Research Article Effectively Detecting Communities by Adjusting Initial Structure via Cores

Community detection is helpful to understand useful information in real-world networks by uncovering their natural structures. In this paper, we propose a simple but effective community detection algorithm, called ACC, which needs no heuristic search but has near-linear time complexity. ACC defines a novel similarity which is different from most common similarity definitions by considering not only common neighbors of two adjacent nodes but also their mutual exclusive degree. According to this similarity, ACC groups nodes together to obtain the initial community structure in the first step. In the second step, ACC adjusts the initial community structure according to cores discovered through a new local density which is defined as the influence of a node on its neighbors. The third step expands communities to yield the final community structure. To comprehensively demonstrate the performance of ACC, we compare it with seven representative state-of-the-art community detection algorithms, on small size networks with ground-truth community structures and relatively big-size networks without ground-truth community structures. Experimental results show that ACC outperforms the seven compared algorithms in most cases.


Introduction
ere are many di erent kinds of networks in the real world, such as biological networks, ecological networks, social networks, etc. Usually, many real-world networks have intrinsic community structures. A community in a network is always expressed as a group of nodes with dense connections within a community but sparse connections with other communities. Community detection can help us to discover the nontrivial structures and topological features of complex networks.
So far, to resolve the community detection problems in complex networks, various algorithms have been developed based on the widely used concept of similarity measures or certain criterions. e well-known FastQ [1] and spectral clustering [2,3], and two new algorithms Attractor [4] and ISCD [5] can be regarded as similaritybased methods [6]. FastQ has introduced a fast greedy strategy for modularity maximization. It e ectively corresponds to a simple nearest neighbor agglomerative clustering of the network where the adhesion coe cient a rs is used as a similarity measure. But if the links between two communities connect low-degree nodes, this approach will fail to detect the communities. Spectral clustering models view the community detection as a graph partitioning, which apply spectral analysis to obtain the cut minimization. However, spectral clustering algorithms are not e cient because their running time is cubic in the size of the input dataset, which often limits the usability of these community detection algorithms only to small networks in practical use. Another limitation of spectral clustering algorithms is that they depend on an input parameter k which is hard to determine. e Attractor converts the edge between two nodes to a distance according to the Jaccard similarity and calculates the graviation between them. But the accuracy of this method needs to be improved. e ISCD uses a common neighbor similarity between two nodes to propose a new evaluation criterion of internal-link compactness, but it also depends on an input parameter k which is hard to determine. e critierion-based community detection methods have proposed several critierions, such as modularity [7], betweenness [8], and minimum-cut [9,10]. But to evaluate the quality of partitioning according to certain criteria often needs to apply optimization methods to detect communities. For example, the most widely used method, modularity maximization, detects communities by searching for one or more divisions with particularly high modularities over possible divisions of a network. Since exhaustively searching over all possible divisions is usually intractable, the time complexity of modularity maximization or minimum-cut in the optimal version of community detection is proved NPcomplete [11]. Besides high time cost, the max modularity does not often result in optimum partitions in real networks [12,13].
Core-based methods [14][15][16] also play an important role in community detection. Algorithms for finding the cores are efficient and amenable for parallelization [17]. But the lack of ability to distinguish influential nodes is a problem to this method [18].
Different from heuristic methods with certain criterion, we propose a simple, yet effective and fast, similarity-based community detection algorithm named ACC (Adjusting initial Community structure via Cores). ACC can overcome the limitations that many existing community detection methods have. In other words, some of the exiting methods are heuristics with heavy computation in practice but ACC are not. e most remarkable characteristic of ACC is the ingenious combination of the advantages of similarity-based methods and core-based methods. Based on a naive fundamental assumption that nodes in the same community are more similar to each other than to those in other communities, ACC first groups nodes together according to a novel similarity to obtain the initial community structure. In particular, the similarity proposed in ACC considers not only the number of common neighbors but also the exclusion degree between two adjacent nodes. en, based on another expression of the assumption that connections of the nodes in the same community are dense while connections of the nodes in different communities are sparse, ACC regards a node with the max local density in an initial community as the core of that community and adjusts the initial communities according to cores.
Our key contributions are as follows. (1) We present a novel similarity which considers not only common neighbors between two adjacent nodes but also their mutual exclusive degree. e threshold of the similarity is easy to be set. (2) We define the influence of a node on its neighbors as its local density, which makes discovering the core in a community much easier. (3) We propose a new community detection algorithm ACC with near-linear time complexity, which can find high-quality communities in different networks.
e remaining sections are organized as follows. We review the related works in Section 2.
en, we give preliminary of ACC in Section 3 and elaborate ACC algorithm in Section 4. After that, in Section 5, we present the performances of ACC on not only networks with groundtruth community structures but also networks without ground-truth community structures, which show how effective our method is, compared to state-of-the-art methods. Finally, Section 6 concludes the work.

