On General Laws of Complex Networks

Transcription

1 On Genera Laws of Compex etwors Wenjun Xiao, Limin Peng, an ehrooz Parhami 3 Schoo of Software Engineering, South China University of Technoogy, Guangzhou 5064, P.R. China wjxiao@scut.eu.cn Department of Computer Science, South China University of Technoogy, Guangzhou 5064, P.R. China pengm86@6.com 3 Dept. Eectrica & Computer Eng., University of Caifornia, Santa arbara, CA , USA parhami@ece.ucsb.eu Abstract. y introucing an anayzing a renormaization proceure, Song et a. [] raw the concusion that many compex networs exhibit sef-repeating patterns on a ength scaes. First, we aim to emonstrate that the aforementione concusion is inaequatey justifie, mainy because their equation (7) on the invariance of egree istribution uner renormaization oes not ho in genera. Secony, arabási an Abert [] fin that many arge networs exhibit a scae-free power-aw istribution of vertex egrees. They show this common feature to be a consequence of two generic mechanisms: (i) networs expan continuousy by the aition of new vertices, an (ii) new vertices attach preferentiay to those that are areay we connecte. We show that when vertex egrees of arge networs foow a scae-free power-aw istribution with the exponent γ, the number of egree- vertices, when nonzero, is of the same orer as the networ size an that the average egree is of orer ess than og. Given that many rea networs satisfy these two conitions, our resuts a another necessary characteristic of the scae-free power-aw istribution of vertex egrees in such networs. Our metho has the benefit of reying on conitions that are static an easiy verifie. They are verifie by many experimenta resuts of iverse rea networs. Keywors: compex networ, sef-simiarity, scae-free, computer networ, computer communication. Introuction Compex systems with many components an associate interactions arise in nature, society, an many human artifacts. Interactions in such systems can be moee by networs compose of vertices an ins, which are in turn abstracte as unirecte or irecte graphs. A graph G, enote as G = (V, E), has a set V of vertices or noes an a set of eges or ins, where each ege is efine by a pair of vertices (orere pair, for irecte graphs) [7]. Compex systems in the three categories of natura, societa, an synthetic incue: J. Zhou (E.): Compex 009, Part I, LICST 4, pp. 8 4, 009. ICST Institute for Computer Sciences, Socia Informatics an Teecommunications Engineering 009

2 On Genera Laws of Compex etwors 9. Protein interactions, metaboic systems, contagious iseases. Acquaintances, movie-actor peer group, research coaborators 3. Power gri, Internet connectivity, Worwie Web inages. Two moes of compex networs have been stuie extensivey [-0]: the sma-wor moe an the scae-free one. The sma-wor moe features ocaize custers that are connecte by occasiona ong-range ins, eaing to an average istance between vertices that grows ogarithmicay with the networ size. Watts an Strogatz [3] investigate mechanisms via which a reguar networ can be transforme into a sma-wor networ, without significanty moifying the vertex-egree istribution, an quantifie the parameters that characterize the resuting structures. Scae-free networs, on the other han, ten to have uneven vertex connectivities, so that a certain fraction of vertices, inepenent of networ size, are highy connecte (the hubs). arabási an Abert [] emonstrate that the scae-free power-aw istribution of vertex egrees in many arge networs is a irect consequence of two generic mechanisms that govern networ formation: (i) etwors expan over time through the aition of new vertices, an (ii) ew vertices attach preferentiay to those that are areay we connecte. It is we-nown that scae-freeom of a networ has significant impications for its iffusion properties an its robustness. In this paper, we focus on scae-free networs. After reviewing the parameters an ey attributes of such networs in Section, we provie necessary characteristic conitions for scae-free compex networs an show that these conitions are both easy to verify an satisfie by many natura an man-mae scae-free networs (Section 3). Section 4 contains our concusions an some irections for further research. On Sef-simiarity of Compex etwors We begin by reviewing some reevant properties of compex networs [-6]. Two moes of rea compex networs have been stuie extensivey: the sma-wor moe an the scae-free one. We sha focus first on scae-free networs. To avoi confusion with equations in this paper, equation numbers in our references wi be encose in square bracets. For many compex networs, the probabiity istribution P() of the number of egree- vertices, aso nown as the egree istribution, can be represente (inepenent of scae) by a power aw with characteristic exponent γ ([] means that this is equation [] in ref. ): P() γ. [] () The renormaization metho, introuce in ref., is as foows. Let G be a networ of egree istribution P() satisfying equation (). Vertices of G are covere by boxes of inear size. oxes are then viewe as vertices of a renormaize networ G, with two such vertices (boxes) connecte in G if an ony if there exists at east one in between their constituent vertices in G. Song et a. [] guie ony by experimenta resuts for WWW, then procee with the assumption that the egree istribution is invariant uner renormaization:

