Commun. Theor. Phys. (Beijing, China) 45 (2006) pp. 950 954 c International Academic Publishers Vol. 45, No. 5, May 15, 2006 Small World Properties Generated by a New Algorithm Under Same Degree of All Nodes LI Yong, FANG Jin-Qing, LIU Qiang, and LIANG Yong China Institute of Atomic Energy, P.O. Box 275-81, Beijing 102413, China (Received August 30, 2005) Abstract Based on the model of the same degree of all nodes we proposed before, a new algorithm, the so-called spread all over vertices (SAV) algorithm, is proposed for generating small-world properties from a regular ring lattices. During randomly rewiring connections the SAV is used to keep the unchanged number of links. Comparing the SAV algorithm with the Watts Strogatz model and the spread all over boundaries algorithm, three methods can have the same topological properties of the small world networks. These results offer diverse formation of small world networks. It is helpful to the research of some applications for dynamics of mutual oscillator inside nodes and interacting automata associated with networks. PACS numbers: 89.75.-k, 89.75.Da, 89.75.Fb Key words: small world network, the same degree of all nodes in the network, spread all over vertices algorithm, average shortest path length, average clustering coefficient 1 Introduction As we all know, the pioneering work of Watts and Strogatz (WS) leads to an avalanche of research on the properties of small-world networks. [1] A much-studied variant of the WS model was proposed by Newman and Watts, [2,3] in which edges are added between randomly chosen pairs of sites, but no edges are removed from the regular lattice. In 1999, Kasturirangan proposed an alternative model to the WS small-world network. [4] The model starts also with one ring lattice, and then a number of extra nodes are added in the middle of the lattices which are connected to a large number of sites chosen randomly on the main lattice. In fact, even in the case where only one extra node is added, the model shows the small-world effect if that node is sufficiently highly connected, which has been solved exactly by Dorogovtsev and Mendes. [5] To investigate the smallworld effect further, Kleinberg has presented a generalization of the WS model which is based on a two-dimensional square lattice. [6] Besides, in order to study other mechanisms for forming small-world networks, Ozik, Hunt and Ott introduced a simple evolution model of growing smallworld networks, in which all connections are made locally to geographically nearby sites. [7] Generally, small-world networks are characterized by three main properties. First, their average path length (APL), which does not increase linearly with the system size, but grows logarithmically with the number of nodes. Second, the average node degree of the network is small. Third, the network has a high average clustering compared to an Erdos Renyi (ER) random network [8,9] of equal size and average node degree. The APL can be used to estimate the average transmission delay. One can always reduce the APL by adding more edges, but this could bring in economical and technical pressures. So the average edge number of nodes must be controlled within an acceptable range. Moreover, the information localizing principle makes the networks with a larger clustering coefficient welcome. In this paper, we focus on the small-world network topology generated by our new algorithm under the same degree of all nodes in the networks. Simulations show that it has the topological properties of small-world networks and develops the style of small-world. It may be helpful to the research of some applications for the dynamics inside nodes of mutual oscillator and of the interacting automata. [10] 2 A New Algorithm for Same Number of Node Degrees To interpolate between regular and random networks, we consider the following random rewiring procedure, as shown in Fig. 1. Starting from a regular ring lattice with N nodes and k edges per vertex, we rewire some edges at random with probability 1/N under our model condition. The number of changed edges is relying on the probability p. This construction allows us to tone the network between regularity ( p = 0) and disorder ( p = 1), and thereby to probe the intermediate region 0 < p < 1 as the WS model. We have proposed an algorithm, the so-called spread all over boundaries (SAB) algorithm, for our model on the same degree of all nodes. [11] In this paper we offer a new algorithm, the so-called spread all over vertices (SAV) algorithm, to rewire connections under unchanged node degrees. The project supported by the Key Projects of National Natural Science Foundation of China under Grant No 70431002, and National Natural Science Foundation of China under Grant Nos. 70371068 and 10247005 E-mail: fangjinqing@gmail.com