Related Works
To date, many different methods have been proposed for community identification. We only report some popular methods. Spectral clustering [3,19] has become one of the most popular clustering algorithms, and it is currently being used in a wide range of applications. It considers the graph as a similarity matrix and solves a data clustering problem where each cluster is a community. Spectral clustering algorithm gets the top-k eigenvectors of eigensystem to form an n × k matrix. en, every column of the matrix is the attributes of the corresponding nodes. It groups these n nodes with k-Means to get the final community structure. Unfortunately, the running time of spectral clustering algorithms might be cubic in the size of the input dataset, which makes it prohibitive to use this approach on very large datasets.
FastQ [1] algorithm is an agglomerative method that merges nodes into bigger and bigger communities hierarchically, using the modularity criterion. It initializes the network from a state in which each node is a sole member of a community. en, it repeatedly merges communities pair, which results in the smallest decrease in modularity and ends up with a state at which all nodes in the network are arranged in a community. e result of FastQ can be represented as a dendrogram. Each level of dendrogram indicates a different community structure. FastQ selects the community structure corresponding to the highest modularity value as the final result. But since FastQ is based on the optimization for modularity, the community it has detected is not always corresponding to the ground-truth community in real application.
Newman2006 [20] algorithm first constructs a modularity matrix for networks. en, it arranges nodes corresponding to the positive element in top eigenvector in a community and other nodes in the opposite community. Newman2006 ends the process when no positive eigenvalue exists.
Louvain method [21] initializes each node as a single community and shifts the community label of each node according to the modularity gain until the labels converge. en, it considers each community as a node to merge some of them again according to the modularity gain.
Infohiermap algorithm [22] is a flow-based community method. It reveals hierarchical organization by multilevel compression of random walks on networks. e label propagation (LPA) approach [23,24] is based on the simple idea that a node should be assigned to the community to which most of its neighbors belong. LPA has been widely concerned. In addition to the advantage of linear time complexity, it does not need to define objective function and the number of community in advance. But since LPA simply updates the label of a node according to the plurality vote of its neighbors, it suffers from the problem of randomness caused by random update order, which affects the accuracy and stability of the community.
In recent years, considerable efforts have been put in improving the effectiveness and efficiency of community detection [25,26]. PPC (Personalized PageRank Clustering) [27] employs the inherent cluster exploratory property of random walks to reveal the clusters of a given graph, which combines random walks and modularity to reveal the clusters of a graph. PPC has a linear time and space complexity. Attractor [4] is a community detection algorithm based on distance dynamics. Attractor converts the edges between two nodes to a distance according to the Jaccard similarity and calculates the graviation between them. e graviation makes the nodes within one community close to each other and the nodes from different communities far away. MEA s -SN [28] is a multiobjective evolutionary algorithm based on similarity to find communities in signed networks. e two objectives to be optimized are based on the concepts of positive and negative cluster similarity between two neighboring nodes. MHGNMF [29] takes higherorder information among the nodes into consideration to enhance the clustering performance. SUM [30] is a similarity-based method which detects communities by suspecting the maximum degree nodes.

Preliminary
In this section, we first prepare the necessary notions about community detection. en, we define a novel similarity and a new local density. Lastly, we introduce the community detecting strategy used in ACC algorithm.

Related Notions.
Let G(V, E) be an undirected graph, where V is the set of nodes, E ⊆ V × V is the set of edges. e � u, v { } ∈ E indicates a connection between the nodes u and v. e number of nodes in a graph can be represented as n � |V|, and the number of edges can be represented as m � |E|.
For a node u ∈ V, the neighbors of node u are the members of set N(u) containing its adjacent nodes which share a common incident edge with u: In this paper, nodes with links to two or more communities are defined as borders.