3 0 W. Xiao, L. Peng, an. Parhami P() P( ) ( ) γ. [7] () An here ies the probem: equation () is neither prove mathematicay nor verifie aequatey by experimentation with ifferent compex networs. Hence, the vaiity of equation () is suspect. We sha eneavor to show equation () invai by examining both the theoretica an experimenta anayses of Song et a []. In the theoretica anaysis that foows, we refer to, an use, four equations from ref., reprouce beow for competeness. Here the networ G is cae sef-simiarity if the equation (3) hos. M ( ) / ( ). [5] (3) = s( ). [6] (4) s( ). [8] (5) γ = + /. [9] (6) In the equations above, M ( ) is the average mass of (or the number of vertices in) a box, is the fracta imension (or box imension) erive from, s( ) < is the scaing of vertex egrees owing to renormaization (i.e., the ratio / ), an is a new exponent characterizing the variation of s with. We prove that equations (4)-(6) can be erive irecty from equation (3), without using the suspect equation (). For this, we set s = ( / ) /( γ), where is the size of /( γ ) G. y equation (3), we have s. Let = / (γ ), we obtain equation (6), which eas to s ; viz., equation (5). Finay, we obtain equation (4) from setting = s, compete our proof. However, we cannot obtain equation () from equation (3). On the other han, it is easiy verifie that equation () is equivaent to n ( ) n(), where n() an n ( ) represent the number of vertices of egrees an in networs G an G, respectivey. It is easiy shown that mutipe renormaizations at fixe can be achieve by an equivaent renormaization. In fact, efining for i = 0,,..., t an = (0), we have (i) / (i+) (i ). (7) / (t) (0) ( t ) (8) ext, setting s (i) = ( (i) / (i+) ) /( γ), we obtain s (i) (i ( i ) /( γ ). Finay, from (i) = (i) / (γ ), we get s (i) ). Taing (0) = an (i+) = s (i) (i) eas to the foowing two equations:

4 On Genera Laws of Compex etwors (t) = s (0)... s (t ). (9) (0) ( t ) s (0)... s (t ).... (0) We have thus emonstrate that the hypothesis which Song et a. [] erive experimentay (namey, that base on Fig. of ref., the egree istribution of the WWW of specia sizes is invariant uner renormaization for ifferent box sizes) is not supporte by theoretica anaysis, eaing to serious oubts regaring its correctness in genera. As a consequence, it is not surprising that many compex networs (Internet, protein interactions, an some ranom networs) ac sef-simiarity, as inicate in the suppementary materias of ref.. This can be expaine theoreticay as foows. y equation (7), we may set thus, eaing to (i) / (i+) (i ) C i. () t / (t) (0) ( ) C 0... C t. () In equation (), the prouct C 0... C t may be an exponentia function of the average istance when C i > an t. This wou impy that the networ size is an exponentia function of the average istance, thus, the networ acs sef-simiarity in this case. 3 ecessary Conitions for Scae-Free etwors In the foowing, we stuy the conitions for vertex egrees of compex networs having scae-free power-aw istribution. We assume that the networ is connecte; simiar arguments appy to isconnecte networs. Let P() be the probabiity istribution of the number of vertices of egree, as previousy efine. Let A enote the average vertex egree an n the number of egree- vertices. We have M = ½ A an n = P(), where an M are numbers of vertices an eges, respectivey. The preceing efinitions impy = n =. (3a) = n = M. (3b) Supposing that n 0, we have n = n P() / P() an = [P() / P()] = / n. (4a) = [ P() / P()] = M / n. (4b)

