Graph operations and synchronization of complex networks

Transcription

1 PHYSICAL REVIEW E 72, Graph operatios ad sychroizatio of complex etworks Fatihca M. Atay* ad Türker Bıyıkoğlu Max Plack Istitute for Mathematics i the Scieces, Iselstr. 22, D Leipzig, Germay Received 23 December 2004; published 25 July 2005 The effects of graph operatios o the sychroizatio of coupled dyamical systems are studied. The operatios rage from additio or deletio of liks to various ways of combiig etworks ad geeratig larger etworks from simpler oes. Methods from graph theory are used to calculate or estimate the eigevalues of the Laplacia operator, which determie the sychroizability of cotiuous or discrete time dyamics evolvig o the etwork. Results are applied to explai umerical observatios o radom, scale-free, ad small-world etworks. A iterestig feature is that, whe two etworks are combied by addig liks betwee them, the sychroizability of the resultig etwork may worse as the sychroizability of the idividual etworks is improved. Similarly, addig liks to a etwork may worse its sychroizability, although it decreases the average distace i the graph. DOI: /PhysRevE PACS umbers: Xt, Ra, Hc, Ox I. INTRODUCTION The sychroizatio of coupled systems is a active field of research with applicatios i may areas of the physical ad biological scieces see Ref. 1 for a geeral itroductio. Sychroizatio is a wide-ragig pheomeo, which ca be observed i systems ragig from pulse-coupled euros 2 to chaotic oscillators 3, ad eve i the presece of delayed iformatio trasmissio 4. I some cases it is a desired pheomeo, such as whe several lasers are coupled to obtai maximum power output, while i other cases it represets a pathology, such as the sychroous eural activity durig epileptic seizures. The coectio structure of a etwork plays a importat effect o its sychroizatio. For diffusively coupled idetical systems, the effects of the etwork topology ca be expressed i terms of the spectrum of a diffusio or Laplacia operator e.g., Refs The spectrum characterizes sychroizatio for a give etwork structure; however, it usually gives little isight ito the effects of chages i the structure. It is ofte difficult to say how some structural chage might affect sychroizatio without calculatig the eigevalues afresh for the etwork. I this paper we study the effects of chages i the etwork structure o the sychroizatio of coupled dyamical systems. The operatios we cosider iclude addig or removig liks from the etwork, combiig two or more etworks ito oe, ad geeratig large etworks from simpler oes. Usig ideas from graph theory, we deduce the spectrum of the resultig etwork from the spectrum of the origial, without resortig to legthy calculatios. For certai operatios the exact values of the eigevalues ca be obtaied, while for others useful estimates are derived. Our results allow a systematic study of the sychroizability of whole classes of etworks, ad offer additioal isight ito the relatio betwee etwork topology ad sychroizatio. *Electroic address: atay@member.ams.org Electroic address: biyikoglu@mis.mpg.de The sychroizatio of coupled chaotic systems depeds typically o a umber of factors, icludig the stregth of the couplig, the coectio topology, ad the dyamical characteristics of the idividual uits, quatified by, e.g., the maximal Lyapuov expoet. To itroduce some otatio, cosider a etwork of idetical systems, idexed by i=1,,, ad govered by the differetial equatios ẋ i t = f x i t j=1 L ij x j t. Here R deotes the couplig stregth. The matrix of couplig coefficiets L=L ij is symmetric, has the diagoal elemets L ii equal to the umber of coectios to uit i, ad the off-diagoal elemets L ij are 1 if the ith ad jth uits are coupled ad zero otherwise. A stadard result is that the eigevalues of L are real ad o-egative, the smallest oe is equal to zero, ad is a simple eigevalue if the etwork is coected. We assume a coected etwork ad order the eigevalues as 0= 1 2. For coveiece we also use the otatio mi ad max for the smallest ad largest ozero eigevalues, 2 ad, respectively. The system 1 is said to sychroize if x i t x j t 0 ast for all i, j, startig from some ope set of iitial coditios. Sychroizatio may occur eve whe the idividual dyamics are chaotic, ad is related to the eigevalues of L. For the system 1, the relevat coditio is of the form 10 mi, where is a quatity that depeds o ad f more precisely, o the maximum Lyapuov expoet of f but ot o the coectio topology. I aother commoly studied system, amely the so-called coupled map lattice x i t +1 = f x i t j=1 L ij f x j t, the sychroizatio coditio takes the form /2005/721/ /$ The America Physical Society

