An Efficient Algorithm for Community Detection in Complex Networks

Transcription

1 An Efficient Algorithm for Community Detection in Complex Networks Qiong Chen School of Computer Science & Engineering South China University of Technology Guangzhou Higher Education Mega Centre Panyu District, Guangdong, China, Ming Fang School of Computer Science & Engineering South China University of Technology Guangzhou Higher Education Mega Centre Panyu District, Guangdong, China, ABSTRACT Community structure detection in complex networks attracts considerable attention in recent years. In this paper we propose an algorithm to detect community structures in very large networks. Basing on the local community detection, this algorithm is able to detect global community structures. We found that the local maximal degree nodes locate dispersedly in networks and can be considered as key nodes of communities. By discovering local communities iteratively from local maximal degree nodes, the global community structures are identified. Only local graph information is required to discover the local maximal degree nodes and their local communities. No priori information (e.g. topology structure of the entire network, the number and sizes of the communities) is needed. The local communities can be detected in parallel by starting simultaneously from several local maximal degree nodes. Comparing to other community detection methods, our algorithm is time efficient and suitable to finding community structures in large real networks. Categories and Subject Descriptors H.2.8 [Information Systems]: Database Applications-data mining General Terms Algorithm,Experimentation Keywords Complex networks, Community detection, Central vertices, Local degree central vertices, Local maximal degree nodes 1. INTRODUCTION There are many complex networks in real world, for example, individual relationship networks, collaboration networks and citation networks, protein interaction networks, and WWW, etc. The communities of complex networks are groups of nodes, with more links connecting nodes of the same group and comparatively less links connecting nodes of different groups [1]. Communities may be groups of related individuals in social networks [5], sets of Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. The 6th SNA-KDD Workshop 12 (SNA-KDD 12), August 12, 2012, Beijing,China. Copyright 2012 ACM $ web pages dealing with the same topic and groups of cells with same or similar functions. The discovery of the communities allows us to understand the attributes of nodes from the network topology alone and to reveal the structure of networks. Identifying community structure in networks attracts much research attention. Many of community detection algorithms have been developed over the last few years. These algorithms use a wide variety of techniques and vary in performance and speed. Most of those methods require knowledge of the entire graph structure. In some cases, the real networks are either too large or too dynamic to be known completely, e.g., the WWW. Some algorithms require prior knowledge of the number or size of communities in the network, the parameters about the structures similarity measure threshold which are unknown beforehand or hard to tune. In this paper, we present a new community detection approach based on key nodes of communities in complex networks. In our work, we find that each community has a few nodes, which are the most key nodes in the corresponding community, and a community is a set of nodes assembling close to the key nodes. Our approach discovers the key nodes in the given network, then starts from the key nodes, iteratively adds the adjacent nodes to the key nodes to obtain the communities. The key nodes in our algorithm are local central vertices. The local central vertices are the local maximal degree nodes whose degrees are larger or equal to all their neighbor nodes and are important in local region of a network. Our approach allows us to identify the community structures without knowing the priori knowledge of the networks. This paper is arranged as follows. In the second section we discuss the related work, in the third section we define the local central vertices and present a new algorithm based on local central vertices, experimental results are reported in the fourth section. In comparison with other known methods, our method is more efficient and accurate. 2. RELATED WORK Detecting community structure in networks is a big challenge. A vast number of community detection algorithms have been developed, including removal of high-betweenness edges [1], optimization of modularity [7], detection of dense subgraphs, statistical inference, random walk [2] and many others. Girvan and Newman proposed modularity Q in [3]. Modularity has become the best known quality function which is used to measure the reasonability of community partitions. The algorithms about

