電氣工程及其自動化專業畢業設計外文翻譯(編輯修改稿)

　

【文章內容簡介】 o. This is because the rewiring of links creates longrange edges (shortcuts) that 1 connects otherwise distant nodes. The effect of the rewiring procedure is highly nonlinear on L, and not only affects the nearest neighbors structure, but it also opens new shortest paths to the nextnearest neighbors and so on. Conversely, an edge redirected from a clustered neighborhood to another node has,at most, a linear effect on C . That is, the transition from a linear to a logarithmic behavior in L(p) is faster than the one associated with the clustering coef?cient C(p). This leads to the appearance of a region of small (but nonzero) values of p, where one has both small path lengths and high clustering. The change in L(p) soon stimulated numerical and analytical work [53,128,129], aimed at inspecting whether the transition to the smallworld regime takes place at a ?nite value of p, or if there is a crossover phenomenon at any ?nite value of N with the transition occurring at p = 0. This latter scenario turned out to be the case. To see this, we follow the arguments by Barrat and Weigt [53], and Newman and Watts [128]. We assume that p is kept ?xed and we inspect the dependency of L(N, p). For small system sizes, the number of shortcuts is less than 1 on average, and the scaling of L(N, p) is linear with the system size. However, for larger values of N , the average number of shortcutseventually bees greater than one and L(N, p) starts scaling as log(N ). Similar to the correlation length behavior in conventional statistical physics, at some intermediate system value N = L, where the transition occurs, we expectL ～ p? . Additionally, close to the transition point, L(N, p) should obey the ?nitesize scaling relation [53,128]: ( , ) ~ ( )L N p p f N p??? where f (x) is a universal scaling function that obeys to: f (x) ～ x if x 1 and f (x) ～ ln x if x?1. Let us suppose now that 1 and assume 1. From Eq. () it follows that 1 11( , ) ~ ( ) ~ l n ( )L N N N f N N N? ? ? ?? ? ? ? ?? ? ? On the other hand, the average number of rewired connections is mN1?1/ , which vanishes as N gro ws. From here, Barrat and Weigt deduced that 1, a result also supported by their numerical simu lations. Newman and Watts corroborated this result by a renormalization group analysis and showed that in fact = 1 [128]. In summary, the model undergoes a continuous phase transition as the density of shortcuts tends to zero with a characteristic length diverging as p?1. The exact form of the scaling function f (x) has been obtained in Refs. [129,131]. Barrat and Weigt have obtained a simple form ula that ?ts well the dependency of C(p) observed in the numerical simulations of the WS model[53]. The formula is based on the fact that for p = 0, C(0) = 3(m ? 1)/2(2m ? 1). Then, because of the fact that with probability (1 ? p) edges are not rewired, two neighbors that were linked together at p = 0 will remain connected with probability (1 ? p)3up to corrections of order 1/N . From here, we get: 33 ( 1 )( ) ~ ( ) (1 )2 ( 2 1 )mC p C p pm ???? where C(p)? is rede?ned as the ratio between the average number of edges between the neighbors of a vertex and theaverage number of possible links between the neighbors of a vertex. As for the degree distribution, when p = 0 it is a delta function positioned at 2m, while for p = 1 it is similar to thatof an ERwork. For intermediate p, the degree distribution is given by [53]: for km, and is equal to zero for k smaller than , Farkas et al. have studied numerically the properties of the spectra in the SW model, showing that the distribution continuously approaches 2 that of a random graph, as p increases [72], and Monasson has studied the spectral properties of the Laplacian operator both numerically and analytically [127].. Static scalefree works,The large amount of works on the characterization of the topological properties of real works has motivated the need to construct graphs with power law degree distributions. Graphs with a powerlaw degree distribution can besimply obtained as a special case of the random graphs with a given degree distribution discussed in Section . We denote such graphs as static scalefree to distinguish them from models of evolving graphs that will be discussed in Section . Aiello et al. have studied a model that involves only two parameters, and , and that assigns uniform probability egree k given by N (k) = e k? [132]. We denote this model as G, since both the total number of nodes N and edges K are fully determined, once and are given. Notice that is the logarithm of the number of nodes with k = 1, while e/is the maximum degree of the graph. Aiello et al. have shown that the Molloy and Reed criterium predicts that G , has a giant ponent when c= . . . , and have proved that the second largest ponents are of size O(log N ) for 2 cand O(1) for 1 2, while the graph is almost surely connected for 0 1. By using Eq. (), Newman has calculated the clustering coef?cient for powerlaw degree distributions ?nding: C～N(3 ?7)/(?1). Thus, C tends to zero in large graphs with 7/3, while for 7/3, C increases with the system size, due to the fact that there can be, in average, more than one edge between two nodes sharing a mon neighbor [4]. Chung and Lu have studied the average distance in random graphs with a powerlaw degree distribution, proving that L scales as log N if 3. Conversely, if 2 3, L is O(log log N ), while the diameter is O(log N ) [125]. Similar results were obtained by Cohen and Havlin [133]. More recently, Chung et al. have shown that the spectrum of the adjacency matrix of random power law graphs follows a power law distribution, while the eigenvalues of the normalized La