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Transport and percolation theory in weighted networks

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Transport and percolation theory in weighted networks
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    a  r   X   i  v  :  c  o  n   d  -  m  a   t   /   0   7   0   1   5   2   6  v   3   [  c  o  n   d  -  m  a   t .  s   t  a   t  -  m  e  c   h   ]   2   7   A  p  r   2   0   0   7 Transport and Percolation Theory in Weighted Networks Guanliang Li, 1 Lidia A. Braunstein, 2,1 Sergey V. Buldyrev, 3,1 Shlomo Havlin, 4,1 and H. Eugene Stanley 1 1 Center for Polymer Studies, Boston University, Boston, Massachusetts 02215, USA 2  Departamento de F´ısica, Facultad de Ciencias Exactas y Naturales,Universidad Nacional de Mar del Plata, Funes 3350, (7600) Mar del Plata, Argentina  3  Department of Physics, Yeshiva University, 500 West 185th Street, New York, New York 10033, USA 4 Minerva Center and Department of Physics, Bar-Ilang University, 52900 Ramal-Gab, Israel  We study the distribution  P  ( σ ) of the equivalent conductance  σ  for Erd˝os-R´enyi (ER) and scale-free (SF) weighted resistor networks with  N   nodes. Each link has conductance  g  ≡  e − ax , where x  is a random number taken from a uniform distribution between 0 and 1 and the parameter  a represents the strength of the disorder. We provide an iterative fast algorithm to obtain  P  ( σ ) andcompare it with the traditional algorithm of solving Kirchhoff equations. We find, both analyticallyand numerically, that  P  ( σ ) for ER networks exhibits two regimes. (i) A low conductance regimefor  σ < e − ap c where  p c  = 1 /  k   is the critical percolation threshold of the network and   k   isaverage degree of the network. In this regime  P  ( σ ) is independent of   N   and follows the power law P  ( σ ) ∼ σ − α , where  α  = 1 − k  /a . (ii) A high conductance regime for  σ > e − ap c in which we findthat  P  ( σ ) has strong  N   dependence and scales as  P  ( σ )  ∼  f  ( σ,ap c /N  1 / 3 ). For SF networks withdegree distribution  P  ( k ) ∼ k − λ ,  k min  ≤ k ≤ k max , we find numerically also two regimes, similar tothose found for ER networks. Recently much attention has been focused on complexnetworks which characterize many biological, social, andcommunication systems [1, 2, 3, 4]. The networks arerepresented by nodes associated with individuals, orga-nizations, or computers and by links representing theirinteractions. In many real networks, each link has an as-sociated weight, the larger the weight, the harder it is totransverse the link. These networks are called “weighted”networks [5, 6].Transport is one of the main functions of networks.While the transport on unweighted networks has beenstudied [7], the effect of disorder on transport in networksis still an open question. Here we study the distribution P  ( σ ) of the equivalent electrical conductance  σ  betweentwo randomly selected nodes  A  and  B  on Erd˝os-R´enyi(ER) [8, 9] and scale-free (SF) [1] weighted networks. Wefirst provide an iterative fast algorithm to obtain  P  ( σ )for disordered resistor networks, and then we develop atheory to explain the behavior of   P  ( σ ). The theory isbased on the percolation theory [10] for a weighted ran-dom network. We model a weighted network by assigningthe conductance of a link connecting node  i  and node  j as in Ref. [11] g ij  ≡ exp[ − ax ij ] ,  (1)where the parameter  a  controls the broadness(“strength”) of the disorder, and  x ij  is a randomnumber taken from a uniform distribution in the range[0,1]. We use this kind of disorder since a recent studyof magnetorresistance in real granular materials systems[11] shows that the conductance is given by Eq. (1).Moreover, a recent study [12] shows that many types of disorder distributions lead to the same universal behav-ior. The range of   a  ≫  1 is called the strong disorder(SD) limit [13, 14]. The special case of unweightednetworks, i.e.,  a  = 0 or  g ij  = 1 for all links have beenstudied