Growth model for complex networks with hierarchical and modular structures

Transcription

1 Growth model for complex networks wth herarchcal and modular structures Q Xuan, Yanjun L,* and Te-Jun Wu Natonal Laboratory of Industral Control Technology, Insttute of Intellgent Systems & Decson Makng, Zhejang Unversty, Hangzhou , Chna Receved 30 August 2005; publshed 3 March 2006 A herarchcal and modular network model s suggested by addng a growth rule along wth the preferental attachment PA rule nto Motter s modelng procedure. The proposed model has an ncreasng number of vertces but a fxed number of modules and herarchcal levels. The vertces form lowest-level modules whch n turn consttute hgher-level modules herarchcally. The creaton of connectons between two vertces n a sngle module or n two dfferent modules of the same level obeys the PA rule. The structural characterstcs of ths model are nvestgated analytcally and numercally. The results show that the degree dstrbuton, the module sze dstrbuton, and the clusterng functon of the model possess a power-law property whch s smlar to that n many real-world networks. The model s then used to predct the growth trends of real-world networks wth modular and herarchcal structures. By comparng ths model wth those real-world networks, an nterestng concluson s found: that many real-world networks are n ther early stages of development, and when the growth tme s large enough, the modules and levels of the networks wll be ultmately merged. DOI: /PhysRevE PACS number s : Fb, Da, b I. INTRODUCTION In the last few years, consderable efforts have been made to understand and model complex networks n the real world 1 4. A number of complex networks have been nvestgated and categorzed, n whch two knds have ganed much attenton due to ther popularty n the real world 2 7. One of the real-world networks whch have been well studed s the scale-free SF network 1,4,7,8. Those complex networks, such as n bologcal 8, socal 4,9, and technologcal systems 10, are found to obey the power-law degree dstrbuton.e., P k k where the degree s the number of edges connectng to a gven vertex. Barabás and Albert BA 4 successfully modeled such networks wth the followng prncple: startng wth a small number m 0 of vertces, addng a new vertex at every tme step, and connectng t to m m m 0 dfferent vertces whch are selected wth a probablty lnearly proportonal to the degree of the target vertex. Such a selecton rule s called the preferental attachment PA rule. Recently, many network models based on BA have been proposed and the dynamc propertes of them have been explored extensvely 1,11. Another knd of real-world networks whch have been well nvestgated s characterzed by a hgh degree of clusterng 2,11 and modular structure 5,12 24, both of whch can be measured by a local clusterng coeffcent 2. The local clusterng coeffcent for vertex wth degree k s defned as C =2n /k k 1, where n s the number of edges between the k neghbors of vertex. A hgh degree of clusterng means that the clusterng coeffcent C.e., the average of C over all the vertces s sgnfcantly hgher for these real-world networks than a regular network of smlar sze. Furthermore, a network can be consdered modular and *Author to whom all correspondence should be addressed. Electronc address: yjlee@pc.zju.edu.cn herarchcal when the clusterng functon C k, the average of C over the vertces wth degree k, satsfes the relatonshp: C k k and C remans fnte for large system sze N 8,13,14,18,22. The BA model obeys the power-law degree dstrbuton wth =3, but C k s ndependent of k and C decreases wth N, because the BA model does not contan modules 18. Recently, Watts et al. 23 and Motter et al. 24 ntroduced socal network models contanng modular and herarchcal structure, where vertces form modules whch are n turn grouped to become bgger modules herarchcally. Ther models, however, are manly focused on networks wth a fxed number of vertces and thus wthout a growng property. Ravasz and Barabás RB ntroduced a herarchcal model n a determnstc way 13,14 ; furthermore, Iguch and Yamada completely analyzed the RB model and removed some flaws n ts arguments 25. Km et al. 18 and Noh et al. 19 ntroduced growng models, where the number of vertces and modules ncreases wth tme. All of the models n the papers 13,14,18,19 obey the power-law degree dstrbuton. In ths paper, we present a growng network model by addng the growth rule and the PA rule n the model of Motter et al. The proposed model s a growth model