212 1201462 1987227 1205996 1987190 1987389 1987451
890340
2111281 1285713 890116
1987527
1987527
2658126 1422221 899761 2412062 801396 1612762 2101633 1256304 2101508 1420867 2782124 2071915 2501534 6108211481117 1534112 2308819 2071702 1480975 1668863 2115323 1562603 1147991 1072646 1481265 1883625 723995 2071918 2120143 1691870 1492536 1905536 1034233 1987391 2625674 1602491 1408975 13759651408884 466613 1810619 1264358 1379264 1775464 1905912 1297432 2091231 734826 1459160 1558007 2354551 1420784 2134243 2248669 1833619 2120126 1368311 1557999 1029079 1396795 1686582 1987521 1177808 1221679 211260 1987240 2335153 2601388 2137044 1987531 2069988 1987357 1987503 1987510 2137239 1987439 19872241987189 1310012 1987366 1987266 1987204 1987554 1860214 1987450 1987484 1987463 1987499 2188016 1987408 1987471 1987250 1987430 1987406 1987393 890186 999432 890340 1987280 1987533 1987535 1987465 19874591987187 1987539 1276075 1987428 1009257 1987416 1518390 1303966 19875751987519 2111281 1987494 19875561987311 1987529 1987382 1987454 1987318 164222 1987256 1987361 1987198 1285713 1987402 1284174 1987385 1987307 1987461 1987452 1437756 1987234 1987542 1987229 1987338 1987288 1987315 1987325 1840963 890116 1987417 2114742 1987558 1987502 1987242 1987188 1987328 1987399 1987545 1351473 1987409 478882 1987232 1987482 1987210 1987253 1987526 1987438 1987202 1987351 1987551 18257291832949 1987467 1987304 1987568 1987314 1987515 1987203 2120776 1987490 1987421 1987447 1987572 19873351987413 1987507 1280211 1987341 48126 18000261563534 1862284 1987260 1987237 1987433 1987573 1987443 1987448 1987387 1987247 1092732 1451970 1987369 1987258 1437258 1987215 1122196 1050829 1987511 1987194 1987372 1987218 1840841 1987321 1987208 1987396 1987498 1987478 1987517 1987426 1165635 1987294 1860586 19872691987196 1987522 618 1987560 1842201 1987332 1520863 1987566 1987434 1987344 1987301 1987513 1987492 1987557 1987276 1987540 1987543 1987271 1987570 1987355 1987263 1987424 1987501 1987291 1987306 1437223 1987547 1987326 1987200 1987486 1987296 1987477 1987469 1987456 1987505 1987279 1987376 1987562 1987473 1987226 1987488 1987524 1987564 1987441 1987216 1987348 1987480 1987508 1987496 1987285 1987274 1987475 1987537 1987436 1987549 1987363 1987282 1987445 1987223 1987244 1987374 1987212 1201462 1987227 1205996 1987190 1987389 1987451 1904930 1795757 1302830 2121579
1987527
1987527
383374 77604 383374
108662
343449
57905
108671
108674
108672
343435
343440
1809
105306 1808
1822
1818
108680
1795 105603
1817
1886687
108665 2779894
1789
820902
2784719
193722
1814
108677
108683
107367 1808
1822
1817
1886687
108665 2779894
1789 108680
1795 105603
64583 105307 1806
108683
1819
346017
108679
1818
937402 2023612
166759 2581248 767662
457400
2010837
2236410
943219
178510
2779893
2581248
343435
1809
1817
108680
803554
346017 52762 943219
530662
537849 300556
457400
178510
812068
590027
590027 253796
76568
704418
704418
1971847
850113 590027
489572
590027
253796
76568
489572
2403112
2403112
253796
76568
489572
704418
704418
2403112 400796
2371909
547543
400796
2371909
235579
547543
2001420
319865
441190 175134 2018382
2208419
400796
2371909
2001420
547543
235579
2001420
3057016
175134 2018382
2538674
744437 2289092
319865
441190
3057016
2208419
2538674 744437 2289092
235579
1995782
2001420
3057016 175134 2018382
2538674
547543 1995782
441190
3057016
400796
2371909 319865
441190
2403112
235579
1995782
1995782 319865
2779893
812068
1971847
850113
253796
76568
489572
2010837
2236410
1971847
850113
1971847
2581248 767662
2779890
803554
2779893
812068
850113
166759
236564
537849 2010837
2041780
633786 178518
300556 2236410
937402 2023612
108670
44707
166759 2581248
767662
457400
178510
820902
2784719
193722 108677
108683
108679
2041780 236564 2779890
78348
1886687
108665 2779894
1789
1795 105603
108676 530662
633786
343440
108682
1807
105306
937402 2023612
108670
44707
178518 2779893
812068
316023 343447
108672
1805 1791
52762
537849 2010837
2236410
1818
216297
108664
1801
1808
1822
300556 457400
108681
38 107367 1814
346017
767662
2779890
803554 537849
300556
178510
1819 820902
2784719
193722 108677
108683
108679
108676
236564
178518
236564 2779890
108680
78348
1886687
108665 2779894
1789
1795 105603
2041780
633786
2041780
633786
803554
1809
1817
343452 268342
108674
108678
64583 105307 1806
343440
108682
1807
1805 1791
166759
530662
44707
343435
2000921 173354
343437268329
545757
107084
1800
188397
106897
57914
343447
316023
226196
108669
216297
108664
530662
44707
178518
108681 108678
108672
1808
1822 937402 2023612
108670
108676
108675 108667
1801
105306
107367 1814
52762 943219
108670
108674
38
1791
52762 943219 108676
268342
222106
343449 268336
2784717 434973
108671 108666 108673
108684
108668
343437268329
545757
107084
1800
108677
346017
108679
820902
2784719
193722
77604
57905
343452
188397
2719322 54270
540124
97541
108662
106897
57914 78348
1809
226196
108669
108667
343440
108682
1807
105306
1805
1805 1791
343435
108675
1801
38
108682 78348
1814
1819
1801 1807
108672
105307 1806
108684
108668
343447
266319 599905 108663
2000921 173354
268336
2784717 434973
108671 108666 108673
222106
343449
540124 57905
216297 316023
383374
77604 2719322 54270
108663 97541
108662
108681 108678
64583 343447
316023
38 107367
108674
107084
1800
216297
108678
105307 1806 1818
383374 266319 599905
268342 343437 268329
108664
108681
107084 64583
173354
343452
188397
545757 106897
57914
343437 268329
545757
108664
1819
226196 108669
108667
2000921
268336
434973
2784717
108666 108673
106897
57914
1800
108671
108684
108668 343452
188397
268342
108669
108667
57905
173354 108675
226196
343449
540124
97541
108662
2000921
268336
434973
2784717
108666 108673
108684
108668 108675
2719322 54270
108663
222106
2719322 54270
540124
222106
599905
266319 77604
266319 599905 108663 97541
175134 2018382
2208419 2538674
744437 2289092
2208419
744437 2289092
1004 1004
1004
2252518
2252502
2252456
2216752
65030
2252471
2252513
2252530
2252450 2251878
2252474
2252500 2252470
2211472
2252459 2252522
22524952252465
2252509
2211390 232966
2252476
2211380 2211389
2252481
2252442 2252515
2211451
2211395
2252500
2252509
2211399 2211437 2211374
2211472
2211390 232966
