Diffusion-annihilation proecesses in weighted scale-free networks with identical degree sequence
Yichao Zhang, Jihong Guan, Zhongzhi Zhang, Shi Zhou, Shuigeng Zhou
aa r X i v : . [ c ond - m a t . d i s - nn ] J un Diffusion-annihilation processes in weightedscale-free networks with identical degree sequence
Yichao Zhang
Department of Computer Science and Technology, Tongji University, 4800 Cao’anRoad, 201804, Shanghai, China
Zhongzhi Zhang
E-mail: [email protected]
School of Computer Science, Fudan University, 200433, Shanghai, ChinaShanghai Key Lab of Intelligent Information Processing, Fudan University, 200433,Shanghai, China
Jihong Guan
E-mail: [email protected]
Department of Computer Science and Technology, Tongji University, 4800 Cao’anRoad, 201804, Shanghai, China
Shuigeng Zhou
E-mail: [email protected]
School of Computer Science, Fudan University, 200433, Shanghai, ChinaShanghai Key Lab of Intelligent Information Processing, Fudan University, 200433,Shanghai, China
Abstract.
The studies based on A + A → ∅ and A + B → ∅ diffusion-annihilationprocesses have so far been studied on weighted uncorrelated scale-free networks andfractal scale-free networks. In the previous reports, it is widely accepted that thesegregation of particles in the processes is introduced by the fractal structure. Inthis paper, we study these processes on a family of weighted scale-free networkswith identical degree sequence. We find that the depletion zone and segregation areessentially caused by the disassortative mixing, namely, high-degree nodes tend toconnect with low-degree nodes. Their influence on the processes is governed by thecorrelation between the weight and degree. Our finding suggests both the weight anddegree distribution don’t suffice to characterize the diffusion-annihilation processes onweighted scale-free networks.
1. Introduction
Complex networks are a powerful and versatile mathematical tool for representingand modeling the the structure of complex systems [1, 2]. Their wide applications in iffusion-annihilation processes in weighted scale-free networks with identical degree sequence P ( k ) ∼ k − γ , leading to the rising of research onour basic understanding of the organization of many real-world systems in nature andsociety [1, 2, 3, 4]. The characteristic exponent γ , usually observed in the range ∈ (2 , A + A → ∅ and different particles A + B → ∅ [8, 9, 10, 11, 12, 13, 14, 15, 16].Unlike diffusion-reaction, the substances in diffusion-annihilation process don’tyield products with mass. In the study of diffusion-annihilation, the density ρ of thesurviving particles is thus a crucial problem since it presents a quantitative descriptionof the reaction process. In the large time limit, ρ behaves as1 ρ ( t ) − ρ (0) = k · t f , (1)where k is the rate constant and ρ (0) is the particle density at t = 0. In themean-field approximation with ρ A (0) = ρ B (0), both processes can be described as dρ ( t ) dt = − const · ρ ( t ) , whose solution is f = 1. The solution is valid in regularlattices of Euclidean space [12] with a spatial dimension d > d c , where d c is the criticaldimension of this process. For the A + A → ∅ process, d c = 2 while for the A + B → ∅ process d c = 4 [13]. Further studies on fractals found that the exponent f = d s for A + B → ∅ [14], where d s is the spectral dimension of the fractal structure.As the existence of the depletion zone ( A + A → ∅ ) [15] and segregation of thereactants ( A + B → ∅ ) [16], the upper bound of the exponent f for the regular latticesis 1. Whereas, when the processes are performed on scale-free networks with identicalnodes and links, f can be considerably higher than 1 [8]. Inspired by the observations,the relation between γ and f on A + A → ∅ was investigated analytically [9] inuncorrelated scale-free networks [17, 18]. Put briefly, the term “uncorrelated” denotesthat no degree-degree correlations among nodes exist in the networks, namely, theconditional probability P ( k ′ | k ) that a node of degree k is connected to a node of degree k ′ can be formalized as k ′ P ( k ′ ) h k i . The analytical solution shows f is only governed by theexponent γ for this class of scale-free networks. Subsequently, an interesting study of A + B → ∅ on fractal scale-free networks shows the segregation