High dimensional Apollonian networks

Transcription

1 High imensional Apollonian networks Zhongzhi Zhang Institute of Systems Engineering, Dalian University of Technology, Dalian 11604, Liaoning, China Francesc Comellas Dep. e Matemàtica Aplicaa IV, EPSC, Universitat Politècnica e Catalunya Av. Canal Olímpic s/n, Castellefels, Barcelona, Catalonia, Spain comellas@mat.upc.es Guillaume Fertin LINA, Université e Nantes, rue e la Houssinière, BP 908, 443 Nantes Ceex 3, France fertin@lina.univ-nantes.fr Lili Rong Institute of Systems Engineering, Dalian University of Technology, Dalian 11604, Liaoning, China llrong@lut.eu.cn Abstract. We propose a simple algorithm which prouces high imensional Apollonian networks with both small-worl an scale-free characteristics. We erive analytical expressions for the egree istribution, the clustering coefficient an the iameter of the networks, which are etermine by their imension. PACS numbers: Hc, Da, 89.0.Hh

2 High imensional Apollonian networks 1. Introuction Since the groun-breaking papers by Watts an Strogatz on small-worl networks [1] an Barabási an Albert on scale-free networks [], the research interest on complex networks as an interisciplinary subject has soare [3, 4, 5]. Complex networks escribe many systems in nature an society, most of which share three apparent features: powerlaw egree istribution, small average path length (APL) an high clustering coefficient. While many moels [3, 4, 5] have been propose to escribe real-life networks, most of them are stochastic. However, new eterministic moels with fixe egree istributions constructe by recursive methos have been recently introuce [6, 7, 8, 9, 1, 10, 11]. Deterministic moels have the strong avantage that it is often possible to compute analytically their properties [13, 14], which may be compare with experimental ata from real an simulate networks. Deterministic networks can be create by various techniques: moification of some regular graphs [15], aition an prouct of graphs [16], ege iterations [18] an other mathematical methos as in [17]. Concerning the problem of Apollonian packing, two groups inepenently introuce the Apollonian networks [19, 0] which have interesting properties like being scale-free, Eucliean, matching, space-filling an can be applie to porous meia, polyisperse packings, roa networks or electrical supply systems [19] an may also help to explain the properties of energy lanscapes an the associate scale-free network of connecte minima [0]. In this paper we present a simple iterative algorithm to generate high-imensional Apollonian networks base on a similar iea as that of the recursive graphs propose in [1]. The introuce algorithm can concretize the problems of abstract high-imensional Apollonian packings. Using the algorithm we etermine relevant characteristics of high-imensional Apollonian networks: the egree istribution, clustering coefficient an iameter, all of which epen on the imension of Apollonian packings. It shoul be pointe out that the concept of high-imensional Apollonian networks was alreay introuce in [19] an [0]. In these works, however, the emphasis is place on two-imensional Apollonian networks an their aim is to aress the behavior of ynamical processes [19] or provie a moel to help unerstan the energy lanscape networks [0, 1, ]. Here, we focus on the proucing algorithm, base on which we provie a etaile calculation of the topology characterization of high-imensional Apollonian networks an we show that it epens on the imension.. The construction of high imensional Apollonian networks From the problem of Apollonian packing, a two-imensional example of which is shown in Fig.1(a), Anrae et al. introuce Apollonian networks [19] which were inepenently propose by Doye an Massen in [0]. Apollonian packing ates back to Apollonius of Perga who live aroun 00 BC. The classic two-imensional Apollonian

