Research on Community Structure in Bus Transport Networks

Transcription

1 Commun. Theor. Phys. (Beijing, China) 52 (2009) pp c Chinese Physical Society and IOP Publishing Ltd Vol. 52, No. 6, December 15, 2009 Research on Community Structure in Bus Transport Networks YANG Xu-Hua, 1,2, WANG Bo, 1,3 and SUN You-Xian 2 1 College of Information Engineering, Zhejiang University of Technology, Hangzhou , China 2 State Key Laboratory of Industrial Control Technology, Zhejiang University, Hangzhou , China 3 Department of Computer Science and Engineering, Yiwu Industrial and Commercial College, Yiwu , China (Received December 24, 2008; Revised April 8, 2009) Abstract We abstract the bus transport networks (BTNs) to two kinds of complex networks with space L and space P methods respectively. Using improved community detecting algorithm (PKM agglomerative algorithm), we analyze the community property of two kinds of BTNs graphs. The results show that the BTNs graph described with space L method have obvious community property, but the other kind of BTNs graph described with space P method have not. The reason is that the BTNs graph described with space P method have the intense overlapping community property and general community division algorithms can not identify this kind of community structure. To overcome this problem, we propose a novel community structure called N-depth community and present a corresponding community detecting algorithm, which can detect overlapping community. Applying the novel community structure and detecting algorithm to a BTN evolution model described with space P, whose network property agrees well with real BTNs, we get obvious community property. PACS numbers: q, Bb, r Key words: bus transport network, overlapping community structure, clique, modularity, N-depth community 1 Introduction Many real complex systems in nature and society can be abstracted to complex networks. [1 3] The small-world network model proposed by Watts and Strogatz in 1998 [4] and the scale-free network model proposed by Barabasi and Albert in 1999 [5] stimulated the researching enthusiasm of complex networks. As the further study on the physical meanings and the mathematical characteristics of the complex networks, researchers found that many practical complex networks have a common feature, which is called community structure. It means that a whole network is composed of several groups or clusters. There is no clear definition of the community within complex network at present. The general definition is that the division of network vertices into groups within which the network connections are dense, but between which they are sparser. [6 8] For example, in Fig. 1, there are three communities, denoted by the dashed circles, which have dense internal edges but between which there are only a lower density of external edges. However, many real community structures are different from the one described in Fig. 1, because there probably exists community overlapping. Namely, some vertices belong to more than one community at the same time. [9 10] In Fig. 2, the small network is also composed of three communities. But the communities overlap with each other; some vertices belong to several different communities. Fig. 1 A small network with three separate communities. Bus transport network (BTN) is a kind of typical complex network. [11 12] Up to now, there are lots of studies about the BTNs. These studies almost concentrate on the statistical characteristics such as network diameter, average path length, clustering coefficient, degree distribution, and so on. There are nearly no papers which study on the community structure within the BTNs. But it is necessary to study the community structure of the BTNs because the community property can affect the network performance. In this paper, the BTNs are abstracted to two kinds of complex networks graph with space L and space P methods respectively. We detect the community structure of the BTNs, which are described with space L and Supported by the National Natural Science Foundation of China under Grant Nos and , the China Postdoctoral Science Foundation Funded Project under Grant No Corresponding author, xhyang@zjut.edu.cn

2 1026 YANG Xu-Hua, WANG Bo, and SUN You-Xian Vol. 52 space P respectively. We find that the BTNs described with space L have obvious community property, but the ones described with space P have not. We analyze the reasons of this phenomenon, propose a novel community structure and present a corresponding community identifying algorithm. At last, applying this novel community structure and identifying algorithm to a BTN evolution model [10] described with space P, whose network property agrees well with real BTNs, we get obvious community property. and overlap with each other in a BTN. [10] Fig. 2 A small network with three overlapping communities. 2 Analysis of Community Structure of BTNs 2.1 Two Kinds of BTNs Graphs Described with Space L and Space P Methods Respectively In this paper, we use two kinds of methods to abstract the BTNs to graphs, which are the space L and the space P method. [11 12] In space L, one vertex represents one station, and one edge represents one link between two vertices if one station of two vertices is the successor of the other on one bus route. While in space P, one vertex represents one station, one edge between two vertices indicates that there is at least one bus route between the two stations. We describe the BTNs by the two methods (space L and space P) and analyze their community property. Figure 3 is a simple BTN which is described in space L and space P. S1 S9 are bus stations, the solid line and dashed line are two bus routes, and S3 is the share station of the two bus routes. In space P, to a bus route, there exists an edge between any two bus stations in the route, there does not exist a station which is not in the route and has links to all the stations which are in the route. Therefore, a bus route is a maximal complete subgraph in a BTN, namely clique. In a BTN, different bus routes maybe have one or more share stations, namely, cliques intensively connect Fig. 3 A simple BTN which is described in space L and space P. 2.2 Community Division of BTNs Recently, there are a lot of studies on the algorithms of detecting community structure in networks. [6 7] These algorithms can be classified into two kinds. One kind divides a network into several separate communities, including Kernighan-Lin algorithm, [13] spectrum algorithm, [14] division algorithm, [15] agglomerative algorithm, [7] and so on. The other divides a network into several overlapping communities, namely one vertex may belong to more than one community at the same time. The typical method is the k-clique community algorithm proposed by Palla et al. [9,16] Using improved community detecting algorithm (PKM agglomerative algorithm), [17] we detect the community structure of the practical BTNs of Hangzhou [18] which is described with space L and space P respectively. Figure 4 shows the variety of the modularity of the BTN of Hangzhou which is described by space L with the algorithm steps. While Fig. 5 shows the variety of the modularity of the BTN of Hangzhou which is described by