2 FATIHCAN M. ATAY AND TÜRKER BIYIKOĞU PHYSICAL REVIEW E 72, mi max. 4 We remark that 4 ca also be the relevat coditio for the sychroizatio of cotiuous-time systems if a more geeral couplig fuctio is used tha the oe appearig i 1. Thus, sychroizatio depeds o the eigevalues mi ad max of L, a larger value of mi or a smaller value of max idicatig that sychroizatio ca be achieved for a larger set of parameter values, e.g., for larger Lyapuov expoets or a wider rage of couplig stregths. The eigevalues i deped oly o the coectio structure ad so ca be itroduced i a graph-theoretic way. Hece, cosider the graph G uderlyig the coupled system. The vertices of G correspod to the dyamical uits, with edges desigatig the iteractio betwee them, which is assumed to be bidirectioal, so G is a udirected graph. All graphs cosidered i this paper are assumed to be coected ad simple i.e., without loops or multiple edges. To avoid some trivial cases, we also assume throughout that they have at least two vertices. We write VG ad EG, respectively, for the vertex ad edge sets of a give graph G. Coversely, we let G=E,V deote the graph formed from give edge ad vertex sets. The otatio VG deotes the umber of vertices of G. The coectio structure of G is described by its adjacecy matrix A=a ij with elemets a ij =1 if the ith ad jth vertices are coected by a edge, ad zero otherwise. Let D be the diagoal matrix of vertex degrees, i.e., its ith diagoal elemet is the umber of edges icidet o the ith vertex. The couplig matrix of the dyamical system is the the Laplacia L=D A of its uderlyig graph. A simple example is give i the Appedix. We also write LG ad i G for the Laplacia ad its eigevalues whe we wish to emphasize the depedece o the particular graph G. The smallest positive eigevalue mi is called the algebraic coectivity of the graph 11 or the spectral gap of the Laplacia. By the coditio 2, it also provides a measure by which various etwork architectures ca be raked with respect to the ease of sychroizatio of cotiuous-time systems 1 defied o them. Hece we ca say that the graph G 1 is a better or poorer sychroizer tha G 2 if mi G 1 is larger or smaller tha mi G 2, respectively. Similarly, etwork architectures ca also be compared with respect to the sychroizability coditio 4 usig the quatity mi / max. This correspodece betwee topology ad dyamics allows the use of mathematical tools from graph theory for the ivestigatio of sychroizatio, which we utilize i the followig sectios. Sectio II cosiders the Cartesia product, the joi, ad the coalescece operatios, which ca be viewed as specific ways of combiig etworks, as well as the more geeral operatios of addig ad removig liks withi a etwork or betwee two etworks. For each operatio, we determie the sychroizability of the resultig graph by calculatig or estimatig the quatities mi ad mi / max from the eigevalues of the origial graph. Aalytical predictios are umerically cofirmed i Sec. III o radom, scale-free, ad small-world etworks. I additio, FIG. 1. The Cartesia product, C 3 P 2 upper row ad P 3 P 4 lower row. several iterestig features observed i the calculatios are explaied usig the theory of Sec. II. II. GRAPH OPERATIONS I this sectio we cosider several graph operatios ad their effects o the eigevalues of the Laplacia. We start by recallig some elemetary estimates o the eigevalues. Sice the diagoal elemets of the Laplacia L are simply the vertex degrees d i, it follows that i=1 i = d i. i=1 Usig the fact that 1 =0, we have i=1 i 1 max.i other words, max 1 d i d avg, 6 1i=1 where d avg deotes