2252476
2252453
2252448
2252526
2252481
2252464
2252450
2252442
1909024 2252515
2252536 2252533
2211451
2211395
2252474
2252500
22524952252465
2252459 2252522 2252509
2211399 2211437 2211374
2211472
2211390 232966
2252476
2252453
2252448
2252526
2252481
2252464
2211380 2211389
2252442
1909024 2252515
2252536 2252533
2211451
2211395
2211381
2252457 2211398 2211453
2252481
965576
2252528 2252532 2252489
2252469
2252466
1990863
2252484
2252466
1990863
2252504
2252484
2252535
2252527 2252442
1909024
2252515
2211451
2211395
2252536 2252533
464385
699348
464385
2023737
2023733
966264
1905522
966264
2176706
1905522
173926 2664957
966264
2664957
546551
546551
1597712 1763623
1441216
204752
2099596
40864 7
360775 982855
1919790
69726
512848 2382063 648274 647115 3044824
3003423 439959 1919887
191978 2
57318
2000741
2151257 1265648 87153 1919865 1256577 4286381102674 1919852 2559307 382299 2369948
2382032
3038848 648248
2164855
57357
2908928
2164869 216481 9
216486 7
2164836 2164871
2164845
3038848 648248
647115 3044824
2164865
216482 3
2164826
1919782
2164855
2164820 2164852 2164830 2164837 57357
2164837
2164819
3038848 648248
1251761
647115 3044824
1184269 2164869
2164867
2164874
2164836 2164871
2164845
2164865
2164823
2164826
1919782
57357
2164833
2164855
2164835
2164841
2164820 2164852 2164830 2164837 718 928982
2164817 151751
57357
2164846 2164858
2164839 2164859 2164853
2164859 2164853
2164822
2164851
2164819
3003430
57466
635211
1184269 2164869
2164867
2164874
2164836
2164823
2164826
2164871
2164845
2164865
Figure 3: Comparison by varying density parameters on Orkut network.
916104
2164876
2164851
2164819
1251761
2789923
2908928
3003429
819303
1919838
2164847 2164843 2164862 1919788 2293036 580482
383610
2164817 151751
916104
2164876
69726
3055530
2164824
2436490
2164846 2164858
1262774
2816043
3003423 439959 1919887
982862 535975 2382080 2164860
2164855
2745022
184431 1503465
3057279
512848 2382063 648274 1655270
2164839
2164822
2164851
3038862
1919888 641990
982866 1326175
57318
2164820 2164852 2164830
1128710
547897 2382002
3003430
2164835
2164841
718 928982
1919785
1897018 598997
2164833
2164859 2164853
916104
2164876
2000741
886362 2793186
1181562 40505 2382026 2590569 2613449 2164349 2381959 2902391 457103 1804362 2624549 964485 3049347 2382263 2382009 29714921919703 2268073 1763064 428670 2171279 80309 1183092 1908248 2930377 2182929 1048180 749339 955779 2730322 1919698 1453217 2382325 198719 982934 2381949982800 578495 546406 642115 6940042995401 921005 588826 612045 918585 28160422347581 1258289 2171274 2382197 1919890 2382172 21766175078 2382022 2381967 2171270 2382284 2555567 408101 2361980 409247 578476 2171286 1919880
2226910
819303
57466
635211
1919790
2745007
2789923
2908928
3003429
2382032
1163206554287 1397156
185366 625756
1919838
2164817 151751
2164846 2164858
408647
2151257 1265648 87153 1919865 1256577 4286381102674 1919852 2559307 382299 2369948
1019303
3055530
2164824
2164847 2164843 2164862 1919788 2293036 580482
383610
57318
2164835
2164841
69726
846320 2781617
360775
1262774
2816043
3003423 439959 1919887
982862 535975 2382080 2164860 2436490
204752
982855
184431 1503465
3057279
512848 2382063 648274
1251761
1597712 1763623
1441216
2745022
547897 2382002 1655270
2164839
2164874
3038862
1919888 641990
819303
3003430
57466 718 928982
2164822
118426 9
2000741
420517
1128710
1897018 598997
2164833
2164859 2164853
916104
2164876
2382032
982866 1326175
2099596
809817
1919785
1181562 40505 2382026 2590569 2613449 2164349 2381959 2902391 457103 1804362 2624549 964485 3049347 2382263 2382009 29714921919703 2268073 1763064 428670 2171279 80309 1183092 1908248 2930377 2182929 1048180 749339 955779 2730322 1919698 1453217 2382325 198719 982934 2381949982800 578495 546406 642115 6940042995401 921005 588826 612045 918585 28160422347581 1258289 2171274 2382197 1919890 2382172 21766175078 2382022 2381967 2171270 2382284 2555567 408101 2361980 409247 578476 2171286 1919880
2226910
2789923
300342 9
216481 7 151751
1919790
2745007
366934
1017023 1925631
2788810
2793186
2151257 1265648 87153 1919865 1256577 4286381102674 1919852 2559307 382299 2369948
1163206554287 1397156 185366 625756
886362
1019303
1262774
1919838
2164847 2164843 2164862 1919788 2293036 580482
383610
57318
846320 2781617
360775
3055530
2164824
2436490
635211
2164851
69726
3003423 439959 1919887
1919782
1251761
204752
2816043
512848 2382063 648274 647115 3044824
2164839
2164822
1597712 1763623
1441216
982855
184431 1503465
3057279
982862 535975 2382080 2164860
2164846 2164858
420517 2745022
3038567
408647
809817
1128710
547897 2382002 1655270
3003430
57466 2164837
3038862
1919888 641990
982866 1326175
2099596
2793186 1919785
768311 1246432
366934
1017023 1925631
2788810
1897018 598997
2164833
71 8 928982
886362
1181562 40505 2382026 2590569 2613449 2164349 2381959 2902391 457103 1804362 2624549 964485 3049347 2382263 2382009 29714921919703 2268073 1763064 428670 2171279 80309 1183092 1908248 2930377 2182929 1048180 749339 955779 2730322 1919698 1453217 2382325 198719 982934 2381949982800 578495 546406 642115 6940042995401 921005 588826 612045 918585 28160422347581 1258289 2171274 2382197 1919890 2382172 21766175078 2382022 2381967 2171270 2382284 2555567 408101 2361980 409247 578476 2171286 1919880
2226910
1028610 3038567
408647
1019303
982855
1919790
2745007
278992 3
2164820 216485 2 2164830
846320 2781617
360775
1163206554287 1397156
185366 625756
81930 3
1919838
2164847 2164843 216486 2 191978 8 38361 0 2293036 580482 2908928 2164835 2164841
63521 1
204752
3055530
2164824 982862 53597 5 2382080 216486 0 2436490
1597712 1763623
1441216
281604 3
3057279
1919888 641990
982866 1326175
547897 2382002 1655270
420517
2793186 809817 1919785 2151257 126564 88715 3 1128710 1919865 1256577 428638110267 4 2745022 1919852 2559307 1019303 382299 236994 8 184431 2382032 1262774 3038862 1503465 3038848 648248
118156 2 40505 2382026 259056 9 2226910 2613449 2164349 2745007 1897018 598997 238195 1804362 9 2902391 457103 2624549 96448 5 3049347 2382263 2382009 29714921919703 2268073 1763064 428670 217127 9 80309 1183092 1908248 2930377 2182929 1048180 74933 9 95577 9 1453217 273032 2 1919698 2382325 198719 98293 4 2381949982800 578495 546406 642115 6940042995401 921005 588826 612045 918585 28160422347581 1258289 2171274 2382197 191989 0 2382172 21766175078 2382022 2381967 2171270 2382284 2555567 408101 2361980 40924 7 578476 2171286 1919880
