Chapter7 Effects of the Smallest Polygons on the Damage Spreading

Chapter 7 Effects of the Smallest Polygons on the Damage SpreadingWe have studied the damage spreading of the Ising model on two sets of trivalentstructures with competing Glauber and Kawasaki dynamics. Further understanding ofthe results is needed. We find that the damage transition temperature T d increases asthe number of the smallest polygons increases at the pure Glauber dynamicscase .This is suggested as a general feature for the hierarchical lattices. A heuristicargument is provided to explain the numerical results from the point of view ofnetwork.7.1 Effects of the Smallest Polygons on the DS: TrianglesWe have discussed the damage spreading for the set of 6,3-12,36-24,3-6-12-48lattices in the above chapters, and we have the result T d (6)<T d (12)<T d (24)<T d (48) forthe pure Glauber dynamics ,where T d (6),T d (12),T d (24) and T d (48) stand for the criticaltemperatures corresponding to the pure hexagonal, the 3-12 , the 3-6-24 and the3-6-12-48 lattice.. For a better understanding of this result, one should pay moreattention to the topology of the structures. The transition from frozen to chaotic statein DS is influenced by two factors: temperature and the competition of the dynamics.In general G-dynamics favors a low energy state while K-dynamics favors a highenergy state. When G-dynamics is dominant (p close to 1), the transition isdetermined mainly by temperature. In this case the transition temperature is higher.But as p decreases, the contribution of K-dynamics plays a more important role,resulting in a lower transition temperature.The relation T d (6)<T d (12)<T d (24)<T d (48) (for p =1) clearly indicates that themore complicated the lattice is, the larger the T d of the damage spreading becomesunder G-dynamics. To understand this trend, the key point may be the numbers oftriangles in the lattices. Note that the number of the triangles in the hexagonal, the3-12, the 3-6-24 and the 3-6-12-48 lattices are respectively 0, 22N , 62N and 182N ,here N is the size of the lattice (the number of the main polygons along one direction).Although the number of the triangles are different among the 3-12, the 3-6-24 and the3-6-12-48 lattices, the concentration of triangles in these lattices are the same (2/3).Since we have neglected the area influence on the DS in the calculation, we considerthe triangle with the same level of importance as other polygons. But the triangle hasonly three edges (the contiguity number f =3) and when it is damaged, less cells willbe affected, so the DS is difficult compared with the 6-gon (f =6), the 12-gon (f =12),the 24-gon (f =24), and the 48-gon (f =48). Thus, for the lattices with larger number oftriangles, bigger thermal noise is needed to reach the equilibrium state (the so-calledchaotic), resulting in higher T d . In pure G-dynamics, the damage is most readilyhealed in the triangle sites. However, in K-dynamics, damaged sites can be healedonly in pairs while they diffuse and also create further damage. The K-dynamicsinvolves two sites while the G-dynamics involves only single site. From Fig.1 we cansee that the roles of the triangles on these two dynamics are very different. The role ofT d1-p the triangles varies with changing p and the number of the triangles determines thetrend of the T d vs (1-p ) curves. From the insert of Fig.1, we can see that three zonesexist for the 3-12, the 3-6-24 and the 3-6-12-48 lattices. In zone A, the triangles hinderthe damage spreading, contrary to the case of zone C, where the triangles enhance thedamage spreading. Zone B is the transition region. It is the K-dynamics that causesthe above change.In order to show more quantitatively the effects of the number of thetriangles on the trend of the T d vs (1-p) curves, we design an artificial trivalent cellularnetwork where the number of the triangles can be changed gradually. Since only thetopology of the structures is emphasized, we consider the simplest star-triangletransformation in a hexagonal lattice. We do the transformation on the vertices of thetopological hexagonal lattice with the dimension of N N (Fig.7.1(a),(b)). Thetransformation is made at the vertices at random. But if we do the transformationregularly at each of the other row or at each row ,we can get the 3-9 lattice (Fig.7.1(c))and the 3-12 lattice (Fig.7.1(d)).We notice that when the number N i of trianglesinserted at vertices increases, the trend of the T d vs (1-p) curves does change. Whenthe number of triangles approaches that of the 3-9 lattice (Fig.7.1(c)), T d decreaseslinearly with 1-p . When the number of triangles is further increased, we see similartrend of T d as in the 3-12 lattice (Fig.7.1(d)). These are clearly shown in Fig.7.2.Notice that we generate N i random points for inserting the triangles. But the realnumber of triangles may be less than N i since the randomly generated points may becoincided. For example, we want to generate 10 triangles but we may get only 9 atlast (see Fig.7.1(b)). On the other hand, the real number of the triangles varies amongconfigurations and our results are averaged for 100 configurations.