An efficient approach of controlling traffic congestion in scale-free networks

An efficient appr oach of controlling traffic congestion in scale-fr ee networks Zongh ua Liu, 1 W eichuan Ma, 2 Huan Zh ang, 1 Y in Sun, 1 and P . M. Hui 3 1 Institute of Theor etical P hysics and Department of Physics, East China Normal U niversity , Shangh ai, 200062 , China 2 Department of Physics, Hubei University , W uhan, 430062 , China 3 Department of Physics, T he Chinese University of Hong Kon g, Shatin, New T erritories, Hong Kon g (Dated: October 31, 2018) W e propose and study a model of traffic in commu nication networks. The underlying network has a struc- ture that is tunable between a sc ale-free gro wing netw ork with preferential attachmen ts and a random growing network. T o model realistic situations where dif ferent nodes in a netw ork may hav e different capab ilit ies, the message or packe t creation and delivering rates at a node are assumed to depend on the degree of t he node. Noting that cong estions are more likely to tak e place at the nodes with high de grees in networks with sc ale-free character , an ef ficient approach of selectively enhancing the message-p rocessing capability of a small fraction (e.g. 3% ) of th e no des is sho wn to perform just as good as enh ancing the capability of all n odes. The interplay between the creation rate and the de livering rate in determining non -congested or congested traffic in a netwo rk is studied more numerically and analytically . P ACS numbe rs: 89.75.Hc, 05.70.Jk INTRODUCTION Operations in the inter net such as browsing webpages in the W orld Wide W eb (WWW), sendin g e-mails, transferring files via ftp, search ing for infor mation, an d electro nic shopping , etc. have bec ome part of daily life for many people. These activities have opened up exciting op portun ities for sharing informa tion, economic transformation, and other acti vities on a global scale [1, 2] . The internet, h owe ver , is not perfect. For example, inte rmittent co ngestion in the internet, s imilar to traffic congestion in highway systems, has been observed [3]. Similar phen omena can be also of relevance in other commu- nication network s, suc h as the transpor tation n etwork in air- lines and the postal serv ice network. A key p roblem in com - munication networks is, therefore, to understand h ow one can control co ngestion a nd m aintain a no rmal and efficient fun c- tioning of the networks. Sev eral mo dels of co mmunicatio ns in a comp uter network have b een extensi vely studied [ 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 1 4, 15, 16]. In these m odels, the in formation proces- sors are ro uters. Their fu nction is to route the data packets to their d estinations. In a computer network, a nod e may b e a host or a ro uter . A host can create messages o r data p ack- ets for targeted destination s and receive packets from other hosts. A rou ter finds the shortest p ath b etween the o rigin of the me ssage and the destination of ea ch p acket a nd f orwards the packet one step closer to the destination alo ng the shortest path in each time step. Th e shortest path is th e path with the smallest n umber of links. Previous studies have mo stly been focused o n three d ifferent com puter network mod els: (i) the nodes at the edge of th e netw or k are hosts and th e inner n odes are routers [9], (ii) all the nodes are both hosts and rou ters [10, 13, 14], and (iii) some o f the nod es are hosts and th e rest are r outers [11, 1 5, 16]. However , these mo dels were studied with the un derlying networks being a two-dimensional lattice [9, 10, 1 1, 15] and a Cayley tree [12, 1 3, 14]. As the inter- net shows a heterogen eous structure with a scale- free degree distribution [17, 18], a more realistic network model for com- munication s should be heterogeneo us. Besides th e internet and its related n etworks such as WWW [19 , 20, 21] and email networks [ 22], ma ny oth er n etworks also show the scale-free behavior . Th ese networks include, for example, the telephone network [23], the biological network in pro teins [24], and th e networks of se xu al contacts [ 25, 26]. Some networks, such as the collaboration network among scientists [27], show a mixed feature o f scale-fr ee and expo nential distributions. I n fact, the study of the science of co mplex networks has b ecome an imp ortant in terdisciplinary area of r esearch. Th e prob lem of efficiency in delivering messages or da ta p ackets in com - munication network s h as been ad dressed recen