A New Similarity.
In general, if two nodes have a number of common neighbors, we believe that the two nodes are similar. For two adjacent nodes u, v ∈ V, their common neighbors are denoted as CN(u, v), and CN(u, v) � N(u) ∩ N(v). erefore, |CN(u, v)| can be used to measure the similarity degree. If there are too many different neighbors between u and v, they will be dissimilar. We call this dissimilarity mutual exclusive degree. Suppose that the degree of node u is smaller than that of v, erefore, the mutual exclusive degree of two adjacent nodes u, v ∈ V can be defined as min (d u Given a graph G(V, E), through the number of the common neighbors and mutual exclusive degree of u and v, the novel reasonable structural similarity sim(u, v) between two adjacent nodes u and v is defined as follows: where N 1 (smalld) is the neighbors with degree 1 of the smaller degree node.

Local Density and Core.
We define the influence of a node on its neighbors as its local density and regard the node with the greatest density in a community as a core. For a given network where p is the number of u's neighbors which have no common neighbors with u and q is the number of u's neighbors which have common neighbors with u. We assume that the influence of u on each of its neighbors is 1. us, the total influence of u on the p neighbors is p 1 � p. e influence of u on each of the q neighbors is weakened by the common neighbors, and the real remaining influence is that 1 subtracts the weakened influence. If we define the weakened influence as the Jaccard [31], then the influence of u on each of the q neighbors is 1 − Jaccard(u, v).
us, the total influence of u on the q neighbors is q (1 − Jaccard(u, v)).
e influence of u on all its neighbors is p v)). erefore, the local density of node u can be expressed as (2)

The ACC Algorithm
We introduce the ACC algorithm in this section. First, we describe the process of community detection of ACC in detail. en, the time complexity of ACC is analyzed.

e ACC Algorithm.
We present our ACC algorithm as follows: Step 1. Obtain the initial community structure: given where the ζ is the threshold of similarity, we put the two nodes into one community. e number of the initial communities discovered in this step is marked as k.
Step 2. Adjust the initial community structure according to cores: we select the top-k nodes with the highest local density as cores of communities. ACC Complexity considers that there is at most one core in one community. us, if there are more than one core in an initial community, the initial community will be broken up into as many communities as the number of the cores, and each core represents a community. en, we rebuild new communities by assigning unlabelled neighbors of each core to the core.
Step 3. Expand communities: we assign each remaining unlabelled node to the community to which its highest density neighbor belongs. If an unlabelled node does not have a highest density neighbor, it will be regarded as an initial community.
Step 4. Merge small communities and reassign borders: for each small community whose size is smaller than 3, we merge it to the community which has most links to the small community. en, we regard nodes having links to two or more different communities as the borders and reassign each border to the community which has most links to the border. Next, considering that reassigning the borders could produce small communities whose size can be smaller than 3, we merge each small community whose size is smaller than 3 to the community which has most links to it.

Time Complexity Analysis.
To invest the initial communities, the similarity of any two linked nodes in a network is required, and thus the time computation of Step 1 of ACC is O(|E|). In Step 2, since ACC needs to get the Jaccard distance and local density, the time complexity is O(d · |E|), where d is the average degree of nodes in network G(V, E). During the process of getting the top-k nodes with the highest local densities, the time complexity is O(|V|) + O(k · log(|V|)). Moreover, to assign the neighbors of cores to the corresponding core, the time complexity is O(k · d).
Step 3 of ACC takes O(q) time to assign the unlabelled nodes, where q is the number of unlabelled nodes, q ≤ |V|. To merge the small-size communities into its most links communities, the time complexity is O(3 · d). e time complexity of reassigning the borders is O(|E|). In total, the time complexity is O(|E| + d · |E| + |V| + k · log(|V|) + k · d + q + 3 · d). Since d ≪ |E|, q ≤ |V|, k ≪ |V|, the time complexity is approximately O(|E| + |V| + log(|V|)).

