Nonlinear Dynamical Behavior in BS Evolution Model Based on Small-World Network Added with Nonlinear Preference

Commun. Theor. Phys. (Beijing, China) 48 (2007) pp. 137 142 c International Academic Publishers Vol. 48, No. 1, July 15, 2007 Nonlinear Dynamical Behavior in BS Evolution Model Based on Small-World Network Added with Nonlinear Preference ZHANG Ying-Yue, YANG Qiu-Ying, and CHEN Tian-Lun Department of Physics, Nankai University, Tianjin 300071, China (Received August 8, 2006) Abstract We introduce a modified small-world network adding new links with nonlinearly preferential connection instead of adding randomly, then we apply Bak Sneppen (BS) evolution model on this network. We study several important structural properties of our network such as the distribution of link-degree, the maximum link-degree, and the length of the shortest path. We further argue several dynamical characteristics of the model such as the important critical value f c, the f 0 avalanche, and the mutating condition, and find that those characteristics show particular behaviors. PACS numbers: 05.65.+b, 45.70.Ht Key words: power-law behavior, small-world network, evolution model, nonlinear preference 1 Introduction In many dynamical models, the underlying networks often have simple topological structure, which makes the simulation easier, such as rings, grids, lattices, etc. However, the real world is very complicated, which has many very complex topological structures. In addition, it was believed that for extensive networks of simple interacting systems, network topology can be as important as the interactions between elements, [1] and much attention was paid to the effects of these complex network topologies. In the previous work of our group, we studied one of the most important components of complex networks 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. [2] In the Watts Strogatz model, [3] smallworld network is built by replacing the original links with random ones. Thus, we can indicate that all nodes are equal in this small-world network model. However, in the real network, not each node is of equality. For instance, when deciding where to link Web page on the Internet, people can choose from a few billion locations. Most of us are familiar with only a tiny fraction of the full Web, and that subset tends to include those more connected sites because they are easier to find. By simply linking to those nodes, people exercise and reinforce a bias toward them. This process of preferential connection occurs elsewhere, including both natural science and social science. [4,5] Thus, the preferential connection is widely existent. To construct a new network, people often determine the probability of the new point added to the former network according to the link-degree of the old points. [6] In our previous paper, [7] we attempted to add mechanism of linear preference to our network. As we know, preferential mechanism can also be nonlinear. A particular nonlinear case was studied, [8] which is power-law preference: g(k) = k y. People studied the network topologies with different non-negative constant y, which only shows the independent nonlinear factor of k 0, k 1, k 2,.... In this paper, we add the exponential preference: g(k) = e βki [9] to our network, for it can be written as the sum of them, so that it may include more complex nonlinear factors. In short, we study a new type of complex network topologies the small-world network with mechanism of exponential preference. We construct the new network by adding new links with mechanism of exponential preference, instead of randomly adding connection in the small-world network. We not only need to know the structure property of the network, but also need a concrete model as the carrier of the network to show the complexity of the network in other respect. In this paper, we emphasize the complexity arisen from the structure of the network as we did in the previous works, so the dynamical behavior of the individual nodes should be as simple as possible. [5] Bak and Sneppen [10] introduced a particularly simple toy model of biological evolution (the BS model). It provides a coarsegrained description of the behavior of interacting species driven by mutation and natural selection. Therefore, we select Bak. [7,11] 2 The Model In the original BS model, [10,12] people consider a dynamical ecosystem of interacting species, which evolve by mutation and natural selection. For simplicity, it is assumed that no species divide into several species and no The project supported by National Natural Science Foundation of China under Grant No. 10675060 and the Doctoral Foundation of the Ministry of Education of China under Grant No. 2002055009 E-mail: zhangyingyue@mail.nankai.edu.cn

