An adaptive routing strategy for packet delivery in complex networks

An adaptiv e routing strategy f or pack et delive ry in complex networks Huan Zhang, 1 Zongh ua Liu, 1 Ming T ang, 1 and P . M. Hui 2 1 Institute of theor etical physics and Department of Physics, East China Normal Univer sity , Shanghai, 200062, P . R. China 2 Department of Physics, The Chinese University of Hon g K ong , Shatin, New T erritories, Hong Kong (Dated: 25 June) W e present an ef ficient routing appro ach for deliv ering p ackets in complex network s. On deliv ering a message from a node to a destination , a node forw ards the message to a neighbor by esti mating the w ait ing time along the shortest path from each of its neighbors to the destination. This projected waiting time is dynamical in nature an d the path thro ugh which a message is deli vered w ould be adapted to the d i stribution of messages in the netw ork. Implementing the a pproach on scale-free networks, we sho w that the present approach performs better th an t he shortest-path approach a nd ano ther appro ach that takes into account o f the waiting time only at the neighboring nod es. Key features in nu merical resu lts are explained by a mean field theory . T he approach h as the merit that messages are distributed amon g the no des according to the capabilities of the nodes in handling messages. P AC S numbers: 89.75.Fb,89.20.-a,05.70.Jk I. INTRODUCTION The proble m of traffic congestions in communication n et- works is u ndou btedly an imp ortant issue. The problem is re- lated to the geometr y of th e un derlying n etwork, the rate that messages ar e genera ted and delivered, and the r outing strat- egy . Many studies ha ve b een focused on spatial structures such as regular lattices and the Cayley tree [1, 2, 3, 4, 5 , 6, 7, 8, 9, 10]. Random networks and scale-free ( SF) n etworks have also be en widely studied. The for mer is homog eneous with a Poisson degree distribution; while the latter typically exhibits a power -law de g ree distribution of the form P ( k ) ∼ k − γ sig- nifying th e existence of nodes with large d egrees. SF ne t- works ar e found in many re al-world networks, such as the Internet, W orld W id e W eb (WWW), and metabolic n etwork [11, 12, 1 3]. A standard model of SF netw orks is the Barab ´ asi and Albert (B A) mo del of g rowing networks with preferential attachments [14, 15]. Th e B A model gives a degree distribu- tion of P ( k ) ∼ k − 3 and is non-assortative [16, 17], i.e., the chance of two node s b eing c onnected is indep endent of the d e- grees of the nodes conce rned. While there a re other variations on the BA mod els that give a degree expon ent that d eviates from 3 [18, 19, 20, 21, 22], the B A model still serves as the basic model for SF networks. In the p resent work , we stud y how a dynamical an d ada ptiv e rou ting strategy would enh ance the p erforma nce in delivering messages ov er a B A scale-free network. In commun ication models, the no des are taken to be b oth hosts and routers, and the links serve as possible path- ways through wh ich messages or packets are for warded to their destination. Early studies assume d a constan t (degree- indepen dent) pac ket generation rate λ and a constant rate of delivering one p acket per time step a t each node . As λ in- creases, the traffic goes from a free-flow ph ase to a congested or jamming phase. Obviously , such models are too simple for real-world networks. More realistic models sho uld incorpo - rate the fact that n odes with highe r degrees w ould ha ve higher capability of handling info rmation packets an d at t he same time g enerate more packets. Mo dels with degree-dep endent packet-delivery rate of th e for m (1 + β k i ) [23, 24] an d degree- depend ent packet-generatin g rate [24] o f the f orm λk i have re- cently been proposed and studied , with routing st rategy based on f orwarding messages th rough the sho rtest path to their des- tination. Im plementing this routing strategy in SF and rand om networks indicate that it is easier to lea d to congestion s in SF networks than in ran dom n etworks [23, 24]. It is b ecause th e nodes with large degree s in SF networks are o n many sho rtest- paths between