A Modified Earthquake Model Based on Generalized Barabási Albert Scale-Free

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1 Commun. Theor. Phys. (Beijing, China) 46 (2006) pp c International Academic Publishers Vol. 46, No. 6, December 15, 2006 A Modified Earthquake Model Based on Generalized Barabási Albert Scale-Free Networks LIN Min, 1, WANG Gang, 2,3 and CHEN Tian-Lun 4 1 Department of Mathematics, Ocean University of China, Qingdao , China 2 Institute of Oceanology, the Chinese Academy of Sciences, Qingdao , China 3 Graduate School, the Chinese Academy of Sciences, Beijing , China 4 Department of Physics, Nankai University, Tianjin , China (Received January 17, 2006) Abstract A modified Olami Feder Christensen model of self-organized criticality on generalized Barabási Albert (GBA) scale-free networks is investigated. We find that our model displays power-law behavior and the avalanche dynamical behavior is sensitive to the topological structure of networks. Furthermore, the exponent τ of the model depends on b, which weights the distance in comparison with the degree in the GBA network evolution. PACS numbers: b, Ht Key words: self-organized criticality, avalanche, GBA scale-free networks 1 Introduction Many social, biological, and communication systems can be modelled as complex networks. In complex systems, the nodes represent individuals or organizations, and the links model the interaction among them. Recently, Barabási and Albert have studied a class of large growing networks whose degree distribution follows a power-law P k k γ. [1,2] Since power-law is free of a characteristic scale, these networks are called scale-free networks. Scientists have found that many real-world networks have scale-free topological structures such as the World-Wide-Web, social acquaintance networks, biological networks. Although the Barabási Albert (BA) scale-free network is a good representation for a huge variety of real networks, it cannot reproduce the characteristics of those systems in which edges are not costless (e.g. power grids or neural networks). To overcome this problem, Cosenza et al. introduced a new network that generalizes the BA scale-free network. Initial condition and growing process of generalized Barabási Albert (GBA for abbreviation) network are the same as of the BA network. The only difference is that 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. [3] For instance, it is more likely that a neuron connects to nearby neurons in a neural network. GBA scale-free network has a precise spatial arrangement. Such a network, because of its plausibility both in static characteristics and in the dynamical evolution, is a good representation for those real networks whose edges are not costless. [3] Another area of active research is related to systems that present self-organized criticality (SOC). The concept of self-organized criticality refers to certain dissipative systems, with many degrees of freedom, naturally evolve to a critical state characterized by power-law distribution in space and time. [4] One of the systems having been studied in connection with SOC is Olami Feder Christensen (OFC) model for earthquakes. Recently, some works have been done to investigate how the topological structure of network affects SOC based on OFC model. As is well known, OFC models of SOC on a regular lattice and on a random graph have been investigated. [5,6] A modified earthquake model of SOC based on small world networks has been discussed in our previous works. [7,8] Zhou et al. introduced an alternative model to mimic the catastrophes in the scale-free network. [9] Naturally, people will ask, whether SOC behaviors can be displayed in OFC model whose network topological architecture is GBA scale-free network. So, based on OFC earthquake model, which is a typical SOC model, we introduce a modified earthquake model in GBA scale-free network. In our model, the distribution of avalanche size shows power-law behavior and the exponent τ depends on b, which weights the distance in comparison with the degree in the GBA network evolution. 2 The Model 2.1 Static Properties of the Graph A generic network is a graph G, in which the N nodes The project supported by National Natural Science Foundation of China under Grant No and the Doctoral Foundation of the Ministry of Education of China linminmin@eyou.com

