Spatial and Temporal Behaviors in a Modified Evolution Model Based on Small World Network

Transcription

1 Commun. Theor. Phys. (Beijing, China) 42 (2004) pp c International Academic Publishers Vol. 42, No. 2, August 15, 2004 Spatial and Temporal Behaviors in a Modified Evolution Model Based on Small World Network ZHAO Xiao-Wei, ZHOU Li-Ming, and CHEN Tian-Lun Department of Physics, Nankai University, Tianjin , China (Received November 4, 2003) Abstract In this paper, we introduce a new modified evolution model on a small world network. In our model, the spatial and temporal correlations and the spatial-temporal evolve pattern of mutating nodes exhibit some particular behaviors different from those of the original BS evolution model. More importantly, these behaviors will change with φ, the density of short paths in our network. PACS numbers: b, Ht Key words: self-organized criticality, small world network, evolution model 1 Introduction The concept of self-organized criticality (SOC) refers to the intrinsic tendency of many extended dissipative dynamical systems to self-organize into a stationary state without spatial and temporal scales. Since the introduction of this idea, it has been a topic with considerable interest, and many simple mathematical models have been introduced to explain the common appearance of spatialtemporal complexity in nature. [1 4] The underlying networks of these models often have simple topological architectures to make the simulation easier, such as rings, grids, lattices, etc. However, the real world is very complicated, there are many very complex topological architectures in it. So some scientists believed that for extensive networks of simple interacting systems,... network topology can be as important as the interactions between elements [5] and they pay attention to the effects of these complex network topologies more and more. One of the most important components of complex networks is small world network, which can possess the characteristics of both regular lattice and random graph, and is believed to lie somewhere between the extremes of order and randomness. [6] Scientists have found that many real-world networks have small world characters. [7] Naturally, people will ask, whether SOC behaviors can be investigated in a model whose network topological architecture is small world network, and whether some particular phenomena will happen in such a model. So, based on Bak Sneppen (BS) evolution model, which is a typical SOC model, and the small world network, we introduce a new evolution model. We can find that our new model can exhibit some particular spatial-temporal behaviors different from the original BS model, and more importantly these behaviors will change with the network topological structures of the model. 2 Model The original BS model is introduced to mimic biological evolution and is one of the simplest SOC models. It has a very simple dynamical mechanism. [4] We will not change this mechanism in our model, because we believe that this simple mechanism can allow us to concentrate mainly on the effect of the small world topology itself without being burdened by any additional complexity of the dynamical mechanism (such as the strength level of interaction, whether energy is conserved or not in some other SOC models). In addition, the original BS model also has a very simple underlying topological architecture, which is a regular ring with periodic boundary condition. Bak et al. wanted to use it to represent the real interacting network among various species in nature (such as food chain ). [4] However, the topology of the real food chain is not so simple. The empirical data show that it is far from a regular ring but has some characteristics of small world. [7] So, in our model, we change the underlying topological architecture of the network from a regular ring into a more complex small world topology. As we know, the main structure of small world model consists of a network of nodes whose topology is that of a regular lattice, but with a low density φ of connections (short paths) between randomly pairs of nodes rewired [8] or added [9]. When φ is small, the network can possess characteristics of regular lattice. When φ is large, it can possess characteristics of random graphs. In this paper, we will use the latter version of the small world model, which is introduced by Newman and Watts. [9] Its construction can be seen as follows. (i) Take a one-dimensional regular lattice with size L and the nearest neighbor connections. The periodic The project supported by National Natural Science Foundation of China and the Doctoral Foundation of the Chinese Education Commission under Grant Nos and xiaoweizhao@eyou.com