the average degree. Aother useful estimate for a graph o vertices is 12 5 i for all i. 7 A. Cartesia product The Cartesia product is oe of the basic operatios o graphs, through which some commo graphs ca be costructed from simpler oes. For istace, regular grids, cubes, ad their couterparts i higher dimesios are obtaied from the Cartesia product of paths liear chais P k. To give a precise defiitio, let G=V,E ad H=W,F be two oempty graphs. The Cartesia product GH is a graph with vertex set VW, ad x 1,x 2 y 1,y 2 is a edge i EGH if ad oly if either x 2 =y 2 ad x 1 y 1 EG or if x 1 =y 1 ad x 2 y 2 EH. Oe may view GH as the graph obtaied from G by replacig each of its vertices with a copy of H ad each of its edges with VH edges joiig correspodig vertices of H i the two copies. For istace, the product of two paths P ad P m yields a m rectagular grid. Some examples are give i Fig. 1. The Cartesia product is a commutative, associative biary operatio o graphs, see e.g., Ref. 13. The eigevalues for the product ca be calculated from the eigevalues for the factor graphs. Propositio 1: The eigevalues of the Laplacia for the

3 GRAPH OPERATIONS AND SYNCHRONIZATION OF PHYSICAL REVIEW E 72, Cartesia product GH satisfy mi GH = mi mi G, mi H, max GH = max G + max H, mi GH max GH mi mig max G, mih max H. Proof: Suppose that G ad H have s ad r vertices, respectively. A result from graph theory implies that the eigevalues of LGH are give by all possible sums i G + j H, 1ir ad 1 js see, for example, Ref. 13. Recallig that 1 is always zero, we have mi GH =mi mi G, mi H. It also follows that max GH = max G+ max H. Fially, assume without loss of geerality that mi G mi H. The max GH mi GH = maxg + max H mi mi G, mi H = maxg mi G + maxh mi G maxg mi G + maxh mi H max maxg mi G, maxh mi H, which completes the proof. Thus, the product GH caot be a better sychroizer tha its factors G ad H, ad i fact, with respect to the coditio 4 it is a strictly poorer sychroizer tha both G ad H. Propositio 1 allows us to coclude simply by visual ispectio that the graphs o the left-had side of the equality sigs i Fig. 1 are better sychroizers tha those o the right-had side. I particular, oe-dimesioal chais are better sychroizers tha two-dimesioal grids, which i tur are better tha three-dimesioal lattices, ad so o. Let us look more closely at the behavior of mi uder the Cartesia product. The product GP 2 is two copies of G with edges betwee the correspodig vertices i each copy. The eigevalues of LP 2 are 0 ad 2, i.e., mi P 2 = max P 2 =2. Propositio 1 implies that = mi GP 2 mig if mi G 2, 8 2 if mi G 2. I other words, mi of the product graph saturates if mi G icreases beyod 2. The more geeral product GH is formed from several copies of G where correspodig vertices i each copy are coected accordig to the structure give by H, ad mi GH is give by the right-had side of 8 with two replaced by mi H. Oe ca fix H ad thus the coectio structure of the copies of G, ad study the resultig sychroizability for differet choices of G. If mi G is icreased for istace, by addig edges withi G, see Sec. II D, we see from the above Propositio that mi GH will icrease as log as mi G mi H, but ot beyod the value mi H. Thus the structure of H sets a upper boud o the sychroizability of GH, ad this boud is quatified by mi H. Sice addig edges to G will ot improve sychroizatio of the product graph beyod FIG. 2. The joi operatio, P 1 C 4 upper row ad P 2 P 3 lower row. this limit, oe might try addig liks across the copies of G istead. For istace, istead of the product GP 2 where each vertex of G is coupled to its twi i the secod copy, oe ca lik each vertex of G to every vertex i the secod copy of G. This is the joi operatio, which is the topic of Sec. II B. There it will be see that mi ca ideed be greatly icreased by this procedure. The situatio is more iterestig with respect to the ratio mi / max. We show i Sec. III that, for fixed H, mi / max ca actually decrease for GH while mi G/ max G icreases. I other words, the better G sychroizes, the worse will the product GH. We will show i Sec. II E that this pheomeo