2 1012 LIN Min, WANG Gang, and CHEN Tian-Lun Vol. 46 represent the basic component of the network and the K edges represent an interaction between them. We represent a network through its adjacency matrix A = (a ij ), whose element a ij is 1 if there is an edge connecting nodes i and j, and 0 otherwise. The network is undirected and has an average degree equal to k = 2K/N. To characterize the structural properties of a graph, we use two parameters as defined in Ref. [10]. The first is the characteristic path length L: i j G L = d ij N(N 1), (1) where d ij is the length of the shortest path connecting nodes i and j. The second parameter is the clustering coefficient C: i G C = C i (2) N with number of edges in C i = G i, (3) k i (k i 1)/2 where k i (k i 1)/2 is the total number of possible edges in G i, which is subgraph of the first neighbors of a node i. The structural properties do not take into account the role that physical distance plays in the formation of new edges between nodes. To evaluate how much the building cost of a network is, we use a cost parameter. [3] The structural cost of the network defined as Cost = a ij l ij, (4) i,j G where a ij is the element ij of the adjacency matrix and l ij is the Euclidean distance between nodes i and j. 2.2 Generalized Barabási Albert (GBA) Scale- Free Networks We generate the network following the prescription of Cosenza et al. in Ref. [3]. We consider nodes placed on a two-dimensional regular lattice with free boundary conditions. But for further discussion, we also use open boundary conditions in subsection 3.3. The algorithm behind the GBA model is the following. (i) Start with a small number (m 0 ) of nodes. (ii) Then we add a new node with m ( m 0 ) edges (that 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, (5) where k h is the degree of node h, l ih is the Euclidean distance between nodes i and h, 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, i.e. L L. In this way, the construction of GBA model is finished. The situation b = 0 corresponds to the original BA scale-free network. An example of a GBA network with N = 100 nodes and m 0 = m = 2 under free boundary conditions is shown in Fig. 1. Fig. 1 Generalized Barabási Albert (GBA) scale-free network for N = 100 nodes, m 0 = m = 2 and the exponent b = 1 with free boundary conditions. Fig. 2 Generalized Barabási Albert scale-free network for N = 100 nodes, m 0 = m = 2. (a) Characteristic path length L; (b) Clustering coefficient C; (c) Structural cost, versus the exponent b. To characterize GBA model, we have calculated the characteristic path length L, the clustering coefficient C, and the structural cost, as functions of b for networks with N = 100 and m 0 = m = 2. In Fig. 2, characteristic path length L increases with the increment of the exponent b. Clustering coefficient C cause a significant increase from b = 0 to b = 4. Moreover, when we increase the exponent

3 No. 6 A Modified Earthquake Model Based on Generalized Barabási Albert Scale-Free Networks 1013 b, long-range connections are hampered and, consequently, the cost decreases from b = 0 to b = Dynamics in the GBA Network Each site of the lattice is associated with a real variable F i, which is initialized with a random value between 0 and a threshold value F th = 1. Then we can describe the dynamical process of our model as follows. (i) Find out the maximal value of all F i, F max, and add F th F max to all sites. When one of them reaches the threshold F th = 1, it becomes unstable. At this point, an avalanche (earthquake) starts. (ii) If any F i F th then redistribute the energy F i on the i-th site to its neighbors: F j F j + α /k i F i, F i 0, (6) for all nodes j adjacent to i, where k i is the degree of node i. (iii) Repeat step (ii) until all sites are stable. Define this process as one avalanche. (iv) Begin step (i) again and another new avalanche begins. The parameter α controls the level of conversation of the dynamics. When α = 1, the model is conservative. Otherwise, the model is nonconservative for α < 1. Let us finally discuss the boundary conditions. The boundary being free means that the blocks in the boundary layer are connected only to blocks within the fault. [5] In our model, we use α bc = α /(k i + 1 α ), except at corner sites where α bc,c = α /(k i +2 2α ). The boundary being open means that the blocks in the boundary layer are coupled to an imaginary boundary block by springs. [5] In our model, we use α bc = α /k i. 3 Simulation Results 3.1 Power-Law Behavior and Influence of the Exponent b Here we use a GBA network of the size 20 20, where α = 0.98, m 0 = m = 2 are fixed. Then we change b, our aim is to investigate the distribution of the avalanche size for different b. To prove the SOC of our system, we measure the probability distribution of the avalanche sizes. The Euclidean lattice mentioned in Fig. 3 is a two-dimensional square lattice and BA network is GBA network with b = 0, m 0 = m = 2 under free boundary conditions. Thus the average degree of the networks is k 4. As shown in Fig. 3, the distribution of avalanche sizes has power-law behaviors, P (S) S τ, which is regarded as fingerprint for SOC. [11] Similar to the result in Ref. [9], the distribution of avalanche size in BA network follows a straight line for more than three decades. With the increment of b, the probability of large size avalanches occurring decreases and the cutoff in the avalanche size distribution decreases. We think that the behaviors are caused by the exponent b in the GBA network. The greater the exponent b is greater the importance of distance (in comparison with the degree) in the GBA network evolution. For b = 0, there are many long-range connections in the network. As shown in Fig. 2, for increasing values of the exponent b, the characteristic path length and the clustering coefficient increases, and the cost decreases. The network tends towards homogeneity and the number of long-range connections decreases. For lattice case, there is no presence of long-range connections. With the increment of b, the number of long-range connections in the network decreases. So the ranges of the particular sites local interaction are reduced. The large size avalanches have less probability to occur and the cutoff in the avalanche distribution decreases. The maximal avalanche size in GBA network is greater than that in Euclidean lattice. Fig. 3 The probability of the avalanche size P (S) as a function of S with system size L = 20, α = 0.98 for GBA network with b = 0, 4, and Euclidean lattice, respectively. At the same time, we present the dependence of the exponent τ on the exponent b. In Fig. 4(a), as can be seen, the value of τ decreases with the increment of b. We also investigate the relation between the average avalanche size S and the exponent b in Fig. 4(b). The average avalanche size S also decreases as b increases. The result is consistent with the conclusion mentioned above. In Fig. 5, we draw the mean size S of avalanches originated from nodes of the GBA network with the same degree for system size L = 20, α = 0.98 with b = 3 and 8, respectively. Overall, S b=3 > S b=8 for the same degree