for complex systems wth a fxed herarchcal and modular structure. And dfferent from other methods, our approach could also be consdered as a naturally generalzed verson of the BA model whose rules are consdered to be very common n our socety. The areas where our model can be more naturally appled are socal systems, wde-area network WAN routng networks, large-scale logstc systems, ntercontnental ar transportaton networks, etc., where the vertces are people, routers, local warehouse, and local arports, respectvely, and the modules of dfferent levels are ctes, provnces, countres for socal systems, local and regonal routng subnets for WAN routng networks, local and transfer networks for logstc systems, local and connectng arlnes for /2006/73 3 / /$ The Amercan Physcal Socety

2 XUAN, LI, AND WU ar transportaton networks, and so on 24. These systems have fxed herarchcal and modular structures whch are all manly determned by many geographc factors, but the vertces of them ncrease very quckly. Our model s desgned accordng to the above system characterstcs and s dfferent from former studes. For example, the model of Watts et al. 23 and the model of Motter et al. 24 do not have the growth property; the RB model 13,14 has totally determnstc vertex growth characterstcs whch are not realstc n the real world; and n the model of Km et al. 18 and the model of Noh et al. 19 both the modules and vertces ncrease wth tme. Therefore, these models cannot explan well the systems mentoned above. In those systems, the creaton of connectons between lower-level modules s easer and more frequent than that at hgher levels. For example, n ar transportaton systems, openng a local arlne s usually easer and more frequent than openng an ntercontnental arlne. In other words, comparng to the rapdly ncreasng number of nfrastructural components, the upper-level organzatons are much more stable, so the number of the upperlevel organzatons can be regarded as constants. In vew of the dynamcs of the vertex degrees and the local clusterng coeffcents, the proposed model exhbts both the propertes of scale-free and modular structure. The degree dstrbuton of the proposed model possesses a power-law dstrbuton wth 2,3 and the clusterng functon C k k, consstent wth most emprcal data. We also brefly study the dynamcs of module sze whch s defned as the number of members n a module. As the proposed model grows, the module sze dstrbuton shows a power-law dstrbuton lke the network model studed prevously 19. The rest of the paper s organzed as follows. In Sec. II, we ntroduce the growng network model. The dynamc propertes of the model can be changed by some parameters. Then analytc results about the dynamcs of vertex degree and module sze are provded n Sec. III, and the numercal smulaton s gven n Sec. IV. The results from the theoretcal analyss and the numercal smulaton are very close. As an applcaton, the proposed model s used n Sec. V to predct the growth trends of a complex network wth modular and herarchcal structure. Our work s summarzed n Sec. VI. II. GROWING NETWORK MODEL In ths secton we present a growng network model wth fxed modular and herarchcal structure. As s shown n Fg. 1, our model has a treelke structure. At the lowest level.e., the frst level of the network, there are several groups of vertces, formng modules of ths level. Those modules are n turn grouped to consttute hgher-level modules and so on. Ths process contnues untl at a certan level there s only one bg module. Ths level s defned as the top level of the network, contanng all the nformaton about the network n a herarchcal way. To smplfy the descrpton, we assume that, at all levels except the top level, every n module of the hth level forms a module of the h+1h level, where n s ndependent of h. Then a network wth M levels wll contan n M 1 lowest-level FIG. 1. The sketch map of the model wth M levels. modules. The parameters M and n are fxed n our model. Two stochastc events are consdered n our model to make the network grow.e., the creaton of a new vertex connected to some other exstng vertces and the creaton of a new connecton attached between two exstng vertces. Accordng to the herarchcal and modular structure of our model, these events can be yelded by two knds of operatons: n-module connecton and between-module connecton. Wth the n-module connecton a new vertex may be created n a module and connected to several vertces n the same module. Wth the between-module connecton two vertces comng from dfferent modules