is ot peculiar to the Cartesia product, ad ca also occur uder quite geeral coectio of etworks. B. Joi Let G 1 =V 1,E 1 ad G 2 =V 2,E 2 be graphs o disjoit sets of ad m vertices, respectively. Their disjoit uio G 1 +G 2 is the graph G 1 +G 2 =V 1 V 2,E 1 E 2, ad their joi G 1 G 2 is the graph o r=+m vertices obtaied from G 1 +G 2 by isertig edges from each vertex of G 1 to each vertex of G 2. See Fig. 2 Propositio 2: Let G ad H be graphs o ad m vertices, respectively. The the eigevalues of the Laplacia for the joi GH satisfy ad mi G H = mi mi G + m, mi H + mi G + mi H max G H = m +. If G ad H have the same umber of vertices, the mi G H max G H Proof: The eigevalues of LGH are give by 0, +m,m+ i G,2i; ad + j H,2jm 13, from which 9 ad 11 follow. By 7, m mi H ad mi G, which imply 10. It follows that whe =m, mi G H max G H = + mi mig, mi H

4 FATIHCAN M. ATAY AND TÜRKER BIYIKOĞU PHYSICAL REVIEW E 72, FIG. 3. All coalesceces of a star with four vertices ad the cycle C 4 upper row, ad all coalesceces of P 3 ad P 3 lower row. Hece, mi GH is always larger tha both mi G ad mi H, ad therefore also larger tha mi GH, by Propositio 1. The situatio is ot so clear cut for the ratio mi / max. For example, for the special case GG, mi / max is usually higher for the joi tha for the origial graph G, but it ca also be smaller. I order to have mi GG/ max GG mi G/ max G, a ecessary coditio from 12 is that mi G max G 1 2, 13 that is, G should already be a good sychroizer. Figure 4 i Sec. III gives a example for the case whe 13 is satisfied ad GG is a slightly poorer sychroizer tha G. Nevertheless, by 12 the joi of two graphs of comparable sizes will yield a good sychroizer, ad for specific graph types it ca be proved that the result will be strictly better tha the idividual graphs. We give a example for the case of trees. Corollary 1: Let T ad S be two trees each havig vertices, the mi T S max T S max mit max T, mis max S. Proof: A tree T with vertices has 1 edges, so the sum of its vertex degrees is 2 1, ad by 6 max T2. Furthermore, it is a kow fact that mi T1 for trees 11. Thus, mi T/ max T1/2. The corollary the follows by 12. C. Coalescece A coalescece of G 1 ad G 2 is ay graph obtaied from the disjoit uio G 1 +G 2 by idetifyig a vertex of G 1 with a vertex of G 2, i.e., mergig oe vertex from each graph ito a sigle vertex. Ulike the Cartesia product, the coalescece geerally does ot yield a uique graph; see Fig. 3. We deote by G 1 G 2 ay coalescece of G 1 ad G 2. We show that GH caot be a better sychroizer tha G or H. Note that the result holds for all possible coalesceces of the pair G,H, ad thus applies to a large umber of graph types. Propositio 3: For ay coalescece GH of G ad H, mi G H mi mi G, mi H, max G H max max G, max H, mi G H max G H mi mig max G, mih max H. Proof: Suppose that b=b 1,,b p ad c=c 1,,c q are sequeces of o-egative real umbers arraged i oicreasig order. We say that b majorizes c if i=1 b i k k i=1 c i,1kmip,q, ad p i=1 b i = q i=1 c i. Let Spec G = max G,, mi G,0 be the sequece of eigevalues of LG i oicreasig order. By a result of Groe ad Merris 14, Spec GH majorizes Spec G+H for ay coalescece GH of G ad H. By majorizatio the result is proved. Applyig Propositio 3 to the graphs i Fig. 3, we ca immediately see without ay calculatios that the star S 5 o 5 vertices caot be a better sychroizer tha the liear chai P 3. Ideed, the ratio mi / max is 1/5 for S 5 ad 1/3 for P 3. Similarly, shorter chais will sychroize better tha loger oes, sice the latter ca be viewed as a coalescece of the former. Similar coclusios ca be draw for the other graphs by visual ispectio. The umerical calculatios i Sec. III cofirm that the coalescece operatio yields reduced sychroizability, ofte by a order of magitude. D. Addig ad removig edges If G=V,E is a graph, the G e deotes the graph obtaied from G by removig the edge eeg. IfeEG, the G+e is the graph obtaied from G by addig a edge e. I geeral, the precise effect o the spectrum of addig or deletig edges is still poorly uderstood. Oe well-kow fact is 12 i G + e i G, 1 i. I particular, mi G+e mi G, so whe edges