4 1014 LIN Min, WANG Gang, and CHEN Tian-Lun Vol. 46 of nodes. It accords with the result that S for b = 3 is larger than S for b = 8 in Fig. 4(b). In Fig. 2, when b is equal to 3, the GBA network has low characteristic path length and high clustering. The power-law degree distribution of GBA network with b = 3 is better preserved. [3] Moreover, for b = 3, the GBA network also meets the requirements of low cost, which is fundamental for real-world networks. From Fig. 4(a), we can see the distribution of avalanche size in b = 3 GBA network follows power-law behavior. In the following, b = 3 will be used. the avalanche distribution does not scale with the system size, it is a localized behavior, otherwise it means SOC. [5] In Fig. 6, we show the results of simulations with b = 3, α = 0.98 for L = 20, 30 and 50, respectively. We can see that the large size cutoff in the avalanche size distribution scale with the system size, which is indicative of a critical state. [5] Fig. 4 (a) The power-law exponent τ of the avalanche size distribution; (b) The avalanche average size S as a function of b for L = 20, α = Fig. 6 The probability of the avalanche size P (S) as a function of size S for α = 0.98, b = 3 with different size L = 20, 30 and 50, respectively. Fig. 5 Mean size S of avalanches originated from node of the network with the same degree for L = 20, α = 0.98 with b = 3 and b = 8, respectively. 3.2 Influence of the Lattice Size In the original OFC model, the lattice size is important. We also investigate the effect of the lattice size on SOC behavior in the model. If the large size cutoff of Fig. 7 Finite-size-scaling with α = 0.98, b = 3, and L = 20, 30, and 50, respectively using P (S, L) L β g(s/l υ ). In order to characterize the critical behavior of the model, the scaling properties of the system are investigated by finite-size-scaling analysis. That is ( S ) P (S, L) L β g L υ, (7)

5 No. 6 A Modified Earthquake Model Based on Generalized Barabási Albert Scale-Free Networks 1015 where g is the so-called universal scaling function and β, υ are critical exponents describing the scaling of the distribution function. The critical index υ expresses how the finite-size cutoff scales with system size, while the critical index β is related to the normalization of the distribution function. [5] If the distribution function is a power-law, it is easy to show the following relation between τ, β and υ, τ = β υ. (8) As can be seen, in Fig. 7, the finite-size-scaling of avalanches for b = 3, α = 0.98 with L = 20, 30, and 50 works well and β 1.45, υ 1.1. The critical exponent τ 1.34 for the system. Its exponents fulfill the relation (8). This phenomenon indicates that the system reaches the SOC state. 3.3 Influence of Boundary Conditions In the OFC model, the existence of criticality depends on the boundary conditions. In fact, criticality in the OFC model on a lattice has been ascribed to a mechanism of partial synchronization. [12] The boundaries act as inhomogeneities that frustrate the natural tendency of the model to synchronize and make the system partial synchronization. So we have investigated the boundary effects of the model. We also check the effects of the boundaries. The probability distribution of the boundary avalanches P (S) for L = 20, α = 0.98, and b = 3 with open boundary conditions and free boundary conditions are shown in Fig. 9. They obey different power-law distributions. We can see the probability of large size avalanches occurring under free boundary conditions is larger than that under open boundary conditions. It is because that the free boundary for system is relatively more conservative than the open boundary. Fig. 9 The probability of the boundary avalanche size P (S) as a function of S for system size L = 20, b = 3, and α = 0.98 with open and free boundary conditions, respectively. 4 Conclusion Fig. 8 Simulation results with all and boundary avalanches for L = 20, b = 3 and α = 0.98 with open boundary conditions. In Fig. 8, we show the silulation results of all and boundary avalanche (avalanches starting from borders) size distributions with open boundary conditions. As shown in Fig. 8, they obey power-law distributions. All avalanches are distributed as P (S) S τ, τ 1.34, and the boundary avalanches are distributed as P (S) S τ, τ It can be treated as evidence that open boundaries create inhomogeneities. In this paper, we provide a modified OFC model based on GBA scale-free network. Taking into account the physical distance between nodes, GBA network is the generalization of the Barabási Albert scale-free network. Our system is different from the previous OFC system, in that the GBA network is a more plausible representation of real-world networks. In our model, we find that the distribution of avalanches sizes display power-law behaviors. More importantly, the exponent τ depends on b, which weights the distance in comparison with the degree in the GBA network evolution. The effect of topological structure of networks to SOC behavior in the model is an interesting and active research area. GBA network is a good representation for the real networks (such as a network of neurons) whose edges are not costless. Our work just attempts to investigate the effects of GBA network to SOC behavior in the OFC model. So we can study the influence of GBA network topology on other dynamical systems in future works.

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