3 No. 6 Research on Community Structure in Bus Transport Networks 1027 space P with the algorithm steps. The modularity Q proposed by Newman measures the quality of the community division of a network and the modularity of a network. [6] Consider a particular division of a network into k communities. They define a k k symmetric matrix e whose element e i.j is the fraction of all edges in the network that link vertices in community i to vertices in community j. The trace of this matrix Tr e = i e i,i gives the fraction of edges in the network that connect vertices in the same community. And they define the row (or column) sums a i = j e i,j, which represent the fraction of edges that connect to vertices in community i. Thus they can define the modularity Q by Q = i (e i,i a 2 i ). The value of the Q decides the quality of a particular community division of a network or the modularity of a network. Generally, it is suggested that a network has the obvious modularity (community property) when the value of Q is greater than 0.3. Fig. 4 The modularity of the BTN of Hangzhou described in space L. Figure 4 shows that the BTN described in space L has the obvious community property. The reason is that each vertex of the BTN described in space L only connects to several other vertices which are its predecessors or successors as the regular lattice. [4] The BTN described in space L can be approximately seen as a regular lattice which has the obvious community property. [19] So the BTN described in space L has the obvious modularity. Figure 5 shows that the BTN described in space P does not have the community property. However, in fact the BTN described in space P is composed of many cliques [10] (a clique is a maximal complete subgraph and a bus route is a clique) which are the communities with the densest connections inside (complete subgraph). Why can not we get the obvious community property from the BTN described in space P? We think the reason is that the community division method which divides a network into several separate communities is not proper to this kind of networks such as the BTN described in space P. The BTN described in space P is composed of many cliques, and these cliques connect each other by the intense overlapping vertices. In this kind of networks, one vertex may belong to several different communities. If one vertex is divided into one community narrowly and can not exist in any other communities, the wrong result that the network has no community property will be obtained and it will disagree with the reality. Fig. 5 The modularity of the BTN of Hangzhou described in space P. For this kind of networks, we expect to identify the overlapping communities. We try using the k-clique community algorithm proposed by Palla. [9] However, using this algorithm, we can not find the k-clique community structure defined by Palla in the BTNs described in space P. We think the reason is that the community definition proposed by Palla, which is the k-clique community composed of some k-cliques that have k 1 common vertices, is too strict to find the communities in the BTNs. The cliques in the BTNs share the vertices variedly, so there are few k-cliques which must share k 1 vertices. Therefore, we can find few k-clique communities in the BTNs, and this algorithm is not proper to detect community in the BTNs described in space P. We expect to design a new community definition and detecting algorithm which are proper to this kind of networks. 3 Definition of N-Depth Community and Detecting Algorithm The standard of the community division is that within communities the network connections are dense, but between communities they are sparser. For BTNs described in space P, within the communities the network connections are dense means that the bus stations in the communities are convenient to transport. While the network connections between the communities are sparse means that any two bus stations between the communities need more transfer times. In BTNs, the communities are overlapping intensely, in other words, one vertex may belong to several different communities at the same time. So, we propose a novel community structure which we called N-depth community. The detecting algorithm of the N-depth community is as follows. First, we find every cliques [9 10,16] in the original network. Then we construct a new network in which