are added respectively, deleted, the sychroizability of the system 1 either icreases respectively, decreases or stays the same. O the other had, it is ot easy to say what will happe to the ratio mi / max, so a similar coclusio caot be draw for the coupled map lattice 3. The computatios i Sec. III verify that, for a fixed umber of vertices, mi is odecreasig as the umber of edges is icreased. However, there are cases where mi / max ca strictly decrease with icreasig edge umber. We study such a case i more detail i the ext sectio, where we cosider the arbitrary coectio of two etworks. E. Coectig two etworks We ow cosider coectig two separate etworks by addig liks betwee them. The ext result gives estimates for the sychroizability of the resultig etwork. Propositio 4: Let G 1,G 2 be two graphs o 1 ad 2 vertices, respectively, ad let H be the graph obtaied by addig k edges betwee G 1 ad G 2. The, mi H 2k mi 1, 2, mi H max H 2k d avg Hmi 1,

5 GRAPH OPERATIONS AND SYNCHRONIZATION OF PHYSICAL REVIEW E 72, FIG. 4. Sychroizability of radom etworks uder graph operatios. The origial graph G is show with the solid lie, while the other curves correspod to the joi GG--, the Cartesia product GP 2 --, ad the coalescece GG --. Proof: The proof makes use of the otio of the isoperimetric umber of a graph. For a subset X of VG, we defie the boudary X as the set of edges e=xv such that xx ad vx. We let S deote the cardiality of a set S. The isoperimetric umber of G is defied as ig=mi XVG X/X, where the miimum is take over all subsets X of the vertex set of G satisfyig 1XVG/2. The quatities mi G ad ig are related by 15 mi G 2iG. Hece, for the graph H, we have mi H 2iH 2 X X, 16 where X is ay subset of VH satisfyig 1XVG/2. I particular, X ca be chose as the smaller of G 1 ad G 2, which yields 14. Usig 6, we obtai 15. Suppose ow that two graphs G 1,G 2 are combied as i the above propositio, ad cosider addig edges to G 1 or G 2 while keepig costat the umber of coectios k betwee them. The effect is to icrease the average degree d avg H of the combied graph ad thus decrease the right-had side of 15. Sice 15 is oly a upper boud, the et effect o mi H/ max H caot be determied for small values of d avg H. However, whe d avg H/k is made sufficietly large by addig eough edges, the 15 implies that mi H/ max H should be small. Therefore, addig more liks i a etwork may impede the sychroizatio, although FIG. 5. Sychroizability of scale-free etworks uder graph operatios. The curves correspod to G,GG--, GP 2 --, ad GG--. it decreases the average distace ad the diameter. Figure 7 i the ext sectio umerically cofirms this observatio. III. NUMERICAL RESULTS We ow use umerical calculatios to obtai more detailed iformatio o the effects of the graph operatios preseted i Sec. II. Our aim is to determie the behavior of several commo architectures, amely radom, scale-free, ad small-world etworks, i relatio to the graph operatios. Furthermore, we wish to explai the umerical observatios usig the foregoig theoretical cosideratios. The calculatios i this sectio are doe o etworks of 500 vertices ad by averagig the results over several realizatios. The radom etworks are costructed startig with a fixed umber of vertices ad addig a edge betwee ay pair of vertices with probability p 16. The scale-free etworks are costructed usig the Barabasi-Albert algorithm for preferetial attachmet 17, startig with m iitial vertices, ad addig a vertex at each step with m liks to existig vertices with probability proportioal to their vertex degrees. For small-world etworks we use the variat proposed i Ref. 18, which is obtaied by radomly addig m liks to a cycle a set of vertices coected to their earest eighbors i circular arragemet. The results are summarized i Figs I each figure, the sychroizability of the graph G is compared to that of the joi GG, the Cartesia product GP 2, ad the coalescece GG. The graph GG is costructed by takig two copies of G ad coalescig a radomly selected vertex from each copy

