Width of Percolation Transition in Complex Networks

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1 APS/23-QED Width of Percoation Transition in Compex Networs Tomer Kaisy, and Reuven Cohen 2 Minerva Center and Department of Physics, Bar-Ian University, Ramat-Gan, Israe 2 Department of Computer Science and Appied Mathematics, Weizmann Institute of Science, Rehovot, Israe Dated: November 30, 2005 It is nown that the critica probabiity for the percoation transition is not a sharp threshod, actuay it is a region of non-zero width for systems of finite size. Here we present evidence that for compex networs pc, where Nν opt is the average ength of the percoation custer, and N is the number of nodes in the networ. For Erdős-Rényi ER graphs ν opt /3, whie for scae-free SF networs with a degree distribution P λ and 3 < λ < 4, ν opt λ 3/λ. We show anayticay and numericay that the survivabiity Sp,, which is the probabiity of a custer to survive chemica shes at probabiity p, behaves near criticaity as Sp, S, exp[p /]. Thus for probabiities inside the region p < / the behavior of the system is indistinguishabe from that of the critica point. PACS numbers: Hc,89.20.Ff I. INTRODUCTION: Recenty the subject of networs has received much attention. It was reaized that many systems in the rea word, such as the Internet, can be successfuy modeed as networs. Other exampes incude socia networs such as the web of socia contacts, and bioogica networs such as the protein interaction networ and metaboic networs [ 3]. The probem of percoation on networs has aso been studied extensivey e.g. [4]. Using percoation theory we can describe the resiience of the networ to breadown of sites or ins [5, 6], epidemic spreading [7, 8], and properties of optima paths in networs with highy fuctuating weights on the ins [9]. A typica percoation system consists of a d- dimensiona grid of ength L, in which the nodes or ins are removed with some probabiity p, or are considered conducting with probabiity p e.g. [0, ]. Beow some critica probabiity the system becomes disconnected into sma custers, i.e., it becomes impossibe to cross from one side of the grid to the other by foowing the conducting ins. Percoation is considered a geometrica phase transition exhibiting universaity, critica exponents, upper critica dimension at d 6 etc. It was noted by Conigiio [2] that for systems of finite size L the transition from connected to disconnected state has a width, where ν is a critica exponent reated L to the correation /ν ength. Percoation on networs was studied aso from a mathematica point of view [4, 3, 4]. It was found that in Erdős-Rényi ER graphs with an average degree the percoation threshod is:. Beow the graph is composed of sma custers most of them trees. As p approaches trees of increasing order appear. At p a giant component emerges and oops of a orders abrupty Eectronic address: aist@mai.biu.ac.i appear. Nevertheess, for graphs of finite size N it was found that the percoation threshod has a finite width N /3 [4], meaning that a attributes of criticaity are present in the system in the range [, + ]. For exampe: The number of oops is negigibe beow + [2]. In this paper we study the Survivabiity of the networ near the critica threshod. The survivabiity Sp, is defined to be the probabiity of a connected custer to survive up to chemica shes in a system with conductance probabiity p [5] i.e the probabiity that there exists at east one node at chemica distance from a randomy chosen node on the same custer. At the critica point, the survivabiity decays as a poweraw: S, x, where x is a universa exponent. Beow the survivabiity decays exponentiay to zero, whie above it decays exponentiay to a constant. Here we wi derive anayticay and numericay the functiona form of the survivabiity above and beow the critica point. We wi show that near the