tly by Arenas et a l. [28, 29, 30], Moreno et al. [31, 32, 3 3], an d Zha o et al. [34]. Arenas et al. focused on finding the optimal network topolog ies for searches in complex networks, while Mo reno et al. stud ied the dependence of the jam ming tran sitions on rout- ing strategies. A common feature in previous studies is th at the cr eation an d delivering r ates of packag es do n ot ch ange from nod e to node. As the nodes in a complex network could have very d ifferent p roperties, e.g., d egrees, a mo re realistic assumption is that the package creatio n ra te and delivering rate at a node be come d egree-depen dent. In the inter net, an importan t site ha s mo re users an d h ence a larger message or package creation and deli vering r ates. A recent s tud y by Zhao et al. [3 4] considered the case o f non-un iform package deliv- ering rates, b ut th e creation rate was tak en to be a con stant. In the p resent work, we study traffic in networks with n on- unifor m packag e cre ation and deliv erin g rates. An imp ortant quantity in communicatio n n etworks is the critical pack age creation rate that signifies a tra nsition fr om a non- congested or free flow regime to a con gested regime. Below the c ritical rate, a non -cong ested steady state is reached after the tr an- 2 sient in which the data packets created can be efficiently han- dled by the n odes. Above the critical rate, a cong ested phase is r eached w here the num ber of packets accum ulated in the system inc reases with time. The v alue of critical r ate thu s measures the cap acity of efficient commu nication inside the network. Here, we stud y the critical rate in n etworks wher e the package creation an d deli vering rates are node-depe ndent. In particular, we present an efficient ap proach to enha nce the capacity of commun ications in scale-free networks. The paper is o rganized as f ollows. Section II d efines th e model with n ode-d ependen t package c reation and d eli vering rates. I n Sectio n III, we pr esent re sults o f nu merical simula- tions an d study the inter play betwe en the critical and deliv- ering rates in determ ining n on-con gested or con gested traffic in a network. Section IV e xp lains the observed features in the numerical results analytically . W e summa rize the paper in Sec. V . MODEL The nod es in a comp lex network such as the internet may represent very different entities. For example, some n odes may just be in dividuals an d oth er may repr esent big co m- panies or univ ersities. Obviously , d ifferent no des will h av e different rates of creatin g m essages. The nodes, d ependin g on their conn ectivity to other nodes and perhap s hardware, also ha ve d ifferent rates of delivering messages. Here, we present a mo re r ealistic mod el of commun ication in c omplex networks that includes no de-depe ndent cr eation an d deliver - ing rates. Ou r model is a modification on several previous models [28, 29, 30, 31, 32, 33, 3 4]. W e assume that for a node i with degree k i , the message c reation rate λk i is proportion al to its degree, with λ being a co nstant. F or message deliver - ing, e ach node sho uld handle at least o ne packet or messag e in each time step. Therefo re, we assume a delivering r ate of 1 + β k i for a nod e with degree k i , with β ≥ 0 being a pa- rameter o f the mod el. Our m odel thus represents the realistic situation that a busy nod e with larger k i has high er rates of generating and delivering messag es. For the u nderlyin g n etwork, we use a model in which the exponent of the degree distribution can be tuned. A scale- free n etwork with P ( k ) ∼ k − γ can b e con structed by incor- porating p referential attachments in a network- growing pr o- cess [35, 36]. The Bara basi a nd Alb ert model [ 36] assumes the prob ability Π i for a node i to attract a link fro m a newly added nod e to be Π i ∼ k i . Th e model g iv es an exp onent γ = 3 [3 6] f or th e degree d istribution. On the o ther h and, a random g rowing n etwork can be constructed by assuming a node-in depend ent Π i . Many ne tworks show characters that are somewhat intermedia te of scale-free and r andom. For them, th e degree distribution shows a mixed feature of the two chara cters [ 27]. This im plies that the probab ility Π i of attracting a ne w link shou ld contain both preferential and ran- dom features. One of the pr esent authors propo sed a hy brid model [37] in which Π i ∼ (1 − p ) k i + p , wher e 0 ≤ p ≤ 1 is a pa rameter r epresenting the proba bility that a newly added node