3 High imensional Apollonian networks 3 Figure 1. (a) A two-imensional Apollonian packing of isks. (b) Construction of two-imensional Apollonian networks, showing the first four iterations steps. packing is constructe by starting with three mutually touching circles, whose interstice is a curvilinear triangle to be fille. Then a circle is inscribe, touching all the sies of this curvilinear triangle. We call this the first iteration t = 1 an the initial configuration is enote by t = 0. For subsequent iterations we inefinitely repeat the process for all the newly generate curvilinear triangles. In the limit of infinite iterations, the well-known two-imensional Apollonian packing is obtaine. From the two-imensional Apollonian packing, one can straightforwarly efine a two-imensional Apollonian network [19, 0], where vertices are associate to the circles an two vertices are connecte if the corresponing circles are tangent. Fig.1(b) shows the network base on the two-imensional Apollonian packing. The two-imensional Apollonian network can be generalize to high-imensions (-imensional, ) [0] associate with other self-similar packings [3]. A comprehensive account is given next. A -imensional ( > ) Apollonian packing can be constructe iteratively in a similar way as shown in Fig. 1(a). Initially, we have + 1 mutually touching -imensional hyperspheres with a curvilinear open -imensional polyheron (polyheron) as their interstice. In the first iteration one -hypersphere is ae to fill the interstice of the initial configuration, such that it shoul touch each of the + 1 -hyperspheres. The process is repeate for all the newly create curvilinear open - imensional polyherons in the successive iterations. In the limit of infinite iterations, the result is a -imensional Apollonian packing. If each -hypersphere correspons to a vertex an vertices are connecte by an ege if the corresponing -hyperspheres are in contact, then one gets a -imensional Apollonian network.

4 High imensional Apollonian networks 4 3. The iterative algorithm of high imensional Apollonian networks In the iterative process for the construction of high-imensional Apollonian networks at each iteration, for each new hypersphere ae, + 1 new interstices are create in the associate Apollonian packing which will be fille in the next iteration. when builing networks, we can say in equivalent wors that for each new vertex ae, + 1 new -polyherons are generate in the network, into which vertices will be inserte in the next iteration. Accoring to this process, we can introuce a general iterative algorithm to create high-imensional Apollonian networks which is similar to the process that allow the construction of the recursive graphs introuce in [1]. Before introucing the algorithm we give the following efinitions. A complete graph K (also referre in the literature as -clique; see [4]) is a graph with vertices, where there is no loop or multiple ege an every vertex is joine to every other by an ege. Generally speaking, two graphs are sai to be isomorphic if the vertices an eges of one graph match up with vertices an eges of other, an the ege matching be consistent with the vertex matching. We enote the -imensional Apollonian network after t iterations by A(, t),, t 0. Then the -imensional Apollonian network at step t is constructe as follows: For t = 0, A(, 0) is the complete graph K +1 (or ( + 1)-clique), an A(, 0) has + 1 vertices an (+1) eges. For t 1, A(, t) is obtaine from A(, t 1) by aing for each of its existing subgraphs isomorphic to a ( + 1)-clique an create at step t 1 a new vertex an joining it to all the vertices of this subgraph (see Fig. 1(b) for the case = ). Then, at t = 1, we a one new vertex an + 1 new eges to the graph, creating + 1 new K +1 cliques an resulting in the complete graph with + vertices, enote K +. At t = we a + 1 new vertices, each of them connecte to all the vertices of one of the + 1 cliques K +1 create at t = 1 introucing ( + 1) new eges, an so on. Note that the aition of each new vertex leas to + 1 new ( + 1)-cliques an +1 new eges. So the number of new vertices an eges at step t i is L v (t i ) = (+1) t i 1 an L e (t i ) = ( + 1) t i, respectively. Therefore, similarly to many real-life networks such as the Worl Wie Web, the -imensional Apollonian network is a growing network, whose number of vertices increases exponentially with time. Thus we can easily see that at step t, the Apollonian network A(, t) has eges vertices an N t = ( + 1) + E t = ( + 1) t t i =1 L v (t i ) = ( + 1)t 1 t + L e (t i ) = t i =1 ( + 1) (1) + ( + 1)t+1 1 ()

