Complex Network Visualization based on Voronoi Diagram and Smoothed-particle Hydrodynamics Zhao Wenbin 1, Zhao Zhengxu 2 1 School of Instrument Science and Engineering, Southeast University, Nanjing, Jiangsu 210096, China 2 School of Information Science and Technology, Shijiazhuang Tiedao University, Shijiazhuang, Hebei 050043, China zhaowb19851015@gmail.com, zhaozx@stdu.edu.cn Abstract. With rapid development of complex networks, it is difficult to only use figures and charts for their management and planning. Visualization technology can offers an effective approach to understand the structure and characteristics of complex networks, so as to mine more valuable information. This paper presents a complex network visualization method, which is based on smoothed-particle hydrodynamics (SPH) and Voronoi diagram. Using statistical characteristics of complex network, this method determines the position of complex network nodes in a two-dimensional plane, and divides this plane into Voronoi diagram, then analyzes the structure and characteristics of complex network. In this paper, the experiment analyzes engineering software format conversion network and realizes its visualization. Keywords: complex network, visualization technology, Voronoi diagram, layout algorithm, smoothed-particle hydrodynamics. 1 Introduction In recent years, with rapid development of various information systems, such as World Wide Web [1], telecommunications network, mobile communications network, etc, the scale of data increases gradually, consequently, traditional technology and method cannot meet the management and operation of these complex networks [2, 3]. By research of web network, social network, biological network, etc, it is found that these networks have some properties in common, such as self-organization, selfsimilarity, attractor, small world [4], scale-free network [5], etc, so a network which has more or all properties is called complex network. Due to the complexity of complex network structure, it is difficult to understand complex network with figures and charts, and to express a lot of information contained in complex network. Visualization method can expresses complex network conveniently and intuitively. The research of complex network further promotes the development of network visualization technology, and simultaneously put forward higher demand for visualization technology. With intuitive accurate display of network, complex network visualization is helpful for understanding its internal structure and mining 55
valuable information [6]. This paper introduces the research of complex network visualization and provides a visualization method of complex network which is based on Voronoi diagram and smoothed-particle hydrodynamics (SPH). 2 Related Works Complex network visualization can express complex network conveniently and intuitively. Its research involves complex system, graph theory, statistics, data mining, information visualization, human-machine interaction, etc. Layout algorithm attracts more concern in complex network visualization. Classic layout algorithms include ring layout algorithm, orthogonal layout algorithm, hierarchical layout algorithm, tree graph layout algorithm, etc. These algorithms have better layout for simple network which has less nodes, but they are not suitable for network which has more nodes and heterogeneous connection. Force-directed algorithm (FDA) [7, 8] was provided by P. Eades in 1984. Its basic idea is that a graph is considered as a physical system, which takes node as steel ring, edge as spring, the system is set at the initial state, so the force of spring causes the movement of steel ring, when the total energy of the system decreases into the minimum value, the movement stops. T. M. J. Fruchterman and E. M. Reingold provided an improved FDA algorithm, FR algorithm [9], which follows that nodes which are connected by edges should be close to each other but not too close. D. S. Chan and K. S. Chua propose an ODL algorithm [10], which is different from traditional Force-directed algorithm, it uses ranking algorithm to reduce the complexity of algorithm and improve efficiency, so as to meet large complex network. A. Ahmed provided a 2.5-dimentional visualization method for scale-free network [11], which puts different level nodes in appropriate plane and uses improved Fast FDA algorithm in each plane, so as to get good visualization effect. 3 Voronoi Diagram of Complex Network This paper presents a complex network visualization method, which determines the position of network nodes in the two-dimensional plane with SPH algorithm, and divides this plane into Voronoi diagram. 3.1 Layout Algorithm With SPH layout algorithm, visualization method arranges network nodes by the statistical characteristics of complex network and determines their position in the twodimensional plane with point-by-point insertion method. SPH algorithm is a widely used mesh-free method, which is used in many fields. Its basic idea is the fluid is divided into a set of discrete particles. These particles have 56