critica point Sp, S, exp[p / ]. Thus, given a system which has a maxima chemica ength at the percoation threshod, for probabiities inside the range p < pc the behavior of the system is indistinguishabe from that of the critica point. Hence we get pc. The maxima chemica ength at the critica threshod, i.e. the ength of the percoation custer, was found to be: N νopt [9] where N is the number of nodes in the networ. For Erdős-Rényi ER graphs ν opt /3, whie for scae-free SF networs with a degree distribution P λ and 3 < λ < 4, ν opt λ 3/λ. II. GENERAL FORMALISM: Consider a random graph with a degree distribution P, i.e., a randomy chosen node has a probabiity P to have ins. The probabiity to reach a node of degree by foowing a randomy chosen in is P P [6] where is the average degree. Accord-

2 2 ingy, we write the two corresponding probabiity generating functions e.g. [8, 6]: and: G 0 x G x G 0x G 0 P x 0 P x P x 2 Where G x describes the probabiity that a node reached by foowing a random in has outgoing ins, not incuding the incoming in. For exampe, in ER graphs: G 0 x G x e x. After randomy removing a fraction p of the ins bond percoation, the probabiity for a randomy chosen node to have remaining ins in the diuted graph is given by [6]: P 0 0 P 0 p p 0 3 The corresponding probabiity generating functions G 0 x P 0 x and G x P x in the diuted graph are [22]: [ ] 0 G 0 x P 0 p p 0 x And [6]: P 0 0 P 0 p + px 0 00 xp p 0 G 0 p + px 4 G x G 0 x G 0 pg 0 p + px pg 0 G p + px 5 For exampe, in ER graphs, G0 x G x e [ p+px] e px. We next define M x m 0 +m x+m 2 x to be the generating function for the number of sites that exists on ayer i.e. chemica she starting from a random node on the diuted graph, and N x n 0 + n x + n 2 x to be the corresponding function for the number of sites that exists on ayer from a node reached by foowing a random in. In order to find M L x for some ayer L we can write the foowing recursive reations [6, 7]: For < L : N x G x 6 N + x G N x 7... FIG. : A graphica setch of the recursion reation 7. Given a conduction probabiity p, the probabiity to reach a branch having i nodes at ayer + may be represented as the sum of probabiities to reach a singe node, to reach a node connected to a singe branch having i nodes at ayer, to reach a node connected to two branches having a tota of i nodes at ayer, etc. see section II. And simiary, for the fina ayer: M L x G 0 N L x 8 Eq. 7 means that the probabiity n + i for reaching a branch having i nodes at ayer + is composed of the probabiity of reaching a node by foowing a in, and then reaching i nodes at ayer by foowing a possibe branches emerging from that node - see setch in Fig.. As a simpe demonstration, et us evauate the probabiity n + 0 to encounter zero nodes at ayer + of a branch. Taing the zeroth power in Eq. 7 we have: n + 0 P + P 2 n 0 + P [ ] 2 3 n , which means that the probabiity to reach zero nodes at ayer + by foowing a in is composed of the probabiity P to reach a node with no emerging branch, the probabiity P 2 n 0 to reach a node that has a singe emerging branch with zero nodes at ayer, the probabiity P [ ] 2 3 to reach a node having two branches n 0 such that both of them have zero nodes at ayer etc. see Fig.. Simiary, Eq. 8 refers to M L x, which gives the probabiity for the number of nodes at ayer L reached by starting from a random node, rather than by foowing a random in [6]. It can be seen that M L 0 m 0 is the probabiity that there are 0 nodes at ayer L from a random node, i.e., the probabiity to die before ayer L. Thus ǫ L M L 0 is the probabiity to survive up to ayer L. Simiary, ǫ N 0 where L is the probabiity for a branch to survive up to ayer. From Eq. 7 we have: N + 0 G N 0 9 ǫ + G ǫ G p + p[ ǫ ] 0 Thus for < L : ǫ + G pǫ And for the fina ayer L we have [Eq. 8]: ǫ L G 0 pǫ L 2 Which gives the survivabiity at ayer L [7].