establishes its new links by r andom attac hments an d (1 − p ) is the prob ability that ne w link s a re established by preferen tial attachments. The degree distribution was shown to b e [3 7] P ( k ) ∼ [ k + p/ (1 − p )] − γ ( p ) with an expo nent γ ( p ) = 3 + p/ [ m (1 − p )] , wh ere m is the number of new links p er node. The p = 0 limit redu ces to the P ( k ) ∼ k − 3 behavior an d the p → 1 limit gives the rand om g rowing net- work behavior of P ( k ) ∼ e − k/m . Here, we u se this model as our under lying network for studying commun ications in net- works. Once the network of a certain value of p is con structed, the dynamics of creating and d eliv ering messages i s implemen ted as f ollows. Each node pla ys the dual role of a host an d a router, with its crea tion and deliv erin g rates assigned acco rd- ing to its d egree. Details of th e dynamic s are listed as follows. (1) At each time step , a node i has a p robab ility λk i of cr e- ating a new message or pac ket with a r andomly chosen desti- nation. If the n ode has som e messages waiting to b e sent, the newly created message will be placed at the end of the queue. The queuing m essages may be created at some previous time steps or received from near by nodes as messages are being sent along their paths to the destinations. (2) Once a packet is created with a chosen destina tion, the node (router) will ide ntify the shortest path towards the desti- nation. If ther e exist several shortest p aths to th e destination, the path is chosen in such a way that the packa ge is sent to a node that has the instantaneou s s ho rtest queue. (3) At each time step, a nod e i has the ability to f orward (1 + β k i ) packets in the q ueue at the nod e on a first-in-first-ou t basis to its ne ighbor s which are a long the path to the d estina- tions. No ting th at β k i may be an integer p lus a fractional part, the f ractional part is implem ented as the probab ility of deliv- ering additional packets in a time step. (4) Messages ar riving at a no de are queu ed up for furthe r de- li vering . When a message arrives at its destination, it is re - moved f rom the system. The step s ar e carr ied o ut fo r every node at the same time. If λk i is replaced by λ and β k i is rep laced by th e integral part int [ β k i ] , the above algorithm will b e equ i valent to th at of Ref. [34]. He re, th e f ractional p art of β k i and λk i are implemen ted in a probab ilistic way . Furthermo re, if we take β = 0 , th e above algorithm will be equiv alent to th at of Refs. [ 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 28, 29, 30, 31, 32, 33]. The par ameters λ and β thus co ntrol the number of mes- sages or packages in the system. A small (h igh) v alue of λ cor- respond s to fewer (more) packets. For simplicity , e ach packet is labeled by two p ieces of in formatio n: the tim e of creation and its destination. Qualitatively , the tota l nu mber of p ack- ets created at each time step is P N i =1 λk i ≈ P N i =1 λ h k i = 2 mλN for a g rowing network of m newly adde d links per node and a total of N nodes. At the same time, the maximu m number of packets processed by the nodes is P N i =1 (1+ β k i ) ≈ (1 + 2 m β ) N . When the nu mber of new packets added to the system equals the number of packages removed up on arriv al at each tim e step, the network runs in the range of Little law [38] 3 and there is no congestion. For scale-fre e n etworks, the struc- ture is heterogeneo us in the sense that some nodes have many more links. As m essages are sen t via the shortest paths, they are likely to pass thr ough the n odes with m ore links. If ev ery node has the same delivering rate, these node s will h av e more accumulated messages and a high chan ce of jamming. In con- trast, ran dom n etworks d o n ot h av e hubs with h igh degrees and thu s the structure is relativ ely “hom ogeneo us”. There- fore, cong estion is easier to occu r in scale-f ree networks than random networks. As most of the real-life network s ar e scale- free, the contr ol o f co ngestion in these networks is of critical importan ce. For examp le, an intuitive but not so efficient ap- proach is to in crease the value o f β for the no des. W e will refer to this appr oach of having identical values of β for a ll nodes as the normal appr oach . The num ber o f nodes in a com munication network is typ - ically very large and there is no central o rganizer to man- age the development of th e wh ole network. It is, theref ore, very difficult to h av e a realistic mech anism