2099596
809817
886362
846320 278161 7
1163206554287 1397156 18536 6 625756
546551 303305
1028610 768311 1246432
366934
1017023 1925631
2788810
1017023 1925631
2788810 420517
1991097
3038567
3038567
2060203
303305 1028610 768311 1246432
1028610 366934
3042443
2464073
3042443
2464073
1991097
303305 546551
768311 124643 2
2176706
1905522
173926
2060203 1991097
1991097
303305
966264
2664957
2664957
3042443
2464073
2023733
731152
2023756
2060203
2060203
2652509
2176706
1905522
731152
2023756
2025841
704284
173926
731152
3042443
2464073
2023737
2023733
173926 2023756
2219157
2025841
704284
2652509
2023737
2023733
2252531 2219157
2025841
704284
2652509
2176706
731152
2023756
699348
2252531 2219157
2025841
704284
2652509
2049979 464385
699348
2252531 2219157
2252504
2252486 2271672
2049979
699348
2252531
2252489
2252484
2252504
2271672
2049979
2252528 2252532 2252469
2252466
1990863 2252484 2252486
2271672
2049979
2023737
2252466
535254
2252489
2252469
1990863
2252504
2252453 2252464
2211378
1909024
2252528 2252532
535254
2252486
2252486 2271672 464385
2252489
2252469
965481 2252443
2211399 2211437 2211374
2252536 2252533
965576
2252528 2252532
535254
60325 2211372 947637
2211388
2252464
2211378
965576 965576
535254
2211424
1937262 2211447
65007
2252448
2252526
2252453
2211380 2211389
965483
2211472
2252511 2252534 2252524
2252499 2252476
2211399 2211437 2211374
2211390 232966
2251860 2252463
2251862
965481 2252443
2211388 2252535
2252527
2211378
2252509
2252488
2252529 2252454
2252459 2252522
22524952252465 2211424
1937262 2211447
2252457 2211398 2211453 60325 2211372 947637
2252500
2251844
965483
2211381
2252507
2252474 2252470
2252487
65007 2252511 2252534 2252524
2252499
2251878
2252446
2252530
2252462
2252488
2251860 2252463
2251862
2252471
2252513
2252450
2252529 2252454
2251844
2252487
2211424 965481 2252443
2211388 2252535
2252527
2211378
2251878
2211380 2211389
1937262 2211447
2252457 2211398 2211453 60325 2211372 947637
2252456 65030
2252530
2252521
2252520
2252446 2252507
965483
2211381
65007 2252511 2252534 2252524
2252471
2252513
2252470
2251860 2252463
2251862 2252499
65030 2252462
2252488
2252529 2252454
2252459 2252522
22524952252465
965481 2252443
2211388 2252535
2252527
2252474
2251844
2252487
2211424
1937262 2211447
2252457 2211398 2211453 60325 2211372 947637
2252448
2252526
2251878
965483
2211381
65007 2252511 2252534 2252524
2252499
2252530 2252507
2252470
2251860 2252463
2251862
2252471
2252513
2252450
2252488
2252529 2252454
2251844
2252487
65030 2252462
2252507
2216752
2252520
2252446
2252477
2252492 2252494
2252478 2252521 2252456
2216752
2252520
2252446
22524972252467 2252477
2252492 2252494
2252478
2252518
2252502
2252518 22524972252467
2252521 2252456
2216752
2252520 2252462
2252492 2252494
2252478
2252502
2252477
22524972252467
2252521
1004
2252518
2252502
2252477
22524972252467 2252492 2252494
2252478
1184269 2164869
2164867
2164874
2164836 2164871
2164845
2164865
2164823
2164826
3003429
2000741
8
Complexity 2167209
867756 453835 1930314
2167550
2627886
1915102 2683105 1909267 724582 1974004 347252 370284 2159180 578522 2684597 370438 1899299 2337498 351280 699079 351406 705277 2899547 370411 350116 1885100 1874568 1915579 867862 2624671 1965576 3019969 32769 46340749144 350542364807 364847 2164103 869979 1951893 1884045 806358 96793518831662164336 806458 707769 1937967 369314 2183234 1909345 348839 2154232 2568937 806486 2664372 867601 806474 2435272 1937955 8309101921843 728599 776589 806545 3069776 806408 454083 46454 83349628210412568915 347928 2257729 2627791 2568900 2206381 184625 2291509 1962422 8123 2684521 806547 1915048 2164094 2568936 2596049 1930860 1883999 867566 547590 1962235 1957581 1937799 349484 866836 806382 806689 25049923475733069785 424606 1937557 1867785 466957 969471 2568903 1902498 977046 1953312 2174079 350087 867255 2568895 728517 369388 2206391806384 3003140 367966 728523 867164 2684588 224752 41058 2269305 1951132 835799 839123 23304 2568869 2568889 2568896 2568925 1959839 372360 2684571 2730269 368521 2206383 834739 578515 347368 2568876 22668882821237 367928 454096 349176 1915849 2568873 3069777 2681959 350777 2568885 1953847 350407 728585 367229 1929277 1949828 779952 728522 2568898 2691438 226374 806466 1915049 353546 2568913 2160570 353217 347487 2411838 1939142 2684603 1948019 453913 377143 446626 369487 869161 967944 347499 2 758790 1930588 1961896 2441153 1946733 367562 2633614 350548 372390 2821219 2041061 369430348645 526275 2684642 2152870 1298220 830428 368612 350454 2899529 336318 729780 368814 2684538 1948721 865042 833412 1956874 7285672568917 348320 728549 30697813069787 368049 2684618 2155795 2568919 350303 88156 349711 3069760 2568926 351590 2629492 975385 2568907 2266811 2568941 226702 149684 695345 2568892 193754583778 2164102 728583 349074 695472 2200468 371308 2865827 700985 1883982 367687 127952 349517 806501 352382 806364 352823 967442 728516 919975 349509 348472 1941756 2684644 2352234 116298 909485369469 372508 608477 349594 353245 3069767 1946504 352978 337929 2498495 351154 2031030 7337152009329 1937296 19484811949007 1880990 296082 8195 1921398 683365 349585 347633 357619 2037147 832181 695397 226920 30635 71780 368794 371142 866844 2205710 1908108 1949269 94807 376677 366751 967848 2154219 577275 369574 436887901020 2150922 1942228 295702 2009330 367422 370642 149617 353681 1913475 1867789 797865 641645 728541247155 353447 563626 2630493 367067 46277 194697 2024676 446922 1937665 834058 245239 837241929805 695443 2524148 2568868 866810 347603 10941 698926 1954771 352750 346867 6957042568884 1915088 370514 2570417 1942141 867445 1915878 350340577274 361256697295 697535 806506 793687 83596 1957123 648159 801903 368736 1910956 1927286 342895 370128 2174407 834191 1940756 2314177 226808 348431 420386 80148 95031 367388 377661 698914 837964 480345 1896257 968494 698913 94780 349968 777196 2174473 9867642247234 727917 83684 475867 349683 369921 366628 477696 456859 2174837 348258 347211 351573 10909 727922 771463 147061 697520 1037407 7750252188124 1929140 352224 85745 83534 370626 2174340 699703 695654 784004617618 2886476 1916792 2305396 353282 2995740 2710857 2438849 350133