two arb itrarily chosen no des, i.e., large be- tweenness [ 25]. Many packets will be passing by and qu eue- ing up a t these no des enroute to their destination. In a random network, jamming is harder to occu r as the packets tend to b e distributed quite uniformly to each node. A g ood routing alg orithm is essential for sustaining the proper fun ctioning of a network [2 6, 27, 28, 29]. The sh ortest- path rou ting ap proach is b ased on static information, i.e ., on ce the network is co nstructed, the sho rtest-paths a re fixed. T o im- prove routing efficiency , Echen ique et al. [30, 3 1] p roposed an approa ch in which a node would ch oose a neig hborin g nod e to deli ver a packet by considerin g the shortest-pa th fro m the neighbo ring node to the d estination and the waiting time at th e neighbo ring node . The waiting tim e depends on the numb er of packets in the que ue at a neighb oring nod e at the time of d e- cision and th us corresponds to a dynamical or time- depend ent informa tion. This algorithm p erform s be tter than the shortest- path a pproach , as packets may b e delivered not necessarily throug h the shortest-path and thus the loading at th e high er degree n odes in a SF network is red uced. T he approac h has also been applied to networks with degree-depen dent packet generation rate [32 ]. Recently , W an g et a l. [33] prop osed an algorithm that ten ds to spread the packets evenly to nodes by considerin g inf ormation on nearest neighbors. Ho wever , the delivering time turns out to b e mu ch long er than that in the shortest-path approach as the packets tend to w ander around the network. In th e present work, we p ropose an ef ficient routin g strategy that is based on th e projected waiting time along the shor test- 2 path from a ne ighbor ing node to the destination . The algo - rithm is im plemented in B A scale-fr ee netw o rks, with d egree- depend ent packet generatin g and delivering rates. Results show that jamming is harder to occur using the present strat- egy , when co mpared with both the sho rtest-path app roach an d the Ech enique’ s ap proach. Key f eatures o bserved in n umeri- cal results a re explained within a m ean field treatmen t. The present approach has the ad vantage of spreading the packets among th e no des accordin g to the degrees of th e nod es. In this way , ev e ry n ode can c ontribute to the packet d eliv er y process. The paper is organized as follows. T he m odel, including the un derlying network , the packet gen eration and deli very mechanisms, and routin g strategy , is introd uced in Sec.II . In Sec.III, we pr esent numerical results and co mpared them with those of the o ther routing strategies. W e also expla in key fea- tures within a mean field theory . W e sum marize our results in Sec.IV . II. MODEL The un derlyin g netw o rk structur e is taken to be the Barabasi-Albert (B A) scale-free growing network with N nodes [15]. Startin g with m 0 nodes, each n ew n ode enter- ing th e network is allowed to establish m ne w link s to exist- ing nod es. Prefer ential attach ment wher eby an existing node i with a hig her degree k i has a highe r pr obability Π i ∼ k i to attract a new link is imposed. The mean degree of th e net- work is h k i = 2 m and the degree distribution P ( k ) f ollows a power - law beha v ior of the form P ( k ) ∼ k − 3 . The dynam ics o f packet generation and delivery is imp le- mented as follows. Due to the inhomoge neous nature of the B A network, it is more natur al to impo se a packet genera- tion r ate that is p ropor tional to th e degree of a node. At each time step, a nod e i creates λk i new packets. Th e fra ctional part of λk i is impleme nted pro babilistically . A destination is random ly assigned to each created pac ket. The newly cre- ated packets will be put in a que ue at the nod e and delivery will be made on the fir st-in-first-out basis. The packets in the queue may consist of those wh ich are created at pre v ious time steps and re ceiv ed fr om n eighbo ring nod es enro ute to their destination. W e also assume a packet deli very rate that is p ro- portion al to the d egree of a no de [24]. At each time step, a node i delivers at most (1 + β k i ) packets to its n eighbor s. The fractional p art of (1 + β k i ) is implemented prob abilistically . A larger β implies a higher packet-han dling capab ility , b ut it would translate into highe r cost or capital. Here , the p aram- eters λ and β ar e taken to be node-ind ependen t. A packet is removed from the system upo n arriv al at its destination. For a given generatio n rate ch aracterized by λ , there exists a crit- ical value of the delivery r ate β c such that for delivery rates β < β c , packets tend to accumu late in the n etwork resulting in a jam ming p hase; while for β > β c , a non- jamming phase results as there are as many p ackets deli vered to their destina- tion as created . A better performan ce is thus characterized by a smaller value of β c . The n ovel featu re of th e present work is the r outing strategy or the selection of a neig hbor in delivering a p acket. T he idea is to choose a neighbor that would g iv e the sho rtest time, in- cluding waiting time, to deliver the packets along the shortest path from the cho sen neigh bor to the d estination. Consider a packet with destination nod e j leaving node i . E ach of the k i neighbo rs of node i has a shortest path to the destination node j . Th e shortest path refers to the smallest nu mber o f links from a node to an other . However , due to the possible accu- mulation o f p ackets at each node, the numb er of time steps it takes to d eli ver the message may be d ifferent from the n umber of links along the shortest path. Consider a n eighbor labelled ℓ of the node i . W e label the shortest path fro m node ℓ to j by { S P : ℓ, j } . Alo ng this path, we e valuate the following quantity for the node ℓ : d ( ℓ ) = X s ∈{ S P : ℓ,j } n s 1 + β k s , (1) where the sum is over the nodes along the shortest path { S P : ℓ, j } , e x cluding th e destinatio n. Here, n s is the n umber of packets accumulate d at nod e s , at the m oment of decision. Thus, d ( ℓ ) is an estimate of the time that a packet would take to go fro m no de ℓ to the destination j thr ough the shor test path. Node i would choose a neighboring n ode with the min i- mum d ( ℓ ) to forward the packet, i.e., the selection is based on min { d ( ℓ ) , ℓ ∈ { i }} , where { i } is the set o f k i nodes co nsist- ing of the neigh bors of node i . This p rocedur e is repeated for each nod e an d eac h packet in e very time step. For a n etwork far from j amming, each node can handle all th e packets in e v- ery tim e step. In this free-flo w situation, the q uantity d ( ℓ ) sim- ply measures the shortest path d ℓ,j from ℓ to j . When packets are queu eing up at the nod es, howe ver , a deli very m echanism based on d ( ℓ ) takes into accoun t of th e que ueing time an d may not pass the p acket to a neighboring n ode that is clo sest to the destination. T o justify our routing scheme, we will compare results with two other ro uting strategies widely stud ied in the literature. Using the sam e packet generatin g m echanism, the shortest- path app roach selects a neighbo r with the shor test path to the destination for forwarding a packet. Echeniqu e et al. [30, 31] propo sed an appr oach that takes into acc ount of the waiting time. For a deli vering rate o f one packet per time step, they propo sed to choose a neighbo r th at has a minimum value o f hd ℓ,j + (1 − h ) n ℓ , where d ℓ,j is the shortest path length from node ℓ to j . The parameter h is a weighin g factor, which can be taken as a variational parameter and h ≈ 0 . 8 is fo und to giv e the b est perfo rmance. The Ech enique’ s appro ach thus accounts for the waiting time only at the neigh boring no des. For a deli very rate of (1 + β k i ) , a modified Echen ique’ s ap- proach is to cho ose a neighboring node with a minimum value of δ ℓ = hd ℓ,j + (1 − h ) n ℓ 1 + β k ℓ . (2) W e ha ve checked that for a given value of λ , the smallest value of β c is attained for values of h ∼ 0 . 8 to 0 . 8 5 . I n what follows, we will use a value of h = 0 . 8 for the Ech enique’ s appro ach giv en by Eq.(2). 3 0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 0 1 2 3 0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 0 5 1 0 1 5 2 0 ( a ) t ( b ) t FIG. 1: (a) Numerical results for the average number of packets per node < n ( t ) > and (b) the average deli vering time < T > as a function of time for λ = 0 . 