wll be connected. All the operatons are conducted n a random way: the connecton type s determned wth a predefned n-moduleconnecton probablty q h for the hth-level modules, h=1,2,...,m, and the vertex selecton follows the PA rule. The whole process of network growng can be descrbed wth the followng top-down vew. Intally we assume that there are m 0 fully connected vertces n each lowest-level module. We set h=m and let Q h be the currently selected module n the hth level n whch a network growth operaton wll be conducted. Of course Q M s the sole module at the top level. Connecton-type selecton. For the submodules of module Q h at the hth level, the n-module connecton type s selected wth the probablty q h 1 or the between-module connecton type s selected wth the probablty 1 q h 1. Module selecton. A h 1h-level submodule of the module Q h s selected wth a unform probablty f the nmodule connecton type s chosen or two h 1h-level submodule of the module Q h are selected also wth a unform probablty f the between-module connecton type s chosen. Vertex connecton. In the case of n-module connecton, a new vertex s created f the h 1h-level submodule s at the lowest level and then connected to m exstng vertces n ths submodule by followng the PA rule. Otherwse, f the h 1h level s not the lowest level, substtute h by h 1 and let ths submodule be the selected module Q h ; then, return to the frst step connecton type selecton. In the case

3 GROWTH MODEL FOR COMPLEX NETWORKS WITH¼ In ths secton we wll nvestgate some mathematcal propertes of the proposed model on the bass of the smplfed assumpton presented n Sec. II;.e., except the lowest level n the model, all the modules at each level are composed of the same fxed number of ther lower-level modules. We would lke to nvestgate the characterstcs of the network model when new vertces and connectons are allowed to add nto the network randomly. In ths stuaton, the degree of each vertex and the number of vertces n each module n the model ncrease as the network grows, presentng a dynamcal prospect. In our model, all the modules at each level have the same unform selecton probablty, and the n-module connecton probablty q h, h=1,2,...,m 1, satsfes nequalty 1. As a result, the modules at the hth level can be consdered to have the same sze of module degree, denoted by k j h M, where k j s the degree of vertex j n a module and M s the total number of levels n the network. We can easly get the relatonshp between the module degree of the hth level and that of the h+1h level: FIG. 2. Color onlne The network model wth M =2, n=3, m=m 0 =2, q 1 =0.9, and growth tme T=30. It s grown up to N=31 vertces. of between-module connecton, on the other hand, two exstng vertces are selected, respectvely, from the submodules determned n step 2 module selecton n terms of the PA rule and then connected each other. The above procedures can be repeated for a number of tmes to emulate network growng processes n the real world. Fgure 2 demonstrates a confguraton of our model wth M =2, n=3, m=m 0 =2, q 1 =0.9, T=30, and grown up to N=31 where N s the number of total vertces. As s dscussed n Sec. I, n real-world networks, the creaton of connectons between lower-level modules s easer and more frequent than that at hgher levels. Ths fact can be expressed n our model by the followng nequalty: 1 q h+1 q h+1 1 q h. 1 Inequalty 1 can be regarded as a constrant on the nmodule connecton probabltes q h, h=1,2,...,m 1. Two necessary condtons for nequalty 1 holdng are that "h, q h q h+1 ;.e., the n-module connecton probablty q 1,q 2,...,q M must be a monotoncally ncreasng sequence and "h 1, q h 0.5. These condtons mply that, n order to preserve a modular and herarchcal structure, the events of vertex creaton n a sngle module must occur much more frequently n the network-growng process than the creaton of a connecton attached between two exstng vertces n dfferent modules; otherwse, the clusters and herarches of the network wll vansh soon. The dynamcal propertes of the network model wll be analyzed n the next two sectons. M M k j h+1 = n k j h. 2 Accordng to the connecton generaton prncple presented n Sec. II, the vertex degree dynamcs n the hth-level module.e., the ncreasng rate of the connectons of a vertex n the selected module at the hth level denoted by k / M h, satsfes the followng recursve equatons: k M k 3 h+1 = q h k M + 2 h k M = 1 1 n M 1 n M h 1 q h mk k j 1 M. k j h M, The frst term on the rght-hand sde of Eq. 3 represents the