480565
700570
2671767
2671706
370874 926394 2655654 2671725 352529 969777 969826 95311 2633171 970220 970044
970477970182 970172 2190207 970311 969773 2570520 348045 9675441968227 969799 2887917 2671745 970619969790 970280 969814 2581808 2655508 350968 2671776 970246 970223 2159233 2953795 969791 41085 2671772 969786 1883995 182396 970123 1883406 970170 2632477 347199 1635578 1931806 453819 95257 347441 2986851
2468261 2468756 2151401 2453532 1898695 2153515
(a) Middle cluster early result 2167209
2167209
867756 453835 1930314
2167550
2627886
1915102 2683105 1909267 724582 1974004 347252 370284 2159180 578522 2684597 370438 1899299 2337498 351280 699079 351406 705277 2899547 370411 350116 1885100 1874568 1915579 867862 2624671 1965576 3019969 32769 46340749144 350542364807 364847 2164103 869979 1951893 1884045 806358 96793518831662164336 806458 707769 1937967 369314 2183234 1909345 348839 2154232 2568937 806486 2664372 867601 806474 2435272 1937955 8309101921843 728599 776589 806545 3069776 806408 454083 46454 83349628210412568915 347928 2257729 2627791 2568900 2206381 184625 2291509 1962422 8123 2684521 806547 1915048 2164094 2568936 2596049 1930860 1883999 867566 547590 1962235 1957581 1937799 349484 866836 806382 806689 25049923475733069785 424606 1937557 1867785 466957 969471 2568903 1902498 977046 1953312 2174079 350087 867255 2568895 728517 369388 2206391806384 3003140 367966 728523 867164 2684588 224752 41058 2269305 1951132 835799 839123 23304 2568869 2568889 2568896 2568925 1959839 372360 2684571 2730269 368521 2206383 834739 578515 2568876 347368 22668882821237 367928 454096 349176 1915849 2568873 3069777 2681959 350777 2568885 1953847 350407 728585 367229 1929277 1949828 779952 728522 2568898 2691438 226374 806466 1915049 353546 2568913 2160570 353217 347487 2411838 1939142 2684603 1948019 453913 377143 446626 869161 967944 347499369487 2758790 1930588 1961896 2441153 1946733 367562 2633614 372390 2821219 2041061 369430348645 526275 2684642 2152870 1298220 830428 368612 350454 2899529 336318 350548 729780 368814 2684538 1948721 865042 833412 1956874 7285672568917 348320 728549 30697813069787 368049 2684618 2155795 2568919 350303 88156 349711 3069760 2568926 351590 2629492 975385 2568907 2266811 2568941 226702 149684 695345 2568892 193754583778 2164102 728583 349074 695472 2200468 371308 2865827 700985 1883982 367687 127952 349517 806501 352382 806364 352823 967442 728516 919975 349509 348472 1941756 2684644 2352234 116298 909485369469 372508 608477 349594 353245 3069767 1946504 352978 337929 2498495 351154 2031030 7337152009329 1937296 19484811949007 1880990 296082 8195 1921398 683365 349585 347633 357619 2037147 832181 695397 226920 30635 71780 368794 371142 866844 2205710 1908108 1949269 94807 376677 366751 967848 2154219 577275 369574 436887901020 2150922 1942228 295702 2009330 367422 370642 149617 353681 1913475 1867789 797865 641645 728541247155 353447 563626 2630493 367067 46277 194697 2024676 446922 1937665 834058 245239 837241929805 695443 2524148 2568868 866810 347603 698926 10941 1954771 352750 695704 2568884 346867 1915088 370514 2570417 1942141 867445 1915878 350340577274 361256697295 697535 806506 793687 83596 1957123 648159 801903 368736 1910956 1927286 342895 370128 2174407 834191 1940756 2314177 226808 348431 420386 80148 95031 367388 377661 698914 837964 480345 1896257 968494 698913 94780 349968 777196 2174473 9867642247234 727917 83684 475867 349683 369921 366628 456859 477696 2174837 348258 347211 351573 10909 727922 771463 147061 697520 1037407 7750252188124 1929140 352224 85745 83534 370626 2174340 699703 695654 784004617618 2886476 1916792 2305396 353282 2995740 2710857 2438849 350133
480565
700570
2671767
2671706
370874 926394 2655654 2671725 352529 969777 969826 95311 2633171 970220 970044
970477970182 970172 2190207 970311 969773 2570520 348045 9675441968227 969799 2887917 2671745 970619969790 970280 969814 2581808 2655508 350968 2671776 970246 970223 2159233 2953795 969791 41085 2671772 969786 1883995 182396 970123 1883406 970170 2632477 347199 1635578 1931806 453819 95257 347441 2986851
2468261 2468756 2151401 2453532 1898695 2153515
(b) Middle cluster halfway result
867756 453835 1930314
2167550
2627886
480565 1915102 2683105 1909267 724582 700570 1974004 347252 370284 2159180 578522 2684597 370438 1899299 2337498 351280 699079 351406 705277 2899547 370411 350116 1885100 1874568 1915579 867862 2624671 1965576 3019969 32769 46340749144 350542364807 364847 2164103 869979 1951893 806358 96793518831661884045 806458 707769 1937967 2164336 369314 2183234 1909345 348839 2154232 2568937 806486 2664372 867601 806474 2435272 1937955 8309101921843 728599 776589 806545 3069776 806408 454083 46454 83349628210412568915 347928 2257729 2627791 2568900 2206381 184625 2291509 1962422 8123 2684521 806547 1915048 2164094 2568936 2596049 1930860 1883999 867566 547590 1962235 1957581 1937799 349484 866836 806382 806689 25049923475733069785 424606 1937557 1867785 466957 969471 2568903 1902498 977046 1953312 2174079 350087 867255 2568895 728517 369388 2206391 367966 806384 728523 867164 2684588 224752 41058 22693053003140 1951132 835799 839123 23304 2568869 2568889 2568896 2568925 1959839 372360 2684571 2730269 368521 834739 2206383 578515 2568876 347368 22668882821237 367928 454096 349176 1915849 2568873 3069777 2681959 350777 2568885 1953847 350407 728585 367229 1929277 1949828 779952 728522 2568898 2691438 226374 806466 1915049 353546 2568913 2160570 353217 347487 2411838 1939142 2684603 1948019 453913 377143 446626 369487 869161 347499 2758790 1930588 1961896 2441153 367562 372390 26336141946733 2041061 2821219 369430348645 526275 967944 2684642 2152870 1298220 368612 350454 2899529 336318 350548 830428 729780 368814 2684538 1948721 865042 833412 1956874 7285672568917 348320 728549 30697813069787 368049 2684618 2155795 2568919 350303 88156 349711 3069760 351590 2629492 2568926 975385 2568907 2266811 2568941 226702 149684 695345 2568892 193754583778 2164102 728583 349074 695472 