138 ZHANG Ying-Yue, YANG Qiu-Ying, and CHEN Tian-Lun Vol. 48 species become extinct. Thus the only effect of evolution is adaptation to the environment. The original BS model is defined and simulated as follows: (i) N species are arranged on a one-dimensional line with periodic boundary conditions. (ii) A random barrier, B i, equally distributed between 0 and 1, is assigned to each species. At each time step, the ecology is updated by (iii) locating the site with the lowest barrier and mutating it by assigning a new random number to that site, and (iv) changing the landscapes of the two neighbors to the right and left, respectively, by assigning new random numbers to those sites, too. The topological structure of original small-world model is a regular ring with L nodes, but with a low density of connections (short paths) between randomly pairs of nodes rewired or added. In this paper, the nonlinearly preferential connection is applied, instead of the principle of randomly adding connection in the small-world network. Based on the new principle, we construct a new network. The way of constructing network is given as follows. In the fundamental structure of small-world network, L connected nodes make up a ring. The link-degree (the number of edges) of every node is two, which is the same as the others. (i) Randomly choose two nodes in the ring and place a connection (short path) between them, then the different link-degrees appear in the network. (ii) We call the nodes that were added with new connection as fixed points, and the others as unfixed points. Randomly choose a node in the unfixed points, then choose a node in the fixed points according to their respective link-degrees, the possibility that we choose the node i is P i (k) = e βki / j (β is a constant). Connect the two chosen nodes. eβkj (iii) Repeat step (ii) until l short paths have been added in the network. Here l is the total number of the new added links. All the above is the construction based on the mechanism of nonlinear preference. This construction allows us to tune the graph between the situation β=0 and β=1. When β=0, new links are added to several different points; when β=1, nearly all new links are added to the same point. Similarly as the former work [7,11] of our group, we also 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 f i of the i-th species. Similarly, in this model, we give each node of 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 node with minimum barrier f i 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 mutating neighbor in the food chain. (iv) Repeat steps (ii) and (iii) respectively. 3 Simulation Results 3.1 Structural Properties of Network In the previous section, we point out that the method of construction in our paper allows us to tune the graph between the situation β = 0 and β = 1. In this section, we quantify the structural properties of these graphs by their maximum node degree k max (β) (Fig. 1), link-degree distribution function P k (β) (Fig. 2), and shortest path length L(β) (Fig. 3). Fig. 1 The maximum link-degree k max of network with different β. Fig. 2 Distribution of link-degree of network with β = 0, 0.285, 1 (When β = 0.285, the maximum link-degree of network is 51).

No. 1 Nonlinear Dynamical Behavior in BS Evolution Model Based on Small-World Network Added with 139 function increases fast with β in the corresponding section (comparing Fig. 1 to Fig. 4). Fig. 3 The shortest path length L of network with different β. 3.1.1 Maximum Link-Degree and Distribution of Link-Degree Not all nodes in a network have the same number of edge (same node degree). The spread in the node degrees is characterized by a distribution function P (k), which gives the probability that a randomly selected node has exactly k edges. The maximum of the node degrees is k max. To reduce the effect of fluctuation on calculated results, for every β, k max is averaged over 200 independent runs for each network constructed with the same β. The maximum degree k max increases with the increase of the parameter β (Fig. 1). The distribution of linkdegree becomes to concentrate on the small link-degree, and the middle-value link-degree disappears gradually (Fig. 2). We divide the increasing progress of k max into three sections: the slow section, the fast section, and the slow section. The first slow section corresponds that the function g(k) = e βk increases slowly with β in the corresponding section; the fast section corresponds that the Fig. 4 This graph corresponds to the part of data in Table 1 (β is between 0 and 0.31). Thus, the increasing of maximum degree k max shows the nonlinear effect of the function g(k) = e βk. Table 1 shows the values of e β 15 with different β. In fact, while constructing a network, there are different link-degrees. The situation with link-degree k = 15 roughly shows the changing rate of exponential function g(k) = e βk with increase of β. Figure 4 only gives the part that β is between 0 and 0.31, for the reason that the value of e β 15 with larger β is too large to show the difference when β is between 0 and 0.31 (see Table 1). The larger the parameter β, the nearer the maximum degree to 100. Thus, when β increases to a certain value, the maximum degree begins to increase slowly as shown in the last slow section of Fig. 2. Table 1 β 0 0.1 0.13 0.16 0.205 0.24 e β 15 1 4.4817 7.0287 11.023 21.65 36.598 0.263 0.285 0.31 0.345 0.4 0.5 0.6 51.676 71.88 104.58 176.8 403.43 1808 8103.1 0.7 0.8 0.9 1 36 316 1.63 10 5 7.29 10 5 3.27 10 6 3.1.2 Length of Shortest Path A fundamental concept in graph theory is the geodesic, or shortest path of vertices. The distance between two vertices of a network is the length of the shortest path between them. The average length of the shortest path l is a natural characteristic distance existing in a network, which allows us to evaluate the linear size of a network. Figure 3 shows that the shortest path l decreases with the increase of β. The value of l varies from 11 to 15, which shows the small effect of parameter β on the average length of the shortest path l. Namely, no matter what the parameter β is, the networks with the exponential preference almost have the same size. 3.2 Dynamical Behavior of BS Model In BS model, the all-time maximum G(t) of the minimum fitness increases with time. G(t) is an envelope function that tracks the increasing peaks in f min, and it is called the gap. [13] When the gap jumps to a larger value, all random numbers are uniformly distributed above this gap. [13,14] G(t) is defined definitely as that, G(0) = f min (0), and the current G(t) is the maximum of all the minimum random numbers, f min (t ), for all 0 t t.