6 FATIHCAN M. ATAY AND TÜRKER BIYIKOĞU PHYSICAL REVIEW E 72, FIG. 7. The lower solid curve is the ratio mi / max for two copies of a radom graph G of 500 vertices coected to each other by 20 liks. The upper dotted curve gives the same ratio for the sigle graph G. FIG. 6. Sychroizability of small-world etworks uder graph operatios. The curves correspod to G,GG--, GP 2 --, ad GG--. I the upper figure, the solid lie correspodig to G is idistiguishable sice mi G coicides with mi GP 2. Some geeral features of the graph operatios are apparet from the Figs For istace, the joi GG typically yields improved sychroizability compared to the origial etwork G, whereas the Cartesia product GP 2 results i reduced sychroizability, ad the coalescece GG has the worst sychroizability, as measured by mi or mi / max. A exceptio is the radom graph where mi / max for GG is about the same as ad ca i fact be slightly less tha the correspodig ratio for G whe G itself is a sufficietly good sychroizer. Note that this happes whe mi G/ max G becomes larger tha 1/2, i agreemet with 13. As the vertical scales i the figures are logarithmic, the sychroizability of the etworks ca ofte differ by several orders of magitude after the graph operatios. We ow use the results of the precedig sectios to give a theoretical uderstadig of several iterestig observatios from the figures. We first ote that the value of mi for the Cartesia product GP 2 saturates to 2 i the radom ad scale-free etworks of Figs. 4 ad 5. This is a cosequece of 8 ad the fact that mi G icreases as more edges are added to G show by the solid curve i the figures. Othe other had, for the small-world etwork of Fig. 6, the curve mi GP 2 coicides with mi G, i agreemet with 8 sice mi G is less tha 2 i this case. With respect to the joi operatio, we have mi GG =500+ mi G, as predicted by 9 ad verified by the figures. I the scale-free ad small-world etworks mi GG appears almost as a horizotal lie at the value 500 sice mi G is relatively small i these cases. We also ote that i most cases sychroizability icreases as edges are added i.e., as p or m is icreased. However, a otable exceptio occurs for the Cartesia product GP 2 of radom ad scale-free etworks. Here, over a large iterval the ratio mi / max mootoically decreases for GP 2, although it icreases for G, as more edges are added. I other words, the better the sychroizability of G, the worse is the sychroizability of GP 2. This iterestig situatio is most promiet i purely radom etworks Fig. 4, somewhat less cospicuous i scale-free etworks Fig. 5, ad abset i small-world etworks Fig. 6. We use the theory of Sec. II to give a quatitative accout. Thus, by Propositio 1, Eq. 8, ad the fact that max P 2 =2, we have mi GP 2 max GP 2 = 2 if mi G max G As edges are added to G, max G will geerally icrease as metioed i Sec. II D, so mi / max will decrease over the rage where mi G2. This latter iequality holds for the radom graph of Fig. 4 throughout the rage of parameters used, so mi / max decreases mootoically, while the opposite is true for the small-world etwork of Fig. 6. Furthermore, for the scale-free etwork of Fig. 5, mi / max iitially icreases, but begis to decrease at the poit where mi G reaches 2. The pheomeo of decreased sychroizability with icreased coectivity ca also be observed i the geeral couplig of two etworks. Ideed, if two copies of a graph G havig vertices are coected by addig k liks betwee them, the for the resultig graph H we estimate mi H max H 2k 17 d avg H usig 15. Ifk is small compared to the size of G, the d avg Hd avg G, i.e., about twice the umber of edges of G. Thus, the right-had side of 17 ca be viewed as the ratio of the umber of liks betwee the two copies of G to the umber of liks withi G. By icreasig the average degree withi G, mi H/ max H ca be made smaller. Figure 7 shows that mi / max decreases for the combied graph H while it icreases for the idividual graph G. By a straight