can also be found in thescale-free networks [10]. Influenced by the segregation, the reaction process is hamperedapparently.Very recently, considering heterogeneous distributions of weights [19, 20], a heuristicresearch on the weighted uncorrelated scale-free networks analytically present a morerealistic conclusion [11]. In this work, the weight of links is defined as w ij = ( k i k j ) θ with the degree k i and k j of both nodes, where θ is the network’s weightiness parameter iffusion-annihilation processes in weighted scale-free networks with identical degree sequence θ = 0, there is no dependence between link weight and node degree, all linkweights are equal with one, and the network becomes an unweighted network. When θ >
0, it is a weighted network where links have different weights. The larger θ and thewider difference between links. Based on the mean-field rate equation for the averagedensity ρ k of a node with degree k , for the A + A → ∅ process, the authors showed ρ ∼ t − θ < γ − t − θγ − θ − γ − ≤ θ < γ − e − t θ ≥ γ − , (2)in asymptotically large networks. For the A + B → ∅ process, inserting the mappingrelation, they claimed ρ ∼ t − θ < γ − ( t ln t ) − θ = γ − t − γ − θ − γ − < θ < γ − . (3)It has been shown that, f is only governed by the weight and degree distribution.In this paper, we study a family of weighted scale-free networks with theidentical degree sequence (weighted IDS-SF networks), the reaction processes are vastlydifferent from the previous reports [8, 9, 10, 11]. To this end, we briefly introduceweighted random diffusion in Section 2. Section 3 is devoted to explicit the IDS-SFnetworks. In Section 4, our extensive numerical simulations are compared with previousanalytic results of the diffusion-annihilation processes running on top of the weighteduncorrelated scale-free networks [11]. Finally, our conclusions are presented in Section 5.Our findings indicate that the disassortative mixing of the nodes is the essential reasonfor generation of the depletion zone and segregation in this class of scale-free networks.
2. WEIGHTED RANDOM DIFFUSION
Before introducing the construction of the networks, we briefly introduce the generalrandom walk on weighted networks to clarify the influence of high-degree nodes (hubs) onthe weighted random diffusion on scale-free networks. Random walk is a mathematicalformalization of a trajectory that consists of taking successive random steps. A familiarexample is the random walk phenomenon in a liquid or gas, known as Brownianmotion [21, 22]. Random walk is also a fundamental dynamic process on complexnetworks [23]. Random walk in networks has many practical applications, such asnavigation and search of information on the World Wide Web and routing on theInternet [24, 25, 26, 27, 28].Let’s consider a weighted random walker starting from node i at step t = 0 anddenote P im ( t ) as the probability of finding the walker at node m at step t . Theprobability of finding the walker at node j at the next step is P ij ( t + 1) = X m a mj · Π m → j · P im ( t ) , (4) iffusion-annihilation processes in weighted scale-free networks with identical degree sequence a mj is an element of the network’s adjacent matrix.In this case, we define the weight of a link between nodes i and j as w ij = w ji = k i k j ) θ link i-j exists , (5)where k i and k j denote the degree of node i and j respectively. On the other hand, thestrength of node i is defined as s i = X j ∈ Γ( i ) w ij = X j ∈ Γ( i ) ( k i k j ) θ , (6)Thus the probability P ij ( t ) for the walker to travel from node i to node j in t steps is P ij ( t ) = X m ,...,m t − w im s i × w m m s m × . . . × w m t − j s m t − . (7)In other words, P ij ( t ) = P m ,...,m t − P im P m m · · · P m t − j . Comparing the expressionsfor P ij and P ji one can see that s i P ij ( t ) = s j P ji ( t ). This is a direct consequence of theundirectedness of the network. For the stationary solution, one obtains P ∞ i = s i /Z with Z = P i s i . Note the stationary distribution is, up to normalization, equal to s i , thestrength of the node i . This means the higher strength a node has, the more frequentlyit tends to be visited by a walker. Notably, for degree uncorrelated networks [29], s i inthe steady state scales with k i as s i ∼ k θ +1 i [20].