2200468 371308 2865827 700985 1883982 367687 127952 349517 806501 352382 806364 352823 967442 728516 919975 349509 348472 1941756 2684644 2352234 116298 909485369469 372508 608477 349594 353245 1946504 3069767 352978 337929 2498495 351154 2031030 7337152009329 1937296 19484811949007 1880990 296082 8195 1921398 683365 349585 347633 357619 2037147 832181 695397 226920 30635 71780 368794 371142 866844 2205710 1908108 1949269 94807 376677 366751 967848 2154219 577275 369574 436887901020 2150922 1942228 295702 2009330 367422 370642 149617 353681 1913475 1867789 797865 641645 728541247155 353447 563626 2630493 367067 46277 194697 2024676 446922 1937665 834058 245239 837241929805 695443 2568868 2524148 866810 347603 10941 698926 1954771 352750 346867 6957042568884 1915088 370514 2570417 1942141 867445 1915878 350340577274 361256697295 697535 806506 793687 83596 1957123 648159 801903 368736 1910956 1927286 342895 370128 2174407 834191 1940756 2314177 226808 348431 420386 80148 95031 367388 377661 698914 837964 480345 1896257 968494 698913 94780 349968 777196 2174473 9867642247234 727917 83684 475867 349683 369921 366628 477696 456859 2174837 348258 347211 351573 10909 727922 771463 147061 697520 1037407 7750252188124 1929140 352224 85745 83534 370626 2174340 699703 695654 784004617618 2886476 1916792 2305396 353282 2995740 2710857 2438849 350133
2671767
2671706
370874 926394 2655654 2671725 352529 969777 969826 95311 2633171 970220 970044
970477970182 970172 2190207 970311 969773 2570520 348045 9675441968227 969799 2887917 2671745 970619969790 970280 969814 2581808 2655508 350968 2671776 970246 970223 2159233 2953795 969791 41085 2671772 969786 1883995 182396 970123 970170 1883406 2632477 347199 1635578 1931806 453819 95257 347441 2468261 2986851 2468756 2151401 2453532
(d) Middle cluster early result
1898695 2153515
2167209
2167209
867756 453835 1930314
2627886
1915102 2683105 1909267 724582 1974004 347252 370284 2159180 578522 2684597 370438 1899299 2337498 351280 699079 351406 705277 2899547 370411 350116 1885100 1874568 1915579 867862 2624671 1965576 3019969 32769 46340749144 350542364807 364847 2164103 869979 1951893 806358 96793518831661884045 806458 707769 1937967 2164336 369314 2183234 1909345 348839 2154232 2568937 806486 2664372 867601 806474 2435272 1937955 776589 8309101921843 806545 3069776 806408 454083 46454 83349628210412568915 347928 728599 2257729 2627791 2568900 2206381 184625 2291509 1962422 8123 2684521 806547 1915048 2164094 2568936 2596049 1930860 1883999 867566 547590 1962235 1957581 19377991867785 349484806382 866836 806689 25049923475733069785 424606 1937557 466957 969471 2568903 1902498 977046 1953312 2174079 350087 867255 2568895 728517 369388 2206391806384 3003140 367966 728523 867164 2684588 224752 41058 2269305 1951132 835799 839123 23304 2568869 2568889 2568896 2568925 1959839 372360 2684571 2730269 368521 2206383 834739 578515 2568876 347368 22668882821237 367928 454096 349176 1915849 2568873 3069777 2681959 350777 2568885 1953847 350407 728585 367229 1929277 1949828 779952 728522 2568898 2691438 226374 806466 1915049 353546 2568913 2160570 353217 347487 2411838 1939142 2684603 1948019 453913 377143 446626 369487 869161 967944 347499 2 758790 1930588 2441153 1961896 367562 26336141946733 350548 372390 2821219 2041061 369430348645 526275 2684642 2152870 1298220 830428 368612 350454 2899529 336318 729780 368814 2684538 1948721 865042 833412 1956874 7285672568917 348320 728549 30697813069787 368049 2684618 2155795 2568919 350303 88156 349711 3069760 2568926 351590 2629492 975385 2568907 2266811 2568941 226702 149684 695345 2568892 193754583778 2164102 728583 349074 695472 2200468 371308 2865827 700985 1883982 367687 127952 349517 806501 352382 806364 352823 967442 728516 919975 349509 348472 1941756 2684644 2352234 116298 909485369469 372508 608477 349594 353245 3069767 1946504 352978 337929 2498495 351154 2031030 7337152009329 1937296 19484811949007 1880990 296082 8195 1921398 683365 349585 347633 357619 2037147 832181 695397 226920 30635 71780 368794 371142 866844 2205710 1908108 1949269 94807 376677 366751 967848 2154219 577275 369574 436887901020 2150922 1942228 295702 2009330 367422 370642 149617 353681 1913475 1867789 797865 641645 728541247155 353447 563626 2630493 367067 46277 194697 2024676 446922 1937665 834058 245239 837241929805 695443 2524148 2568868 866810 347603 10941 698926 1954771 352750 346867 695704 2568884 1915088 370514 2570417 1942141 867445 1915878 350340577274 361256697295 697535 806506 793687 83596 1957123 648159 801903 368736 1910956 1927286 342895 370128 2174407 834191 1940756 2314177 226808 348431 420386 80148 95031 367388 377661 698914 837964 480345 1896257 968494 698913 94780 349968 777196 2174473 9867642247234 727917 83684 475867 349683 369921 366628 456859 477696 2174837 348258 347211 351573 10909 727922 771463 147061 697520 1037407 1929140 352224 85745775025 2188124 83534 370626 2174340 699703 695654 784004617618 2886476 1916792 2305396 353282 2995740 2710857 2438849 350133 700570
2671767
2671706
370874 926394 2655654 2671725 352529 969777 969826 95311 2633171 970220 970044
970477970182 970172 2190207 970311 969773 2570520 348045 9675441968227 969799 2887917 2671745 970619969790 970280 969814 2581808 2655508 350968 2671776 970246 970223 2159233 2953795 969791 41085 2671772 969786 1883995 182396 970123 1883406 970170 2632477 347199 1635578 1931806 453819 95257 347441 2986851
2468261 2468756 2151401 2453532 1898695 2153515
(c) AH-KSC method result 1909241 1726372
1909241 1726372
1909241 1726372 2167550
1909241 1726372
1909241 1726372
1909241 1726372 2167550 480565
867756 453835 1930314
2167550
2627886