4 1028 YANG Xu-Hua, WANG Bo, and SUN You-Xian Vol. 52 a vertex represents one clique, and there exists s an edge if two cliques share at least one vertex of the original network. We call the new network 1-depth clique network. The 2-depth clique network can be gotten by making the same operations stated above on the basis of the 1-depth clique network. So we can get the N-depth clique network by making the operation N times. Namely, the N-depth clique network is composed of the cliques found in the N 1-depth clique network. We define the N-depth communities of the original network. A vertex of the N-depth clique network corresponds to an N-depth community of the original network. Namely, those vertices, in the original network, which belong to an N-depth community, are mapped to a vertex of the N-depth clique network. There is an especial clique network the 0-depth clique network which is the original network, and a 0-depth community is a vertex of the original network. The definition of the N-depth community has the practical physical meaning. The community diameter of an N-depth community is N (as the network diameter, the community diameter means the longest distance between any two vertices within the community). We can get the maximal N by doing the operations stated above ceaselessly until getting an N-depth clique network which is composed of only one vertex. This vertex of the N-depth clique network corresponds to the N-depth community of the original network. In this case, the N-depth community includes all the vertices of the original network, and the community diameter N is also the diameter of the original network. Figure 6 shows the process of the N-depth community division of an ideal network (a vertex at least belongs to one clique in an ideal network). The 0-depth clique network is the original ideal network which is composed of 5 cliques denoted by the dashed circles with different colors. The 5 vertices of the 1-depth clique represent 5 cliques of the 0-depth clique network (the original network), and there is an edge between two vertices if these two cliques share at least one vertex of the original network. The color of the dashed circles in the 0-depth clique network corresponds to the color of the vertices in the 1- depth clique. We can get the 2-depth clique network and 3-depthe clique network by doing the operation repeatedly. The 3-depth clique network is composed of only one vertex, which means that 3 is the maximum value of N and the 3-depth community includes all the vertices of the original network. And 3 is also the diameter of the original network. So the ideal original network has five 1-depth communities, three 2-depth communities and one 3-depth community. The communities are overlapping intensely, which means some vertices can belong to more than one community. The N-depth community division we have discussed above happens in the ideal situation which means the N- depth networks of every layer can form cliques naturally (any vertex at least belong to one clique). And this kind of ideal network is existent such as the real BTNs. [10] However, if we use this community definition and detecting algorithm on the general networks, there will be some different from the ideal situation. In a general network, not all of the vertices can be divided into at least one clique, some vertices do not belong to any clique. So, these vertices are eliminated in the process of finding cliques. The N-depth communities can not include all the vertices of the original network. In other words, some vertices do not belong to any N-depth community. And it is reasonable that some vertices of a networks do not belong to any community in some particular divisions. [9] Certainly, In this case the maximum value of N is not equal to the original network diameter. Fig. 6 The process of the N-depth community division of an ideal network. Figure 7 shows the process of the N-depth community division of a general network. This general network (0-depth clique network) has two solid vertices which do not belong to any clique. These two vertices are eliminated in the finding clique process because they do not belong to any 1-depth community. We can see that vertices may be eliminated in the finding cliques process that