2 the optimization of modularity include greedy optimization, simulated annealing, extremal optimization, spectral optimization [8]. These algorithms are able to find fairly good approximations of the modularity maximum in a reasonable time. An exhaustive optimization of Q is impossible, as it has been recently proved that modularity optimization is an NP-complete problem [13], Modularity-based methods are by far the most popular class of methods to detect the community in graphs and require the entire topological knowledge of the networks. Some local community detection methods are proposed because of the fact that not all global information of networks is always available [9,10,16,18]. The algorithms construct a cluster by starting at a seed node and updating it by adding or deleting one vertex at a time as long as the density metric strictly improves. The community detection algorithms based on local community detection were proposed. Jiyang Chen et al. use the method that is similar to Clauset s to discover the local community and then start from a neighbor of the local community to discover another local community. The whole community structures of the graph can be discovered through recursively applying the local community detection algorithm[5]. Khorasgani proposed a TopLeader algorithm to discover the top k local communities. This method requires the number of communities which is unknown in advance in most cases[4]. Raghavan proposed a community detection algorithm based on label propagation[17]. Each node is initialized with a unique label and at every iteration of the algorithm, each node adopts a label that a maximum number of its neighbors have. At the end of the iterative process nodes with the same label are grouped together as communities. In [13], local maximal degree nodes are used as the seed nodes to detect the community structure of the network. The method in [13] is similar to the K- means algorithm., Let the local maximal degree nodes be the temporary centers, add the neighbor nodes to the nearest local maximal degree nodes, then choose the new centers for each clusters and add the neighbor nodes to the new centers. Repeat the process until the centers of clusters are no longer changed. 3. VERTEX CENTRALITY Vertex centrality measures central property of a vertex in networks and reflects the importance of the vertex in the networks. A lot of centrality measures are proposed, including closeness centrality, graph centrality, betweenness centrality, degree centrality, etc. The entire topological information of networks is required for the calculation of closeness centrality, graph centrality and betweenness centrality. Only the local connecting information is required to compute the degree centrality which is defined as the number of links incident upon a node. C ( v ) = deg( v ) d i i where deg(v i ) is the degree of node v i. Global Central vertices: Rank vertices according to the centrality measures, the top k vertices are called global central vertices of the network. Local central vertex: When the centrality of a vertex is not less than the centrality of all its neighbor vertices, the vertex is called a local central vertex of this network. When we use degree centrality to measure the influence of nodes in networks, the top k vertices are global degree central vertices, the local central vertices are local maximal degree nodes [6]. Given a node v in graph G, the degree of the node v is greater (4) than or equal to those of all its neighbor nodes, the node v is called as the local maximal degree node. The local maximal degree node has the following property: In graph G, v a and v b are two local maximal-degree nodes, if deg(v a ) deg(v b ),then v a and v b are not adjacent. Therefore, local maximal-degree nodes have large degree and most of them are located dispersedly in network. The two local maximal-degree nodes are adjacent only if they have same degree. 3.1 Distribution of Degree Central Vertices We explored the distribution of degree central vertices and relationship between the degree central vertices and community structures in LFR benchmark datasets [14] and real data sets. The real datasets are Zachary Karate Club network, Dolphin network, Political Books Network and College Football Network [12]. The LFR benchmark datasets has 200 nodes which are divided into 9 communities. The distribution of global degree central vertices (GDCV) and local degree central vertices (LDCV) are listed in Table 1. Table 1. The distribution of degree central vertices in LFR datasets Community No GDCV(Top 3) GDCV(Top 9) LDCV Shown in table 1, the top 3 global degree central nodes are in same community No.5, there are 9 local degree central nodes which are located in different community. We also list the top 9 global degree central nodes because there are 9 communities in the network. In real networks, the number of communities is not known and the distribution of global central nodes is random, so we do not know how many global central nodes should be chosen. The distributions of local degree central vertices in Zachary Club Network and Dolphin Network are shown in Figure 1. Blue vertices are the local degree central vertices. (a) Zachary Karate club network (b) Dolphin network Figure 1. The distribution of local degree central vertices 4. LCE ALGORITHM Global degree central vertices are used as the initial nodes to find the community structure of networks [5]. Starting at the global degree central vertices, the adjacent vertices are added to the communities by an association method and temporary communities are formed. Each of temporary community contains one global central vertex. However, the large community may contain more than one global central vertex and small communities may not contain global central vertices at all, so this method may result in incorrect community structure. In general, the number of communities is unknown in advance, so that it is hard to determine how many global central vertices are selected as the starting nodes to discover the communities.