earlier [7].To construct ER networks of size  N  , we randomly con-nect nodes with   k  N/ 2 links, where   k   is the averagedegree of the network. To construct SF networks, inwhich the degree distribution follows a power law, weemploy the Molloy-Reed algorithm [15]. The traditionalalgorithm to calculate the probability density function(pdf)  P  ( σ ) is to compute  σ  between two nodes  A  and  B by solving the Kirchhoff equations with fixed potential V  A  = 1 and  V  B  = 0 and compute  P  ( σ ) dσ , which givesthe probability that two nodes in the network have con-ductance between  σ  and  σ  +  dσ . However, this methodis time consuming and limited to relatively small net-works. Here we also use an iteration algorithm proposedby Grimmett and Kesten [16] to calculate  P  ( σ ) and showthat it gives the same results as the traditional Kirchhoff method. R 2 R 1 r i Ignored loop linesInfinitely   Distant Nodes C Branch i A B FIG. 1: Schematic Iteration model. In this example  R 1  isinfinite, so it is not taken into account in the sum in  R i  of Eq. (2). In the limit  N   →∞ we ignore the loops between 2 ran-domly chosen nodes because the probability to have loopsis very small. Hence the resistivity  R i  of a randomly se-lected branch  i  connecting a node with infinitely distantnodes satisfies  R i  =  r i  + 1 / (  k − 1 j =1  R − 1 j  ), where  r i  =  e ax i is the random resistance of the link outgoing from this  2node and  k  is a random number taken from the distri-bution ˜  p k  =  p k  · k/  k  , which is the probability that arandomly selected link ends in a node of degree  k , where  p k  is the srcinal degree distribution. In Fig. 1, we showthe schematic iteration method. The randomly selectednodes A and B are connected to the infinitely distantnodes C. When we calculate  R AC  , the resistance betweenA and C, we perform the iterative steps as follows:First we calculate the distribution of resistivities of thebranches connecting node A with C. We start with  N  branches having resistivities  R (0) i  = 0 ( i  = 1 , 2 ,...,  N  ),where  N   is an arbitrary large number. Thus, initiallythe histogram of these resistivities  P  0 ( R ) =  δ  ( R ). At theiterative step  n +1, we compute a new histogram  P  n +1 ( R )knowing the histogram  P  n ( R ). In order to do this wegenerate a new set of resistivities  R ( n +1) i  by connecting inparallel  k − 1 outgoing branches coming from a randomlyselected node of degree  k  obtained from the distribution˜  p k  =  p k  · k/  k  . Then we compute the resistivity of abranch going through this node via an incoming link witha random resistivity  r ( n ) i  taken from the link resistivitydistribution, R ( n +1) i  =  r ( n ) i  + 1  k − 1 j =1  1 /R ( n ) j .  (2)In Eq. (2), if at least one of the terms  R ( n ) i  = 0, we take R ( n +1) i  =  r ( n ) i  . Thus after the first iterative step  P  1 ( R )coincides with the distribution of link resistivities.According to the theorem proved in [16], as  n  → ∞ , P  n ( R ) converges to a distribution of the resistivities of abranch connecting a node to the infinitely distant nodes.The resistivity between a randomly selected node of de-gree  k  and the infinitely distant nodes is defined by˜ R ( i )  = 1  kj =1  1 /R j ,  (3)where  k  is selected from the srcinal degree distribution  p k  and  R j  is selected from  P  n →∞ ( R ).Finally, to compute the resistivity  R ij  between tworandomly selected nodes  i  and  j  (for example  A  and  B in Fig. 1), we compute  R ij  = ˜ R ( i )  + ˜ R ( j ) , where ˜ R ( i )  and˜ R ( j )  are randomly selected resistivities between a nodeand the infinitely distant nodes. If   N   is a sufficientlylarge number, we find the conductance distribution  P  ( σ )between any two randomly selected nodes.In Figs. 2(a) and 2(b) we show the results of   P  ( σ ) usingthe traditional method of solving Kirchhoff’s equationsfor different values of   N   and the iterative method with N   →∞  for both ER and SF networks. We see that themain part of the