7 GRAPH OPERATIONS AND SYNCHRONIZATION OF forward extesio of this argumet, a similar coclusio ca be draw whe two differet graphs G 1 ad G 2 are coected by addig liks betwee them. IV. CONCLUSION We have cosidered some commo operatios o graphs ad studied their effects o the sychroizatio of coupled dyamical systems. For the Cartesia product ad the joi operatios, the eigevalues of the Laplacia for the resultig graph ca be directly determied from those for the origial graphs, which gives a method for determiig sychroizability without legthy calculatios. For the other operatios the eigevalues ca be estimated, providig useful isight ito the relatio betwee sychroizatio ad coectio topology. I simpler cases, the results allow oe to determie which etwork is a better sychroizer simply by visual ispectio of its structure. Such heuristics should be useful i desig procedures. We have illustrated our results umerically o radom, scale-free, ad small-world etworks, ad used the theory to explai several features observed i umerical calculatios. For istace, we have show that addig liks to a graph may improve, saturate, or worse its sychroizability, although the average distace of the graph decreases. A related observatio is that, whe two etworks are combied by addig liks betwee them, the sychroizability of the resultig etwork ca worse as that of the idividual etworks is improved. Usig the theoretical results we are able to explai ad predict whe such situatios ca arise. Clearly, such aalytical tools ca help us better uderstad the structure ad dyamics of complex etworks. For example, it has recetly bee show that the degree distributio of a etwork geerally does ot determie its sychroizability 19, a sigificat fact which is difficult to establish o the basis of umerical simulatios aloe. Several of the ideas preseted here ca be exteded to more geeral couplig operators, such as weighted coectio matrices. These results will be reported i a future work. APPENDIX The followig is a short example illustratig the otatio ad the relevat matrices for a simple graph, amely a liear chai of three vertices, deoted by P 3 ad depicted i the secod row of Fig. 3. Labelig the vertices liearly i a obvious way, the vertex degrees umber of eighbors of each vertex are 1, 2, ad 1, which form the diagoal etries of the matrix D=diag1,2,1. The eighborhood relatio of the graph is cotaied i the adjacecy matrix ad the Laplacia is give by PHYSICAL REVIEW E 72, A = , L = D A = The eigevalues of L are 0, 1, ad 3. Disregardig the trivial eigevalue, we have mi =1 ad max =3 i our otatio. Usig the graph operatios o P 3, oe obtais several graphs whose sychroizability ca be raked by the ratio mi / max of their respective Laplacias. The Cartesia product P 3 P 3 isa33 rectagular grid with mi / max =1/6, so is a poorer sychroizer tha P 3. The joi P 3 P 3 has mi / max =4/6, implyig improved sychroizability over P 3. These values ca be directly foud from Propositios 1 ad 2. All coalesceces of P 3 P 3 are show i the secod row of Fig. 3, ad for these graphs the ratio mi / max takes the values 0.106, 0.124, ad 0.2 from left to right, all showig decreased sychroizability over P 3. O the other had, addig a edge to P 3 gives the cycle C 3, depicted i the first row of Fig. 1, which is a complete graph ad has mi / max =3/3, the maximum possible value for ay graph. 1 A. Pikovsky, M. Roseblum, ad J. Kurths, Sychroizatio A Uiversal Cocept i Noliear Sciece Cambridge Uiversity Press, Cambridge, R. E. Mirollo ad S. H. Strogatz, SIAM J. Appl. Math. 50, L. M. Pecora ad T. L. Carroll, Phys. Rev. Lett. 64, F. M. Atay, J. Jost, ad A. Wede, Phys. Rev. Lett P. M. Gade, H. A. Cerdeira, ad R. Ramaswamy, Phys. Rev. E P. M. Gade, Phys. Rev. E L. M. Pecora ad T. L. Carroll, Phys. Rev. Lett K. S. Fik, G. Johso, T. Carroll, D. Mar, ad L. Pecora, Phys. Rev. E J. Jost ad M. P. Joy, Phys. Rev. E X. Li ad G. Che, IEEE Tras. Circuits Syst., I: Fudam. Theory Appl M. Fiedler, Czech. Math. J., C. Godsil ad G. Royle, Algebraic Graph Theory Spriger- Verlag, New York, R. Merris, Liear Algebr. Appl., 278, R. Groe ad R. Merris, Liear Multiliear Algebra, B. Mohar, J. Comb. Theory, Ser. B P. Erdos ad A. Réyi, Publ. Math. Debrece, A. L. Barabasi ad R. A. Albert, Sciece M. E. J. Newma ad D. J. Watts, Phys. Lett. A F. M. Atay, T. Bıyıkoğlu, ad J. Jost IEEE Tras. Circuits Syst., I: Fudam. Theory Appl. to be published