復(fù)雜網(wǎng)絡(luò)上的損傷擴(kuò)散研究

1、復(fù)雜網(wǎng)絡(luò)上的損傷擴(kuò)散研究損傷擴(kuò)散技術(shù)動力學(xué)規(guī)則損傷擴(kuò)散技術(shù)的演化模式二維Ising點(diǎn)陣上的損傷擴(kuò)散復(fù)雜網(wǎng)絡(luò)Complex networks三角形小世界網(wǎng)絡(luò)的構(gòu)成 三角形小世界網(wǎng)絡(luò)上的Damage spreading 內(nèi)容提要Damage SpreadingRandom growth processesThe shape of a snowflake, the roughness of a crack surface, corroding process in iron, virus spreading, economic and social phenomena Growth models:

2、percolation, the damage spreading (DS) The damage spreading (DS)Simulation of the time evolution of perturbation spreading throughout a cooperative system 損 傷 擴(kuò) 散 方 法Damage spreading(DS)方法通過比較處于相同熱噪聲的兩個系統(tǒng)（A、B，開始時存在微小差別）的時間演化的差別來研究影響系統(tǒng)演化的關(guān)鍵因素。DS技術(shù)的核心思想是通過兩個系統(tǒng)上節(jié)點(diǎn)或單元（cell）的差別（或節(jié)點(diǎn)或單元上自旋的差別）來定義損傷，它的主要測度是

3、系統(tǒng)的損傷密度，即Hamming距離（一般為平均值）。TheoryThe Hamiltonian : The Hamming distance : Glauber-Kawasaki dynamicsDamage Spreading : what have we learnt?1.Td and Tc :the heat-bath dynamics Td coincides with Tc whereas for Glauber and Metropolis dynamics Td is near but smaller than Tc 2. Elements considered:the int

4、eractions (ferro, antiferro , spin glass ,etc.), the Monte Carlo rules (heat bath , Glauber , Metropolis , etc.), the lattice geometry ( square , triangle , cubic , etc.), the symmetry of the spin variables and the external conditions (e.g., magnetic field ). 3. DS is less sensitive to statistical f

5、luctuations 4.DS is sensitive to the ways of updating 5. DS is sensitive to dynamics Damage Spreading: problems unsolved1.Td and Tc:explanation2.How to determine Td (Tc:Binders cumulants)3. It appears that DS is uniquely defined only if one specifies the Hamiltonian and the dynamic rule; there are a

6、lso new and interesting phenomena that cannot be measured by Hamming distance . Hamming distance is too much averaged4. The networks studied are far away from the realistic social networks. 規(guī)則網(wǎng)絡(luò)（點(diǎn)陣）上的損傷擴(kuò)散triangular square 3-6 hexagonal Bethe Some two-dimensional structures Trivalent structures (hexa

7、gonal Bethe) are of great importance due to their topological stability and general occurrence in nature Hierarchical trivalent structures generated by the star-triangle transformation: (a) the pure hexagonal lattice; (b) the 3-12 lattice generated by the first-order star-triangle transformation; (c

8、) the primary cell of the 3-12 lattice; (d) the primary cell of the 3-6-24 lattice; (e) the primary cell of the 3-6-12-48 lattice (a)(b)(c)(d)(e) 3-12 4-8 4-10 從網(wǎng)絡(luò)的角度The topological hexagonal lattice(b)The star-triangle transformation (c)The 3-9 lattice (d)The 3-12 lattice(a)(b)(c)(d)The topological

9、 hexagonal latticeThe 4-8 lattice (a)(b)Soap froth and Voronoi structuresK.Y.Szeto,Xiujun Fu,W.Y.Tam,Phys.Rev.Lett.88,138302(2002).Damage Spreading on two-dimensional trivalent structures and Complex NetworksG-dynamicsThe critical temperature of damage spreading with 1-p, in which N=25,25,15,15 for

10、the hexagonal,the3-12,the3-6-24 and the3-6-12-48 lattice respectively. Left:The Curie temperatures of various structures versus 1-p Right:Binders cumulants as the function of temperature for the 3-12 lattice Left:The transition temperature of the 4-8 lattice as a function of 1-p for various f: only

11、nearest-neighbor interactions consideredRight:The transition temperature of the 4-8 lattice as a function of 1-p for various f: up to next nearest-neighbor interactions The neighbors of the 8-gon (left) and 4-gon (right): solid : nearest ; dot: next-nearest f=F2/F1 復(fù)雜網(wǎng)絡(luò)上的損傷擴(kuò)散隨機(jī)圖random graph隨機(jī)圖由N個節(jié)點(diǎn)構(gòu)

12、成，而每個節(jié)點(diǎn)對應(yīng)于一個社會個體。設(shè)節(jié)點(diǎn)的度滿足冪率分布，分布指數(shù)為 ，分布范圍從dmin 到 dmax并滿足條件， 這里dmin 、dmax和ds都是正的常數(shù)，dmin dmaxShortcutsAirplane route network:a scale-free networkExp. vs. Scale-FreePoisson distributionExponential NetworkPower-law distributionScale-free NetworkTwo important observations(1) The number of nodes (N) is NOT

13、 fixed. Networks continuously expand by the addition of new nodesExamples: WWW : addition of new documents Citation : publication of new papers(2) The attachment is NOT uniform.A node is linked with higher probability to a node that already has a large number of links.Examples : WWW : new documents

14、link to well known sites (CNN, YAHOO, NewYork Times, etc) Citation : well cited papers are more likely to be cited againOrigins SFPerspectives: Weighted networksScientific collaborationsInternetEmailsAirports networkFinance, economic networks.= are weighted networks !Approaches for the studying of c

15、omplex networksGraph theoryComputer simulationThe model consists of N individuals and is parameterized by two probably distributions: (1) degree distribution: each individual of the N numbers has degree between dmin and dmax following a power law distribution with exponent . Here is required with ds

16、, dmin and dmax are all constants which satisfy dmin dmaxds. (2) acquaintance probability x: individuals are connected only if they are acquainted (with probability x), this parameter is one of the main characteristics of highly clustered networks introduced by Newman DS on random graphThe averaged damage as the function of temperatures for different x and p(a)x=0.1(b)x=0.5(c)x=1.0 The averaged damage as the function of temperatures for different x and p(a)x=0.1(b)x=0.5(c)x=1.0(Right)The averaged damage spreading as the funct