to incre ase β fo r all no des at th e same time . Realistically , larger comp anies and academic in stitutions could increase their local v alue of β more readily . Noting that cong estion is m ore likely to occu r at the nodes with many links in scale-fr ee networks, one m ay significantly r educe congestion by selectively increasing th e delivering rates of the nodes with high degrees. Th erefore , we suggest th at a po ssible co st-effecti ve way to control co nges- tion is to a sk the node s with lar ger links to increase their v alue of β . Here, we stud y a mod el in which a fractio n f of n odes with high degrees are assigned a finite value o f β > 0 , and the rest are a ssigned β = 0 . Th is models the high er message- processing capab ility of the hubs in a network . W e refer to this model as the efficient a ppr oach . In this model, the maxim um number of packets processed in a time step is ( N + P i ∈ f β k i ) , where the sum is over the n odes with finite β . Results of nu- merical simulations show th at the e fficient ap proach performs compara bly w ith the norm al approa ch. In the fo llowing sec- tions, we co mpare re sults o f the two a pproach es and explain the results analytically . NUMERICAL RESUL TS The n etwork is constructed as de scribed in Sec. II an d in Ref.[37], with N = 10 00 , m = 3 and different values of p . The dyn amics of p ackage d eliv ering is then imp lemented o n the network. W e first consider the normal appr oach in which all the no des hav e the same value of β . Intu iti vely , a larger β can assure free traffic flow fo r a larger crea tion rate sign i- fied by a larger value of λ . Here, we fix λ = 0 . 01 and take β = 0 , 0 . 05 , and 0 . 1 to illustrate the effects. For β = 0 , ev ery node has a cr eation rate λk i depend ing on k i , but the deliver - ing rate o f fo rwarding at m ost one message per time step ap- plies to all no des. T o und erstand how cong estion occur s, we calculate the average numb er o f packets h n ( k ) i on the nod es with a g iv en num ber of links k . This quan tity serves to sh ow where are the longest queues. Numeric al simulatio ns show 0 5 10 15 20 25 0 100 200 300 k 0 30 60 90 0 1000 2000 3000 k 0 5 10 15 0 2 4 k 0 10 20 0 1 2 k (a) (b) FIG. 1: The a verage number of packets h n ( k ) i as a function of the number of links k in networks characterized by m = 3 and N = 1000 for (a) random growing networks ( p = 1 ) an d (b) scale- free networks ( p = 0 ). The parameter characterizing the message creation rate is λ = 0 . 01 . Results are obtained after t = 500 time steps and a veraging ov er 1 00 dif ferent rea lizations fo r a gi ven set of parameters. Dif ferent symbols label dif ferent pack et-processing ca- pabilities: β = 0 (circles), β = 0 . 05 (squares), and β = 0 . 1 (stars). that they are th e nodes with more link s, as sho wn in Fig.1 (cir- cles) for systems after t = 500 time steps. Th e results are ob- tained by a veraging over 100 d ifferent re alizations for a given set of pa rameters. Fig.1(a) shows the results fo r random grow- ing networks ( p = 1 ) and Fig.1(b) shows the resu lts for scale- free network ( p = 0 ). It is c lear that in b oth c ases, the node s with large nu mber of lin ks are mo re likely to be cong ested. Comparing the results in Fig .1(a) with (b) , one sees that the accumulatio n of packets in scale-free networks is m uch pro - nounc ed than that in ran dom network s. The results indicate that conge stion is m uch easier to occur in scale-free networks. For β = 0 . 05 (squares in Fig. 1), the a ccumulation o f pa ckets is greatly suppressed in both lim its of the und erlying network. In particular, congestion almost disappeared in the case of ran- dom growing networks. For even high er m essage-pro cessing capability β = 0 . 1 (stars in Fig.1), congestion disap peared in both the ran dom and scale-f ree n etworks. These resu lts also indicate that ther e exists a cr itical value β c for a given λ so that cong estion o ccurs for β < β c . W e will study the d epen- dence of β c on λ in n etworks of different un derlyin g stru ctures characterized by p . Another p oint that is worth n oticing is that the n odes in a range of small to interm ediate d egrees (see inset in Fig.1) in the normal approach usually carry fewer messages than the unifor m delivering cap acity . The result implies that it is un- necessary for these nodes to have a higher delivering capa- bility ch aracterized by a finite β . It leads us to consider the efficient appr oach . I n the scale-free ( p = 0 ) limit as shown in Fig.1(b), the degree distribution is a power law and the no des that carry k ≥ 20 acco unt fo r only 3% of all the N = 1 000 nodes. T o illustrate the idea of the efficient approach, we take f = 3% , i.e., we assign a non-vanishing β only to nodes 4 0 30 60 90 0 200 400 600 800 k 0 30 60 90 0 5 10 k (a) (b) FIG. 2: h n ( k ) i as a function of k for scale-free networks ( p = 0 ) with N = 1000 and m = 3 for the efficient approach (circles) with f = 3% and the normal approach (stars). The parameter characteriz- ing the message creation rate is λ = 0 . 