3 3 III. ERDŐS-RÉNYI GRAPHS: For ER graphs: G 0 x G x e x and Eq. gives: ǫ + e [ pǫ] e pǫ [ pǫ + p2 2 ] ǫ p ǫ p2 2 ǫ Where. Setting δ p, we get: ǫ + + δ ǫ + δ ǫ ǫ + δ ǫ 2 ǫ2 4 Where we have eft ony terms of second order in ǫ, δ [23]. We thus get: dǫ d ǫ + ǫ 2 ǫ2 + δ ǫ 5 At criticaity, δ 0 and the soution to this equation is: ǫ. The additiona term suggests the foowing soution near criticaity: ǫ exp δ. Note that for ER graphs Equations and 2 are the same, and thus the survivabiity ǫ L at the fina iteration aso has the same form: ǫ L L exp δl. The above resut can be written as: Sp, S, exp p. 6 In order to chec this resut we numericay soved the survivabiity Sp, near according to the exact enumeration method presented in [7]. Fig. 2a shows the survivabiity Sp, for different vaues of p. For p the survivabiity decays as a power aw, whie above and beow there is an exponentia decay, either to zero for p < or to a constant for p >. Fig. 2b shows that a curves of the survivabiity Sp, from a can be rescaed such that they a coapse. Moreover, scaed survivabiities from a different graphs with different vaues of i.e., different vaues of aso coapse on the same curve. However, equation 6 is true ony beow the percoation threshod where there is no giant component. Above the percoation threshod there is an exponentia decay to a non-zero constant, and the generaized expression is: Sp, S, exp p + P, 7 Where P is the probabiity for a randomy chosen site to be inside the percoation custer [24]. Indeed, setting ǫ + ǫ in the recursive reation ǫ + e pǫ, the resuting steady state soution is ǫ P [4]. S,p a [S,p S,p]/S, b p p / c FIG. 2: Coor onine a The survivabiity Sp, for an ER graphs with 5, numericay cacuated for different vaues of p:, ± 5 0 4, ± 3 0 4, ± 0 4, ± , and ± For p the survivabiity decays to zero according to a power aw: S,. For p <, Sp, 0, whie for p >, Sp, Const. The decay is exponentia to zero or to a constant according to equations 6 and 7. b Scaing of the survivabiity for different vaues of p,, and. Shown is Sp, Sp, S, vs. p for ER graphs with 5 unfied symbos and 0 fied symbos. The coapse of a curves on an exponentia function for arge shows that indeed the scaing reations 6 and 7 are correct. IV. SCALE-FREE GRAPHS: Scae-free graphs can be taen to have a degree distribution of the form P c λ where c λ m λ [6]. In order to sove equation we have to evauate: G pǫ P pǫ. 8 Expanding by powers of ǫ, and inserting P c λ with 3 < λ < 4, we get [8, 9] [25]: P ǫ ǫ+ c 2 Γ4 λǫλ 2. Thus equation becomes: 9 ǫ + [ pǫ + c 2 Γ4 λ pǫ λ 2] p ǫ c 2 Γ4 λ pλ 2 ǫ λ 2 20 Where [6]. Taing p + δ we get: ǫ + + δ ǫ c 2 Γ4 λ + δ λ 2 ǫ λ 2 ǫ + δ ǫ c Γ4 λpλ 2 c 2 + δ λ 2 ǫ2 λ 2

4 4 S,p a [S,p S,p]/S, b icaity: ǫ x exp δ. The ast iteration [Eq. 2] can be shown to give the same behavior for ǫ L. A simiar form can be found aso for λ > 4 [26]. The scaing form for SF networs is confirmed by numerica simuations as shown in Figures 3a and b p / FIG. 3: Coor onine a The survivabiity Sp, for a SF networ with λ 3.5, numericay cacuated for different vaues of p:, ± 6 0 2, ± 4 0 2, ± 2 0 2, , and ± For p the survivabiity decays to zero according to a power aw: S, 2. For p <, Sp, 0, whie for p >, Sp, Const. The decay is exponentia to zero or to a constant according to equations 6 and 7. b Scaing of the survivabiity for different vaues of p,, and λ. Shown is Sp, Sp, S, vs. p for SF graphs with λ 3.5 fied symbos and λ 5 unfied symbos. Due