5 High imensional Apollonian networks 5 The average egree k t is then k t = E t N t = ( + 1)t ( + 1) t For large t it is approximately ( + 1). We can see when t is large enough the resulting networks are sparse graphs as many real-worl networks whose vertices have many fewer connections than is possible. (3) 4. Relevant characteristics of high imensional Apollonian networks Below we will fin that the imension is a tunable parameter controlling all the relevant characteristics of the -imensional Apollonian network Degree istribution When a new vertex i is ae to the graph at step t i (t i 1), it has egree + 1 an forms + 1 new ( + 1)-cliques. From the iterative algorithm, we can see that each new neighbor of i generate new ( + 1)-cliques with i as one vertex of them. In the next iteration, these ( + 1)-cliques will introuce new vertices that are connecte to the vertex i. Let k i (t) be the egree of i at step t (t > t i + 1). Then, as in [0], k i (t) = k i (t) k i (t 1) = k i (t 1) (4) combining the initial conition k i (t i ) = + 1 an k i (t i + 1) = + 1, we obtain k i (t) = ( + 1) t t i 1 an the egree of vertex i becomes k i (t) = k i (t i ) + t t m=t i +1 ( ) t t i 1 k i (t m ) =( + 1) The istribution of all vertices an their egrees at step t is given in Table 1. It shoul be mentione that here we on t give the etaile egree evolution process of the +1 initial vertices create at step 0 an just list the evolution result, which is ifferent form others vertices. But when the network become very large, these few initial vertices have almost nothing effect on the network topology characteristics. From Table 1 we can see that the egree spectrum of the graph is iscrete an some values of the egree are absent. To relate the exponent of this iscrete egree istribution to the stanar γ exponent as efine for continuous egree istribution, we use a cumulative istribution P cum (k) k k N(k, t)/n t k 1 γ. Here k an k are points of the iscrete egree spectrum. The analytic computation etails are given as follows. For a egree k ( t l ) 1 k = ( + 1) there are ( + 1) l 1 vertices with this exact egree, all of which were born at step l. (5) (6)

6 High imensional Apollonian networks 6 Table 1. Distribution of vertices an their egrees for A(, t) at step t. Number of vertices Degree t j=0 j + 1 ( + 1)( t j=0 j + 1) + 1 ( + 1)( t 3 j=0 j + 1) ( + 1) ( + 1)( t 4 j=0 j + 1) ( + 1) t 3 ( + 1)( ) ( + 1) t ( + 1)( + 1) ( + 1) t All vertices introuce at time l or earlier have this an a higher egree. So we have l N(k, t) = ( + 1) + L v (s) = ( + 1)l k k As the total number of vertices at step t is given in Eq. (1) we have [ ] 1 γ ( + 1)( t l (+1) 1 l ) = (+1) t = ( + 1)l + ( + 1) 1 ( + 1) t + ( + 1) 1 Therefore, for large t we obtain ( t l ) 1 γ = ( + 1) l t s=1 an ln( + 1) γ 1 + (8) ln so that < γ < We notice that this value has been obtaine previously by Doye an Massen in [0]. Also, notice that when t gets large, the maximal egree of a vertex roughly equals to t 1 ln / ln(+1) Nt = N 1/(γ 1) t. 4.. Clustering istribution The clustering coefficient [1] of a given vertex is the ratio of the total number of eges that actually exist between all its k nearest neighbors an the potential number of eges k(k 1)/ between them. The clustering coefficient of the whole network is obtaine averaging over all its vertices. We can erive analytical expressions for the clustering C(k) for any vertex with egree k. When a vertex is create it is connecte to all the vertices of a ( + 1)-clique whose vertices are completely interconnecte. It follows that a vertex with egree k = + 1 has a clustering coefficient of one because all the ( + 1)/ possible links between its (7)