01 . Results are obtained after t = 500 time steps and by av eraging over 100 different realizations for a gi ven set of p arameters. T wo v alues of β are us ed: (a) β = 0 . 05 and (b) β = 0 . 1 . that have k ≥ 20 . Figure 2 compares r esults of the normal approa ch (stars) and the efficient approac h (circles) for two values of β . Obviously the difference in h n ( k ) i between the two app roaches is small in scale-free n etworks. As th e e ffi- cient appr oach does not require all nodes to be equ ipped with the sam e capability , it represents a mor e p ractical and cost- effecti ve way to avoid jamming . In contrast, we note that the efficient approach do es not work so w ell in ran dom networks ( p = 1 ). It is beca use the degree distribution is narrower com - pared with the p = 0 case. Ther efore, the qu eues are more ev enly distributed amo ng the nod es a nd con gestion is not r e- stricted to the n odes amon g the highest degrees. It is thus necessary to assign a finite f to a larger fraction of n odes to av oid congestion , and the two approache s becom e similar . W e defin e h n 1 ( t ) i to b e the average num ber of messages per node. In the cong ested regime, h n 1 ( t ) i increases with time t . In th e no n-con gested regime, h n 1 ( t ) i flu ctuates arou nd a constant. The slope of < n 1 ( t ) > after th e tran sient can thu s be used to d etermine β c [30, 3 2]. For a given λ , the slop e gradua lly d ecreases as β incr eases. The value of β th at the slope becom es zero gives β c . For β > β c , the slope rem ains zero. Figu re 3(a) shows typ ical results with λ = 0 . 01 for thre e different values of β within the efficient appro ach ( f = 3% ). For β = 0 . 0 5 , h n 1 ( t ) i inc reases with tim e witho ut boun d. The critical value is found to b e β c = 0 . 059 where th e slope vanishes. For β = 0 . 7 > β c , the slop e remains zero . As congestion mainly occu rs at the no des with large degrees, the number of message s h n 2 ( t ) i a verage d over the 3% o f nodes should also sho w a similar behavior with time. It is inde ed the case (see Fig.3(b)). Next, we stud y the depend ence of β c on the creation rate characterized by λ in scale-free an d ran dom growing n et- works. Fig.4(a) shows the results in the scale-free limit 0 100 200 300 400 500 0 0.5 1 t 0 100 200 300 400 500 0 10 20 30 t (a) (b) FIG. 3 : (a) The average number of m essages per node an d (b) the a v- erage number of messag es among the top 3% nodes with the highest degree s as a function of ti me in a scale-free n etwork ( p = 0 ) for th ree differe nt v alues of β . The lines from t op to bottom refer to β = 0 . 05 , 0 . 059 , and 0 . 07 , respective ly . The other parameters are λ = 0 . 01 , f = 3% , N = 1000 , and m = 3 . ( p = 0 ) for both the n ormal (circ les) and efficient (stars) approa ches. For small λ , β c vanishes as the default deliver - ing rate of o ne m essage per time step is already sufficient to handle the small m essage cr eation rate. For the rang e of λ shown in the figu re, β c increases linearly with λ and the two approa ches giv e similar results. This again shows that the ef- ficient appr oach perfo rms as go od as ad justing β across the whole network. It should be n oted that f or larger values of λ (beyond the range shown h ere), assigning a finite β to only the to p 3% of n odes may not be suffi cient to av oid conges- tion. For r andom growing ne tworks ( p = 1 ), we show β c ( λ ) in Fig.4(b) only for the nor mal app roach, as the efficient ap- proach becomes similar to the n ormal appro ach. Qualitatively , β c ( λ ) sh ows a similar behavior to that in scale-free networks. Quantitatively , β c = 0 for a larger rang e of λ in rand om networks and th e slo pe of the linear d ependen ce in β c ( λ ) is smaller . It is because the no des in ran dom networks are mo re “homog eneous” and a q ueue will n ot emerge at th e hub s for small λ as in the case of scale-f ree networks. The function β c ( λ ) in Fig.4(a) also d ivides the β - λ space in to two regions. The r egion above th e line rep resents a n on-con gested or free flow regime and that belo w the line rep resents a congested regime. Thus for given λ , one ca n go fro m a con gested to a non-co ngested regime by increasing β . Similar ly , for a g iv en β , one can go from a n on-con gested regime to a con gested regime by in creasing λ . Although we only present resu lts for networks with N = 1000 nodes, we have checked that the lin- ear dependence o f β c on λ also holds for netw ork s with larger N . 