distribution (low conductances) doesnot depend on  N  , and only the high conductance taildepends on  N  .The behavior of the two regimes, low conductance andhigh conductance, can be understood qualitatively asfollows: For strong disorder  a  ≫  1 all the current be-tween two nodes follows the optimal path between them. 10 -6 10 -4 10 -2 10 0 σ 10 -4 10 0 10 4      P    (      σ    )    N  =256  N  =1024  N  =4096  N  →∞ (a) ER e -ap c -0.8 10 -8 10 -6 10 -4 10 -2 10 0 σ 10 -4 10 0 10 4 10 8      P    (      σ    )  N  =512  N  =2048  N  =8192  N  →∞ (b) SF -0.83 FIG. 2: Plots of   P  ( σ ) for several values of   N  . The symbols arefor the Kirchhoff method and the solid line is for the iterativemethod with  N   → ∞ . (a) ER networks with fixed   k   = 3and  a  = 15. (b) SF networks with fixed  λ  = 3 . 5,  k min  = 2,  k  ≈  3 . 33 and  a  = 20. The dashed line slopes are from theprediction of Eq.(11) or (13). The problem of the optimal path in a random graph inthe strong disorder limit can be mapped onto a perco-lation problem on a Cayley tree with a degree distribu-tion identical to the random graph and with a fraction  p of its edges conducting [17]. However, the conductanceon this path is determined by the bond with the lowestconductance  e − ax max , where  x max  is the maximum ran-dom number along the path. In the majority of cases x max  > p c , where  p c  is the critical percolation thresholdof the network, and only when the two nodes both belongto the incipient infinite percolation cluster (IIPC) [10], x max  < p c . Since the size of the IIPC scales as  N  2 / 3 , theprobability of randomly selecting a node inside the IIPCis proportional to  N  2 / 3 /N   =  N  − 1 / 3 [8, 9, 10]. Then theprobability of randomly selecting a pair inside the IIPCis proportional to ( N  − 1 / 3 ) 2 =  N  − 2 / 3 . These nodes con-tribute to the high conductance range  σ > e − ap c of   P  ( σ ).The low conductance regime is determined by the distri-bution of   x max , that follows the behavior of the orderparameter  P  ∞ (  p ) (for  p > p c ) in the percolation problemwhich is independent of   N   [17]. (This will be explainedlater in the theoretical approach for the low conductanceregime.)We call the low conductance regime a  non-percolation regime   and the high conductance regime a  percolation regime  . In contrast, the property of existing two regimesdoes not show up in the optimal path length [18, 19] andonly the scaling regime with  N   appears. This is sincethe path length for almost all pairs is dominated by theIIPC [19].In Figs. 3(a) and 3(b) we plot for a given  N   only thenon-percolation part of   P  ( σ ) as a function of   σ  for fixedvalues of    k  /a  and different   k   and  a  values for ERnetworks. We see that it obeys a power law with theslope  k  /a − 1 for  σ < e − ap c . Note that for ER networks  p c  = 1 /  k   [8, 9]. In Fig. 3(c), we plot the conductancedistribution for SF networks for fixed values of   k  /a . Wecan see the non-percolation part seems to obey the samepower law as ER networks.Next we present an analytical approach for the formof   P  ( σ ) for low conductance regime. The distributionof the maximal random number  x max  along the optimal  3 10 -10 10 -8 10 -6 10 -4 10 -2 σ 10 0 10 4 10 8      P    (      σ    )   a =15,  N  =4096 a =20,  N  =4096 a =25,  N  =4096 a =25,  N  →∞ e -ap c 〈 k  〉  /  a =0.2-0.8  (a) ER 10 -2 10 -1 10 0 σ 10 -4 10 -2 10 0      P    (      σ    ) a =5,  N  =4096 a =10,  N  =4096 a =20,  N  =4096 a =20,  N  →∞  e -ap c 〈 k  〉  /  a =1.50.5 (b) ER 10 -12 10 -9 10 -6 10 -3 σ 10 -4 10 0 10 4 10 8 10 12      P    (      σ    ) a =17,  N  =8192 a =25.5,  N  =8192 a =34,  N  =8192 a =34,  N  →∞ -0.65  λ =2.5 〈 k  〉  /  a ≈ 0.35 (c) SF FIG. 3: Plots of   P  ( σ ) for fixed   k  /a . The symbols are forthe Kirchhoff method and the solid line is for the iterativemethod. For the same   k  /a , the iterative method for dif-ferent  a  shows the same  P  ( σ ) except that the lower cutoff isdifferent. (a) ER network with   k  /a  = 0 . 