Commun. Theor. Phys. (Beijing, China) 47 (2007) pp. 512 516 c International Academic Publishers Vol. 47, No. 3, March 15, 2007 Self-organized Criticality in a Modified Evolution Model on Generalized Barabási Albert Scale-Free Networks LIN Min, 1, WANG Gang, 2 and CHEN Tian-Lun 3 1 Department of Mathematics, Ocean University of China, Qingdao 266071, China 2 First Institute of Oceanography, State Oceanic Administration, Qingdao 266061, China 3 Department of Physics, Nankai University, Tianjin 300071, China (Received March 23, 2003) Abstract A modified evolution model of self-organized criticality on generalized Barabási Albert (GBA) scale-free networks is investigated. In our model, we find that spatial and temporal correlations exhibit critical behaviors. More importantly, these critical behaviors change with the parameter b, which weights the distance in comparison with the degree in the GBA network evolution. PACS numbers: 05.65.+b, 45.70.Ht Key words: self-organized criticality, evolution model, GBA scale-free networks 1 Introduction In 1987, Bak, Tang, and Wiesenfeld introduced a concept of self-organized criticality (SOC). [1] It is shown that the extended nonequlibrium system can organize into a scale-invariant critical state spontaneously, without fine tuning of a control parameter. This critical state is characterized by a power-law distribution of avalanche size, which is regarded as fingerprint for SOC. The phenomenon of SOC has been observed in many extended dissipative dynamical systems, such as earthquakes, [2] biology evolution, [3] forest fires, [4] and so on. Recent studies have devoted particular attention to the large evolving complex networks. One of the most important components of complex networks is scale-free network, defined as the network whose degree distribution follows the power-law behavior, which has been an interesting and significant research area. [5] Many real networks, including citation network, the internet, WWW, and food webs, have a power-law degree distribution, which characterizes the scale-free structure of complex networks and can be explained by the Barabási Albert (BA) model. Barabási and Albert presented the scale-free model (BA model) with the mechanism called linear preferential attachment. The generalized Barabási Albert (GBA for abbreviation) scale-free network introduces the parameter b, which weights the distance in comparison with the degree in the GBA network evolution. [6] The only difference in comparison with the original BA model is in the preferential attachment, GBA model takes into account the physical distance between nodes, which in most real cases is an important parameter in the network evolution. In recent years, there has been extensive study on the effects of the topology of the network on the SOC behavior. The Bak Sneppen (BS) evolution model is one of the simplest models giving rise to SOC behavior. The BS model has been extensively investigated for regular networks. [3,7] Boer et al. have studied the BS model on annealed random network. [8] Moreno et al. have proposed the BS model on scale-free networks. [9] However, the topology of the real network is not so simple. The GBA network is a new generalization of the BA model for networks with a precise spatial arrangement. The GBA, being a dynamical model, is a more plausible representation of real-world networks. For b > 2, the GBA model also meets the requirements of low cost, which is fundamental for real-world networks. [6] It is then natural to ask whether SOC behaviors can be investigated in a model whose network topology is GBA scale-free network, and whether and to what extent the topology of GBA network would affect many of the results obtained in the original BS model. So in this paper, we investigate the dynamics of the BS model on GBA scale-free networks. We study the space and time correlations and find our model exhibits some particular spatial and temporal behaviors different from the original BS model. More importantly, the dynamical behaviors of the system change with the parameter b, which weights the distance in comparison with the degree in the GBA network evolution. 2 The Model We consider nodes placed on a one-dimensional lattice with period boundary conditions. We generate the under- The project supported by National Natural Science Foundation of China under Grant No. 90203008 and the Doctoral Foundation of Ministry of Education of China E-mail: linminmin@eyou.com
No. 3 Self-organized Criticality in a Modified Evolution Model on Generalized Barabási Albert Scale-Free Networks 513 lying networks following the prescription in Refs. [6] and [10]. Its construction can be seen as follows. (i) Start with a small number (m 0 ) of nodes. (ii) Then we add a new node with m(< m 0 ) edges, which will be connected to the nodes already present in the system. The probability for a new node i to be connected with an already present node j is (j) = k j l b ij 1 h G k h/lih b where k h is the degree of node h, l ih is the Euclidean distance between nodes i and h in a network G, and b is an exponent that weights the distance in comparison with the degree. (iii) Repeat step (ii) until the number of nodes is N. In this way, the construction of GBA scale-free network is finished. It is obvious that b = 0 corresponds to the BA scale-free network. In the graph, any node is connected with different numbers k of other nodes. Two connected nodes are indicated as the nearest neighbor. Now, let us define and simulate the dynamical mechanism of our model as follows. (i) The model consists of an N-site GBA network with period boundary conditions, where each site represents a species. Each species has associated a real variable f i, 0 f i 1, that measures the relative fitness barrier. (ii) At each update t, the least survivable species (the node with minimum barrier f min (t)) is identified and assigned a new random number between 0 and 1. In the original BS model, this change is thought of as the species undergoing a mutation. (iii) At the same update t, all the nodes k connected with the extremal node are assigned new random barriers uniformly distributed between 0 and 1 too. (iv) Repeat steps (ii) and (iii) definitely. 