2 No. 2 Spatial and Temporal Behaviors in a Modified Evolution Model Based on Small World Network 243 boundary condition is used in the lattice, so it is a ring with L connections. (ii) Randomly choose two nodes of the ring and place a connection (short path) between them. (iii) The step (ii) repeats until φ L short paths have been added in the network. Then the construction of the small world network is finished. Here, the steps (ii) and (iii) mean that we add with probability φ one short path for each connection on the original ring. In the graph, any node is connected with different numbers Z of other nodes (including its two nearest neighbors), but on average, a node is connected with Z = 2(1 + φ) other nodes. It is obvious that φ = 0 corresponds to the simple regular ring of original BS model and large φ corresponds to random graph. Now, let us define and simulate the dynamical mechanism of our model as follows. (i) In the original BS model, each node of the ring represents a species in the food chain, and is assigned a random number (barrier) as a measure of the survivability of the species. Similarly, in this model, we give each node of the network a random barrier f i, which is chosen from a flat distribution between 0 and 1. (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 Z nodes connected with the extremal node are assigned new random barriers uniformly distributed between 0 and 1 too. Just as in the original BS model, this step can represent that species survivability might be affected by a mutating neighbor in the food chain. (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 what is called f 0 avalanche, which we will discuss with our new model in an other paper (in preparation), the other is the spatial and temporal behaviors of nodes with minimum barrier (such as spatial and temporal correlation functions between them). Some scientists argue that, except a power law distribution of avalanches size, an SOC system must have simultaneously power law correlations (with appropriate exponents) both in space and in time. [10] In this paper, we will not discuss whether this view is right or not. We just want to investigate how the spatial and temporal correlation functions change with the probability φ. We hope the study will help us to understand the effect of topological structure to the spatial and temporal behaviors in our modified BS model. 3.1 Spatial Correlation First, we will study the spatial correlation between nodes with minimum barrier (the mutating nodes) in our new model. In the original BS evolution model, the probability that the mutating nodes at two successive updates will be separated by X sites is given by a power law P (X) X φ, φ 3.1 [4] (Fig. 1). This is a kind of scale-invariant behavior. Here, we also study the distribution P (X) of the distance X between successive mutating nodes in our modified model. Fig. 1 The probability distribution P (X) of spatial distances X between successive mutating nodes of our modified BS model with different numbers of short paths (φ L = 1, 10, 1000). We also draw P (X) of the original and the random neighbor BS model. For all the cases, the network size L is In Fig. 1, we draw the distribution P (X) of our model with system size L = 1000 and different numbers of short paths. We can find that P (X) is very different from that of the original BS model: when there is only one short path in our network, although P (X) still obeys a power law behavior, the slope (φ) of P (X) is larger than that of the original BS model. And more importantly, P (X) has a peak in a special distance X. We believe that this phenomenon is due to a long-range spatial correlation existing between the two separate parts of the network. After careful investigation, we can find that the value of X is just equal to the distance between two ends of the short path in the model. This fact makes us guess that the short path plays a very important role on the emergence of the longrange spatial correlation. In addition, from Fig. 1, we can also see that, with the increment of the number of short paths in our network, the number of peaks also increases. It implies that the long-range spatial correlation between separate parts of the network becomes more and more important, and the randomness of our model becomes more and more obvious. When the number of short paths becomes very large (e.g. φl = 1000), we can find P (X) is spatially uniformly distributed along X except where X is very small. This distribution is similar to that of the random neighbor BS model. (See Fig. 1 and Ref. [10]). In the random neighbor (RN) model, there is a significant probability that the minimum barrier will be at the same node at two successive updates, which means X = 0 and we do not draw it in Fig. 1, and except X = 0, P (X) is spatial-uniformly distributed along X. We think that it is

3 244 ZHAO Xiao-Wei, ZHOU Li-Ming, and CHEN Tian-Lun Vol. 42 the randomness, which is introduced via the short paths in our model, that makes our model have the similar spatial correlation behaviors to RN model. It is worth while noting that the P (X) of our model with large φ still has some differences from that of the RN model. From Fig. 1, we can see that P (X) of our model with φ = 1 still obeys some sort of power law behavior when the distance is small (X 1, 2, 3). However, for the RN model, except when X = 0, P (X) is spatial-uniformly distributed. We think that the difference is because of different randomness existing in the two models. The randomness in the RN evolution model is in fact a kind of annealed randomness [11] (i.e., in the next time the same mutating node triggers two other nodes to evolve, they are chosen at random anew), but the randomness in our model is quenched because the spatial structure of the network is fixed during the whole evolving process. So our model has more spatial structure and order than the RN model. There are some notable differences between these two models. 3.2 Temporal Correlations There are two important temporal correlation functions between the nodes with minimum barrier studied in evolution models: the first return probability P first (t) and the all-return probability P all (t). P first (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. P all (t) is the probability that this node will also undergoes a mutation at t 0 + t regardless of what happens at intermediate steps. model with different φ respectively. When φ is small, the two temporal correlation functions of our model have similar power law behaviors to those of the original BS model. However, as φ, and with it the number of short paths increases, the two correlation distributions all change from the original power law distributions. At last, when φ is very large, they are similar to those of RN evolution model. We think that this behavior is also the result of increment of the randomness of our model. With the increasing of φ, the randomness of our model also becomes more and more obvious, so the dynamics of our model also becomes closer and closer to that of RN evolution model, which is completely random. Fig. 3 The all return probability distribution of our model with different numbers of short paths (φl = 1, 10, 1000). We also draw the distribution of the original and the random neighbors BS model. For all the cases, the network size L is Fig. 2 The first return probability distribution of our model with different numbers of short paths (φl = 1, 10, 1000). We also draw the distribution of the original and the random neighbor BS model. For all the cases, the network size L is In the original BS model, both of the two temporal correlations obey the power law: for the first return probability, P first (t) t τ first, τ first 1.58; for the all-return probability, P all (t) t τ all, τ all [12] In Figs. 2 and 3, we can see P first (t) and P all (t) of our modified evolution 3.3 Spatial-Temporal Evolving Pattern of Nodes with Minimum Barrier According to the dynamics of evolution models, at any given step, there is one and only one node where the mutation happens (the node s barrier is the global minimum). In the study of evolution models, how the location of mutating node evolves with the time is also very interesting. From Fig. 4, we can see that, in the original BS model, it forms a so-called spatial-temporal fractal pattern. [12] There appears to be some spatial correlation between mutating nodes at successive time steps. The neighbors of the mutating nodes at this time have large probabilities to undergo mutation at the next step. In addition, the locations of mutating nodes form many clusters with different sizes in the figure. At the same time, if we focus on a given node, we can find a punctuated equilibrium behavior. That means the node is in a state with long periods of calm (The node s barrier is not the minimum) interrupted by sudden bursts of mutation. Here, we study the spatial-temporal pattern of our modified model. In Fig. 5, we show the pattern of our model with one short path (φl = 1, here L = 1000). We can see that, although there are still spatial correlations