Experiments and Analysis
In this section, we evaluate our proposed algorithm ACC on real-world networks to demonstrate its benefits.

Baselines.
To evaluate the performance of ACC, we compare it with several representative state-of-the-art community detection algorithms.
Spectral clustering (SC) [2] is one of the most popular community detection algorithms and is currently being used in a wide range of applications. It is based on the graph p-Laplacian.
Newman2006 [20] is a splitting algorithm, which uses the maximum eigenvalue of a matrix and recursively splits a network until the final results are obtained.
Infohiermap [22] is a well-known approach, which requires a strong information-theoretic background and reveals multilevel structures in networks.
FastQ [1] is a bottom-up algorithm based on optimizing, which selects the result from the tree where the Modularity Q is the maximum.
LPA [23] is a fast community detection algorithm with a linear time complexity, which is based on label propagation.
PPC [27] is an efficient graph clustering algorithm, which employs random walks to detect communities.
Attractor [4] is one of the current most popular community detection algorithms, which is based on distance dynamics.
ACC is implemented in Python 2.7 environment. FastQ and LPA are obtained from igraph which is Python module, and their running environment is also Python 2.7. e source codes of Spectral clustering, Newman2006, Infohiermap, PPC, and Attractor are provided by their authors.

Benchmarks.
We evaluate the performances of different community detection algorithms on real-world networks, including small-size networks with class information and relatively large size networks without class information.
e basic statistical information of the networks is listed in Table 1. In all the networks used in this paper, ζ of ACC algorithm is 0, except Polbooks network in which ζ of ACC is − 1.
Karate [8,32] is a famous network, derived by Zachary's observation on a karate club. It is composed of trainees, the club's instructor, and the club's administrator. Because of the dispute between the node 34 and node 1, the network is split into two small communities. e ground-truth community structure is shown in Figure 1(a).
Dolphins [33] is constructed on the basis of observations of 62 bottlenose dolphins over 7 years by Lusseau and Newman. e nodes and the edges of Dolphins network indicate bottlenoses and communications between two bottlenoses respectively. is network could be divided into four small communities. e ground-truth community structure is shown in Figure 2(a).
Risk Map [8] is a network of adventure game Risk 2, which is originated from a turn-based game for two to six players. It is composed of seven groups. e ground-truth community structure is shown in Figure 3(a).
Polbooks [7] is a network, derived from the politic books about US politics published around the time of the 2004 presidential election, which consists of 105 nodes and 441 edges. Nodes represent books sold by the online bookseller Amazon.com. Edges represent frequent copurchasing of books by the same buyers. It is composed of three communities. e ground-truth community structure is shown in Figure 4(a).
Football [8] is      6 Complexity regular-season games between the two teams they connect. e teams are divided into 12 conferences containing around 8-12 teams each, and thereby the real number of communities is also 12.
e ground-truth community structure is shown in Figure 5(a).
Santa Fe [8] is a network which consists of 118 nodes and 197 edges. Each node represents a scientist who works at Santa Fe Institute, and each edge indicates the collaborations of scientists. ere will be an edge if the two scientists have collaborated with each other at least once. It is composed of six communities. e ground-truth community structure is shown in Figure 6(a).
Email [34] is a network without ground-truth community structure, which consists of 5,451 emails. Nodes in the network are employees and edges are emails.
CA-GrQc [35] is an author collaboration network from the e-print arXiv and covers scientific collaborations between authors whose papers are submitted to General Relativity and Quantum Cosmology category. An edge between two authors represents a common publication. e network has 5,242 nodes and 14,496 edges, spanning from January 1993 to April 2003.
CA-Hep [35] is an author collaboration network from the e-print arXiv and covers scientific collaborations between authors whose papers are submitted to High Energy Physics-eory category. An edge between two authors represents a common publication. e network has 9,877 nodes and 25,998 edges, spanning from January 1993 to April 2003.
DBLP [36] is a coauthorship network where two authors are connected if they publish at least one paper together. e DBLP computer science bibliography provides a comprehensive list of research papers in computer science. Publication venue, e.g, journal or conference, defines an individual ground-truth community; authors who published to a certain journal or conference form a community.
Amazon [36] is a network which was collected by crawling Amazon website. It is based on customers who bought an item and also bought a feature of the Amazon      Complexity website. If a product i is frequently copurchased with product j, the graph contains an undirected edge from i to j.