3 Simulation Results In the study of BS evolution model, there are two widely discussed topics: one is the dynamics of f 0 avalanche, the other is the spatial and temporal behaviors of the nodes with minimum barrier. To determine whether the system attains a self-organized critical state, we analyze the following quantities: the minimum barrier distribution and f 0 avalanche size distribution in the critical state, the spatial correlation C(x) and the first return time distribution P f (t). We will investigate how these quantities change with the parameter b, which weights the distance in comparison with the degree in the GBA network evolution. We hope the study will help us to understand the topological structure of networks affects self-organized criticality in our modified BS model., 3.1 Power-Law Behavior of f 0 (b) Avalanche Here we use a GBA network with N = 1000 nodes and the average degree is k = 2. Then we change b, our aim is to investigate avalanche dynamical behaviors for different b. Starting from a random initial condition, we let the system evolve. After a transient, the system reaches a highly correlated stationary state. All the minimum barriers f i (t) are lower than what is called the self-organized threshold f c in Ref. [11]. A self-organized threshold also exists in our model. From Fig. 1, for a certain b, we can see that the distribution of the lower barriers in the critical state vanishes at and above the corresponding f c (b). We call the self-organized threshold for a certain b in our model f c (b). Our simulations show that the distribution of the minimum fitness follows a different pattern. Indeed, figure 1 shows that self-organized threshold for b = 2.5 is bigger than the one found for the random neighbor (RN) model. [8] As can be seen, by increasing the value of b, the threshold f c (b) moves towards the threshold value f c 0.667 obtained in the original BS model. It can be explained that with the increment of b, the importance of distance is greater than the degree in the GBA model evolution. The number of long-range connections decreases and the network tends toward homogeneity. In this case, the scopes of the particular sites decrease. So the speed of collective dynamics decreases and the system exhibits higher barriers. Fig. 1 Distribution of minimum barrier values for four different values of b: 0, 1, 2.5, and 10. All cases have N = 1000. In the BS model, an avalanche is defined as the sequence of time steps for which the minimal site has a barrier value smaller than a threshold f 0. Similar to those used in Refs. [7], [12], and [13], for a certain b, we present the definition of the f 0 (b) avalanche, where f 0 (b) [0 < f 0 (b) < f c (b)] is an auxiliary parameter used to define the avalanche. Suppose that at time s, the smallest
514 LIN Min, WANG Gang, and CHEN Tian-Lun Vol. 47 random number in the system is larger than f 0 (b). According to the rules of the model, if, at time step s + 1, the lowest of the new random numbers selected is less than f 0 (b), a f 0 (b) avalanche begins. The avalanche continues to run if the lowest random barrier is lower than f 0 (b). The avalanche stops, say at time s + S, when the lowest number is larger than f 0 (b) for the first time. The f 0 (b) avalanche size is defined as the duration of the avalanche S. Figure 2 shows the distribution of avalanche duration for b = 3. The avalanche follows a power-law distribution P (S) S τ, τ 0.85. The power-law behavior is the essence of self-organized criticality. reaches a saturation value α = 3.2 for b > 4, in agreement with the results observed in the original BS model. [14] Fig. 3 The probability distribution C(x) of spatial distance x between successive mutating nodes of our modified BS model with different b. The system size is N = 1000. Fig. 2 Power-law distribution of f 0 avalanche in an N = 1000, b = 3 system. 3.2 Spatial Correlation Power-law distribution of avalanche is a first evidence of SOC dynamics. Additional information can be obtained from the study of spatial correlation. Following Bak and Sneppen, we investigate the spatial correlation between nodes with minimum barrier in our new model. In Fig. 3, we present the distribution C(x) of the distances x between subsequent mutations with system size N = 1000 and different b. The simulations show C(x) follows power-law distribution C(x) x α. C(x) is very different from that of the BS model for small values of the parameter b. When b 4, although C(x) still obey powerlaw behaviors, the slopes α of C(x) are smaller than that (α = 3.15 ± 0.05) of the original BS model. For b = 0, C(x) displays a non-trivial power-law decay with x, unlike the RN model for which C(x) is constant. The exponent α for the law C(x) x α, α 2.21 with b = 2.5 dose not coincide with the one they obtained (3.15 ± 0.05) in the original BS model, indicating that our model belongs to some other universality class. In Fig. 4, we present how α changes with b. From Figs. 3 and 4, we can see that the slope α of C(x) increases with the increment of b until it Fig. 4 Spatial correlation exponent α of the power-law in Fig. 3 as a function of b. 3.3 Temporal Correlation In the BS model, there are two temporal correlation functions between the minimum barriers investigated: the first and all return time probability distributions. The first return probability P f (t) is defined as the probability distribution that, if a given node undergoes a mutation (with the minimum barrier) at step t 0, it will again undergo mutation for the first time at step t 0 + t. The all return probability P a (t) is the probability that this node will also undergo a mutation at t 0 + t regardless of what happens at intermediate steps. Next we discuss the first return time distribution P f (t). In Fig. 5, we present the first return probability distribution for different b. The first return probability P f (t)