7 High imensional Apollonian networks 7 neighbors actually exist. After that, if its egree increases by one, then its new neighbor must link to its existing neighbors. Thus for a vertex v of egree k, the exact expression for its clustering coefficient is C(k) = (+1) + (k 1) = k(k 1) (k +1 ) k(k 1) epening on egree k an imension. Using this result, we can compute now the clustering of the graph at step t, it is C t = S t /N t, where N t is number of vertices at step t which is provie by Eq. (1) an S t represents the sum of clustering coefficient for all vertices given by S t = ( + 1) (D 0 +1 ) D 0 (D 0 1) + t q=1 (D q +1 )L v(q) D q (D q 1) where D 0 = t 1 + an D 1 q = ( + 1) ( t q ) given by Eq. (6) are the egrees of 1 the vertices create at steps 0 an q, respectively. One can easily prove that for t 7 an for any the following relation hol true. C t > 3 (11) 3 1 Therefore the clustering coefficient of high imension Apollonian networks is very large. Similarly to the power exponent γ of the egree istribution, the clustering is also tunable simply by changing the value of control parameter. From Eq. (11), one can see that the clustering coefficient increases with an approaches a limit of 1 when gets large. In the special cases = an = 3, C equals to constant asymptotic values (see also [19]) an 0.885, respectively Diameter The iameter of a network characterizes the maximum communication elay in the network an is efine as the longest shortest path between all pairs of vertices. In what follows, the notations x an x express the integers obtaine by rouning x to the nearest integers towars infinity an minus infinity, respectively. Now we compute the iameter of A(, t), enote iam(a(, t)) for : Step 0 an 1. The iameter is 1. Steps to + 1. In this case, the iameter is, since any new vertex is by construction connecte to a ( + 1)-clique, an since any ( + )-clique uring those steps contains at least the vertex create at step 1, which is from the initial clique K + or A(, 1) obtaine after step 1, thus the iameter is. Steps + to +. In any of those steps, some newly ae vertices might not share a neighbor in the original clique K + obtaine after step 1; however, any newly ae vertex is connecte to at least one vertex of the initial clique K +. Thus, the iameter is equal to 3. Further steps. Clearly, at each step t + 3, the iameter always lies between a pair of vertices that have just been create at this step. We will call the newly (9) (10)

8 High imensional Apollonian networks 8 create vertices outer vertices. At any step t + 3, we note that an outer vertex cannot be connecte with two or more vertices that were create uring the same step 0 < t t 1. Moreover, by construction no two vertices that were create uring a given step are neighbors, thus they cannot be part of the same ( + )-clique. Thus, for any step t + 3, some outer vertices are connecte with vertices that appeare at pairwise ifferent steps. Thus, there exists an outer vertex v t create at step t, which is connecte to vertices v i, 1 i t 1, which all are pairwise istinct. We conclue that v t is necessarily connecte to a vertex that was create at a step t 0 t 1. If we repeat this argument, then we obtain an upper boun on the istance from v t to the initial clique K +. Let t = α( + 1) + p, where p +. Then, we see that v t is at istance at most α + 1 from a vertex in K +. Hence any two vertices v t an w t in A(, t) lie at istance at most (α+1)+1; however, epening on p, this istance can be reuce by 1, since when p + 1, we know that two vertices create at step p share at least a neighbor in K +. Thus, when p + 1, iam(a(, t)) (α + 1), while when + p +, iam(a(, t)) (α + 1) + 1. One can see that these istance bouns can be reache by pairs of outer vertices create at step t. More precisely, those two vertices v t an w t share the property that they are connecte to vertices that appeare respectively at steps t 1, t,... t 1. Base on the above arguments, one can easily see that for t > +, the iameter increases by every + 1 steps. More precisely, we have the following result, for any an t (when t = 1, the iameter is clearly equal to 1): iam(a(, t)) = ( t + 1) + f(, t) + 1 where f(, t) = 0 if t t ( + 1) + 1, an 1 otherwise. +1 In the limit of large t, iam(a(, t)) t, while N +1 t (+1) t 1, thus the iameter is small an scales logarithmically with the network size. 5. Conclusion an iscussion In conclusion, we have propose a general iterative algorithm to prouce high imensional Apollonian networks associate with high imensional packings. The networks present the typical characteristics of real-life networks in nature an society as they are small-worl an have a power-law egree istribution. We compute analytical expressions for the egree istribution, the clustering coefficient, an the iameter of the networks, all of which are etermine by the imension of the associate Apollonian packings. The high imensional Apollonian networks introuce here, an the consieration of the metho presente in Ref. [5], allow the construction of high imensional ranom Apollonian networks [6]. In aition, it shoul be worth stuying in etail physical moels such as Ising moels [7] an processes such as percolation, spreaing, searching an iffusion that take place on the higher-imensional Apollonian networks to know also their relation with the imension.