3 The local maximal-degree nodes have large degree and locate dispersedly in networks. We propose an algorithm for community detection based on local maximal-degree nodes. We first find local maximal degree nodes. Let the local maximal degree nodes be seeds, starting from these seeds to expand communities, we can find the local communities to which the seed nodes belong. In this step we identify some communities of networks. Maybe these are not all communities in networks because there are some communities which have no local maximal central nodes. In order to solve the problem, we remove the communities which have been identified from the network and find local maximal degree nodes in the remaining parts of the network. These local maximal degree nodes are also the key nodes of communities. Starting from these key nodes, some communities are discovered. Repeat the above steps until no more community can be found. We call the algorithm as LCE algorithm. Our algorithm has the following steps: Algorithm1. LCE Algorithm Input: A social network G = (V, E) Output: A communities set C Step 1. Find the local maximal degree nodes in G, put the local maximal degree nodes in set H. Step 2. For each node h i in H, using Algorithm 2 to discover a local community Ci. put all the identified local communities into the set C. Step 3. If the identified local communities do not cover the whole network, remove C from G. Let H be empty set and go back to Step 1. Step 4. If the identified local communities cover the whole network, merge the communities with high similarity in C. Step 5. Return C as the communities set of network. 4.1 Finding Local Maximal-Degree Nodes Given a graph G, V is the set of nodes, suppose the node v has the largest degree, v is a local maximal-degree node, because the degrees of the nodes in v s neighbors set is smaller or equal to v s degree. Put v into the local maximal degree nodes set H and remove v and its neighbors with degree smaller than v from V. In the remaining nodes in V, if the node with the largest degree is a local maximal degree node, remove the node and its neighbors whose degree is smaller than the local maximal degree node. If the nodes with the largest degree is not a local maximal degree node, remove it from V. Repeat the process, until all local maximal nodes are found and not any node is left in V. We should not compare every node and its neighbor to determine whether it is a local maximal degree node. We do not to analyze the nodes which are in the local maximal degree node s neighbor set and have smaller degree. The worst time complexity is O(dn) to finding all the local maximal degree nodes, where n is the number of nodes in network and d is the average degree of nodes. 4.2 Local Community Detection For each node in local maximal-degree nodes set H, we detect the local community by iteratively adding adjacent nodes to the community. There are many methods to be used to expand the communities, such as M method [9], R method [10], F method [16], and label propagation [17]. A modified R method is proposed to expand community in our algorithm. The R metric is defined as: R = E E where E I is the number of edges with one endpoint in core and one endpoint in boundary, E T is the number of edges with one or more endpoints in boundary [9]. For a node h i in H, we look for a node v i in the neighbor set Γ( h i ) of hi and add it to the community, the node v i should has the greatest common neighbors with h i and adds v i to the community that results in the greatest increase of the R metric. We continue this process until there are no nodes left that could give the increase of R. See Algorithm2. Algorithm 2. Local Community Detection Algorithm Input: A social network G and a local maximal degree node h i. Output: A local community C i for h i Step 1. Put h i in C i, Step 2. Add to Ci the neighbor node v i of h i that results in the largest increase in R and has the greatest common neighbors with h i, Step 3. Add to Ci the neighbor node v i of Ci that results in the largest increase in R, Step 4. Repeat Step 3 until there are no nodes left that increase R when the node was added to the community. Step 5. Return Ci as h i s local communities. In Algorithm 2, we put a local maximal node in the community, and then add its adjacent nodes to the community. The R method in paper [9] is sensitive to the degree distribution of the source nodes neighbors: when the source degree is high, the algorithm will first explore its low degree neighbors. So the first node that was added to the community is the node with the lowest degree. The node with the smallest degree is added to the community and the algorithm may expand community along this node, but the node may not belong to the same community with the starting node. In this case, R method may not discover the starting node s local community correctly. When the node with the lowest degree has no common neighbors with the starting node, we do not add the node to the community. When two adjacent nodes have more common neighbors, they are more likely in the same community. Therefore, we firstly add the node which has more common neighbors with the starting node and results in the largest increase in R. 4.3 Merge Communities Each local maximal degree