480565 1915102 2683105 1909267 724582 700570 1974004 347252 370284 2159180 578522 2684597 370438 1899299 2337498 351280 699079 351406 705277 2899547 370411 350116 1885100 1874568 1915579 867862 2624671 1965576 3019969 32769 46340749144 350542364807 364847 2164103 869979 1951893 806358 96793518831661884045 806458 707769 1937967 2164336 369314 2183234 1909345 348839 2154232 2568937 806486 2664372 867601 806474 2435272 1937955 8309101921843 728599 776589 806545 3069776 806408 454083 46454 83349628210412568915 347928 2257729 2627791 2568900 2206381 184625 2291509 1962422 8123 2684521 806547 1915048 2164094 2568936 2596049 1930860 1883999 867566 547590 1962235 1957581 1937799 349484 866836 806382 806689 25049923475733069785 424606 1937557 1867785 466957 969471 2568903 1902498 977046 1953312 2174079 350087 867255 2568895 728517 369388 2206391806384 367966 728523 867164 2684588 224752 41058 22693053003140 1951132 835799 839123 23304 2568869 2568889 2568896 2568925 1959839 372360 2684571 2730269 368521 834739 2206383 578515 347368 2568876 22668882821237 367928 454096 349176 1915849 2568873 3069777 2681959 350777 2568885 1953847 350407 728585 367229 1929277 1949828 779952 728522 2568898 2691438 226374 806466 1915049 353546 2568913 2160570 353217 347487 2411838 1939142 2684603 1948019 453913 377143 446626 369487 869161 347499 2758790 1930588 1961896 2441153 367562 372390 26336141946733 2821219 2041061 369430348645 526275 967944 2684642 2152870 1298220 368612 350454 2899529 336318 350548 830428 729780 368814 2684538 1948721 865042 833412 1956874 7285672568917 348320 728549 30697813069787 368049 2684618 2155795 2568919 350303 88156 349711 3069760 2568926 351590 2629492 975385 2568907 2266811 2568941 226702 149684 695345 2568892 193754583778 2164102 728583 349074 695472 2200468 371308 2865827 700985 1883982 367687 127952 349517 806501 352382 806364 352823 967442 728516 919975 349509 348472 1941756 2684644 2352234 116298 909485369469 372508 608477 349594 353245 3069767 1946504 352978 337929 2498495 351154 2031030 7337152009329 1937296 19484811949007 1880990 296082 8195 1921398 683365 349585 347633 357619 2037147 832181 695397 226920 30635 71780 368794 371142 866844 2205710 1908108 1949269 94807 376677 366751 967848 2154219 577275 369574 436887901020 2150922 1942228 295702 2009330 370642 149617 367422 353681 1913475 1867789 797865 641645 728541247155 353447 563626 2630493 367067 46277 194697 2024676 446922 1937665 834058 245239 837241929805 695443 2568868 2524148 866810 347603 698926 10941 1954771 352750 346867 6957042568884 1915088 370514 2570417 1942141 867445 1915878 350340577274 361256697295 697535 806506 793687 83596 1957123 648159 801903 368736 1910956 1927286 342895 370128 2174407 834191 1940756 2314177 226808 348431 420386 80148 95031 367388 377661 698914 837964 480345 1896257 968494 698913 349968 94780 777196 2174473 9867642247234 727917 83684 475867 349683 369921 366628 456859 477696 2174837 348258 347211 351573 10909 727922 771463 147061 697520 1037407 7750252188124 1929140 352224 85745 83534 370626 2174340 699703 695654 784004617618 2886476 1916792 2305396 353282 2995740 2710857 2438849 350133
2671767
2671706
370874 926394 2655654 2671725 352529 969777 969826 95311 2633171 970220 970044
970477970182 970172 2190207 970311 969773 2570520 348045 9675441968227 969799 2887917 2671745 970619969790 970280 969814 2581808 2655508 350968 2671776 970246 970223 2159233 2953795 969791 41085 2671772 969786 1883995 182396 970123 1883406 970170 2632477 347199 1635578 1931806 453819 95257 347441 2468261 2986851 2468756 2151401 2453532
(e) Middle cluster halfway result
1898695 2153515
2167209
867756 453835 1930314
2627886
480565 1915102 2683105 1909267 724582 700570 1974004 347252 370284 2159180 578522 2684597 370438 1899299 2337498 351280 699079 351406 705277 2899547 370411 350116 1885100 1874568 1915579 867862 2624671 1965576 3019969 32769 46340749144 350542364807 364847 2164103 869979 1951893 806358 96793518831661884045 806458 707769 1937967 2164336 369314 2183234 1909345 348839 2154232 2568937 806486 2664372 867601 806474 2435272 1937955 8309101921843 728599 776589 806545 3069776 806408 454083 46454 83349628210412568915 347928 2257729 2627791 2568900 2206381 184625 2291509 1962422 8123 2684521 806547 1915048 2164094 2568936 2596049 1930860 1883999 867566 547590 1962235 1957581 1937799 349484 866836 806382 806689 25049923475733069785 424606 1937557 1867785 466957 969471 2568903 1902498 977046 1953312 2174079 350087 867255 2568895 728517 369388 2206391806384 367966 728523 867164 2684588 224752 41058 22693053003140 1951132 835799 839123 23304 2568869 2568889 2568896 2568925 1959839 372360 2684571 2730269 368521 2206383 834739 578515 347368 2568876 22668882821237 367928 454096 349176 1915849 2568873 3069777 2681959 350777 2568885 1953847 350407 728585 367229 1929277 1949828 779952 728522 2568898 2691438 226374 806466 1915049 353546 2568913 2160570 353217 347487 2411838 1939142 2684603 1948019 453913 377143 446626 369487 869161 347499 2758790 1930588 2441153 1961896 367562 372390 26336141946733 2821219 2041061 369430348645 526275 967944 2684642 2152870 1298220 368612 350454 2899529 336318 350548 830428 729780 368814 2684538 1948721 865042 833412 1956874 7285672568917 348320 728549 30697813069787 368049 2684618 2155795 2568919 350303 88156 349711 3069760 2568926 351590 2629492 975385 2568907 2266811 2568941 226702 149684 695345 2568892 193754583778 2164102 728583 349074 695472 2200468 371308 2865827 700985 1883982 367687 127952 349517 806501 352382 806364 352823 967442 728516 919975 349509 348472 1941756 2684644 2352234 116298 909485369469 372508 608477 353245 349594 3069767 1946504 352978 337929 2498495 351154 2031030 7337152009329 1937296 19484811949007 1880990 296082 8195 1921398 683365 349585 347633 357619 2037147 832181 695397 226920 30635 71780 368794 371142 866844 2205710 1908108 1949269 94807 376677 366751 967848 2154219 577275 369574 436887901020 2150922 1942228 295702 2009330 370642 149617 367422 353681 1913475 1867789 797865 641645 728541247155 353447 563626 2630493 367067 46277 194697 2024676 446922 1937665 834058 245239 837241929805 695443 2568868 2524148 866810 347603 10941 698926 1954771 352750 346867 695704 2568884 1915088 370514 2570417 1942141 867445 1915878 350340577274 361256697295 697535 806506 793687 83596 1957123 648159 801903 368736 1910956 1927286 342895 370128 2174407 834191 1940756 2314177 226808 348431 420386 80148 95031 367388 377661 698914 837964 480345 1896257 968494 698913 94780 349968 777196 2174473 9867642247234 727917 83684 475867 349683 369921 366628 477696 456859 2174837 348258 347211 351573 10909 727922 771463 147061 697520 1037407 7750252188124 1929140 352224 85745 83534 370626 2174340 699703 695654 784004617618 2886476 1916792 2305396 353282 2995740 2710857 2438849 350133