No. 3 Self-organized Criticality in a Modified Evolution Model on Generalized Barabási Albert Scale-Free Networks 515 obeys power-law behaviors P f (t) t τ f. This power law means that there is no temporal scale that would control the dynamics, and in this sense our model is clearly SOC. [15] In Fig. 6, we show the dependence of the first return probability exponent τ f on the exponent b. For b = 0, the first return probability distribution obeys power-law behavior and τ f 0.8, displaying the different behavior found in the RN BS model with K = 3, where τ f = 1.5 exactly. [8] This result is also different from that in an evolution model with long-range interactions for α = 0. [14] We think that the difference is caused by different randomness. The randomness in the RN model is in fact a kind of annealed randomness, [8] but the randomness in our model is quenched, that is, the spatial structure of the network is fixed. From Figs. 5 and 6, we can see that the exponent τ f increases as b increases. For b > 4, the value of τ f attains a saturation value τ f 1.56 in agreement with the value observed in the original BS model where τ f 1.58. For increasing values of the exponent b, the order of our model becomes more and more obvious, so the dynamics of our model becomes closer and closer to that of the original BS model. different scales. [1] Fig. 6 The first return probability exponent τ f as a function of b. Fig. 7 The first and all return probability distributions of our model with b = 4, for the system size N = 1000. Fig. 5 The first return probability distribution of our model with different b. The system size is N = 1000. In Fig. 7, we show the first and all return probability distributions for our model with b = 4. Both of the two temporal correlations follow the power-law behaviors: for the first return probability, P f (t) t τ f, τ f 1.52; for the all return probability, P a (t) t τa, τ a 0.44. The relation is satisfied with τ f + τ a 2, which coincides with the result obtained in the original BS model. The spatial and temporal distributions of avalanches have been well described by power laws, indicating that system is in a critical state and that the dynamics can be seen at many 4 Conclusion In summary, we have extended the one-dimensional Bak Sneppen model to generalized Barabási Albert (GBA) scale-free networks. In our model, we find that spatial and temporal correlations exhibit critical behaviors. We find that the dynamical behaviors are strongly correlated with the topology of the network. More importantly, these behaviors change with the parameter b, which weights the distance in comparison with the degree in the GBA network evolution. For 0 b 4, the system exhibits non-universal SOC, i.e., the associated critical exponents depend strongly on b. For 0 b 1, C(x) displays a power-law decay with x, unlike the RN model for which C(x) is constant. The exponent for the first return time probability distribution is different from the one for
516 LIN Min, WANG Gang, and CHEN Tian-Lun Vol. 47 the RN model with τ f = 1.5 or the evolution model with long-range interaction for 0 < α < 1 (τ f = 1.5). [14] By increasing the value of b, the values of critical exponents increases. For b > 4, we have a short-range critical regime, where the system presents SOC. The associated critical exponents are independent of b, obtaining the values observed in the original BS model. Our work just investigates the influence of GBA scale-free network topology to SOC behavior in our modified evolution model. So we can study the effects of GBA network topology on other dynamical systems in future works. References [1] P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59 (1987) 381; Phys. Rev. A 38 (1988) 364. [2] Z. Olami, S. Feder, and K. Christensen, Phys. Rev. Lett. 68 (1992) 1244; K. Christensen and Z. Olami, Phys. Rev. A 46 (1992) 1829. [3] P. Bak and K. Sneppen, Phys. Rev. Lett. 71 (1993) 4083. [4] K. Christensen, H. Flyvbjerg, and Z. Olami, Phys. Rev. Lett. 71 (1993) 2737. [5] A.L. Barabási and R. Albert, Science 286 (1999) 509; A.L. Barabási, R. Albert, and H. Jeong, Physica A 272 (1999) 173. [6] S. Cosenza, et al., Mathematical Biosciences and Engineering 2 (2005) 53. [7] M. Paczuski, S. Maslov and P. Bak, Phys. Rev. E 53 (1996) 414. [8] J. de Boer, A.D. Jackson, and T. Wettig, Phys. Rev. E 51 (1995) 1059. [9] Y. Moreno and A. Vazquez, arxiv.cond-mat/0108494. [10] M. Lin, G. Wang, and T.L. Chen, Commun. Theor. Phys. (Beijing, China) 46 (2006) 1011. [11] L. da Silva et al., Phys. Lett. A 242 (1998) 343. [12] M. Lin and T.L. Chen, Phys. Rev. E 71 (2005) 016133. [13] M. Lin, G. Wang, and T.L. Chen, Commun. Theor. Phys. (Beijing, China) 46 (2006) 362. [14] P.M. Gleiser, F.A. Tamarit, and S.A. Cannas, Physica A 275 (2000) 272. [15] R.V. Solé and S.C. Manrubia, Phys. Rev. E 54 (1996) R42.