9 High imensional Apollonian networks 9 Acknowlegment This research was supporte by the Natural Science Founation of China (Grant No ). Support for F.C. was provie by the Secretaria e Estao e Universiaes e Investigación (Ministerio e Eucación y Ciencia), Spain, an the European Regional Development Fun (ERDF) uner project TIC References [1] D.J. Watts an H. Strogatz, Nature (Lonon) 393, 440 (1998). [] A.-L. Barabási an R. Albert, Science 86, 509 (1999). [3] R. Albert an A.-L. Barabási, Rev. Mo. Phys. 74, 47 (00). [4] S.N. Dorogvtsev an J.F.F. Menes, Av. Phys. 51, 1079 (00). [5] M.E.J. Newman, SIAM Review 45, 167 (003). [6] A.-L. Barabási, E. Ravasz, an T. Vicsek, Physica A 99, 559 (001). [7] K. Iguchi an H. Yamaa, Phys. Rev. E 71, (005). [8] S.N. Dorogovtsev, A.V. Goltsev, an J.F.F. Menes, Phys. Rev. E 65, 0661 (00). [9] S. Jung, S. Kim, an B. Kahng, Phys. Rev. E 65, (00). [10] E. Ravasz an A.-L. Barabási, Phys. Rev. E 67, 0611 (003). [11] J.D. Noh, Phys. Rev. E 67, (003). [1] F. Comellas, G. Fertin an A. Raspau, Phys. Rev. E 69, (004). [13] H. Rozenfel; J. Kirk; E. Bollt an D. ben-avraham, J. Phys. A 38, 4589 (005). [14] E. Bollt, D. ben-avraham, New Journal of Physics 7, 6 (005) [15] F. Comellas, J. Ozón, an J.G. Peters, Inf. Process. Lett. 76, 83 (000). [16] F. Comellas an M. Sampels, Physica A 309, 31 (00). [17] T. Zhou, B.H. Wang, P. Q. Jiang, P. M. Hui an K. P. Chan, e-print con-mat/ [18] Z.Z. Zhang, L.L. Rong an C.H. Guo, e-print con-mat/ (Physica A in press). [19] J.S. Anrae Jr., H.J. Herrmann, R.F.S. Anrae an L.R.a Silva, Phys. Rev. Lett. 94, (005). [0] J.P.K. Doye an C.P. Massen. Phys. Rev. E 71, (005). [1] J.P.K. Doye. Phys. Rev. Lett. 88, (00). [] J.P.K. Doye an C.P. Massen. J. Chem. Phys. 1, (005). [3] R. Mahmooi Baram, H.J. Herrmann, an N.Rivier, Phys. Rev. Lett. 9, (004). [4] D.B. West, Introuction to Graph Theory (Prentice-Hall, Upper Sale River, NJ, 001). [5] T. Zhou, G. Yan, an B.H. Wang, Phys. Rev. E 71, (005). [6] Z.Z. Zhang, L.L Rong an F. Comellas, e-print con-mat/ (Physica A in press). [7] R.F.S. Anrae an H.J. Herrmann. Phys. Rev. E 71, (005).