2671767
2671706
370874 926394 2655654 2671725 352529 969777 969826 95311 2633171 970220 970044
970477970182 970172 2190207 970311 969773 2570520 348045 9675441968227 969799 2887917 2671745 970619969790 970280 969814 2581808 2655508 3509682671776 970246 970223 2159233 2953795 969791 41085 2671772 969786 1883995 182396 970123 1883406 970170 2632477 347199 1635578 1931806 453819 95257 347441 2468261 2986851 2468756 2151401 2453532 1898695 2153515
(f) Proposed method result
Figure 4: Analysis of agglomeration process. (a, b, c) AH-KSC method. (d, e, f) Proposed method.
(at the 17th iteration). The early stage of AH-KSC was successfully clustering colored in red but in the halfway stage (at 26th iteration), the middle community was agglomerated together with some nodes that were included in the left side community, even though there were no direct connections to the nodes. In the late stage of agglomeration process (at the 30th iteration), the middle community was clustered with the right side community even though the right side community was the satisfied community, as shown in Figure 4(c). Figure 4(d) is the early stage of the proposed method. Like AH-KSC, the agglomeration process of the middle community is well clustered (at the 23rd iteration). In the halfway stage of agglomeration (at the 26th iteration), the middle community was clustered successfully because only directly connected nodes are agglomerated according to (5), where the number of edges between the nodes is added to the similarity function. At the late stage (at the 39th iteration), the right side community is clustered accurately because the ratio of the inner and outer edge connection is applied to (7) so that the smallest community on the right side has stopped agglomerating. Figure 4(f) shows the final result of the algorithm. This study compared the accuracy of detection of the proposed method with AH-KSC and KSC using the groundtruth LiveJournal network. To show the comparison conveniently, only four parts of the network are used because the network is too large, that is, more than 4 million nodes. As shown in Figure 5, there are four subnetworks with different structures. The 1st network has 292 nodes and 1858 edges, and ground-truth community, to which the seed node belongs, has 24 nodes. The second network has 356 nodes and 33616 edges with a ground-truth community of 52 nodes. The third network has 652 nodes and 63044 edges, and the groundtruth community has 22 nodes. The last network has 119 nodes and 866 edges with a ground-truth community of 15 nodes. The 2nd and 3rd networks are so complex that it is difficult to detect communities while the 1st and 4th networks are well structured, that is, average in difficulty. In Figure 5, the light yellow colored node groups in the first column
Table 1: Overall accuracy of community detection for the proposed method, AH-KSC, and KSC. Precision Recall 𝐹-score
Proposed method 0.64 0.95 0.75
AH-KSC 0.57 0.70 0.61
KSC 0.55 0.82 0.62
are the ground-truth community, the green colored node groups in the second column are the detected community from the proposed method, and the red colored node groups are the result of detected community from AH-KSC and KSC, respectively. From the observation of the experiment, AH-KSC agglomerates the neighbor nodes successfully in the early stages of agglomeration, as mentioned above, but it failed to terminate the agglomeration, as shown in the first and last networks in Figure 4 due to the lack of termination criteria. In addition, when the networks are too complex, such as the 2nd and 3rd networks, clustering is not done efficiently. KSC also produces similar results to AH-KSC but it clusters better than AH-KSC for the case in which the network is well organized, as with the 4th network. AH-KSC provides a better result than KSC when the network is very complex, such as the 2nd and 3rd network. Table 1 shows overall accuracy of community detection using LiveJournal ground-truth network with respect to the precision, recall, and 𝐹-score for the proposed method, AH-KSC, and KSC. For AH-KSC, the average precision, recall, and 𝐹-score were 0.57, 0.7, and 0.61, respectively. For KSC, the average precision, recall, and 𝐹-score were 0.55, 0.82, and 0.62, respectively. For the proposed method, the average precision, recall, and 𝐹-score were 0.64, 0.95, and 0.75, respectively. The overall precision of AH-KSC and KSC were similar with a 2 percent difference but KSC has higher performance in the overall recall with more than 12 percent, which means the KSC detected more true positive nodes than AH-KSC from the network. The average 𝐹-score for AH-KSC and KSC is similar with only a 1% difference. The proposed
Complexity Ground-Truth result
9 Proposed method result
AH-KSC method
KSC method
Figure 5: Example of detected community comparison.
method outperformed AH-KSC and KSC in all evaluation metrics. In average precision, the proposed method improved 7 to 9% compared to AH-KSC and KSC. In the average recall, the proposed method had the highest improvement with 25 to 13%. The average 𝐹-score of the proposed method is improved by 14%.
5. Conclusion This paper introduced an agglomerative spectral clustering with conductance and edge weight for detecting communities. The proposed method projects the original points into eigenvector feature space in the first stage. In the second
stage, the eigenvector space and the number of edges between nodes are used to evaluate the similarity between nodes. Each node finds candidate for the most similar nodes. The third stage finds the conductance between the node and its candidate. If only the conductance improves, the nodes are agglomerated. The three-stage process is iterated until the network requires no further agglomeration. The time complexity of the proposed method is increased compared to AH-KSC because we check the conductance of each agglomerated node but it is necessary for more accurate detection. From the analysis of the experiment, the proposed method outperformed the AH-KSC and KSC using a real life network, LiveJournal.