5 No. 6 Research on Community Structure in Bus Transport Networks 1029 the N 1-depth clique network transform to the N-depth clique network. So, there are some vertices which belong to some N 1-depth communities but do not belong to any N-depth community. Fig. 7 The process of the N-depth community division of a general network. 4 Application 4.1 A BTN Evolution Model [10] We apply this community definition and division algorithm to a BTN evolution model [10] described with space P, whose network property agrees wells with real BTNs, in which a vertex stands for a bus station and a clique stands for a bus route. The model is as follows. Starting with a clique whose size is a random number which agrees with the Gaussian distribution N(µ, σ) at t = 0, networks expand continuously by the addition of a new clique in each time step. The size of the new clique is a random number which agrees with the Gaussian distribution N(µ, σ). The new clique is composed of two parts which are some new nodes and some old nodes that already exists in the network, respectively. These old nodes will be randomly selected from those nodes which were added at previous time steps. The number of the new nodes and the old nodes agrees with the Gaussian distribution N(µ 1, σ 1 ) and N(µ 2, σ 2 ), respectively, with µ 1 +µ 2 = µ and σ 2 1 +σ 2 2 = σ 2. We connect any two nodes in the coming new clique with an edge, repeatedly connect if there already have been one or more edges between the two nodes, and these nodes will constitute a new maximal complete subgraph, namely the new clique. In this BTN evolution model, [10] we consider multiple edges. However, in the novel community definition and division algorithm, we only need the connectivity information and do not consider the level of busyness of a road. Therefore, in the study of this paper, in this BTN evolution model, we substitute single edge for multiple edges. 4.2 Application on BTN Model We use the BTN model to produce a network, which has 30 bus routes and 136 bus stations. And we apply this community definition and division algorithm to the network. First, we make the 1-depth community division on this BTN. Thirty 1-depth communities are obtained which are just the 30 bus routes. It means that a 1-depth community is composed of the vertices of corresponding bus routes. Repeating the operation on the 1-depth clique network, we can get seventeen 2-depth communities. A 2- depth community is composed of several bus routes. Repeating this operation, the 3-depth clique network is obtained which contains only one vertex, which means the BTN has only one 3-depth community and all the bus stations belong to this 3-depth community. It means this BTN model is an ideal network as stated in Sec. 3 and the diameter of the BTN model is equal to the deepest community diameter which is 3. Because the diameter of the N-depth community is N, in the BTN the N-depth community division also shows the transport convenience between two stations. The most transfer times between any two stations which belong to an N-depth community is N 1. Therefore if two stations belong to a common N-depth community, the smaller the value of N, the more convenient between these two stations. Therefore, the most transfer times between any two stations in above BTN model is 2. 5 Conclusion and Discussion Many practical complex networks have a common feature of community structure. Recently, there are a lot of studies on detecting community algorithms in networks. However, all these existing algorithm are not proper to the typical practical network: real BTNs. So, we propose a novel community structure called N-depth community and present the method of finding this kind of communities in networks. This community definition and division algorithm is not only proper to the BTNs but also most collaboration networks which are composed of many cliques such as scientific collaboration network [20] (one paper is one clique), actor collaboration network [21] (one movie is one clique), Chinese medical prescriptions network [22] (one medical prescription is one clique), and so on. This community division algorithm also shows the degree of relationship between any two vertices if they belong to a same N-depth community, the smaller the value of N the closer relationship between the two vertices.

6 1030 YANG Xu-Hua, WANG Bo, and SUN You-Xian Vol. 52 Reversing the process of detecting the N-depth community, we can obtain a new network evolving algorithm. The diameter of the network model created by this algorithm can be controlled. It means that we can create a controlled diameter network. From this viewpoint, it also has very good practical meaning. For this novel community structure and detecting algorithm, we just make an exploratory research. It is not the best definition and algorithm, but for collaboration networks such as BTNs it has enormous advantages. We expect that this community definition and detecting algorithm will be applied to more collaboration networks and even many real complex networks. We hope that the new controlled diameter network model can be carried out and applied. These are also our future research directions. References [1] R. Albert and A.L. Barabasi, Rev. Mod. Phys. 74 (2002) 47. [2] M.E.J. Newman, SIAM Review 45 (2003) 167. [3] T. Zhou, W.J. Bai, B.H. Wang, et al., Physics 34 (2005) 31. [4] D.J. Watts and S.H. Strogatz, Nature (London) 393 (1998) 440. [5] A.L. Barabasi and R. Albert, Science 286 (1999) 509. [6] M.E.J. Newman and M. Girvan, Phys. Rev. E 69 (2004) [7] A. Clauset, M.E.J. Newman, and C. Moore, Phys. Rev. E 70 (2004) [8] Z. Xie and X.F. Wang, Complex Systems and Complex Science 2 (2005) 1. [9] G. Palla, I. Derenyi, I. Farkas, et al., Nature (London) 435 (2005) 814. [10] X.H. Yang, B. Wang, W.L. Wang, et al., Commun. Theor. Phys. 50 (2008) [11] X.P. Xu, J.H. Hu, F. Liu, et al., Phys. A 374 (2007) 441. [12] Y.Z. Chen, N. Li, and D.R. He, Physica A 376 (2007) 747. [13] B.W. Kernighan and S. Lin, Bell System Technical Journal 49 (1970) 291. [14] F. Wu and B.A. Huberman, Eur. Phys. J. B 38 (2004) 331. [15] M. Girvan and M.E.J. Newman, Proc. Natl. Acad. Sci. 99 (2001) [16] I. Derenyi, G. Palla, and T. Vicsek, Phys. Rev. Lett. 94 (2005) [17] H.F. Du, W.F. Marcus, S.Z. Li, et al., Complexity J. 12 (2007) 53. [18] [19] H.F. Du, S.Z. Li, W.F. Marcus, et al., Acta Phys. Sin. 56 (2007) [20] A. Cardillo, S. Scellato, and V. Latora, Phys. A 372 (2006) 333. [21] N. He, W.Y. Gan, D.Y. Li, et al., Complex Systems and Complexity Science 3 (2006) 1. [22] Y. He, P.P. Zhang, J.Y. Tang, et al., Science & Technology Review 23 (2005) 36.