4 No. 2 Spatial and Temporal Behaviors in a Modified Evolution Model Based on Small World Network 245 between successive mutating nodes and clusters formed in the pattern, something different happens. From the figure, we can find that the locations of mutating nodes (black dots) form two main clusters near two special nodes. It tells us that, mutation has every large probability to occur near the two nodes, and we can find that the two nodes are just the two ends of the short path in our network. In addition, from the figure, we can see that there are many large bulks existing near the areas far from the two special nodes. It tells us that the nodes far from the ends of the short path have very low possibilities to undergo mutation. modified model; and the two attractors have certain attract scopes. Once the global minimum falls on the nodes within the scope of any of the two attractors, it is hard to jump out from the scope. In addition, we guess that there is a long range spatial correlation emerging between the two attractors via the short path, so the global minimum can jump back and forth between nodes belonging to different attract scopes. Thus the nodes near the two attractors undergo mutation much more frequently than the rest of the network. Fig. 4 The spatial-temporal pattern of the location of mutating nodes evolving with time of the original BS model. The system size L is The time is measured as the number of update steps, t. Fig. 6 The spatial-temporal evolving pattern of our modified BS model with two short paths. The system size L is Fig. 7 The spatial-temporal evolving pattern of our modified BS model with 1000 short paths. The system size L is 1000, so φ is equal to 1. Fig. 5 The spatial-temporal evolving pattern of our modified BS model with one short path. The system size L is The two ends of the short path are located on nodes of the network Nos. 178 and 415. There are two main clusters emerged near the two nodes. This pattern looks like that there are two spatial attractors forming at the ends of the short path in our We also study the spatial-temporal patterns of our modified model with different φ, and with it the number of short paths. From Fig. 6, we can see that there are four clusters emerging near the four ends of the two short paths when φl = 2. It makes us guess that attractor can emerge near each end of each short path in our model, and the number of attractors will become large as φ increases. In addition, when φ is very large (φ = 1), we can see that

5 246 ZHAO Xiao-Wei, ZHOU Li-Ming, and CHEN Tian-Lun Vol. 42 the pattern exhibits a disorder state, and the locations of mutating nodes seem to be random (Fig. 7). We think that the reason of this behavior is the existence of a large number of short paths and the randomness introduced by these short paths. When φ is large, there are many short paths in our network; and near each end of each short path, there is an attractor. So there are many attractors emerging in the whole network. Obviously, the attract scope of these attractors will overlap with each other. At this time, the randomness becomes very important in our model, and the global minimum will jump among attractors via the short paths in an indefinite way. Therefore the whole pattern exhibits a very disorder state. This state is similar to that of the RN model. 4 Conclusion In this paper, we study a modified evolution model based on small world topology. Our model is different from previous evolution models, in that a kind of quenched randomness is introduced into our model via a relative complex network topological structure. Our model will degenerate to the original BS model when density φ = 0. If φ is not equal to 0, our model can exhibit some particular spatial-temporal behaviors different from the original BS model. More importantly these behaviors will change with φ, and with it the network topological structures of the model also change: with the increment of φ, all of these behaviors exhibit more and more disorders. Our model is obviously a more generalized model. Of course, this work is just a preliminary study. Many other important questions concerned with the model need to be studied in future works. References [1] P. Bak, C. Tang, and K. Wiesenfield, Phys. Rev. A38 (1988) 364. [2] Z. Olami, S. Feder, and K. Christensen, Phys. Rev. Lett. 68 (1992) 1244; K. Christense and Z. Olami, Phys. Rev. A46 (1992) [3] K. Christensen, H. Flyvbjerg, and Z. Olami, Phys. Rev. Lett. 71 (1993) [4] P. Bak and K. Sneppen, Phys. Rev. Lett. 71 (1993) [5] K. Ziemelis, Nature (London) 410 (2001) 241. [6] S.N. Dorogovtsev and J.F.F. Mendes, Advances in Physics 51 (2002) [7] R. Albert and A.L. Barabási, arxiv.cond-mat/ [8] D.J. Watts and S.H. Strogatz, Nature (London) 393 (1998) 440. [9] M.E.J. Newman and D.J. Watts, Phys. Rev. E60 (1999) [10] J. de Boer, A.D. Jackson, and T. Wetting, Phys. Rev. E51 (1995) [11] H. Flyvbjerg, K. Sneppen, and P. Bak, Phys. Rev. Lett. 71 (1993) [12] M. Paczuski, S. Maslov and P. Bak, Phys. Rev. E53 (1996) 414.