3. The scale-free networks with identical degree sequence (IDS-SFnetworks)
The scale-free networks with identical degree sequence are a common topic in complexnetworks, which offer researchers a platform to understand how the dynamical behaviorsare influenced by the degree heterogeneity of networks [30, 31]. As a class of the thesenetworks [32, 33], the construction of the present model is controlled by a parameter q [32, 33] as shown in Fig. 1, evolving in a recursive way. We denote the network after n iterations by G ( n ), n ≥
0. Then the networks are constructed as follows. For n = 0,the initial network G (0) consists of two nodes connected to each other by a link. For t ≥ G ( n ) is obtained from G ( n − G ( n ), one can add threelinks to each link existing in G ( n −
1) (as shown on the left of Fig. 1) with probability q , or replace it with a quadrangle (as shown on the right of Fig. 1) with complementaryprobability 1 − q . In Fig. 2, next, we present the first three iterations of two specialnetworks corresponding to two limiting cases q = 0 and q = 1, respectively.As discussed in the reference [32], these two limiting cases and the middle cases(0 < q <
1) exhibit many interesting properties. For instance, the same degree sequenceindependent of parameter q , the identical degree distributions, and no triangles [34]formed by connections among the neighbors. Note that, as shown in Fig. 3 the Pearson iffusion-annihilation processes in weighted scale-free networks with identical degree sequence Figure 1. (Color online) Iterative method of the network construction. Each edge isreplaced by the connected clusters on the left-hand side with a certain probability q ,otherwise by the one on the right-hand side, where red squares represent new nodes. Figure 2. (Color online) Illustration of the first three iterations of the network forthe particular cases q = 0 and q = 1. coefficient increases with q generally, indicating the IDS-SF networks are disassortativefor q = 0 (the index tends to − . N → ∞ [35]) and uncorrelated for q = 1 [36].Hence, for q = 1, the topological structure of network satisfies the conditions of applying P ( k | k ′ ) = kP ( k ) / h k i [11] and mean field approximation well, which will be discussedin Section 4.1 in detail. Adopting several q values from 0 to 1, one can generatevarious networks, for example, fractal ( q = 0) and non-fractal ( q = 1) networks. Theseparticular features have the kinetics taking place upon the model be distinct from thewell known results for other networks, e.g., the Barab´asi-Albert (BA) graph [9, 37] anduncorrelated configuration networks [9, 31]. In the following, we will show a number ofinteresting behaviors of Diffusion-annihilation processes on the networks.
4. Diffusion-annihilation processes on the weighted IDS-SF networks
According to the conclusion on the weighted random diffusion operating in the weighteduncorrelated scale-free networks in Section 2, it is easily seen that P ∞ i = k θ +1 i Σ i k θ +1 i . Thus,for θ >
0, particles move towards hubs with time gradually. As hubs are the minority iffusion-annihilation processes in weighted scale-free networks with identical degree sequence Pearson coefficientDiameter
Average path length
Figure 3.
Pearson correlation coefficient, average path length, and diameter versus q ranging from 0 to 1 for the IDS-SF networks. Each data point corresponds to tenindependent realizations of the network for n = 7. of the population, moving to them means getting concentrated actually. At these hubs,particles have thus a high probability to collide and react with each other, leading to ahigher reaction rate than that in homogeneous networks [9]. For θ <
0, conversely, theparticles are repelled by the hubs. In this case, the particles are getting dispersed on thelow-degree nodes with time, which are also called leaves. Seen in this light, reaction rateof diffusion-annihilation tends to decrease with θ for the two processes. In what follows,we will show the diffusion tendency mentioned above is correct, but the influence of θ on the reaction rate is not monotonic, which depends not only on degree distributionbut also other topological features of the network.We first generate a special IDS-SF structure through an iterative way with n = 7.The simulation results are obtained on IDS-SF networks with 10 ,