Evaluation Matrices.
On the networks which already have known community structures, the performances of ACC and the seven baselines are quantitatively measured by three widely used external evaluation measures: Adjusted Rand Index (ARI), Normalized Mutual Information (NMI) and Accuracy, and one popular internal measure Modularity (Q).
On the networks without ground-truth community structures, to compare the performances of distinct algorithms in an objective way is a nontrivial task. But since Modularity is the most common internal measure in community detection, we use it to evaluate the quality of communities produced by ACC and the seven baselines.

Networks with Ground-Truth Community Structures.
To demonstrate that ACC can be applied to networks containing different communities, the results of all the algorithms on the networks with ground-truth community structures are exhibited in Figures 1-4, and 6, where colors of nodes indicate different detected communities. e quantitative performances of ACC and its baselines are summarized in Table 2, which are also shown at the bottom of the corresponding figures respectively to give visual explanations. For ground-truth communities of each network, since the ARI, NMI, and Accuracy are always 1, we only show Q on the Bar chart. Note that for LPA, the ARI, NMI, and Accuracy are the average values of thirty times running, and the figures demonstrated are the best picked from figures of thirty times. Figure 3 shows the results of ACC and its baselines on Risk Map network. From Figure 3(b), we can see that ACC correctly identifies the community structure. Figures 3(e) and 3(g) show that Infohiermap and PPC also get the right partitions. From Table 2, we can see that the performances of FastQ and LPA are slightly behind that of ACC, Infohiermap, and PPC. However, Spectral clustering, Newman2006, and Attractor identify the community structure incorrectly, which results in a relatively low value of Accuracy, ARI, and NMI. Figure 2 exhibits the results of ACC and the compared algorithms on Dolphins network. Figure 2(b) shows that ACC successfully finds out the four communities, except for the node sn89. e two compared algorithms PPC and Infohiermap also find the basic community structure. But as demonstrated in Figures 2(g) and 2(e), respectively, PPC wrongly assigns six nodes and Infohiermap wrongly assigns seven nodes. e community structures identified by the other five comparing algorithms are not so clear as the above three. From Table 2, we can also know that ACC achieves the best performance with high values of Accuracy, NMI, and ARI. Hence, on this network, ACC outperforms the seven baselines. Figure 1(b) illustrates the clustering results of our algorithm ACC and the other seven algorithms on Zachary Karate network. Our algorithm successfully identifies the two communities and divides this network into two communities clearly. Attractor obtains the second best result with only one wrongly assigned node. Additionally, there is no clear boundary and internal structure between communities obtained by Spectral clustering. e partition found by Newman2006 contains four communities. e results detected by FastQ, Infohiermap, and PPC show that they divide this network into three communities. Figure 4 shows the results of applying our algorithm ACC and its baselines to Polbooks network. From both the Bar charts and community structure in Figure 4(b), we can know that ACC detects the basically correct community structure and gets the best results with the highest Accuracy. Infohiermap achieves the best performances on NMI and ARI. FastQ and LPA also get relatively good partitions. e results of the other algorithms are not as good as that of the four methods. Figure 6 shows the results of ACC and its seven baselines on the network Santa Fe. We can see that the result of ACC is much closer to the ground-truth than the results of the baselines, although it does not show very good performance on this network. Table 2 also shows that ACC achieves the highest ARI, NMI, and Accuracy of all the methods. Figure 5 demonstrates the results of ACC and the compared algorithms on the network Football. From Figure 5(b), we can see that ACC basically identifies the community structure, except for further partitioning of two of the total twelve communities. For algorithms Attractor and Infohiermap, they obtain the correct community except for a wrongly assigned node. LPA and Spectral clustering also yield a comparable result, while Newman2006, PPC, and FastQ produce a relatively bad grouping on this network.

Comprehensive Assessment on Networks with Ground-Truth Community Structures.
To comprehensively analyze the performances of ACC and the seven baselines, we use box plot as a descriptive statistics means. us, to fairly evaluate each algorithm, the box plots of the four evaluation indexes are shown in Figure 7 respectively. For ARI, NMI, and Accuracy, ACC shows the best performance by a landslide in quartile, median, minimum, and maximum, respectively. erefore, ACC shows the best statistics performance on networks with ground-truth community structures.