node belongs to a community, but a community may contain more than one local maximal degree node. The communities detected from different local maximal node may be identical or similar, in which case, they should be merged into one community. Given two communities C 1 and C 2, if these two communities can be merged into one community C, it must satisfy Formula (6). C1 C2 (6) > ξ min{ C, C ) 1 2 where ξ (0<ξ<1) is a threshold, here let ξ=0.5, meanings that the most members of the small community are in the large community, the two communities can be merged into one. The nodes in C 1 C 2 are overlapping nodes when Formula (6) is not satisfied. In our approach, Algorithm 2 has not deleting step for detecting local community, the time complexity of detecting local community is O(kd), where k is the number of vertices to be explored and d is the average degree of the vertices. Appling I T (5)

4 Algorithm 1 iteratively to detect the whole structure of the network, the time complexity is O(nd), where n is the total number of vertices in the networks. 5. EXPERIMENTS We evaluated our approach in four data sets: Zachary Karate Club Network, Dolphin Network, Political Books Network and College Football Network [12] and LFR benchmark data sets[14]. The true community partitions of the four data sets and LFR benchmark data sets are available. We compare our results to the true community partitions and evaluate the performance of our algorithm. 5.1 Evaluations and Comparisons Modularity Q is the most popular quality function used by researchers to access the performance of an algorithm. A high modularity value indicates more intracommunity edges than it would be expected by chance. The original modularity measure is defined only for disjoint communities, but Nicosia et al [19] presented a modified modularity for overlapping communities. The communities detected by our method may have overlapping. We use this overlap modularity measure for the experiments in this paper. The modified Q ov is: Q 1 kk = [ A ]s i j ov ij ij 2m c C i, j V 2m where s ij is the sum of the products of the belonging coefficients of i and j in communities to which they both belong. Its value depends on the number of communities to which each vertex belongs and the strength of its membership to each community. For the data sets with ground truth and benchmark data sets, we use Omega Index [18], an overlapping version of the Adjusted Rand Index to evaluate the results. Let V be the node set of a network. Given two partitions of a network, R and T, R is the resulting partition and T is the true partition. The Omega Index is defined as: ωu( RT, ) ωe( RT, ) ω ( RT, ) = 1 ωe ( RT, ) mmax mmax 1 1 ω ( R, T u ) = t j( R ) t j( T ), ωe( R, T ) = t 2 j( R ) t j( T) N j= 0 N j= 0 (8) where ω u (R,T) is the fraction of pairs that occur together in the same number of communities in both, ω ε (R,T) is the expected value of this fraction in the null model, t j (R) is the set of pairs of items that appear together exactly in j communities in partition C. In our algorithm, when the local maximal degree nodes are discovered, some expanding methods can be used to detect the community. We implement two local community expanding method and do experiments to compare the partition results of the two methods. LCE-R uses the modified R method and LCE-F uses LFM method. We also compare our algorithm with some wellknown algorithms: Newman s algorithm [7] based on modularity, ILE [5], COPRA [17], LFM[16], etc. The modularity and the Omega index values are shown in Table 2. Table 3 lists the values of modularity for different algorithms. Shown in Table 2 and Table 3, our method gets good partitions in the four data sets with ground truth. For the other methods, the results are good in some data sets and are not good in other data sets. (7) Table 2. The Omega index in real data sets Algorithm Karate Club Dolphin Political books NCAA Football LCE-R LCE-F Newman LFM ILE COPRA Table 3. The modularity Q ov in real data sets Algorithm Karate Club Dolphin Political books NCAA Football LCE-R LCE-F Newman LFM ILE COPRA In order to further assess the performance of LCE, we tested its ability to recover the input modules of LFR benchmark graphs. The benchmark networks possess properties found in real networks, such as heterogeneous distributions of degree and community size. In our experiments, the size of networks is 1000, 10000, 50000, and 0 respectively, the size of communities is from 10 to 150. The average degree of nodes <k> is 10, the maximal degree of nodes k max is 30. The results are shown in the following tables. The figure after the LFR is the size of the benchmark network. The Omega index cannot be calculated in networks with 1M nodes because of memory. Table 4 and Table 5 illustrate the Omega index and Modularity in LFR graphs. Our algorithm can work as well as the other algorithms. Newman s algorithm cannot run in our computer when the size of community is large. Table 4. The Omega index in LFR data sets Algorithm LFR-1000 LFR LFR LFR- LCE-R LCE-F Newman LFM ILE COPRA Table 5. The modularity Q ov in LFR data sets Algorithm LFR-1000 LFR LFR- LFR- 0 LCE-R LCE-F Newman LFM ILE COPRA Table 6.The number of communities detected in LFR data sets LFR-1000 LFR LFR- LFR Number. of Communities LCE-R LCE-F Newman LFM ILE COPRA

