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Journal of Software Engineering and Applications, 20 12, 5, 42-45 doi:10.4236/jsea.2012.512b009 Published Online December 2012 (http://www.SciRP.org/journal/jsea) Copyright © 2012 SciRes. JSEA Dimension al Measurement of Com pl et e-connective Network under the Condition of Particle’s Fission and Growth at a Constant Rate* Jinsong Wang, Beibei Hu Department of management sci ence and engin eer ing, Bei jing University of Technology, Beijing100124, P.R. China. Email: xizixinxiang@163.com Received 2012 ABSTRACT We construct a complete-connective regular network based on Self-replication Space and the structural principles of cantor set and Koch curve. A new definition of dimension is proposed in the paper, and we also investigate a simplified method to calculate the dimension of two regular networks. By the study results, we can get a extension: the formation of Euclidean space may be built by the process of the Big Bang’s continuously growing at a constant rate of three times. Keywords: Particle’S Fis sio n; Regular Frac ta ls; the Complete-Connective Network; N etwor k Dimension 1. Introduction The recent research shows that the fractal structure exists in massive networks, and there are different methods to calculate the fractal dimension of the network. Song et al. [1-3] adopted the Box counting method and Burning al- gorithm to measure the fractal dimensions of the net- wor ks; Ki m et al . [4] also applied the Box counting method to the measurement of fractal dimensions of scale-free networks. However, bui lding the ne twork is stil l an under explored field which has not yet been adequatel y s t ud ied , especially by using a fractal set to identify the network and calculate its dimension. In this paper, we attempt to make the further analysis on the construction of the network in the perspective of using the nodes of so me regular fractals to build the net- works. The study is based on the theory of self-replication algebraic space, and assumes the par- ticle’s fission grow at a constant rate. To build the net- wor k, we take the nodes from some regular fractals, such as the end point of cantor set and the apex of modified Koc h curve . T he st ud y result s su gges t tha t the j oi nt fu nc- tion of the correlation and connectivity among network nodes might influence the measurement of network di- mensi on and sp ace str ucture . Thus, thi s work c ontrib utes to re-measuring the dimension of the network from the perspective of the joint function of correlation and con- necti vi t y amo ng ne t wor k no de s. 2. Problem Introduction If a particle A generates m particles after one fission, and get m2 particles after twice fissions, , there will be mn articles after n times of fissio ns, and these n m particles could form a space. In this space, the n m particles are taken as the nodes, which mean the salient points, and the network will form through the interaction among the m particles. We use n m m C to describe the total correlation degr ee of t he net work, so the next step is to calculate the dimension of the constructed regular network space. To calculate the dimension from a new perspective of corre- lation and connectivity among network nodes, three hy- potheses are introduced: 1) Self-replication Space: Particle A generates N par- ticles after n times fission. The particles are independent mutually in the proceeding of fission, at the same proba- bility, and the correlation degrees among these particles are also the same with each other. T hus, the space formed by the N particles is regarded as a Self-replication Space. 2) Network Connectivity: Regular fractals, such as the cantor set and the Koch curve, describe the fragmented and singular graphics which are self-similar with unli- mited multi-levels. Following the principle of the cantor set, and by using its end point as the nodes, correlation is established among these nodes and then a complete- connective network is formed, as shown in Figure 1-((a)(b)). And in Figure 1-((c)(d)), the paper rebuilds the Koch curve based on its basic principles and then takes the apex of the modified Koch curve as the nodes. Finally, a complete-connective network is also formed *P roject supported by the Major Program of the National Natural Science Foundation of China (Grant No. 71261026) National Key Techn ology R & D Program of China (Gran t No. 201 2BAJ11B03) Dimensional Measurement of Complete-connective Network under the Condition of Particle’s Fission and Growth at a Constant Rate Copyright © 2012 SciRes. JSEA 43 thro ugh t he co rre lation among these kinds o f node s . 