(a) (b) (c) (d)Fig.7.1 (a) The topological hexagonal lattice; (b) The star-triangle transformation (N i =10) made at the vertices randomly ;(c) The 3-9 lattice ; (d) The 3-12 latticeT d1-pFig.7.2 The effect of the numbers of the triangles on the trend of the T d vs (1-p ) curves(N =20Finally, when Kawasaki is dominant, damage always spreads, resulting in no transition. This agrees with the conclusion stated in last chapter, when K-dynamics is dominant in an Ising model. Thus, in the case of pure K-dynamics, we define T d =0. This agrees with our simulation result in Fig.6.1, where T d of all of the structures approaches to zero and no transition occurs, when p approaches 0.7.2 Effects of the Smallest Polygons on the DS: RectanglesThe DS dependence on the topology of the structures was investigated above. One can demonstrate that by considering two kinds of trivalent structures. One kind of the structures are the random lattices such as soap froth and V oronoi structures and the other kind of the structures are a set of hierarchical regular lattices-the 3-12, the 3-6-24 and the 3-6-12-48 lattices which are generated from the hexagonal lattice by the star-triangle transformation. It was shown that for the hierarchical lattices the smallest polygons (here we refer to the smallest polygon as the polygon in which the number of sides is the smallest) in the lattice play a very great role-the transition temperature T d increases with the increasing of the number of the smallest polygons N s when the system is under the pure Glauber (G-) dynamics and T d decreases fast as N s increases when Kawasaki (K-) dynamics becomes important. In this paper we study another set of trivalent structures- the 4-6-7, the 4-8 and the 4-10 lattices generated from the topological hexagonal lattice(see chapter 2 for figures) to see if this phenomenon is a general feature for the hierarchical lattices. The dynamics we used here is the competing Glauber and Kawasaki (G-K) dynamics, which has been proved of great importance in describing the time evolution of many dynamical systems.We do the investigations on these hierarchical trivalent structures by using Monte Carlo method. We put the spin on the center of each polygon of the pattern thus form the Ising model .Fig.7.3 The phase transition temperatures of the various trivalent structures as thefunction of 1-pThe results of the phase transition temperatures for our various trivalent structures as the function of 1-p are shown in Fig.7.3. Here, the phase transition temperatures are determined in the same way as before.For the pure G-dynamics case(p=1),we have T d(4-6-7)<T d(4-8)<T d(4-10), here T d(4-6-7),T d(4-8) and T d(4-10) stand for the DS transition temperatures corresponding to the 4-6-7, the 4-8, and the 4-10 lattice. This phenomenon is similar to the DS on the set of 3-12,3-6-24,3-6-12-48 lattices. We can understand this trend by considering the effects of the polygons with the smallest side number, the rectangles here. Note that the number of the rectangles in corresponding the 4-6-7, the 4-8,the 4-10 lattice is N2/4, N2 and 2N2, respectively, here N is the size of the lattice (the number of the main polygons along one direction, see Fig.2.18(a)). Since we have neglected the area influence on the DS in the calculation, we give the rectangles the same level of importance as other polygons. Now when a rectangle is damaged, at most four neighboring cells will be affected, so that the spreading of damage is more difficult compared with the cases when other cells with higher edge number, such as the 6-gon, the 7-gon, the 8-gon, or the 10-gon are damaged. In other words, for the lattices with many rectangles, higher temperatures are needed to reach the equilibrium state(the so-called chaotic state), resulting in the larger T d .The transition from frozen to chaotic is subjected to two factors: temperature and the competition of dynamics. In general, G-dynamics favors a low-energy state while K-dynamics favors a high-energy state. When G-dynamics is dominant, the transition is mainly determined by temperature. In this case the transition temperature is higher. But as p decreases, the contribution of K-dynamics plays a much greater role, resulting in a lower transition temperature.As has been stated above, the lattices with many rectangles need higher temperatures to reach the equilibrium state under the pure G-dynamics, resulting in the larger T d . This phenomenon can be understood in the view of damage healing. For the G-dynamics, the damage is healed in the rectangle sites, so the damage is not easy to spread out. But the case is different for the situation in which the K-dynamics is dominant. In K-dynamics, damaged sites can heal only in pairs while they diffuse along and also create further damage. The K-dynamics is a non-local algorithm while the G-dynamics is a local one. The density of sites affects the non-local process. Originally, K-dynamics was used to simulate the diffusion process of spins and as we know, diffusion is affected strongly by the density of spins. For a given N, the number of sites for the topological hexagonal, the 4-6-7,4-8,4-10 lattice is N2, 5N2/4,<D>T <D >T<D >T d 2N 2 and 3N 2,respectively. The increase of the lattice sites equivalents to the increase of spin density .The more the lattice contains the rectangles, the bigger the spin density becomes. Thus the spins have more chance to exchange and the Kawasaki process is enhanced. This is similar to that of the case of Potts model with the non-local Swendson-Wang dynamics in which larger q has lower transition temperature 19. Maybe this is the reason why the DS transition temperature