Visualization of Complex Networks Based on Dyadic Curvelet Transform

Interdisciplinary Description of Complex Systems 4(1), 51-62, 2006 VISUALIZATION OF COMPLEX NETWORKS BASED ON DYADIC CURVELET TRANSFORM Marjan Sedighi Anaraki1,*, Fangyan Dong1, Hajime Nobuhara2 and Kaoru Hirota1 1 Department of Computational Intelligence & Systems Science, Tokyo Institute of Technology Yokohama – Tokyo, Japan 2 Department of Intelligent Interaction Technologies, Graduate School of Systems and Information Engineering, University of Tsukuba Ibaraki, Japan Regular paper Received: 28 April, 2006. Accepted: 5 July, 2006. SUMMARY A visualization method is proposed for understanding the structure of complex networks based on an extended Curvelet transform named Dyadic Curvelet Transform (DClet). The proposed visualization method comes to answer specific questions about structures of complex networks by mapping data into orthogonal localized events with a directional component via the Cartesian sampling sets of detail coefficients. It behaves in the same matter as human visual system, seeing in terms of segments and distinguishing them by scale and orientation. Compressing the network is another fact. The performance of the proposed method is evaluated by two different networks with structural properties of small world networks with N = 16 vertices, and a globally coupled network with size N = 1024 and 523 776 edges. As the most large scale real networks are not fully connected, it is tested on the telecommunication network of Iran as a real extremely complex network with 92 intercity switching vertices, 706 350 E1 traffic channels and 315 525 transmission channels. It is shown that the proposed method performs as a simulation tool for successfully design of network and establishing the necessary group sizes. It can clue the network designer in on all structural properties that network has. KEY WORDS visualization, complex network, human visual system CLASSIFICATION PACS: 89.75.-k *Corresponding author, η: ; +81 45 924 5686, +81 45 924 5682; G3-49, 4259 Nagatsuta, Midori-ku, Yokohama-city 226-8502, Japan M. Sedighi Anaraki, F. Dong, H. Nobuhara and K. Hirota INTRODUCTION Complex networks are being studied across many fields of science [1 – 4] inspired by empirical studies of networked systems such as the internet, biological networks like brain neural networks and so on. Scientists think they know most of the elements and forces of networks [5, 6]. But networks are inherently difficult to understand because of structural complexity. For a network with millions of vertices direct analyzing by sketching the structure of network and looking at them by eye is hopeless. Having some static displays for instance vertex map or matrix, data filtering as an interactive control, or smooth zooming can be useful. The recent development of statistical methods for quantifying huge networks is to a large extent an attempt to find something to play the part played by eye for complex networks. Furthermore it usually is assumed that the network architecture is static. These simplifications allow us to sidestep any issues of structural complexity and propose a visualization method based on an extended Curvelet transform named Dyadic Curvelet Transform (DClet) for understanding the topology of complex networks. The proposed method gives us the matrix of network in dyadic scales, filters data by the coarse to fine strategy and brings us smooth zoom without losing the structural properties of a network. This modification is done to solve the Curvelet transform [7, 8] inconvenience of decomposition into components at different scales. One potential of this proposal might be the optimization of a network by removing redundant edges in the network without changing the desired properties of it. The aim of this research is showing that the proposed visualization method with directional sensitivity can be considered as an efficient tool for investigating the structural properties of complex networks. It can be considered as a simulation tool, or a compression method. It does not have the geographical displays of a network. As the topology of network is robust in any scales of the DClet, it can be worked on a smaller matrix and modeled the real network with the DClet reconstruction. The performance of the proposed method is evaluated in two different networks with structural properties of small world networks with N = 16 vertices, and a globally coupled network with size N = 1024 and 523 776 edges. As the most large scale real networks are not fully connected, it is tested on the telecommunication network of Iran as a real extremely complex network with 92 intercity switching vertices, 706 350 E1 traffic channels and 315 525 transmission channels. It is presented that the proposed method provides a natural way of analyzing a complex network due to analyzing the network at a coarse level and then increasing the resolution in dyadic scale, if necessary. The experiments have been done in Matlab on a personal computer with processor Pentium IV 2.4 MHz, RAM 256 M, and HDD 60GB. The details of evaluation of the proposed method on the telecommunication network of Iran is not shown here just the final outputs are considered as an Annex. The proposed method based on the DClet is presented in second section. In third section, experimental results are shown. The article is concluded with some discussion and an outlook. VISUALIZATION METHOD BASED ON THE DCLET The idea of the DClet is first to decompose the input data into a set of wavelets bands and analyze each band by a non-redundancy Finite RIdgelet Transform (FRIT) [9]. This idea makes a directional wavelet that outperforms wavelet for orientation. It is scale sample of wavelet transform following a geometric sequence of ratio 2. The input is decomposed into smooth blocks of side length l units in such a way that adjacent blocks are square array of 52 Visualization of Complex Networks Based on Dyadic Curvelet Transform size l × l although it is possible to be rectangular array of size l × l/2 without overlapping. Fig. 1 shows the general concept of the DClet. It achieves non-redundancy transformation with invertibility via the FRIT. The FRIT is ridgelet transform based on the Finite RAdon Transform (FRAT) that uses the orthogonal symmlet wavelet with four vanishing moments. For the FRAT, first the set of normal vectors is obtained which indicate the represented directions. The decomposition transform for an input matrix with size M × M is done according to the following procedure: First, the 2D wavelet transforms (•)WAr(b) of a matrix A with size M × M, A ∈ ZM × M at scale r and translation b with the mother wavelet ψ is given by: (•) WAr(b) = A(p1, p2)∗(•)ψbr(p1, p2), (1) where b = {1, 2, …, M/2r} and (•) shows the approximation or details and ∗ denotes convolution operation: M M WAr(b) = ∑ ∑ A( p1, p2 ) (•)ψbr(p1, p2). (•) (2) p1 =1 p 2 =1 Then, subband decomposition: A( p1, p2 ) ⇒ (LL)WAr (b) + (LH )WAr (b) + ( HL)WAr (...truncated)