5 The number of communities detected by our algorithm LCE-R and LCE-F are closer to the true number of communities in LFR graphs, shown in Table 6. Local maximal-degree nodes locate dispersedly in networks, local community detection procedures can start simultaneously from different local maximal degree nodes. Our method can deal with large networks efficiently, because our algorithm can be applied parallelly to discover the community structures from the local maximal degree nodes. We compare the running time of our algorithm with other algorithms in LFR benchmark networks. See Figure 2. For each local maximum degree node, a thread is created to expand the community. We do our experiments in a computer of dual-core CPU, four threads execute in parallel. The running time of LCE-R and LCE-F are shown in Figure 2. If the algorithm can be run on a computer of multi-core CPU, more threads can execute in parallel, the algorithm running time will be less. If the local communities expands in parallel by starting simultaneously from all local maximal degree nodes then the running time will be even less. Therefore, our algorithm is efficiency for the community detection in large networks. 8. REFERENCES [1] M. Girvan, M. E. J. Newman Community structure in social and biological networks, Proc. Natl. Acad. Sci. USA 99, [2] P. Pons, M. Latapy Computing communities in large networks using random walks. Physics and Society (physics.soc-ph). [3] M. E. J. Newman Modularity and community structure in networks. PNAS. [4] R.R. Khorasgani, J.Y.Chen, and O. R.Zaiane Top Leaders Community Detecting Approach in Information Networks. KDD 10 July, Washington, DC, USA [5] J. Y. Chen, O. R. Zaiane, R. Goebel Detecting Communities in Large Networks by Iterative Local Expansion International Conference on Computational Aspects of Social Networks. [6] Q. Chen, T.T. Wu A Method for Local Community Detection by Finding Maximal-degree Nodes. International Conference on Machine Learning and Cybernetics (ICMLC),, July [7] M. E. J.Newman Fast algorithm for detecting community structure in networks. Phys. Rev. E [8] S. Fortunato Community detection in graphs, Phys. Rep [9] A. Clauset Finding local community structure in networks, Physical Review E, vol. 72, p [10] F. Luo, J. Z. Wang, and E. Promislow Exploring local community structures in large networks, in WI 06: Proceedings of the 2006 IEEE/WIC/ACM International Conference on Web Intelligence, pp [11] J. Y. Chen, O. R. Zaiane, R. Goebel Local Community Identification in Social Networks, Proceeding of the 2009 International Conference on Advances in Social Network Analysis and Mining-Volume, p [12] M.Newman. Figure 2. Running time of algorithms on LFR networks: k=10,k max =30,μ=0.1,C min =10, C max =0.3*N, τ 1 = -2,τ 2 = -1,O n =O m =0. [13] U. Brandes, D. Delling, M. Gaertler, R. Gorke, M. Hoefer, Z. Nikoloski, and D. Wagner Maximizing modularity is 6. CONCLUSIONS hard. Arxiv preprint physics/ This paper presents a global community detection algorithm [14] Q. Chen, M. Fang Community Detection Based on based on local community detection that started at local degree Local Central Vertices of Complex Networks, International central vertices and gives the experiment results on typical Conference on Machine Learning and Cybernetics (ICMLC), datasets and benchmark data sets. The experiments show that the Volume 2,P local maximal degree nodes can be considered as the key nodes of [15] A. Lancichinetti, S. Fortunato,and F. Radicchi communities and starting from local maximal degree nodes we Benchmark graphs for testing community detection can discover the community structures of networks. The number algorithms, Physical Review E, 78, , of communities in network is not required in advance. Only [16] A.Lancichinetti, S.Fortnato, J.Kertesz Detecting the partial information of networks is required to discover the local overlapping and hierarchical community structure of maximal degree nodes and to expand the local communities, the complex networks. arxiv: v2 11 global information of entire network need not be known in [17] U.N. Raghavan, R. Albert, and S.Kumara Near linear advance. The overlapping communities can also be identified by time algorithm to detect community structures in large-scale using our method. The local communities can be detected in networks. Phys. Rev. E parallel. The experiment results show that the identification of [18] F. Havemann, M. Heinz, and A. Struck, et al community structure obtained by our algorithm is as good as the Identification of overlapping communities and their other algorithms, while the running time of our algorithm is less hierarchy by locally calculating community-changing than that of the other algorithms. Therefore, our algorithm is time resolution levels. arxiv: v1 6 efficiency and suitable to finding community structures in large [19] V. Nicosia, G. Mangioni, V. Carchiolo and M. Malgeri, real networks Extending the definition of modularity to directed graphs with overlapping communities, Journal of Statistical Mechanics: Theory and Experiment, Vol ACKNOWLEDGEMENTS This work was supported by the National Natural Science Foundation of China (No ) and the Nature Science Foundation of Guangdong Province (No ).