2. (b) ER networkwith   k  /a  = 1 . 5. (c) SF network with   k  /a ≈ 0 . 35,  λ  = 2 . 5.The dashed line slopes are from the prediction of Eq.(11) or(13). path can be expressed in terms of the order parameter P  ∞ (  p ) in the percolation problem on the Cayley tree,where  P  ∞ (  p ) is the probability that a randomly chosennode on the Cayley tree belongs to the IIPC [17]. Fora random graph with degree distribution  p k , the prob-ability to arrive at a node with  k  outgoing branches byfollowing a randomly chosen link is ( k  + 1)  p k /  k   [20].The probability that starting at a randomly chosen  link on a Cayley tree one can reach the  ℓ th generation is f  ℓ (  p ) ≡ f  ℓ  = 1 − ∞  k =1  p k k (1 −  pf  ℓ − 1 ) k − 1  k   ,  (4)where  f  0  = 1. Slightly different from  f  ℓ  is the probabilitythat starting at a randomly chosen  node  one can reachthe  n th generation,˜ f  n  = 1 − ∞  k =0  p k (1 −  pf  n − 1 ) k .  (5)In the asymptotic limit  f  ℓ  converges to  P  ∞  for a givenvalue of   p , f  ℓ  → P  ∞ (  p ) = 1 − ∞  k =1  p k k (1 −  pP  ∞ ) k − 1  k   .  (6)In this limit we have a pair of nodes on a random graphseparated by a very long path of length  n . The proba-bility that two nodes will be connected (conducting) atgiven  p , can be approximated by the probability thatboth of them belong to the IIPC [16]:Π(  p ) =   ˜ P  ∞ (  p )˜ P  ∞ (1)  2 ,  (7)where ˜ P  ∞ (  p )  ≡  lim n →∞  ˜ f  n  = 1 −  ∞ k =0  p k (1 −  pP  ∞ ) k .Note that the negative derivative of Π(  p ) with respectto  p  is the distribution of   x max  and thus gives  P  ( σ ) inthe SD limit. In our case  σ  =  e − ap , so replacing  p  by  p  =  − ln σ/a  in Eq. (7) and differentiating with respectto  σ , we obtain the distribution of conductance in the SDlimit when the source and sink are far apart ( n →∞ ), P  ( σ ) = −  ddσ Π( σ ) = 2 ˜ P  ∞ (  p ) σa [ ˜ P  ∞ (1)] 2  ·  ∂   ˜ P  ∞ (  p ) ∂p  |  p = − ln σ/a  . (8)For ER networks the degree distribution is a Poissondistribution with  p k  =  k  k e − k  /k ! [8, 9] and thus  P  ∞ (  p )satisfies P  ∞ (  p ) = 1 − e − k   pP  ∞ (  p ) ,  (9)which has a positive root  P  ∞  for  p > p c  = 1 /  k  . And˜ P  ∞ (  p ) =  P  ∞ (  p ), thus P  ( σ ) = 2 P  ∞ (  p ) σa [ P  ∞ (1)] 2  ·  ∂P  ∞ (  p ) ∂p  |  p = − ln σ/a ,  (10)where  P  ∞ (  p ) and  P  ∞ (1) are the solutions of Eq. (9).We test the analytical result Eq. (10) by comparingthe numerical solution of Eqs. (9) and (10) with the sim-ulations on actual random graphs by solving Kirchhoff equations (Figs. 2 and 3). The agreement between thesimulations and the theoretical prediction is perfect inthe SD limit, i.e. when   k  /a  is small. 10 -4 10 -3 10 -2 10 -1 10 0 σ 10 -2 10 0 10 2 10 4      P     p    (      σ    )    N  =256  N  =1024  N  =4096 a  /   N  1/3 =1.5 e -9.45  p c e -15  p c e -23.8  p c (a) 10 0 10 1 10 2 σ  /  〈σ〉 10 -2 10 0        〈     σ       〉      P     p    (      σ    ) a  /   N  1/3 =0.5 a  /   N  1/3 =1.5 a  /   N  1/3 =2.5 (b) FIG. 4: Kirchhoff method results of the percolation part of ER networks with the same value of   p c  = 1 /  k   = 0 . 33. (a)Normalized  P  p ( σ ) for fixed  a/N  1 / 3 = 1 . 5. (b) Scaled plot of   σ  P  ( σ ) as function of   σ/  σ   for three values of   a/N  1 / 3 . Foreach value of   a/N  1 / 3 , the thick line is for  N   = 256 and thethin line is for  N   = 1024. Next we simplify  P  ( σ ) from Eq. (10). Assuming that P  ∞ (1) ≈ 1 which is true for large  k  and approximatinga slow varying function  P  ∞ (  p ) by  P  ∞ (1) we obtain P  ( σ ) ≈ 2  k  a σ  k  /a − 1 ,  (11)for the range  e − a ≤  σ  ≪  e − ap c with  p c  = 1 /  k  . InFigs. 2 and 3 we also show the results predicted byEq. (11). For an infinite network, for  p  ≤  p c  = 1 /  k  , P  ∞ (  p ) = 0, and hence, the distribution of conductancesmust have a cutoff at  σ  =  e − ap c . Indeed, in Fig. 2(a) andFigs. 3(a) and 3(b) we see that the upper cutoff of theiterative curves is close to  e − ap c .  