drops so sharply for the hierarchical lattices as the factor of K-dynamics is weighted more and more.In order to show the effects of the smallest polygons , the rectangles here, we consider the case of Fig.2.18(e) and calculate the T d with increasing N i . We notice that, for the case of p =1, when the number N i of the inserted rectangles increases, T d also increases(Fig.4), which agrees with the conclusion we got before :T d (4-6-7)<T d (4-8)<T d (4-10). But when p <1, T d decreases as N i increases(see Fig.5). That is due to the increases of the density of spins caused by inserting of rectangles.Notice that N i is in fact the operating number for inserting the rectangles. The real number of rectangles may be less than N i since the randomly generated points may be coincided. For example, when N i =20000, the real inserted number is just 1536, and it is unchanged as N i increases. On the other hand, the real number of the rectangles varies among configurations. When N i is smaller, the real inserted number of rectangles changes at each averaging, for example, when N i =10, the real number may be 5,6,5,7,…. Our results are averaged for 100 configurations. But the maximum number we can insert into the topological hexagonal lattice is fixed, that is why we see a saturated phenomenon in the figure of T d vs. N i (Fig.7.5 and Fig.7.6). The realistic inserted numbers become constant and the lattice becomes the 4-10 lattice at last.Finally, when K-dynamics is dominant, damage always spreads, resulting in no transition. This has been stated before. Thus, in the case of pure K-dynamics, we define T d =0. This case is shown in our simulation result of Fig.7.3, where T d of all of the structures approach to zero and no transition occurs, when p approaches to zero.Fig.7.4(a) Fig.7.4(b)T d N iN i C l u s t e r i n g C o e f f i c i e n t s N i /N tot.Fig.7.4(c)Fig.7.5Fig.7.6 Fig.7.4 Typical phase transition curves of the random lattices under pure Glauber dynamics with N =48 and (a) N i =10;(b) N i =100;(c) N i =200 Fig.7.5The change of the phase transition temperature vs. N i at the case of p =1 Fig.7.6 The change of the phase transition temperature vs. N i at the case of p =0.6 7.3 Explanation from the Point of View of Network What is the key point of the evolution of the networks shown in Fig.7.1(b)? We expect that clustering is an important topological property that greatly affects the evolution of damage spreading in the network. According to Newman, two measures can be defined to describe the clustering of a network (called the clustering coefficients),i.e.,(3-3)(3-4). In the following discussion on the relation between clustering effect and damage spreading, we focus on C 2 since the dynamics of damage spreading is affected by the local clustering, and the proper average should be performed on each given vertex i . In any case, we have calculated the clustering coefficients of the networks in our study (the networks are formed by inserting triangles at random on the vertices of the hexagonal lattice, see Fig.7.1(b)) and the results are shown in Fig.7.7.Fig.7.7 The clustering coefficients of the network of Fig.7.1 (b) (N=40) versus the number of inserted triangles (square) and the ratio of the inserted triangles to the total bubbles (hollow circle)We also calculated these two values for V oronoi and soap froth structures studied here for comparison, which are respectively C1=0.293,C2=0.425 for V oronoi and C1=0.296 ,C2=0.427 for soap froth structure. Since our spins are put at the centers of the bubbles, the networks of spins are formed actually by connecting the centers of the bubbles (the dual lattice). When there is no inserted triangle, the hexagonal lattice has similar clustering property to V oronio and soap structures. As we increase the number of the inserted triangles at the vertices of the hexagons, the average network density C2 is also increased. The increase in network density C2implies that the increased effect of pairwise damage spreading via the neighbors of a given vertex i through Kawasaki dynamics. With the star-triangle transformation going, more and more triangles are inserted, leading to the increasing of network density. For example, after the first-order star-triangle transformation, one bubble turns into 3 bubbles (one hexagon into one 12-gon and two triangles). For a given N, each insertion of triangle results in an increase of network density locally. This is equivalent to an increase of spin density, resulting in higher probability for spins to exchange with neighboring ones. In short, increased network density enhances the Kawasaki process. This is similar to the case of Potts model with the Swendson-Wang dynamics in which larger q has lower transition temperature [20]. From this simple topological investigation, we provide a heuristic understanding for the comparatively rapid drop in DS transition temperature for the hierarchical lattice as the effect of K-dynamics on DS is weighted more (1-p increases) for these structures (due to increased in network density), as compared to the rather stable DS transition temperature in the soap and V oronoi cases. ReferencesThis chapter is written mainly according to Guo Z. Z. and Szeto Kwok Yip, Damage spreading in two-dimensional trivalent cellular structures with competing Glauber and Kawasaki dynamics, Phys.Rev.E, 2005,71,066115.Z.Z.Guo,Xiaowei Wu,Int.J.Mod.Phys.B,2007.。