3) Net work tr ansfor ms based on the c antor set a nd the Koch curve: It has been revealed that some of the regular fractals have a feature of the particle’s fission at a con- stant rate. For example, the end points of the cantor set always grow at a constant fission rate of two times, and the apex of the modified Koch curve with the constant rate of four times. Therefore, on the premise of retaining the topological feature of fractals, the networks built in this paper are still based on the fundamentals of the can- tor set and the Koch curve, and gain through the 2 or 3 times fission of particle A which still exists during the fission. More specifically, we regard that all particles after fi ss io n own t he homo ge n eo us mut ua l c or r el ati o n, so the particles A ,1 A ,2 A , or A ,1 A , 2 A , 3 A will be gained after the first fission of particle A ,…, continuously, the new particles will be generated after one more iteration, and finally there are N particles after several fissions. These N particles will form a complete-connective regu- lar network. However, there are two additional assumed cond itions: a) in o rde r to get the net work with sel f- simi- larity from N particles, we first generate the network in accordance with an identically independently-distributed process with the same probability. (b)Additio nally, when the N particles are taken as the network nodes to build regular network, there is random distance between each nodes. 3. A New Definition of Dimension and Mea- surement of Network Dimension Definition 1. We assume there are N nodes, and they compose a network based on the interaction of the mul- tiple nodes. The relevance K is used to describe the corre- lation degree of this network space, and its dimension d is represented as: ln lim ln n K dN →∞ = . (1) Theorem1. If particle A generates m particles 1 A, 2 A , ···, m A ( 2m≥ ) during the time 1 t where the correlation degree among these m particles is totally 1 =C m m K ; and then after time 2 t , each of the particle i A ( ,1, 2,,ij m= ) generates m particles ij mA ( ) ,1, 2,ij m= with 2 2m m KC= ,···, continuing to re- peat to timen t, there are = n n Nm particles from A with n m nm KC= . Accordingly, the dimension d of the regular network here is formed by the n N particles generated from particle A duri ng time n t : ln ln lim lim ln ln n m nm nn n C K dm N nm →∞ →∞ === (2) Proof: Mathematical induction is applied to solve this problem. Firstly, we have an explanation, i d represents the dimensions of network space composed by n i particles. (1) When m is equal to 2. Through the first fission, there are 2 particles. Thus, the total correlation degree 1 K is 2 2 C . Through the second fission, there are 2 2 particles. Thus, the total correlation degree 2 K is 2 2 2 C . Through the n times fission, there will be 2 n par- ticles. Thus, t he total corr e la tion degree n K is 2 2 n C . ( )( ) 2 -1 2 22-1 ===22-1 2 n nn nn n KC Figure 1. Fractal and comple te -connective net work graph. Current Distortion Evaluation in Traction 4Q Constant Switching Frequency Con verters Copyright © 2012 SciRes. JSEA 44 According to the Equation (2), we can attain: ( ) ( ) ( ) -1 2 ln22 -1 ln = limlim ln ln 2 -1ln2+ ln2-1 = lim=2= ln 2 nn nn nn n n n K dN nm n →∞ →∞ →∞ = (2) (2) When m is equal to 3. Through the first fission, there are 3 particles. Thus, the total correlation degree 1 K is 3 3 C . Through the second fission, there are 2 3 particles. Thus, the total correlation degree 2 K is 2 3 3 C . Through the n times fission, there will be 3n par- ticles. Thus, t he total corr e la tion degree n K is 3 3 n C . ( )( )() 3 -1 3 33 -1 ===33-1 3-2 3 n nn nn n n KC According to the Equation (2), we can attain: ( )() ( ) ( )() -1 2 ln33-1 3-2 ln = limlim ln ln3 -1 ln3+ln3 -1 +ln3 -2 = =3= ln3 nn n nn nn n nn K dN nm n →∞ →∞ = (4) (3) For the same reason, when m is equal to k. Through the n times fission, the dimensions d of network space composed by n k particles are equal to k.That’s to say, = k dk . When = +1mk Then ( ) ( ) +1 +1 =n k nk KC , ( ) = +1 n n Nk First ly, we k now when n→∞ , ( ) ( ) +1 -1 .. +1 nk as n Kk→ (5) Therefore, we can attain: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) () () +1 +1 +1 +1 +1 +1 +1 ln ln = lim= lim ln ln +1 lnln ln = limlnln+1 ln ln ln lim ln+1 ln ln+1 -1ln+1 = limln+1-1 ln 1 =1+=+1= n nn n n n k k n kn nn n k k kk k nk k k k nk n C K dNk C kC nk kC kC kkC k nkk kk nkk k km k →∞ →∞ →∞ →∞ →∞ ⋅⋅ ⋅⋅ ⋅ = ⋅⋅ ⋅⋅ ⋅⋅⋅ (6) We have got the conclusion that, through n times fis- sion, the dime nsion d of the network space composed by the n m new particles is equal to m. The demonstration is completel y shown in the Equa tion(2) . 