02 . Lines from top to bottom correspond to β = 0 . 04 , 0 . 048 and 0 . 07 . III. RESUL TS AND DISCUSSION The different p hases in a n etwork can b e illustrated by lo ok- ing at the average number of packets per node at a g iv en time h n ( t ) i and th e average time for a packet to remain in th e net- work or the delivering time h T i . W e take m 0 = 3 an d m = 3 and construc t a B A scale-f ree network of N = 100 0 nodes. Figure 1 sh ows the results of h n ( t ) i and h T i as a f unction of time for a fixed value o f λ = 0 . 02 . As β increases, th ere are distinct behavior . For values of β smaller than some critical value β c ( λ ) , h n ( t ) i grows almo st linearly with tim e after the transient ( see Fig .1(a)). This corr esponds to a jammin g ph ase. As β increases, the slope in the long time behavior decreases, indicating a slo wer accumulation of packets in the netw ork as the ability of handling packets β increases. For β > β c ( λ ) , h n ( t ) i becomes indepe ndent of time in the long time limit. This correspon ds to a non- jamming p hase. Similarly behav- ior is exhib ited in h T i . In the jamming p hase, h T i increases with time mono tonically , du e to the incre asing waiting time in the queues at in termediate nodes as a packet is for warded to its destination . Fewer packets are delivered to th eir destina- tion than generated . In th e no n-jammin g phase, h T i becomes indepen dent of time in the long time limit. In this re gime, fur - ther increasing β will lead to smaller h n ( t ) i and sho rter h T i in the long time limit until the se quantities satur ate. T his is possible since a no n-jamming phase correspo nds either to the case in which all th e packets at the nod es are forwarded e v- ery time step or steady queues of packets exist at the nodes. In both cases, the numb er of p ackets does n ot increase in th e long time limit. The f ormer case is the free -flow phase, while the latter is reminiscent o f the synchr onized p hase in vehicu lar traffic flows [34] in wh ich th e packets un dergo a stop-an d-go behavior . For β = 0 . 07 > β c for example, h T i ≈ 9 . 5 , which is some wh at larger than the a verage sh ortest distance or diam- eter D ≈ 3 . 332 o f the network. This in dicates tha t, due to the rou ting strategy in forwarding a packet, th e dyn amics in the fr ee-flow phase is different from that of the shortest-p ath 0 .0 2 0 .0 3 0 .0 4 0 .0 5 0 .0 6 0 .0 7 0 .0 8 0 .0 0 0 .0 5 0 .1 0 0 .1 5 0 .2 0 FIG. 2: Numerical results for the quantity η as f unction of β for λ = 0 . 02 . The crit ical va lue β c separates the behavior of η = 0 for β > β c and η 6 = 0 fo r β < β c . 0 .0 0 0 .0 1 0 .0 2 0 .0 3 0 .0 0 .1 0 .2 0 .3 c FIG. 3: The crit ical v alue β c ( λ ) for three different routing ap- proaches for forwarding packe ts: the present approach (circles), the Echenique’ s approach with h = 0 . 8 (stars), and the shortest-path approach (squares). The lines are guides to eye. approa ch. The critical value β c ( λ ) can be determined by co nsidering the quantity η = lim t →∞ 1 2 mλ < ∆ n > ∆ t , (3) where ∆ n = n ( t + ∆ t ) − n ( t ) an d th e average is over all the n odes at a time t . This quan tity η ∈ [0 , 1] is basically the slope o f h n ( t ) i in the long time limit. In the n on-jamm ing phase, the slo pe vanishes and η = 0 ; while in the jamming phase, η > 0 . Figure 2 shows η as a func tion of β , for a fixed value of λ = 0 . 0 2 . Th e critical v alue β c can b e iden tified as the value that sep arates the η = 0 and η 6 = 0 b ehavior . W e carried out similar calculations f or different values of λ and determined β c ( λ ) . The r esults are shown in Fig.3 (circles) . W e will explain the fo rm of β c ( λ ) using a m ean field theory . The cur ve β c ( λ ) c an also be regarded as a p hase b ound ary in the λ - β plan e, separating the jamm ing pha se below the cur ve and the non-jam ming phase above the curve. T o show the superio r performance of ou r routing strategy , 4 0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 0 2 0 0 4 0 0 6 0 0 0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 0 2 0 4 0 6 0 ( b ) k ( a ) k FIG. 4: The average number of packets n k at a node of degree k as a f unction of k for λ = 0 . 