924 nodes and 16 , w ij s i from a node i to a randomly chosen nearest neighbor j . If it isempty, the particle fills it, leaving i empty. If j is occupied, the two particles annihilate,leaving both nodes empty. An initial fraction ρ (0) of nodes in the networks is randomlychosen, which is occupied by an A particle with probability 0 . A + B → ∅ process, the initial densities of A and B are equal, i.e., ρ B (0) = ρ A (0).For a convenience of discussion, we define f as the first order derivative of ρ ( t ) , where ρ ( t ) = ρ A ( t ) for A + A → ∅ and ρ ( t ) = ρ A ( t ) + ρ B ( t ) for the A + B → ∅ process.In the cases among 0 < q <
1, each plot corresponds to 100 simulations that are tenruns for ten independent realizations of the network with the same parameters. For thetwo limiting cases q = 0 ,
1, each plot corresponds to 100 runs for the two deterministic iffusion-annihilation processes in weighted scale-free networks with identical degree sequence q ∈ [0 ,
1] are the same, inwhich γ = 3 [32], Eq. (2) and Eq. (3) can be rewritten as ρ ∼ t − θ < t − θ − θ ≤ θ < e − t θ ≥ . (8)For the A + B → ∅ process, inserting γ = 3, one can also obtain ρ ∼ t − θ < t ln t ) − θ = 0 t − − θ < θ < . (9)For a convenience, we define two quantities as follow: Q AA = N AA ( t ) M ( t )( M ( t ) − , (10) Q AB = N AB ( t ) M ( t )( M ( t ) − , (11)where N AA ( t ) denotes the number of close contacts between two nodes with the identicalparticles for the A + A → ∅ process. N AB ( t ) denotes the number of contacts betweenthe distinct particles for the A + B → ∅ process at time t [38]. M ( t ) denotes the totalnumber of particles at time t . q = 1As shown in Fig. 2, in the case q = 1, the networks are reduced to the (1 , d f can beobtained by d f = lim n →∞ (cid:18) ln N n ln l n (cid:19) , (12)where N n and l n are the size and diameter of G n respectively. Inserting N n = (4 n + 2)and l n = 2 n [35] into Eq. (12), we have d f = lim n →∞ (3 ln 2 n ) . (13)Obviously, the net is infinite-dimensional, namely, a non-fractal network. For A + A → ∅ ,Fig. 4 shows the relation between f and time t for θ = − , ,
1, where red lines are thepower-law fittings of the plots. Note that all the plots about the dependence of f and Q AA ( AB ) on time t are logarithmically binned in this paper. Concretely, f t = ρ ( t − ρ ( t t − t and Q t = Q t − Q t t − t , where the time interval log ( t ) − log ( t ) = 0 .
1. For each panel,the curve is relatively stable in the beginning and fluctuates radically in the end. Thenumerical results show the reaction processes are vastly different from the previousanalytical predictions on the weighted uncorrelated networks denoted by the dashedlines [11]. iffusion-annihilation processes in weighted scale-free networks with identical degree sequence -2 f~t TimeTimeTime f~t f=1 f~t -0.38 f=1 -4 -3 -4 -4 (c) =-1(a) =1 (b) =0 Figure 4. (Color online) f and Q AA as a function of time t for A + A → ∅ with q = 1. The first and second row denote f and Q AA versus t respectively. The dashedlines correspond to the mean field prediction: For the A + A → ∅ process, f = 1 when θ = 0 , −
1, and f = exp ( t ) when θ = 1. Compared with the mean-field prediction, one can observe many discrepancies inFig. 4. Here, we only focus on looking for some common reasons. Note that, for θ ≥ f = e t is much higher than our results in Fig. 4(a). To show theplots clearly, we omit the dashed line in this panel. The apparent discrepancy is mainlycaused by the approximation N g → ∞ . In this condition, h k θ i → ∞ , which makes thedifferential equation solvable. For 0 < θ <
1, on the other hand, the approximation inthe literature [11] omits the reaction running on the low-degree nodes, which causes thepredicted reaction rate is lower than our observation. For θ = 0, our results in Fig. 4(b)roughly match the conclusion in the reference [11] in term of the scaling of f . But, thevalue of f is a bit higher than the prediction in that the global mean first-passage timeof random walks G in the mean field prediction of Eq. (8) is proportional to N g [40]while G ∼ N ln ln g [33]. So that, one can expect a larger deviation in the IDS-SF networkswith n >