ncreasng rate of the connectons wthn the selected module of the hth level, based on the n-module-connecton probablty q h, and the second term concerns the ncreasng rate of the connectons between those hth-level modules that are ncluded n a module of the h+1h level, on the bass of the between-module-connecton probablty 1 q h. Equaton 4 mples that the connectons n any of the lowest-level modules obey the PA rule. But from Eqs. 2 4 one s unable to get the module degree dynamcs k / M M from the top-level vew, because the total number of connectons, k j M M, n the network s unknown. Correspondng to Eq. 3, however, k j h+1 h+1 can also be dvded nto two parts: 4 h+1 h k j h+1 = qh k j +2 1 qh h, 5 k j 1 1 =2mt, where t s the network growth tme. If we defne 6 III. THEORETICAL ANALYSIS a 1 = m, b 1 =2m, a h+1 = q h a h +2 1 q h, 7 8 9

4 XUAN, LI, AND WU FIG. 3. Plots of the module sze dstrbuton P S s of the model wth M =10, n=3, and 8 vertces n each smallest module. The slope of the sold lne s 1.0. b h+1 = q h b h +2 1 q h, 10 then from Eqs. 2 6, the degree dynamcs of the network can be easly descrbed as k M = a M k M b M t, 11 where a M and b M can be derved from Eqs The result allows us to conclude that the dstrbuton functon P M k k M.e., the power-law dstrbuton, wth exponent M =1+b M /a M. 12 Because t satsfes 1 b M /a M 2, we can fnd that the exponent M 2,3 whch fts most emprcal data n 4,8 10,26. The fact that the exponent M s between 2 and 3 can be easly understood. When the number of new connectons created wthn the lowest-level modules are far more than those between dfferent modules.e., the n-module-connecton probablty q h 1, for all h=1,2,...,m 1 the model wll be close to the BA model and the exponent M 3. Whereas f at some level say, the hth level most of the new connectons are created between modules.e., q h 0 or the number of levels n the network s very large.e., M then, M 2. Snce the modules at any level have an dentcal selecton probablty P h S, all the modules of ths level can be consdered to have the same module sze. We defne S h and R h as the module sze and the number of modules at the hth level, respectvely. From the model structure, t can be easly derved that S h+1 = ns h, R h+1 = 1 n R h From Eqs. 13 and 14, we can get ln S h = lnr h + a 0, 15 where a 0 s ndependent of h. And accordng to the defntons of S h and R h, we have P S h R h. 16 Combnng Eq. 15 wth Eq. 16, we know that the module sze obeys the power-law dstrbuton P S h S 1 h. 17 Numercal analyss also shows that P S s almost satsfes P S s s 1, 18 whch s shown n Fg. 3. Such a result s smlar to that of some real-world networks 19,27. IV. NUMERICAL SIMULATION The analytc result n Sec. III has been valdated by numercal smulaton. In the frst example, we set M =4, n=3, q 1 =0.7, q 2 =0.8, q 3 =0.9, m=m 0 =2, and T=7000, and n the second example, M =5, n=3, q 1 =0.7, q 2 =0.9, q 3 =0.95, q 4 =0.995, m=m 0 =2, and T=6000. We present the numercal data n Fg. 4. It shows that the results of the two examples satsfy the power-law degree dstrbuton wth a dfferent exponent; the frst has the exponent 2.5 and the second the exponent 2.6, whch are very close to the analytc results from Eq. 12. We also nvestgated the clusterng functon C k of the proposed model wth varous parameters. Roughly speakng, C k s lkely to obey the dstrbuton of k. In the early stage of the growng process.e., each smallest module has a relatve small number of vertces, s close to 1, as s shown n Fgs. 5 a and 5 d. As the network grows, decreases from 1 to 0, shown n Fgs. 5 a 5 c. The reason s that as the network grows, the connecton between modules

5 GROWTH MODEL FOR COMPLEX NETWORKS WITH¼ FIG. 4. Plots of the degree dstrbuton P k : a The frst example wth M =4, n=3, q 1 =0.7, q 2 =0.8, q 3 =0.9, m=m 0 =2, T=7000, and N=3562. The slope of the sold lne s 2.5. b The second example wth M =5, n=3, q 1 =0.7, q 2 =0.9, q 3 =0.95, q 4 =0.995, m=m 0 =2, T=6000, and N=3775. The slope of the sold lne s 2.6. becomes stronger and the connecton wthn the lowest-level modules s weaker due to the ncreasng number of vertces. So the clusterng phenomenon n the network s not so clear as n the early growng stage. The result means that many of the real-world networks are n ther early growng stage because s close to 1 for them 18. The length of the early stage s manly determned by the number of levels, M; the n-module-connecton probablty q h ; and the number of connectons, m, added whle a vertex s created. The larger the parameters, the