02 for (a) β = 0 . 06( > β c ) and (b) β = 0 . 04( < β c ) . In each case, results are sho wn at three dif ferent times of t = 100 (circles), 200 (stars) and 300 (squares) time steps. we also p erform ed calculations using the shortest-path ap- proach and th e Echeniq ue’ s approach with h = 0 . 8 in Eq.(2) . The same d egree-depen dent pac ket gener ating mech anism is used. Results of β c ( λ ) for these two models are also shown in Fig.3 for c omparison . The present approach gi ves the best perf ormanc e. For a gi ven λ , we see the im provement in perfo rmance from th e shortest-path ap proach throug h the Echeniqu e’ s ap proach to the p resent ap proach, signified by the dr op of β c . F or th e shor test-path app roach, it has been shown [24] that β c ( λ ) follows the functional form of β S P c = αD ( λ − λ S P min ) , (4) where α ≈ 2 and λ S P min = 1 / ( αD k max ) with D be ing the di- ameter and k max the m aximum degree of the network. For λ < λ S P min , β S P c = 0 . W ith the present app roach, β c ( λ ) follows a similar f unctiona l f orm, but with a higher value o f λ min and a smaller p refactor that gives the slop e. Both the present app roach and the Echeniqu e’ s app roach p erform bet- ter than the shortest-appr oach appr oach because packets a re re-directed to other nodes when there are long queues at the hubs. The better performance of th e present approach is ac hieved by spreading the pa ckets among the nodes so that the n umber of p ackets at a nod e is proportional to the degree k of th e node in the free-flow phase. W e use a m ean field ap proach to illustrate this poin t. Let n k be th e average number of packets at th e nodes with degree k . In the free-flo w ph ase wh ere n k < 1 + β k , we have dn k ( t ) dt = λk − n k ( t ) + k k max X k ′ = k min P ( k ′ | k ) n k ′ ( t ) k ′ − λ h k i . (5) The first an d s econd terms denote th e packets genera ted at the node and de li vered to neig hborin g nodes, respe ctiv ely . The third ter m acco unts fo r the packets delivered into the no de from its neighbo ring nodes. Here P ( k ′ | k ) is the cond itional probab ility that a node of degree k has a neighb or of degree k ′ and the sum ru ns from the minimum degree k min to k max in the network. In the free- flow regime, the pac kets that are removed upon arri val a t th eir destina tion can be assum ed to be k -indep endent and ap proxima ted by the term λ h k i . The non- assortativ e feature o f BA networks [16 , 17] gives P ( k ′ | k ) = k ′ P ( k ′ ) / h k i , where P ( k ′ ) is the d egree distribution. After the transient behavior , dn k /dt = 0 and we have n k = ( λ + < n > < k > ) k − λ h k i , (6) where h n i = P k max k ′ = k min P ( k ′ ) n k ′ is the mean numbe r of packets per nod e. Thus for k > h k i = 6 , n k ∼ k in the free-flow pha se after th e transient. Figure 4 (a) shows the nu - merical results obtained by a veragin g the numb er of p ackets on th e no des with degree k at different tim es ( time t = 1 00 , 200 , 3 00 tim e steps) o f a run . In the f ree-flow phase, n k ∼ k and beco mes time-in depend ent after the transient, as s hown in Fig.4(a) for the case of β = 0 . 06 an d λ = 0 . 02 . This beha v ior is consistent with that in Eq.(6). For the jamming p hase, numer ical results (see Fig. 4(b)) show th at (i) n k ∼ k at a fixed instant and (ii) n k increases with time for fixed value of k . Th is behavior can b e under- stood provided that the packets are s till distributed among the nodes in proportio n to the degree k of a nod e via our strate g y . In this phase, th e long time behavior is characte rized by an increasing ac cumulation of packets an d th e delivery to desti- nations becomes ne gligible compared with pac ket generation. W ith n k > 1 + β k for all n odes and ignor ing the re moval of packets, Eq.