5 W. Xiao, L. Peng, an. Parhami For scae-free networs, we have P() = P() γ, which eas to = γ = / n. Therefore, assuming γ, which is nown to ho for many scae-free networs [4,5], we have / n = γ = = π /6. (5) This eas to the concusion n. Let f() = P() / P(). When n, equations (4) yie = f(). (6a) = f() A. (6b) Assuming that a ogarithms are in base, equations (6) yie < A og, given that og. Thus, we have prove that for many rea compex networs of = scae-free power-aw istribution with γ, the number of egree- vertices, when nonzero, is of the same orer as the networ size an that the average egree is of orer ess than og. On the other han, if og og A og, then equations (6) impy f () γ. (7) We now eaborate on equation (7). There are three canonica cases for the function f() when og og A og : (a) f() = an f() a constant for ; (b) f() = e ; an (c) f () γ. For case (c), equations (6) ho when γ < 3. However, for cases (a) an (b), equation (6b) is not satisfie if equation (6a) hos. ote that more compex functions satisfying equations (6) exist. For exampe, one can efine the function f () e, if og ; f () γ, otherwise. (8) which satisfies equations (6). As a consequence of our resuts above, the scae-free property of compex networs must be viewe as an approximate or fuzzy property. The preceing eas to a moe for scae-free networs, nown to satisfy γ = n / n. Taing this equaity to be exact, an noting that the right-han sie is a rationa number, we can reaiy prove that γ must be an integer that ivies n. Then, n must be ivisibe by the east common mutipe of γ γ γ,,...,, where = < <... < is the egree sequence of the networ. This is a genera moe for scae-free networs that we now aim to stuy further. γ γ γ Let n = c,,..., ] 0, where c is constant. If γ then [ = n i= γ π 5 i 6 n < 3 n = i= an γ M n n og og. i

6 On Genera Laws of Compex etwors 3 ow we give a sufficient an necessary conition that a connecte graph is a tree. Proposition. Assume that G is connecte an scae-free, then G is a tree γ =. n γ n γ γ Proof. G is a tree = P() = = n. Tabe. umber of vertices (), number of eges (M), average egree (A), an characteristic exponent (γ) in some compex networs etwor M A γ Internet 0,687 3, Fim actors 449,93 5,56, Metaboic networ Protein interactions Tabe ists the parameters, M, A, an γ for severa rea scae-free networs [4,5]. We note that A ½ og (respectivey, 5 og, og, an og og ) for the Internet (fim actors, metaboic, an protein interaction) networ. 4 Concusion We have shown that for many rea networs of scae-free power-aw egree istribution with the exponent γ, the number of egree- vertices, when nonzero, is of the same orer as the networ size an that the average egree is of orer ess than og. Our metho has the benefit of reying on conitions that are static an easiy verifie for any networ. Such istributions are nown to be appicabe to iverse fies of stuy, incuing computer communication an software architecture. However it is worth to further research networs of scae-free power-aw egree istribution with the exponent γ <. Such extensions an variations wi further broaen the appications of our resuts in iverse subfies within computing, communication, bioogy, an the socia sciences. Acnowegements. Research of W. Xiao is supporte by Guangong Key Laboratory of Computer etwor (CCL00705) an Laboratory of asic Software an Appie Construction Technoogy of Guangong( ). References. Song, C., Havin, S., Mase, H.A.: Sef-simiarity of compex networs. ature 433, (005). arabási, A.-L., Abert, R.: Emergence of scaing in ranom networs. Science 86, (999)

7 4 W. Xiao, L. Peng, an. Parhami 3. Watts, D.J., Strogatz, S.H.: Coective ynamics of sma-wor networs. ature 393, (998) 4. Abert, R., arabási, A.-L.: Statistica mechanics of compex networs. Rev. Mo. Phys. 74, (00) 5. ewman, M.E.J.: The structure an function of compex networs. SIAM Rev. 45, (003) 6. Xiao, W.J., Parhami,.: Cayey Graphs as Moes of Deterministic Sma-Wor etwors. Information Processing Letters 97, 5 7 (006) 7. iggs,.: Agebraic Graph Theory. Cambrige University Press, Cambrige (993) 8. Myers, C.R.: Software systems as compex networs: Structure, function, an evovabiity of software coaboration graphs. Physica Review E 68, 0466 (003) 9. Montoya, J.M., Soe, R.V.: Sma Wor Patterns in Foo Webs. Journa of Theoretica ioogy 4, (00)