5 0 0.002 0.004 0.006 0.008 0.01 0.012 0 0.02 0.04 0.06 0.08 λ β c 0 0.005 0.01 0.015 0.02 0 0.02 0.04 0.06 0.08 λ β c (a) (b) FIG. 4: β c as a f unction of λ for (a) scale-free networks ( p = 0 ) of N = 1000 and m = 3 within the normal (circle) and ef ficient (stars) approaches ( f = 3% ); and (b) random gro wing networks ( p = 1 ). THEORETICAL EXPLANA TION In this section, we aim at explainin g the behavior of β c ( λ ) quantitatively . As discu ssed, h n 1 ( t ) i behaves differently for β < β c and β ≥ β c . In th e no n-cong ested or free flow r egime, the steady state satisfies the Little’ s law [38], which states th at the numb er of delivered messages is ba lanced by the num ber of n ewly created m essages. Th is sugge sts a way to estimate β c ( λ ) for a g i ven under lying network characterized by the p a- rameter p . Consider the n ode with the highe st degree k max ( p ) where messages are mo st likely to accumulate. At the critical value β c , the node can h andle 1 + β c k max ( p ) p ackets per time step, while th e numb er is smaller for oth er nod es. For scale-free networks ( p = 0 ), k max ≈ mN 1 / ( γ − 1) [39]. W e note that the pac kets at a no de i origina te f rom two different sources: those c reated at no de i and those passing by n ode i . The cre- ation rate λk i is linear in k i . The packets passing by are more likely to go through the nodes with higher de gre es, and hence the number of packets passing by a node will b e some n onlin- ear fu nction o f its d egree. W ith these co nsideration s, we ap- proxim ate the av erag e nu mber o f p ackets at the critical value β c at some node i as α ( k i , p )(1 + β c k max ( p )) k i /k max ( p ) , where 0 < α ( k i , p ) ≤ 1 and α ( k max , p ) = 1 is a n onlin- ear decreasing fu nction of k i that reflects the contribution o f messages passing by the nod e. For the case of p = 0 , no ting that ther e are only 1 + β c k max (0) packets at the node s with k max , the average numbe r of packets at th e nodes with small and interm ediate d egrees will b e less than one . This implies that there is not enoug h pa ckets for the par ameter β to take effect at these nod es. Therefore, we expect the expression α ( k i , p )(1 + β c k max ( p )) k i /k max ( p ) to be a good approxima- tion for both the normal an d ef ficient approach es in scale-f ree networks. On the other hand , there are 2 mλN ne wly created packets in each time step. Let h ( p ) be th e diameter of the network which measur es the av erag e nu mber of nodes that a packet passes thr ough on its way to its destinatio n, includ ing the destination itself. If the system is in the non-cong ested regime, ther e are a total of h ( p )2 mλN m essages in the sys- tem. T o avoid a queu e at any node and hen ce conge stion, all the messages should be han dled by the no des in a time step. Thus, we have h ( p )2 mλN = N X i =1 α ( k i , p )(1 + β c k max ( p )) k i /k max ( p ) . (1) Writing P N i =1 α ( k i , p )(1 + β c k max ( p )) k i /k max ( p ) ≡ α 1 ( p ) P N i =1 (1 + β c k max ( p )) k i /k max ( p ) , we then have β c ( λ ) = h ( p ) λ α 1 ( p ) − 1 k max ( p ) . (2) From Eq.(2), it f ollows th at (i) β c > 0 only when λ is su ffi- ciently large, (ii) for a given stru cture of the network (fixed p ), β c increases linearly with λ , and (iii) th e slope of β c ( λ ) d e- pends on the under lying network structure c haracterized by p . All these fe atures agree with those observed in the numeric al results (see Fig.4). Equation (2) can be ap plied to estimate β c , if we know h ( p ) , k max ( p ) , an d α 1 ( p ) . The diameter h ( p ) can be cal- culated using the m ethod in Ref. [35]. Figur e 5(a) shows h ( p ) over the whole range o f p . It increases only slightly as p increases. On the other hand, k max ( p ) drops sensitively with p as shown in Fig. 5(b). Since α 1 ( p ) depen ds only on p , it can be determined by using numer ical results of β c for gi ven λ . For example, β c ( λ = 0 . 