(6) is modified to dn k ( t ) dt = λk − (1 + β k ) + k k max X k ′ = k min P ( k ′ | k ) 1 + β k ′ k ′ = − 1 + k ( λ + 1 < k > ) . (7) It follows that n k ( t ) increases with time t as n k ( t ) = n k (0) + ( k ( λ + 1 < k > ) − 1) t, (8) which describ es very well the feature s in Fig.4(b). Thus, the present app roach h as the effect o f reducin g ( increasing) the probab ility o f passing packets to neig hbors with high ( low) degrees when there are lon g (no or shor t) queues, resulting in a distribution of packets accor ding to th e degrees of the n odes. A rough estimate of β c ( λ ) can be obtaine d by eq uating n k in the free-flow phase to (1 + β c k ) . In particular, taking n k max = 1 + β c k max , we get from Eq.(6) that β c = λ − λ h k i k max + < n > < k > − 1 k max = λ − λ h k i k max + ( D − 1) P λk i N < k > − 1 k max =  D − h k i k max   λ − 1 ( D − h k i / k max ) k max  ≈ D  λ − 1 D k max  , (9) 5 where D is the average number of nod es that a packet p asses throug h from its or igin to the destination, which is the diame- ter of the netw ork in the free-flo w p hase. The last lin e is v alid for k max ≫ h k i . Comparing with Eq.(4) for t he shortest-path approa ch, we no te that λ min = 1 / ( D k max ) > λ S P min and the prefactor D , which gives the slope in Fig. 3, is smaller than that in the sho rtest-path appro ach. These feature s are con- sistent with numerica l results. In par ticular , for N = 1000 nodes, we fo und that D ≈ 3 . 332 an d k max ≈ 85 , giving λ min ≈ 0 . 007 , which is in reasonable agr eement wit h numer- ical results in Fig.3. IV . SUMMAR Y In summary , we ha ve proposed an ef ficient routing strategy on forwarding packets in a scale-fr ee network. The strategy accounts not only for the phy sical sep aration from th e des- tination but also on the w aitin g time along possible paths. W e showed th at o ur strate gy p erform s better than bo th t he shortest-path app roach and the Echeniqu e’ s approac h. Ana- lytically , we construct a mea n field treatme nt whic h g i ves re- sults in a greement with observed features in n umerical results. Our r outing stra tegy has the merit of distributing the packets among the nodes according to the degre e, and hence handling capability , of the no des. Althoug h our discussion was carried out on B A networks, we b eliev e that o ur app roach is also ap- plicable in other spatial structures. W e end by co mparing the three d ifferent routing strategies in m ore general ter ms. Th e sho rtest-path ap proach dep ends entirely on geometric al info rmation that is static . Once the origin an d th e destinatio n of a packet is known, the shortest- path is fixed. This strategy is non -adap tive , i.e., it will not be change with time. The Echenique’ s app roach co nsiders b oth geometrica l an d local dynamical information. By conside ring the w aitin g time at a neigh boring no de, a packet fr om a node i to a destination j will not always follow the sam e path. Thus, the Ec henique ’ s a pproach is a strategy that is a daptive , i. e., a decision based on th e curren t situation. The presen t strategy , like the Ec henique’ s approach, is also adaptive and makes u se of global inform ation in wh ich all the waiting times along a path a re taken into conside ration. W e see that b y allowing for adaptive strategies an d ta king more infor mation in to con- sideration, a better perform ance results. This line of th ough t is in acco rdance with that in complex adaptive systems [35] whereby active agents ma y adapt, inte ract, an d learn fro m p ast experience. It should b e, ho wever , no ted t hat it pays to be bet- ter . The shortest-path approach does not require update of the routing strategy . The Echen ique’ s approa ch and the present approa ch require co ntinuing upda te of the num ber of packets accumulated at the nodes. Such updating plays the role of a cost, with the pay off bein g the better performan ce. Practical implementatio n would hav e to consider the balance be tween the cost and the payoff. This work was suppo rted by the NNSF o f China under Grant No. 1047 5027 and No. 1063 5040 , by the PPS under Grant No. 05PJ1403 6, and by SPS unde r Grant No. 05SG2 7. P .M.H. acknowledges the sup port f rom the Resear ch Gr ants Council of th e Hong Kong SAR Government und er g rant number CUHK-40100 5. Email: zhliu@phy .ecnu.e du.cn Su ggested Referees: [1] H. Li, M. 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