7. Notably, this observation in this case is inconsistent with the previousconclusion on finite size effects, i.e., ρ ( t ) ∼ N − γ t for γ ≤ θ <
0, our resultsin Fig. 4(c) are basically lower than the prediction. This deviation is caused by theapproximation in Taylor expansion. As is known, ρ ( t ) can only be omitted at the end iffusion-annihilation processes in weighted scale-free networks with identical degree sequence -3 -2 f ~ t Time (a) =1 (c) =-1 f ~ t -0.20 f=ln(t)+1 f ~ t -0.66 f=1 -4 -4 (b) =0 Figure 5. (Color online) f and Q AB as a function of time t for A + B → ∅ with q = 1.The first and second row denote f and Q AB versus t respectively. The dashed linescorrespond to the mean field prediction: For A + B → ∅ , f = 1 when θ = − f = ln ( t ) + 1 when θ = 0. of the reaction, where it is close to 0.For the A + B → ∅ process, we measure the relation between the total particledensity ρ ( t ) = ρ A ( t ) + ρ B ( t ) and time t as shown in Fig. 5. Our observation exhibits thesimilar behaviors with A + A → ∅ . As the probability of collision between two identicalparticles is equal to that for distinct ones, one can find Q AB in Fig. 5 is about half ofthe corresponding Q AA . Thus, the reaction rate of the A + B → ∅ process is naturallymuch lower than that of A + A → ∅ .It should be mentioned that the (1 , q = 0. iffusion-annihilation processes in weighted scale-free networks with identical degree sequence -4 -4 Time f ~ t -0.23 f ~ t -0.13 f ~ t -0.38 -4 -4 -4 -4 (c) =-1(a) =1 (b) =0 Figure 6. (Color online) f and Q AA as a function of time t for A + A → ∅ with q = 0.The first and second row denote f and Q AA versus t respectively. q = 0Unlike the case of q = 1 addressed above, for q = 0, the networks are reduced to the(2 , d f = ln 4ln 2 = 2 [35]. By definition, the fractal network is a networksatisfying the fractal scaling N B ( l B ) ∼ l d f B , where N B is the number of boxes needed tocover the entire network with boxes of size l B . Note that the fractal scaling d f holdsin the system where hubs are located separately from each other [41, 42]. As is known,the mean-field theory can only be applicable when the nets have infinite dimensionalitybut not in the fractal ones [35, 10]. Thus, the discrepancies between the mean fieldprediction and our results are not unexpected. However, the weighted networks havetheir unique subtle properties, which gives rise to many interesting dynamical behaviorsdistinct from the previous unweighted fractal nets.For the A + A → ∅ process in Fig. 6, we also measure the relation between f and t for the set of θ . In Fig. 6, one can observe that f decays with time t in all the threepanels. In the case of θ = 1 in Fig. 6(a), the exponent f decreases with t abnormallyand exhibits a contrary behavior with the case of q = 1, in which f increases with t .For the A + B → ∅ process shown in Fig. 7, one can observe a similar phenomenon with iffusion-annihilation processes in weighted scale-free networks with identical degree sequence -4 -4 f ~ t -0.40 Time (c) =-1(b) =0 f ~ t -0.32 (a) =1 f ~ t -0.50 -4 -4 -4 -4 Figure 7. (Color online) f and Q AB as a function of time t for A + B → ∅ with q = 0.The first and second row denote f and Q AB versus t respectively. the A + A → ∅ process as well. Because of Q AA ∼ Q AB , f in this case is much lowerthan the A + A → ∅ as well. Notably, for the unweighted case, i.e., θ = 0, f shown inFig. 7(b) is not a constant 0 . A and B , ρ A (0) = ρ B (0), local hubs and the random fluctuation in the initial particle number generate thesegregation of distinct particles, which drastically slows down the reaction rate. Usually,for the unweighted uncorrelated scale-free networks, it is hard for a large number ofparticles to form a close formation that cannot be penetrated by the other speciesbecause of a short diameter. However, for q = 0, the influence of disassortative mixingis enhanced by the high heterogeneous weight distribution as shown in Fig. 6(a) andFig. 