01) = 0 . 059 in the scale-free ( p = 0 ) limit. T ogeth er with h (0) = 3 . 3 2 and k max (0) = 85 (see Fig. 5), Eq.( 2) gives α 1 (0) ≈ 0 . 452 2 and the slop e of the line β c ( λ ) is h ( p ) / α 1 ( p ) = 7 . 34 . Similarly , β c ( λ = 0 . 01 2) = 0 . 0 27 in the rando m network limit ( p = 1 ). T ogeth er with h (1) = 3 . 82 and k max (1) = 25 (see Fig.5), Eq.(2) gives α 1 (1) ≈ 0 . 6842 an d the slope of the line β c ( λ ) to be 5 . 58 . These values are in reason able a greement with the slop es in the plots in Fig.4. Eq uation ( 2) also shows that β c = 0 f or λ < λ min = α 1 ( p ) / [ h ( p ) k max ( p )] . Using the extracted v alues of the par ameters, we g et λ min = 0 . 0016 for p = 0 an d λ min = 0 . 007 2 fo r p = 1 . T hese values are consistent with the results in Fig.4. CONCLUSIONS In network c ommun ications, a simple way to con trol n et- work traffic is to limit the len gth o f the que ues [40], e.g . by source quenching, rand om dr opping , fair q ueueing , etc. This will, however , in crease the av erag e d eliv ering time. As many rea l-life networks are heterogen eous networks and many shortest pa ths between any two no des p ass th rough the nodes with hig h d egrees, it will b e these nod es that control the n etwork traffic. Thus we study the strategy of enh ancing the me ssage d eliv erin g cap ability selectively at the n odes with high degrees. W e fo und that the strategy work s well in net- works with scale-free chara cter and it is a hig hly co st-effecti ve 6 0 0.2 0.4 0.6 0.8 1 3.2 3.4 3.6 3.8 4 p h(p) 0 0.2 0.4 0.6 0.8 1 20 40 60 80 100 p k max (p) (a) (b) FIG. 5: (a) The diameter h ( p ) and (b) the high est degree k max ( p ) i n a network as a function of the underlying network structure charac- terized by the parameter p . The network s have N = 1000 nodes and m = 3 . way to a void network congestion. This idea is in line with the recent results in Ref.[41, 42]. The major difference in n etwork cong estion in a scale-free network and a rando m growing network is that the scale-fr ee network h as h ubs, i.e., nodes that are co nnected to many othe r nodes. T he degree distribution in a scale- free network fol- lows a p ower law for large networks. In a rando m n etwork, the degree distribution is r elativ ely narrower and the degrees of the nod es do not differ by mu ch. For identical message creation rate an d deliv ering rate at the no des, it is then ex- pected that co ngestion will take place m ostly at the n odes of high degree s in a scale-f ree network. For a ran dom n etwork, congestion may take place at more places across the network. Strategically enh ancing the message- processing c apability at the h igh-degree no des in a scale-fr ee network as in the e f- ficient approach studied in the present work will greatly en- hance network traffic. This strategy also makes go od u se of the power-la w degree distribution in that it is sufficient to al- locate r esources to enhan ce the ca pability of a small frac tion of nodes with high degrees in a netw ork in order to avoid traf- fic congestion . If we car ry ou t the same strategy to a rando m growing ne twork, a much larger f raction of nodes will be in- volved a nd hence the cost-effecti veness will be lowered. In summ ary , we have constru cted and studied a m odel of commun ications in complex network s. W e u se a network model that can be tuned fro m the scale-free preferential grow- ing n etwork limit to the ran dom growing network limit. Our model assumes a message creation rate λk i that depe nds on the degree of a no de. Each node a lso h as a message deli vering rates of 1 + β k i . The m odel thus re presents a step towards a more realistic mod elling of tr affic cong estion in comm uni- cation network s in tha t it incorpo rates the different capacities of the nod es in creating and handling messages. In particu lar , we studie d an efficient ap proach that increases the commu ni- cation capacity in scale-free networks. Numerical re sults in - dicate that o ur efficient approach of selecti vely enhancin g the delivering rate in a small f raction of nodes p erforms as good as enhancin g the capability of a ll the nodes in the network. Con- sidering the c ost of en hancing the d eliv erin g rate a t a node, the present schem e will be highly cost-effecti ve. W e also studied the depen dence of the critical value of β , which ch aracterizes the message delivering rate, on the par ameter ch aracterizing the message creation rate λ . 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