No. 5 Small World Properties Generated by a New Algorithm Under Same Degree of All Nodes 951 Fig. 1 Illustration of generating small world properties by a new algorithm scheme in our model based on the same degree of all nodes, which interpolate between a regular ring lattice and a random network, starting with a ring of N nodes, each connected to its k nearest neighbors by undirected nodes. For clarity, here N = 20 and k = 4 but much larger N and k are used in the rest of this paper. We choose a node and the edge that connects it to its nearest neighbor in a clockwise sense. Fig. 2 Illustration of SAV algorithm scheme in our model based on the same degree of all nodes. Solid lines denote original ruled edges. Dashed lines denote the removing edges between the selected nodes and dash-dot lines denote new edges according to the SAV algorithm. Let us introduce the SAV algorithm for constructing small-world network, as shown in Fig. 2. We start with a ring of N nodes, each connected to its k nearest neighbors by undirected edges. For clarity, N = 10 and k = 4 are taken as the schematic examples here. At each time step we perform one of the following three operations: (i) We randomly select a node i. From the nodes connected with the node i, we randomly select a node j and remove the edge connected to the node i. (ii) From the nodes unconnected with the node i, we randomly select a node u and connect a new edge to the node i. (iii) From the nodes connected with the node u, we randomly select a node v and remove the edge connected to the node u. (iv) For the node v we connect a new edge to the node j. At each increment of time, four nodes are involved and two original ruled edges are removed, which have never been selected before, two new edges are connected, which will never be changed later. The growth process is repeated until the network grows to the desired percentage. Note that in every step self-connections and duplicate edges are excluded. The above network develops by the successive change based on the above procedure. According to different percentage, which is the proportion of successfully changed node number to the total node number, simulations show that it can generate the small-world effect. The approach we propose is similar to the well-known WS model, but the present method is quite different as it leaves the number of connections k of each node unchanged, while the WS algorithm gives rise to a Poisson distribution of connectivity. Leaving k unchanged makes us able to study the effect of rewiring nodes on topological property of these networks. According to the WS model, there is a broad interval of p over which average path length L(p) is close to L(1) yet clustering coefficient C(p) C(1), as given in Fig. 3. In the regime the curver originates in a rapid drop of L(p) for small values of p, while C(p) stays almost unchanged, resulting in networks that are clustered but have a small characteristic path length. This coexistence of small L and large C is the character of small-world networks.
952 LI Yong, FANG Jin-Qing, LIU Qiang, and LIANG Yong Vol. 45 Fig. 3 Characteristic average path length L( p) ( ) and clustering coefficient C( p)(0) for our model. The data are normalized by the L(0) and C(0) for a regular lattice. A logarithmic horizontal scale resolves the rapid drop in L( p), corresponding to the onset of the small-world phenomenon. During this drop C( p) remains almost constant, indicating that the transition to a small world is almost undetectable at the local level. Here N = 1000 and k = 4 as the schematic examples. 3 Relevant Characteristics of Small-World Network 3.1 Degree Distribution As is well known, the degree distribution of nodes for the WS model has the Poisson distribution due to the random rewiring procedure of nodes. However, the degree distribution of nodes is keeping a constant in our model. So it is quite different between them. How about the other topological properties? These are focused on in the following subsections. 3.2 Average Path Length (APL) So far the most attention has been focused on the APL of the small-world model. We denote this quantity as L. In our model the network is undirection and without weight. So L( p) is the APL about percentage p: 1 n L(p) = d(p) ij, N(N + 1)/2 where d( p) ij is the shortest distance from node i to node j. The origin of the rapid drop in L is the appearance of shortcuts between nodes. Every shortcut, created at random, is likely to connect widely separated parts of the network, and thus has a significant impact on the characteristic path length of the entire network. Even a relatively low fraction of shortcuts are sufficient to drastically decrease the APL, yet locally the network remains highly ordered. ij Fig. 4 Characteristic L( p)/l(0) verses p for N = 1000 and k = 4. Here L(0) is the APL in a regular lattice. Fig. 5 APL versus N for different p logarithmically. We adopt Floyd algorithm to calculate the APL. Figure 4 shows the relation between L( p)/l(0) and p. From Fig. 4 we can observe that APL rapidly declines when p < 0.1 and it slowly declines when 0.1 < p < 1. The origin of