Networks without Ground-Truth Community Structures.
In this section, we evaluate the performances of ACC and its baselines on relatively large scale networks without groundtruth community structures. As there exist no convincing measures for unlabelled network, we use Modularity (Q) to evaluate these algorithms in an informative way. Networks with high Modularity have dense connections between the nodes within communities but sparse connections between nodes in different communities. e value of the Modularity lies in the range [− 1/2, 1], Q is equal to 0 only if all nodes are put in a single community [37]. Generally speaking, the smaller the value of Modularity is, the more random the distribution of the edges is in the network is and the less clear the community structure is. High value of Modularity indicates a strong community structure. But in practical application, the achievable modularity score depends on the sparseness and size of the network [38]. As shown in the bar graphs of Figures 1-6 and the box plots in Figure 7, the largest Modularity does not necessarily correspond to the best result of community detection, and the community structure corresponding to the largest Modularity often has a certain deviation to the real community structure [12,13]. Usually, a Modularity value larger than 0.3-0.4 is a clear indication that the subgraphs of the corresponding partition are modules [12]. e maximum Modularity of many networks approximately ranges from 0.4 to 0.7 [12]. e quantitative results of each algorithm on each network are listed in Table 3 In total, the experiments on all real-world networks demonstrate that ACC not only allows extracting meaningful communities in networks with ground-truth community structures with high external measures Accuracy, NMI, and ARI, but also scales up large-size networks without ground- truth community structures and yields a good partitioning in terms of the internal measure Modularity. erefore, ACC is an effective community detection method.

reshold of Similarity.
We can see in Algorithm 1, ACC needs a similarity threshold, ζ, which can be set to an integer.
In this section, we perform experiments on LFR benchmark networks with different community structures and different distributions of node degrees to thoroughly investigate the impact of the similarity threshold on the quality of community detection of ACC and further discuss how to choose an appropriate ζ for a network. Besides, as shown in Figure 8, the range of similarity threshold ζ is relatively big under different mixing parameters; especially when mu is 0.3 and 0.4, the community structures are clear. When mu is 0.3 and 0.4, ACC can get the satisfying results under the threshold 0. As the mixing parameter mu increases, the number of common neighbors of two adjacent nodes on LFR networks decreases gradually when the distribution of node degrees in the networks does not change.
erefore, according to equation (1), the value of the similarity will decrease as the mixing parameter increases. Correspondingly, the range of threshold gradually shifts toward the left side of 0 as shown in Figure 8.

On LFR Benchmark Networks with Different Distributions of Node
Degrees. Since our similarity in ACC is determined by the number of common neighbors and the smaller degree of two adjacent nodes, we have performed experiments on LFR benchmark networks with different distributions of node degrees. In this experiment, the size of the networks is still 10,000, the mixing parameter is fixed to 0.4, the average degree of nodes is fixed to 15, and the maximum degrees of nodes are 30, 45, and 60, respectively, on the three networks. We all know that when the average degree of nodes on a network remains unchanged, as the maximum degree of nodes increases, the degrees of the smaller degree nodes will decrease. If the average degree of nodes and the number of nodes are the same, the number of common neighbors of two adjacent nodes on LFR benchmark networks generated under the same mixing parameter will remain little changed. erefore, the similarity will increase according to equation (1). As is shown in Figure 9, the effective range of ζ shifts toward the right side as the maximum degree of nodes increases.
On all the networks used in this paper, ζ is 0 except the network Santa Fe, where ζ is − 1. us, in real applications, we suggest setting ζ to 0. When the appropriate initial communities cannot be obtained, the value of threshold ζ can be reduced to get satisfying partition.