10 The two contributions of this method can be summarized as follows. One is the improvement accuracy compared to related works. The other is that the proposed method is feasible for practical situations because the performance of the method is well suited to real life social networks. On the other hand, the eigenvector space is calculated in every iteration so that the computation time is slower than that of KSC. Our future work will focus on improving the time complexity with a method such as parallel computing.
Conflicts of Interest The authors declare that they have no conflicts of interest.
Acknowledgments This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF2016R1D1A1B03932447).
References [1] S. Fortunato, “Community detection in graphs,” Physics Reports. A Review Section of Physics Letters, vol. 486, no. 3-5, pp. 75–174, 2010. [2] J. Leskovec, K. J. Lang, and M. Mahoney, “Empirical comparison of algorithms for network community detection,” in Proceedings of the 19th International World Wide Web Conference (WWW ’10), pp. 631–640, ACM, New York, NY, USA, April 2010. [3] M. Girvan and M. E. Newman, “Community structure in social and biological networks,” Proceedings of the National Academy of Sciences of the United States of America, vol. 99, no. 12, pp. 7821–7826, 2002. [4] M. E. Newman, “Community detection and graph partitioning,” EPL (Europhysics Letters), vol. 103, no. 2, Article ID 28003, 2013. [5] M. E. Newman, “Spectral methods for community detection and graph partitioning,” Physical Review E, vol. 88, no. 4, 2013. [6] S. C. Johnson, “Hierarchical clustering schemes,” Psychometrika, vol. 32, no. 3, pp. 241–254, 1967. [7] Y. Zhao and G. Karypis, “Hierarchical clustering algorithms for document datasets,” Data Mining and Knowledge Discovery, vol. 10, no. 2, pp. 141–168, 2005. [8] U. von Luxburg, “A tutorial on spectral clustering,” Statistics and Computing, vol. 17, no. 4, pp. 395–416, 2007. [9] Y. Ng. Andrew, M. I. Jordan, and Weiss Y., “On spectral clustering analysis and an algorithm,” in Proceedings of the Advances in Neural Information Processing Systems, British Columbia, Canada, 2001. [10] J. Yang and J. Leskovec, “Defining and evaluating network communities based on ground-truth,” in Proceedings of the 12th IEEE International Conference on Data Mining, ICDM 2012, pp. 745–754, bel, December 2012. [11] G. Karypis and V. Kumar, “A fast and high quality multilevel scheme for partitioning irregular graphs,” SIAM Journal on Scientific Computing, vol. 20, no. 1, pp. 359–392, 1998. [12] B. W. Kernighan and S. Lin, “An efficient heuristic procedure for partitioning graphs,” The Bell System Technical Journal, vol. 49, no. 1, pp. 291–307, 1970.
Complexity [13] H. N. Djidjev, “A scalable multilevel algorithm for graph clustering and community structure detection,” Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 4936, pp. 117–128, 2006. [14] A. Clauset, C. Moore, and M. E. J. Newman, “Hierarchical structure and the prediction of missing links in networks,” Nature, vol. 453, no. 7191, pp. 98–101, 2008. [15] M. C. Ramos, “Divisive and hierarchical clustering techniques to analyse variability of rainfall distribution patterns in a Mediterranean region,” Atmospheric Research, vol. 57, no. 2, pp. 123–138, 2001. [16] L. Lin, T. Luo, J. Fu, Z. Ji, and D. Xiao, “A new community detection based on agglomeration mechanism,” in Proceedings of the IEEE 2nd International Conference on Computing, Control and Industrial Engineering, CCIE 2011, pp. 352–355, chn, August 2011. [17] M. Leng, J. Wang, P. Wang, and X. Chen, “Hierarchical Agglomeration Community Detection Algorithm via Community Similarity Measures,” TELKOMNIKA Indonesian Journal of Electrical Engineering, vol. 10, no. 6, pp. 1510–1518, 2012. [18] I. S. Dhillon, Y. Guan, and B. Kulis, “Kernel k-means, spectral clustering and normalized cuts,” in Proceedings of the KDD-2004 - Proceedings of the 10th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 551–556, Seattle, Wash, USA, August 2004. [19] C. Alzate and J. A. K. Suykens, “Multiway spectral clustering with out-of-sample extensions through weighted kernel PCA,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 32, no. 2, pp. 335–347, 2010. [20] R. Langone, R. Mall, C. Alzate, and J. A. K. Suykens, “Kernel spectral clustering and applications,” Unsupervised Learning Algorithms, pp. 135–161, 2016. [21] R. Langone, C. Alzate, and J. A. K. Suykens, “Kernel spectral clustering for community detection in complex networks,” in Proceedings of the 2012 Annual International Joint Conference on Neural Networks, IJCNN 2012, Part of the 2012 IEEE World Congress on Computational Intelligence, WCCI 2012, Queensland, Australia, June 2012. [22] R. Mall, R. Langone, and J. A. K. Suykens, “Agglomerative hierarchical kernel spectral data clustering,” in Proceedings of the 5th IEEE Symposium on Computational Intelligence and Data Mining, CIDM 2014, pp. 9–16, usa, December 2014. [23] B. Hendrickson and T. G. Kolda, “Graph partitioning models for parallel computing,” Parallel Computing, vol. 26, no. 12, pp. 1519–1534, 2000. [24] C. F. Olson, “Parallel algorithms for hierarchical clustering,” Parallel Computing, vol. 21, no. 8, pp. 1313–1325, 1995. [25] A. R. Brodtkorb, T. R. Hagen, and M. L. Sætra, “Graphics processing unit (GPU) programming strategies and trends in GPU computing,” Journal of Parallel and Distributed Computing, vol. 73, no. 1, pp. 4–13, 2013.
Advances in
Operations Research Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Journal of
Applied Mathematics
Algebra
Hindawi Publishing Corporation http://www.hindawi.com
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Journal of
Probability and Statistics Volume 2014
The Scientific World Journal Hindawi Publishing Corporation http://www.hindawi.com
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at https://www.hindawi.com International Journal of
Advances in
Combinatorics Hindawi Publishing Corporation http://www.hindawi.com
Mathematical Physics Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
International Journal of Mathematics and Mathematical Sciences
Mathematical Problems in Engineering
Journal of
Mathematics Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
#HRBQDSDĮ,@SGDL@SHBR
Journal of
Volume 201
Hindawi Publishing Corporation http://www.hindawi.com
Discrete Dynamics in Nature and Society
Journal of
Function Spaces Hindawi Publishing Corporation http://www.hindawi.com
Abstract and Applied Analysis
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
International Journal of
Journal of
Stochastic Analysis
Optimization
Hindawi Publishing Corporation http://www.hindawi.com
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014