No. 5 Small World Properties Generated by a New Algorithm Under Same Degree of All Nodes 953 rapid drop in L is the appearance of shortcuts between nodes. Every shortcut, created at random, is likely to connect widely separated parts of the network, and thus has a significant impact on the characteristic path length of the entire network. Even a relatively low fraction of shortcuts are sufficient to drastically decrease the APL, yet locally the network remains highly ordered. The characteristic is same between our model and the WS model. Figure 5 gives the relation between APL and logarithmic N according to the SAV algorithm. It is seen that the APL is increased as notes number N is increased for different values of p. When p is small, the APL curve is waved slightly and when p is getting large the curve becomes smooth. The reason for this with known N is that small p can cause L(p) to change acutely as choosing nodes twice randomly. The characteristic of randomness is that its numerical value is ranged around its expectation and is various in each calculation. When the curve is converged to single logarithmic reference frame, the curve becomes linear. So the APL is scaled logarithmically with N as shown in Fig. 5. This is similar to the WS model, in which it is believed that L(0) N/2K and L(1) ln(n)/ ln(k). [12] The model makes shortcut step by step as p is changed and the total number of shortcuts is pnk/2. We can imagine that the number of shortcuts is increased with k increasing, so the L(p)/L(0) will become less. Figure 6 shows the L(p)/L(0) verses p for different k and fixed N = 1000. The greater the p, the less the L(p)/L(0); the less the k, the greater the L(p)/L(0). Figure 7 shows the comparison of our model with the WS model about L(p)/L(0). In our model we take two algorithms to generate the small world network. When 0 < p < 0.1 there exists cross and when p > 0.1 they tend to be consistent. For two models there is distinction when p is little. They all start with regular lattice, but the meaning of p is different. Our p is a percentage and it is determinant, whereas p is probability in the WS model. Fig. 6 L( p)/l(0) verses p for different k and N = 1000. Fig. 7 L( p)/l(0) verses p, comparison of SAB and SAV algorithms with WS model for N = 1000 and k = 4. 3.3 Clustering Coefficient In the same degree k of all nodes for our model, the average clustering coefficient is C( p) C(0)(1 p) 3. Figure 8 shows the average clustering coefficient C, comparing SAB and SAV algorithms with WS model. In a regular lattice ( p = 0) the clustering coefficient does not depend on the size of the lattice but only on its topology. As the changing of the network is randomized, the clustering coefficient remains close to C(0) up to relatively large values of p. Fig. 8 Clustering coefficient C verses P, comparison of SAB and SAV algorithms with the WS model for N = 1000 and k = 4.
954 LI Yong, FANG Jin-Qing, LIU Qiang, and LIANG Yong Vol. 45 4 Conclusion and Discussion In conclusion, we have presented two algorithms (SAB and SAV) for our model based on the same degree of all nodes. And we made comparison of our model with the WS model, which shows that they have the same topological properties of small-world network, small average path length and big clustering coefficient. However, one of advantages for our model is that all degrees of nodes are a constant but can be selected according to practical need. Two algorithms offer good examples of diverse formation of small world networks since their distribution of node degrees are quite different. We believe that our model may help engineers in topology-designing and performance analysis of small world network. It is also useful to the research of some applications for the mutual oscillator kinetics inside nodes and boolean interacting automata related to networks. References [1] D.J. Watts and S.H. Strogatz, Nature (London) 393 (1998) 440. [2] M.E.J. Newman and D.J. Watts, Phys. Lett. A 263 (1999) 341. [3] M.E.J. Newman and D.J. Watts, Phys. Rev. E 60 (1999) 7332. [4] R. Kasturirangan cond-mat/9904055 (1999). [5] S.N. Dorogovtsev and J.F.F. Mendes, Europhys. Lett. 50 (2000) 1. [6] J. Kleinberg, Nature (London) 406 (2000) 845. [7] J. Ozik, B.R. Hunt, and E. Ott, Phys. Rev. E 69 (2004) 026108. [8] P. Erdös and A. R enyi, Publ. Math. 6 (1959) 290. [9] P. Erdös and A. R enyi, Publ. Math. Ins. Hung. Acad. Sci. 5 (1960) 17. [10] R. Serra, M. Villani, and L. Agostini, Complex Systems 15 (2005) 2004. [11] Qing Liu, Jin-Qing Fang, Yong Li, and Yong Liang, Complex System and Complexity Science 2 (2005) 13. [12] R. Albert and A.L. Barabasi, Phys. Rev. Mod. 74 (2002) 47.