4As discussed above, the range of high conductivitiescorresponds to the case where both the source and thesink are on the IIPC. Previously we found this percola-tion part to scale as  N  − 2 / 3 . Using Fig. 2(a), we computethe integral for each  P  ( σ ) from  e − ap c to ∞ , and find thatindeed   ∞ e − apc  P  ( σ )d σ  ∼  N  − 2 / 3 , in good agreement withthe theoretical approach. To show how the percolationpart of   P  ( σ ) is related to the parameters  N  ,  a  and  p c ,we analyze the conductance between pairs on the IIPC,i.e., each link on the optimal path from source to sinkhas  x  less than  p c . We compute  P   p ( σ ) of these pairs onthe IIPC. When we simulate this process, we have only N  − 2 / 3 probability to find this part from the srcinal nor-malized distribution  P  ( σ ). Thus, we normalize  P   p ( σ ) bydividing by  N  − 2 / 3 . Figures 4(a) and 4(b) show the nor-malized  P   p ( σ ) of pairs on the IIPC. In this range, we seethat  P   p ( σ ) is dominated by high conductivities and wefind   σ ≈ e − ap c and  σ  P   p ( σ ) =  f    σ  σ  , ap c N  1 / 3  ,  (12)that is, for fixed  ap c /N  1 / 3 ,  σ  P   p ( σ ) scales with  σ/  σ  asseen in Fig. 4(b). The scaled distributions have the sameshape for the same  ap c /N  1 / 3 which specifies the strengthof disorder similarly to the behavior of the optimal pathlengths [12, 18, 19, 21]. The explanation of this fact forthe distribution of conductances is analogous to the argu-ments presented in Refs. [17] and [18] for the distributionof the optimal path. Thus the position of the maximumof the scaled curves in Fig. 4(b), and the whole shape of the distributions, depend on  ap c /N  1 / 3 .We also find that the extreme high conductivities cor-respond to the case where source and sinks are separatedby only one link. In this case,  P  ( σ ) =   k  aNσ  ∼  σ − 1 ,( σ <  1).In summary, we find that  P  ( σ ) exhibits two regimes.For  σ < e − ap c , we show both analytically and numeri-cally that for ER networks  P  ( σ ) follows a power law, P  ( σ ) ∼ σ − α [ α  = 1 − k  /a ] .  (13)We also find that for SF networks, Eq. (13) seems to bea good approximation, consistent with numerical simula-tions. The distributions of optimal path length and thepath length of the electrical currents in complex weightednetworks [18, 19] have been found to depend on  N   for alllength scales and all types of networks studied. In con-trast, here we find that the  low   conductance tail of   P  ( σ )does not depend on  N   for both ER and SF networks.However, the  high   conductance regime ( σ > e − ap c ) of  P  ( σ ) does depend on  N  , in a similar way to the opti-mal path length and current path length distributions[18, 19].We thank ONR, Dysonet, FONCyt (PICT-O2004/370), FONCyt (PICT-O 2004/370), Israel ScienceFoundation and Conycit for support, and Zhenhua Wufor helpful discussions. [1] R. Albert and A.-L. Barab´asi, Rev. Mod. Phys.  74 , 47(2002).[2] S. N. Dorogovtsev and J. F. F. Mendes,  Evolution of Net-works: From Biological Nets to the Internet and WWW  (Oxford University Press, Oxford, 2003).[3] R. Pastor-Satorras and A. Vespignani,  Structure and Evolution of the Internet: A Statistical Physics Approach  (Cambridge University Press, Cambridge, 2004).[4] R. Cohen and S. Havlin,  Complex networks: Stability,Structure and Function   (Cambridge University Press,Cambridge, In press).[5] L. A. Braunstein  et al. , Phys. Rev. Lett.  91 , 168701(2003).[6] A. Barrat, M. Barth´elemy, R. Pastor-Satorras and A.Vespignani, PNAS  101 , 3747(2004).[7] E. L´opez  et al. , Phys. Rev. Lett.  94 , 248701 (2005).[8] P. Erd˝os and A. R´enyi, Publ. Math. (Debrecen)  6 , 290(1959).[9] P. Erd˝os and A. R´enyi, Publications of the MathematicalInst. of the Hungarian Acad. of Sciences  5 , 17 (1960).[10] A. Bunde and S. Havlin,  Fractals and Disordered Systems  (Springer-Verlag, Heidelberg, 1995).[11] Y. M. Strelniker  et al. , Phys. Rev. E  69 , 065105(R)(2004).[12] Y. Chen  et al. , Phys. Rev. Lett.  96 , 068702 (2006).[13] M. Cieplak  et al. , Phys. Rev. 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