7(a). The tighter local hubs attract the particles, the lower the diffusion rate is.As shown in Fig. 8(a), a hub leads to a fast decay of the local A particle density inthe beginning, followed by a slow decay in the long time regime as shown in Fig. 6(a).Thus, one can clearly observe depletion zones emerging from the intervals among hubsin this panel. In Fig. 6(b), a hub in a A or B -rich domain can give rise to a pure A or B zone after a prompt local annihilation of A and B , leaving a relatively particle-freespace among the hubs. With these segregations, one can observe a slow decay of thereaction rate as shown Fig. 7(a). iffusion-annihilation processes in weighted scale-free networks with identical degree sequence (a) (b) Figure 8. (Color online) Illustration of the A + A → ∅ and A + B → ∅ processes onthe IDS-SF networks with n = 4. (a) A + A → ∅ at t = 10, and (b) A + B → ∅ at t = 15. Red and blue plots denote A and B particles respectively. Interestingly, the depletion zone and segregation also inhibit particles moving fromleaves to hubs when θ <
0. The behavior can be observed by measuring the averagedegree of occupied nodes increases with t as plotted in Fig. 9. Note that the plots arealso logarithmically binned. Recalling the discussion at the beginning of this section,particles are attracted by the leaves in this condition. For q = 1, particles tend toagglomerate around hubs for θ > θ < θ >
0. For θ < θ = 0, theireffect is hardly identified as shown in Fig. 9(b) without the enhancement of weight.Apparently, these interesting behaviors are vastly distinct from the previously reportedresults [10, 11]. < q < < q <
1, the networks are stochastic, which makes them not self-similar [33]. Thusthe networks are non-fractal in this middle case. Thus, in order to discuss the variationin the dependence of f on q , we have performed extensive numerical simulations forvarious q from 0 to 1. The simulation settings were the same as the former cases. When q increasing from 0 to 1, the exponent of global mean first-passage time of random walks G ( N g ), decreases from 1 to ln 3ln 4 [33], which indicates the enhancement of transportingefficiency during the process. At the same time, the diameter of the networks alsodecreases while the disassortative mixing feature disappears.In Fig. 10, one can observe that the segregations among hubs disappear graduallywith the increase of q . Under the influence, the diffusion rate increases drastically,leading to an apparent enhancement of f for θ = 1 (see Fig. 10(a)). Notably, in panel(b), the purely topological segregations for θ = 0 are also observable, although itsinfluence is not as apparent as that in panel (a). Also, comparing q = 0 . q = 0 . iffusion-annihilation processes in weighted scale-free networks with identical degree sequence Time
A+A,q=0, =0 A+A,q=1, =0 A+B,q=0, =0 A+B,q=1, =0 (c)(b) < k > A+A,q=0, =-1 A+A,q=1, =-1 A+B,q=0, =-1 A+B,q=1, =-1
A+A,q=0, =1 A+A,q=1, =1 A+B,q=0, =1 A+B,q=1, =1 (a)
Figure 9. (Color online) h k i as a function of time t for A + A → ∅ and A + B → ∅ for θ = − , , q = 0 , the subtle influence of segregations on the reaction rate in the case of θ = − q , which is consistent with our observation in Section 4.2 .
5. CONCLUSION
In summary, we have investigated the diffusion-annihilation process on a family ofweighted scale-free networks with identical degree sequence (weighted IDS-SF networks),which is controlled by a parameter q ∈ [0 , w ij = ( k i k j ) θ with the degree k i and k j of both nodes, where θ is thenetwork’s weightiness parameter. For a convenience, we define a kinetic exponent f as d ( ρ ( t ) ) dt , where ρ ( t ) = ρ A ( t ) for the A + A → ∅ process and ρ ( t ) = ρ A ( t ) + ρ B ( t )for the A + B → ∅ process. Based on the definition, we provide numerical results tocharacterize the relation between f and the reaction time t for the A + A → ∅ and A + B → ∅ bimolecular reactions.One significant observation is that, in contrast to the commonly accepted conceptionthat the depletion zone and segregation only exist in fractal networks. Our observationshows they can exist in the diffusion-annihilation process on the non-fractal networks as iffusion-annihilation processes in weighted scale-free networks with identical degree sequence