140 ZHANG Ying-Yue, YANG Qiu-Ying, and CHEN Tian-Lun Vol. 48 After millions of updates, the gap G(t) increases up to a critical value f c, so f c has the definition: f c = lim t G(t), where f c is the threshold value, for all the minimum barrier f min is less than f c. There is a widely discussed topic: f 0 avalanche, in which people find the power-law behavior. In this paper, we define the avalanche like that: randomly choose a number f 0, 0 < f 0 < f c ; if f min (t) > f 0, and f min (t + 1) < f 0, then an avalanche begins at t + 1 ; to the step t + t 1, if we have f min (t+t 2 ) < f 0 for all the t+1 t+t 2 t+t 1, the avalanche will continue; to the step t + T, if f min (t + T ) > f 0 for the first time after step t, the avalanche stops. The duration time T of the avalanche is defined as the size of the avalanche, and T is the average size. 3.2.1 The Critical Value f c in G(t) Evolution It is known that f c is an important characteristic property of BS model. People studied the relationship: T f 0 and T L in Ref. [15] (L is number of points in a network), and they got two conclusions about f c : (i) There is a transition between two essentially different regimes, for T (1 f 0 ), and the transition happens when f 0 = f c ; (ii) Only at f c does full power-law behavior occur, for T L. Thus, f c is very important for BS model. Figure 5 shows that the value of f c decreases with the increasing of the value of β. The changing rate of f c similarly corresponds to that of k max (comparing Fig. 5 to Fig. 1). Thus, not only the topology of networks shows the nonlinear effect of the function f(k) = e βk, but also the dynamical parameter f c in BS model shows that effect. The dynamical parameter f c mainly depends on the maximum degree k max, for the relationship between f c and k max is approximately linear: f c k max (Fig. 6). The critical value f c with k max of different net- Fig. 6 works. Fig. 5 The critical value f c with different β. Different values of β lead to different network topologies. BS model on each network has corresponding f c. 3.2.2 f 0 Avalanche The f 0 avalanche distribution of BS model is powerlaw: P (T ) T τ. There is another power-law, it is defined the characteristic spatial size R of an avalanche cluster as the mean square root deviation of the set of active sites in the avalanche from their center of mass. In this definition each site is counted with the weight given by the number of times it was visited by the avalanche. The avalanche mass dimension D is defined by the scaling relation T R R D, connecting the avalanche size T R (temporal duration) to its spatial extent R. In this work, there is still a good power-law behavior when β = 0. With the increase of β, the power-law behavior is broken partly. When the value of β reaches to 1, the power-law behavior disappears completely (Fig. 7). Fig. 7 (a) Distribution of the number of the size of f 0 avalanche of models with different β. (b) The behavior of the different models between T R and R.

No. 1 Nonlinear Dynamical Behavior in BS Evolution Model Based on Small-World Network Added with 141 3.2.3 Mutating Condition As we discussed in the previous section, the maximum link-degree k max is an important structural properties in our network, which is based on the exponential preference. Thus, it is necessary to further argue the effect of the maximum link-degree node in the whole network. Here, we define d as the distance between the maximum link-degree node and the other nodes, which is the shortest path from the other nodes to the maximum link-degree node, and we call the mutation times of each nodes as MT for short. The nodes mainly distribute around a peak, which moves from right to left with the increase of parameter β (Fig. 8). The longer the distance to the maximum link-degree node, the smaller the average mutation probability. The mutation probability is zero when the node is far enough away from the maximum link-degree node (Fig. 9). Thus, the maximum link-degree node plays an important role in our network. Here, we study the mutation case after the transient period. Fig. 8 Distribution of points with the same d (the distance between the maximum link-degree point and the other points) β = 0, 0.205, 0.285, 1. Fig. 9 The relationship of d (the distance between the maximum link-degree point and the other points) and MT (the average mutation times of each points) β = 0, 0.205, 0.285, 1. Graph (b) is interception of graph (a) on 0 1000 of y-axis, so that the difference can show clearly. 4 Conclusion and Discussion In this paper, we study a modified BS evolution model based on the small-world network with exponential preference, which is one particular case of nonlinear preference. Then we have several conclusions. (i) The nonlinear effect of the function f(k) = e βk appears not only in the structural properties the increasing of maximum degree k max, but also in the important critical value f c. (ii) Dynamical behavior f 0 avalanche of our model is different from that of the original BS model. With the increase of the value of β, the power-law behavior of f 0 avalanche of our model is broken gradually. (iii) After we study the dynamical effect of the important structural property the maximum link-degree k max, we conclude that the value of d around which the nodes mainly distribute decreases with the increase of β. The further away from the maximum link-degree node, the smaller the average mutation probability. The node will never mutate, if the node is far enough.

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