to numerica difficuties ony curves with p < are shown. Setting A c 2 Γ4 λpλ 2 c we get: ǫ + ǫ + δ [ ǫ A ǫ + δ ǫ A ǫ Aǫ λ 2 ] [ + λ 2 δpc ǫ λ 2 + δ ] λ 2 ǫ λ 2 + δ [ ǫ Aλ 2 ǫ λ 2 ] 22 For arge, ǫ, and taing into account that λ 2 > we have ǫ λ 2 ǫ. Therefore: dǫ d ǫ + ǫ Aǫ λ 2 + δ ǫ 23 For δ 0 the soution is ǫ x with x λ 3. The additiona term suggests the foowing soution near crit- V. SUMMARY AND CONCLUSIONS We have shown anayticay and numericay the the survivabiity in ER and SF graphs scaes according to equations 6 and 7 near the critica point. Thus, the scaing form of the survivabiity near the critica probabiity obeys the foowing scaing reation for p < : p pc Sp, S, exp, 24 where pc. Given a system with a maxima chemica ength at criticaity, for a vaues of conductivity p inside the range [, + ] the survivabiity behaves simiar to the power aw S, x found at p. Thus, the width of the critica threshod is pc, where is the chemica ength of the percoation custer. For ER graphs, N /3, whie for SF networs with 3 < λ < 4, N λ 3/λ. Acnowedgments We than the ONR, the Israe Science Foundation and the Israei Center for Compexity Science for financia support. We than E. Persman, S. Sreenivasan, L. A. Braunstein, S. V. Budyrev, S. Havin, H. E. Staney, Y. Strenier, A. Samuhin, O. Riordan and P. L. Krapivsy for usefu discussions. RC wishes to than the Pacific Theaters Foundation for support. [] A.-L. Barabási, Lined: The new science of networs Perseus Boos Group, [2] S. N. Dorogovtsev and J. F. F. Mendes, Evoution of Networs - From Bioogica Nets to the Internet and WWW Oxford University Press, [3] R. Pastor-Satorras and A. Vespignani, Evoution and Structure of the Internet : A Statistica Physics Approach Cambridge University Press, [4] R. Abert et a., Rev. of Mod. Phys. 74, [5] D. S. Caaway et a., Phys. Rev. Lett. 85, [6] R. Cohen et a., Phys. Rev. Lett. 85, [7] R. Cohen et a., Phys. Rev. Lett. 90, [8] M. E. J. Newman, Phys. Rev. Lett. 95, [9] L. Braunstein et a., Phys. Rev. Lett. 9, [0] A. Bunde and S. Havin, eds., Fractas and Disordered Systems Springer, New Yor, 996. [] D. Stauffer and A. Aharony, Introduction to Percoation Theory Tayor and Francis, London, 992. [2] A. Conigio, J. Phys. A. 5, [3] P. Erdös and A. Rényi, Pub. Math. Inst. Hungar. Acad. Sci. 5, [4] B. Boobás, Random Graphs Cambridge University Press, 200. [5] F. Tzschichhoz et a., Phys. Rev. A 39, [6] M. E. J. Newman et a., Phys. Rev. E 64, [7] L. A. Braunstein et a., in Lecture Notes in Physics Springer, Berin, 2004, vo. 650, p. 27. [8] R. Cohen and S. Havin, Physica A 336, [9] R. Cohen and S. Havin, Compex Networs: Structure, Stabity and Function Cambridge University Press, to appear, [20] R. Cohen et a., Phys. Rev. E 66,

5 5 [2] O. Riordan and P. L. Krapivsy, private communication. [22] A simiar derivation was done by Newman [8] for a sighty different situation. [23] We assume that p < and thus ǫ for arge. [24] Sp, is the probabiity that if we start from a randomy chosen site, we wi survive an infinite chemica distance. This equas to the probabiity P that the chosen site resides in the giant component. P obeys the transcendenta equation: P e pp. [25] We expand up to second order in ǫ. The ast term is the remainder of the series expansion, R 2 2 ǫ2 2c λ ǫ 3 2 ǫ2 ce 3ǫ 3 λ e ǫ d c 2 Γ4 λǫ λ 2, which arises from the fact that the second derivative diverges see [9] for detais. [26] In this range the behavior is simiar to ER graphs [20].