a spatial distance, over which their properties are "smoothed" by a kernel function. This means that the physical quantity of any particle can be obtained by summing the relevant properties of all the particles which lie within the range of the kernel [12]. In this paper, the nodes of complex network are considered as a series of particles which own quality, speed and energy, their distribution is calculated in a two-dimensional plane. Discrete SPH function is as follow: N 1 u( x) = [ u( xi ) + w( x xi, h)] (1) N i= 1 Where, w(x-x i,h) is the interpolation kernel function of the relevance between node x and x i as well as control factor h, h determines the size of the kernel function s influence domain. This paper adopts B splint function as the kernel function. 3.2 Neighbor particle search Neighbour particle search is an important and the most time-consuming step in SPH method. Each particle can interact with other particles in computation, so neighbour particle search is a N 2 problem. SPH method commonly adopts two neighbour search algorithms: linear search method and tree search method. These algorithms have their own advantages and disadvantages, linear search method is simple but inefficient, and tree search method is efficient but more complex. RNN algorithm is a nearest neighbour search algorithm based on Voronoi diagram [13]. This paper adopts RNN algorithm as neighbour particle search algorithm, Figure 1 show particle d i s influence domain and particles in its influence domain. d i (a) Particle d i s influence domain (b) Particles in Particle d i s influence domain Fig.1. Particles d i s influence domain and particles in its influence domain 3.3 Visualization Method The proposed complex network visualization method includes the following steps: (1) According to complex network, its statistical characteristics are determined, such as directed or undirected, the number of nodes, the number of edges, average degree, average path length, clustering coefficient and so on; 57
(2) Nodes of complex network are ranged by node degree in descending order, and their positions are determined with SPH layout algorithm in the two-dimensional plane; (3) According to the positions of nodes in complex network, the plane is divided into Voronoi diagram. 4 Experiment Experiment takes engineering software as research subject, selects two-dimensional and three-dimensional graphic design software, engineering simulation software and data document processing software, and statistics their input and output formats. Engineering software format conversion network is a network which takes data formats as network nodes and is based on the conversion relationship between data formats of engineering software. Engineering software format conversion network is defined as G=(V, E), which can express topological relationship between engineering software data formats. The node set of engineering software data formats is V={v i i=1,2,,n}, if data format v i V can be converted to data format v j V, a directed edge e(v i, v j ) is create in network graph. Matrix W is the adjacency matrix of network graph G. if a directed edge exist between v i and v j, w ij =1, otherwise w ij =0. According to statistical data formats and conversion relationship, experiment construct engineering software format conversion network, as shown in Figure 2. Fig.2. Engineering software format conversion network Fig.3. the relationship between average path length and the number of nodes Experiment analyzes the statistical characteristics of engineering software format conversion network, such as average path length, clustering coefficient and degree distribution. Table 1 show nodes number, average path length, clustering coefficient and network diameter. Table 1. The basic index of engineering software format conversion network Node Number Average Path Length Clustering Coefficient Network Diameter 54 1.48186 0.49968 2 58
82 1.55829 0.57296 2 106 1.73248 0.54861 3 131 1.81549 0.54213 3 157 1.85644 0.52463 3 186 1.87075 0.57556 4 223 1.88114 0.63686 4 268 1.88823 0.62332 4 306 1.93619 0.58704 4 346 1.97497 0.57292 4 378 2.03986 0.56924 4 397 2.06192 0.56428 4 416 2.07654 0.57589 4 431 2.12055 0.58968 4 456 2.13339 0.60699 4 473 2.13819 0.60957 4 488 2.13865 0.61743 4 Figure 3 show the relationship between average path length and nodes number. The curve s growth become gradually slow, it indicates engineering software format conversion network follows small world property. Fig.4. the relationship between clustering coefficient and the number of nodes Fig.5. the distribution curve of nodes degree Fig.6. the distribution of nodes cumulative degree Fig.7. the fitting curve of nodes cumulative degree Figure 4 show the relationship between clustering coefficient and the number of nodes. When the number of nodes increases, clustering coefficient of network fluctuates in small-scale. Therefore, clustering coefficient is irrelevant to network scale. 59