longer the early stage s. So some real-world networks may have to spend a very long early stage. V. TREND PREDICTION USING OUR MODEL Here we wll study the growth trends of those real-world networks by usng our network model wth correspondng parameters. We consder a medum sze network wth M =4, n=3, q 1 =0.5, q 2 =0.95, and q 3 =0.995 and plot n Fg. 6 the change of the exponents of C k of any module at each level except the frst level, because the modules at the frst level do not have a modular structure wth respect to the ncreasng number of vertces n the network. In Fg. 6 we fnd that as the network grows, lower modules lose ther ntermodular structure quckly, and when the growth tme T s large enough, the modules and consequently the herarches of the network wll eventually merge. That s to say, for example, one day, n human socety the admnstratve organzatons governng vllages, ctes, provnces, and countres wll vansh as a result of the rapd development of communcaton among people. In fact, even at present, we have seen a phenomenon that the boundares of dfferent ctes n the developed countres are gettng more and more blurry wth FIG. 5. The clusterng functon C k wth the parameters m=m 0 =2, q 1 =0.7, q 2 =0.8, and q 3 =0.9: a M =4, n=3, T=400, and N=240; b M =4, n=3, T=1600, and N=859; c M =4, n=3, T=3200, and N=1632; d M =4, n=5, T=1600, and N=1049. The slopes of sold lnes are 1.0 n a and d, 0.5 n b, and0n c

6 XUAN, LI, AND WU FIG. 6. Plots of the trends of exponents for modules at each level except the lowest level n a network wth M =4, n=3, q 1 =0.5, q 2 =0.95, and q 3 = the development of local and long-dstance transportaton systems. VI. SUMMARY Ths paper proposes a herarchcal and modular network model by addng the growth prncple and the PA prncple nto the model of Motter et al. The proposed model has ncreasng vertces, a fxed number of modules, and a herarchcal structure. The vertces of the model form the lowestlevel modules whch n turn consttute hgher-level modules herarchcally. The creaton of connectons between two vertces n a sngle module or n two dfferent modules of the same level obeys the PA rule. Wth theoretcal analyss and numercal smulaton, t s shown that the degree dstrbuton, the module sze dstrbuton, and the clusterng functon of the model possess a power-law property whch s smlar to that n many real-world networks. The model has been used to predct the growth trends of the real-world networks wth modular and herarchcal structures. By comparng ths model wth those real-world networks, an nterestng concluson s found that many real-world networks are n ther early stages of development, and when growth tme s large enough, the modules and levels of the networks wll be ultmately merged. It s noteworthy that, as the lower modules dsappear, the number of network levels wll decrease and the betweenmodule-connecton probablty of orgnal hgher-level modules wll ncrease correspondngly. Ths can be consdered as an nterm process between two relatvely stable stage of the network growng,.e., the ntal stage at whch there s a very clear modular and herarchcal structure and the ultmate stage at whch all the nfrastructures and superstructures vansh. Unfortunately, our model s unable to characterze ths nterm perod. Ths s an nterestng topc about the development of complex networks, and we wll study t n our subsequent work 28. ACKNOWLEDGMENTS We would lke to thank all the members n our research group n the Insttute of Intellgent Systems and Decson Makng, Zhejang Unversty at Yuquan Campus, for valuable dscussons about the deas presented n ths paper. Ths work was supported by the Natonal 973 Program of Chna Grant No. 2002CB R. Albert and A.-L. Barabás, Rev. Mod. Phys. 74, D. J. Watts and S. H. Strogatz, Nature London 393, A. Barrat and M. Wegt, Eur. Phys. J. B 13, A.-L. Barabás and R. Albert, Scence 286, M. Grvan and M. E. J. Newman, Proc. Natl. Acad. Sc. U.S.A. 99, C. M. Song, S. Havln, and H. A. Makse, Nature London 433, L. F. Costa, F. A. Rodrgues, G. Traveso, and P. R. V. Boas, e-prnt cond-mat/ A.-L. Barabás and Z. N. Oltva, Nature London 5, S. Eubank, H. Guclu, V. S. A. Kumar, M. V. Marathe, A. Srnvasan, Z. Toroczka, and N. Wang, Nature London 429, A. Barrat, M. Barthélemy, R. P. Satorras, and A. Vespgnan, Proc. Natl. Acad. Sc. U.S.A. 101, S. A. Pandt and R. E. Amrtkar, Phys. Rev. E 60, R A. Clauset, Phys. Rev. E o be publshed. 13 E. Ravasz, A. L. Somera, D. A. Mongru, Z. N. Oltva, and

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