4. Simple Calculation method of Dimension of Regular Network Based on the New De- finition of Dimension As predicted above, two new particles can be generated from only one particle, and each of the two new ones can generate extra two particles, so the correlation could be established among any two of these 2 2 particles to form a complete-connective regular network(Figure 2-(a)). According to the Equation (2), the dimension of the ne w network here could b e calculated as 2 2 2 ln ln6 1.293 ln 22ln 2 C d== ≈ . Furthermore, if the newly generated 2 2 particles get two more particles separately, the correlation could be established among any two of the 3 2 particles, and the dimension of this complete-connective regular network could be calculated as 3 2 2 3 ln 2 ln71.602 3ln 2 ln 2 C d+ = =≈ . To get correlation among the fresh 4 2 particles split from those 3 2 particles, then we can get another regular network with complete-connectivity, its dimension can be calculated as4 2 2 4 ln 2 ln71.727 4ln 2 ln 2 C d+ ==≈ ;···; To put it briefly, the dimension of the complete-connective regular network will increase gradually, as the number of newly generated particles grows at a constant rate of 2 times. When the number of particles becomes 2n , and n→∞ , the complete-connective regular network has the dimen- sion =2 d . Figure 2. (a) shows Complete-connective regular networks built by particles growing at a constant rate of 2 times. (b) shows complete-connective regular networks built by par- ticles g rowing at a const ant rate of 3 times. Dimensional Measurement of Complete-connective Network under the Condition of Particle’s Fission and Growth at a Constant Rate Copyright © 2012 SciRes. JSEA 45 To consider the situation that one particle can produce 3 new particles, and each of the 3 new ones can have other extra 3 particles, so totally there are 2 3 particles and a complete-connective regular network based on the correlation among these particles (Figure 2-(b)). Also accord ing t o the d e fi niti o n o f di men sio n, we ca n me a sur e the network with dimension 2 3 3 2 ln 2ln 2ln 212.016 2ln3 ln3 C d+ = =≈ . Another network will be formed from the fission of the 2 3, and the dimension of this network with totally 3 3 particles will be calculated as 3 3 3 3 ln 2(ln3 ln5)+ln132.421 3ln3 ln3 C d+ = =≈ . If we continue, there will be the network composed by 4 3 particles with dimension 4 3 3 4 ln 2.584 ln3 C d= ≈ ; ···; In short, the dimension of the complete-connective regular network will increase gradually, as the number of newly generated particles grows at a constant rate of 3 times. When the number of particles becomes 3n and n→∞ , the complete-connective regular network has the dimen- sion =3d . 5. Conclusion Based on the Self-replication Space and the principle of the cantor set and the Koch curve, we establish a com- plete-connective regular network, and propose a new definition of the dimension due to the network correla- tion: ln lim ln n K dN →∞ = . We identify that the number of network nodes estab- lished by particles after fissions at a constant rate of m times is = n n Nm , and the original particle will not exist after the fission. Furthermore, our examination suggests that the total co rrelation degree among no des is = n m nm KC , and as n→∞ , the dimension =dm . Moreover, we also give a simplified method for the measurement of the di- mension of the two regular networks. T hen we can get a extension: the formation of the three-dimension Eucli- dean space may be built by the process of the Big Bang's continuously growing at a constant rate of 3 times, to some extent, this can provide more supportive explana- tion and judgment on the causal relationship between materials and space. REFERENCES [1] Song Chao-ming, Gallos L K, Havlin S,et al. How to calculat e the fractal dimensi on of a complex network: The box covering algorithm [J]. Journal of Statistical Me- chani cs, 2007 [2] Song Chao-ming, Havlin S, Makse H A. Sel f-si milarity of complex networks[J].Nature, 2005, 433: 392-395 [3] Song Chao-ming,Havlin S,Makse H A.Origins of fractal- ity in the growth of complex networks[J].Nature Physics， 2006，2：275-281. [4] Kim J S, Goh K-I, Kahng B, et al. A box-covering algo- rithm for fractal scaling in scale-free networks[J]. Chaos, 2007, 17（2） |