Figure 5 show the probability distribution of nodes degree. Nodes whose degree is less than 180, account for about 90% of the total number of nodes. The average of nodes degree is 63.98. The degree of nodes can express the importance of nodes in a sense. Nodes which have higher degree are less, but they are the key node of network. The cumulative probability distribution is used to describe nodes degree, as shown in the Figure 6. The cumulative degree of nodes in engineering software format conversion network follows exponential distribution. Through curve fitting, the result is as shown in Figure 7. Nodes cumulative degree follows exponential distribution curve P(k)=0.7502e -0.0199k. Figure 8 show Voronoi diagram of engineering software format conversion network. Data formats are divided into several clustering, this shows the proposed visualization method is effective. The distribution of some clustering is denser, and the distribution of other clustering is sparser. This is because denser clustering contains more data formats which can be interconverted, and conversely sparse clustering contains less data formats. In Voronoi diagram the neighbouring relationship between data formats can directly reflect their conversion relationship. Data formats which can be converted into more data formats locate at centre of their distribution, conversely, other data formats locate at edge of their distribution. Fig.8. Voronoi diagram of engineering software format conversion network 5 Conclusion and future work Visualization technology offers an effective method for researching complex network, and makes analysis of complex network more flexible and diversified. This paper proposes a visualization method of complex network which is based on Voronoi diagram and smoothed-particle hydrodynamics (SPH). According to statistical characteristics of complex network, this method uses SPH layout algorithm to 60
determine the position of complex network s nodes in two-dimensional plane and divides this plane into Voronoi diagram. Experiment takes engineering software format conversion relationship as research subject, achieves its complex network visualization and analyzes its structure and properties. At present, complex network mainly focuses on the structure and characteristics of static complex network, but these structure and characteristics cannot precisely describes continuing evolution of complex network in reality. In future, we will study how to achieve visualization of complex network in dynamic evolution process, and provide an effective visualization method to analysis dynamic complex network. Acknowledgments. This study is funded by the National Natural Science Foundation of China under Grant No.60873208. References 1. Albert R, Jeong H, Barabási A L.: Diameter of the world-wide web. J. Nature. vol. 401, No. 6749, pp. 130--131(1999) 2. Faloutsos M, Faloutsos P, Faloutsos C.: On power-law relationship of the Internet topology. J. ACM SIGCOMM Computer Communication Review. vol. 29, No. 4, pp. 251--262(1999) 3. Barabási A L, Bonabeau E.: Scale-free networks. J. Scientific American. vol. 288, pp. 60-- 69(2003) 4. Watts D J, Strogatz S H.: Collective dynamics of smallworld networks. J. Nature. Vol. 393, pp. 440--442(1998) 5. Barabási A L, Albert R.: Emergence of scaling in random networks. J. Science. Vol. 286, No. 5439, pp. 509--512(1999) 6. Adel Ahmed, Tim Dywer, Seok-hee Hong, Colin Murray, Le Song, Ying Xin Wu.: Visualisation and Analysis of Large and Complex Scale-free Networks. In: EUROGRAPHICS-IEEE VGTC Symposium on Visualization, pp. 239--246(2005) 7. Eades P.: A heuristic for graph drawing. J. Congressus Nutnerantiunt. Vol. 42, pp. 149-- 160(1984) 8. Kamada T, Kawai S.: An Algorithm for Drawing General Undirected Graphs. J. Information Processing Letters. Vol. 31, pp. 7--15(1989) 9. Fruchterman, T M J, Reingold, E M.: Graph drawing by force-directed placement. J. Software-Practice and Experience. Vol. 21, No. 11, pp. 1129--1164(1991) 10. Chan D, Chua K, Leckiem C, Parhars A.: Visualisation of Power-Law Network Topologies. In: 11th IEEE International Conference on Networks (ICON2003), Sydney, Australia, pp. 69-74(2003) 11. Quigley A, Eades P.: FADE: Graph drawing, clustering, and visual abstraction. In: International Symposium on Graph Drawing (GD2000), Springer, Vol. 1984/2001,pp. 197--210(2000) 12. Liu G. R, Liu M. B.: Smoothed Particle Hydrodynamics: a meshfree particle method. Singapore: World Scientific (2003) 13. Ioana Stanoi, Mirek Riedewald, Divyakant Agrawal, Amr EI Abbadi.: Discovery of influence sets in frequently updated databases. Proc. Int. Conf. on Very Large Databases (VLDB), pp.99-108(2001) 61