Similarity Evaluation.
To fairly validate the similarity sim that we propose, sim and three other well-known similarity measures are applied to ACC on the same networks.
e three well-known similarity measures are Common Neighbor (CN) [39], Jaccard [31], and Salton and McGill [40], of which Jaccard similarity is a frequently used measure of similarity of two nodes, and it is also a measure of similarity of two links [41].
Given G(V, E), for two adjacent nodes u and v, the three similarity measures are as follows: For CN, Jaccard and Salton, we determine their relative optimal results from plenty of experiments by tuning their corresponding input thresholds. For our similarity sim, the threshold is always set to 0. e relative optimal results of the four similarities for ACC algorithm and their corresponding thresholds are listed in Table 4. We can see from Table 4, by tuning the corresponding thresholds, each similarity can get an effective result, and results of the four similarities are of little differences. Relatively, sim gets the highest ARI, NMI, and Accuracy on networks Risk Map, Dolphins Karate, Polbooks, and Football. On network Santa Fe, sim achieves the highest ARI and NMI, and Salton gets the highest Accuracy. erefore, taking all these factors into account, sim is superior to the other three similarity measures.
5.6. Robustness on Incomplete Networks. We are also interested in probing the efficiency of ACC on incomplete networks as in [42,43]. In this section, we evaluate the robustness of ACC in discovering communities on incomplete networks generated from sampling processes.
To create an incomplete network G′, where G ′ ⊂ G, the edges in G are randomly sampled with a sampling rate 10% Input: G(V, E), ζ: the threshold of similarity Output: C: a set of communities Step 1: Obtain the initial community structure (1) C ⟵ ∅, V′ ⟵ V (2) while V′ ≠ ∅ do (3) select u ∈ V′ (4) for each v ∈ N(u) ∩ V′ do (5) if sim(u, v) > ζ then (6) if u or v has already existed in a community then (7) put u and v to the community (8) else (9) put u, v to a new community CS (10) put CS into C (11) remove u from V′ (12) k � |C| Step 2: Adjust the initial community structure according to cores (1) Take the top-k nodes with the highest local densities as cores (2) R ⟵ ∅ (3) Put all cores into R (4) for each CS ⊂ C do (5) if |R ∩ CS| ≥ 2 then (6) take each core in R ∩ CS as a new community CM (7) remove CS from C (8) put CM to C (9) S ⟵ ∅ (10) for each core u ∈ R ∩ CS do (11) assign all the unlabelled neighbors of u to u Step 3: Expand communities (1) for each unlabelled node v do (2) assign v to its neighbor with the biggest local density Step 4: Merge small communities and reassign borders (3) for each C i ⊂ C do (4) if |C i | ≤ 3 then (5) merge C i to the community with most links with it (6) remove C i from C (7) Assign each border to its neighbor with most links with it (8) for each C i ⊂ C do (9) if |C i | ≤ 3 then (10) merge C i to the community with most links with it (11) remove C i from C (12) Output C ALGORITHM 1: ACC. 16 Complexity  interval from 10% to 100% as suggested in [43]. en, we run ACC on these networks to evaluate its robustness. To highlight the variation in the community detection quality, we average the results over 10 datasets for each sampling rate. Figure 10 shows the robustness results of ACC on incomplete networks. e lines represent the mean NMI on each sampling rate. ACC is robust even when several edges are missing.

e Advantages of ACC.
From the above comprehensive studies and experiments, we can conclude that ACC has the following advantages: (1) ACC can detect communities from different types and scales of networks automatically with high accuracy and stability (2) ACC can discover the core for a community according to the definition of the local density proposed in this paper (3) ACC is a community detection algorithm with nearlinear in the time complexity

Conclusion
In this work, we develop a simple but effective community detection algorithm, named ACC. ACC combines the advantages of similarity-based methods with the advantages of core-based methods. ACC depends on a novel similarity to get initial communities from networks and further adjusts the initial communities according to the cores in them found out through our new definition of local density. en, ACC assigns the remaining unlabelled nodes to their high local density neighbors and adjusts border nodes to their neighbors with most links with them. ACC can overcome the limitation of heuristic search according to certain criteria, i.e., the calculation of heuristic values is time-consuming. We demonstrate the power of ACC on different  networks by comparing it with some state-of-the-art community detection methods. Experimental results provide compelling evidence that ACC is an effective community detection algorithm.
Data Availability e complex network data used to support the findings of this study can be found in the website http://snap.stanford. edu/data/.

Conflicts of Interest
e authors declare that they have no conflicts of interest.