024 10 (c) =-1(b) =0(a) =1 q=0.9q=0.5q=0.1 Time
Figure 10. (Color online) Reaction rate exponent f with θ = − , , q =0 . , . , . A + A → ∅ (black lines) and A + B → ∅ (red lines) processes on theIDS-SF networks respectively. well. This striking feature in scale-free networks was not reported in previous studies.In fact, the depletion zone and segregation can both exists in fractal and non-fractalnetworks no matter whether it is weighted or not. We found that the segregation effectis essentially caused by the disassortative mixing, i.e., high-degree nodes tend to connectwith low-degree nodes. On the weighted networks, its influence on the particles diffusionis highly enhanced by the weight heterogeneity. We have demonstrated that both degreeand weight distribution do not suffice to characterize the diffusion-annihilation processeson weighted scale-free networks. Our observations suggest care should be taken whenmaking general statements about the diffusion-annihilation process in weighted scale-free networks. Acknowledgment
This work was supported by National Natural Science Foundation of China (NSFC)under grants Nos. 60873040, 60873070, 61074119, and the Shuguang Program ofShanghai Municipal Education Commission and Shanghai Education DevelopmentFoundation. iffusion-annihilation processes in weighted scale-free networks with identical degree sequence References [1] R. Albert and A.-L. Barab´asi, Rev. Mod. Phys. , 47 (2002).[2] S. N. Dorogvtsev and J. F. F. Mendes, Adv. Phys. , 1079 (2002).[3] M. E. J. Newman, SIAM Rev. , 167 (2003).[4] S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D.-U. Hwanga, Phys. Rep. , 175 (2006).[5] F. M. Atay, T. Biyikoˇglu, and J. Jost, IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. ,92 (2006).[6] A. Hagberg, P. J. Swart, and D. A. Schult, Phys. Rev. E , 056116 (2006).[7] V. M. Egu´ıluz and K. Klemm, Phys. Rev. Lett. , 108701 (2002).[8] L. K. Gallos and P. Argyrakis, Phys. Rev. Lett. , 138301 (2004).[9] M. Catanzaro, M. Bogu˜n´a, and R. Pastor-Satorras, Phys. Rev. E , 056104 (2005).[10] C. K. Yun, B. Kahng and D. Kim, New J. Phys. , 063025 (2009).[11] S. Kwon, W. Choi, and Y. Kim, Phys. Rev. E , 021108 (2010).[12] B. P. Lee, J. Phys. A , 2633 (1994).[13] U. C. T¨auber, M. Howard, and B. P. Lee, J. Phys. A , R79 (2005).[14] F. Leyvraz and S. Redner, Phys. Rev. A , 3132 (1992)[15] D. C. Torney and H. M. McConnell, J. Phys. Chem. , 1941 (1983); Proc. R. Soc. London A , 147 (1983).[16] D. Toussaint and F. Wilczek, J. Chem. Phys. , 2642 (1983).[17] R. Pastor-Satorras, A. Vzquez, and A. Vespignani, Phys. Rev. Lett. , 258701 (2001).[18] A. V´azquez, R. Pastor-Satorras, and A. Vespignani, Phys. Rev. E , 066130 (2002).[19] A. Barrat, M. Barth´elemy, R. Pastor-Satorras, and A. Vespignani, Proc. Natl. Acad. Sci. U.S.A. , 3747 (2004).[20] A. Fronczak and P. Fronczak, Phys. Rev. E , 016107 (2009).[21] A. Einstein, Ann. Phys. (Leipzig) , 549 (1905); , 371 (1906).[22] M. Smoluchowski, Ann. Phys. (Leipzig) , 756 (1906).[23] J. D. Noh and H. Rieger, Phys. Rev. Lett. , 118701 (2004).[24] F. Jasch and A. Blumen, Phys. Rev. E , 041108 (2001).[25] L. A. Adamic, R. M. Lukose, A. R. Puniyani, and B. A. Huberman, Phys. Rev. E , 046135(2001).[26] R. Guimer´a, A. D´ıaz-Guilera, F. Vega-Redondo, A. Cabrales, and A. Arenas, Phys. Rev. Lett. ,248701 (2002).[27] S. Lee, S.-H. Yook, and Yup Kim, Phys. Rev. E , 046118 (2006).[28] S. Lee, S.-H. Yook, and Yup Kim, Physica A , 743 (2007).[29] A. Fronczak and P. Fronczak, Phys. Rev. E , 026121 (2006).[30] M. Molloy and B. Reed, Random Struct. Algorithms , 161 (1995).[31] M. Catanzaro, M. Bogu˜n´a, and R. Pastor-Satorras, Phys. Rev. E , 027103 (2005).[32] Z. Zhang, S. Zhou, T. Zou, L. Chen, and J. Guan, Phys. Rev. E , 031110 (2009).[33] Z. Zhang, S. Zhou, W. Xie, M. Li, and J. Guan, Phys. Rev. E , 061111 (2009).[34] M. E. J. Newman, SIAM Rev., 061111 (2009).[34] M. E. J. Newman, SIAM Rev.