Congestion phenomena on complex networks

Congestion phenomena on complex net w orks Daniele De Martino a , Luca Dall’Asta b , Ginestr a Bianconi b , and Matteo Marsili b a International Scho ol for A dvanc e d Studies SISSA and IN FN, via Beirut 2-4,34014 T rieste, I taly, b The A b dus Salam I C TP, Str ada Costier a 11, 34014, T rieste, Italy (Dated: Octob er 29, 2018) W e d efine a minimal mo del of traffic flo ws in complex netw orks containing t he most relev ant features of real rout ing schemes, i.e. a trade–off strategy b etw een topological-based and traffic-based routing. The resulting collecti ve b ehavior, obtained analytically for the ensem ble of uncorrelated netw orks, is physical ly very rich and reproduces results recently observed in traffic simulatio n s on scale-free netw orks. W e fin d that traffic con trol is useless in homogeneous graphs but may improv es global performance in inhomogeneous netw orks, enlarging the free-flo w region in parameter space. T raffic control also introduces non-linear effects and, beyond a critical strength, ma y trigger the app earance of a congested phase in a discontin uous manner. P ACS n umbers: 02.50.Ey , 68.35.Rh, 89.20.Ff, 89.75.Fb The fir st identified Internet’s congestio n co llapse da tes back to Octob er 1986, when data troughput from Lawrence Berkeley Natio nal Lab or atory to the Univ er- sity o f California in Ber keley dro pped from 3 2 Kbps to 40 bps. After that initial even t, traffic cong estion con- tin ued to threaten In terne t’s practitioners, even after the implemen ta tion of c ongestion c ont r ol algo rithms able to recov er the s y stem in case of traffic overloads [1]. Com- puter scientists hav e also elab orated several schemes of c ongestion avoidanc e , that should prevent co ngestion to o ccur by keeping the system far fro m high levels of traffic [2]. Congestion av o idance and co n tr ol are p erfor med by contin uously up dating the dynamics of end-to- end flo ws in res po nse to the v ar iation of the loa d level in the net- work. Their functioning dep end on the average r ound- trip-time (R TT) of the Ac knowledgement signals (ACKs) used to ex change infor ma tion be t ween ro uters. F or this reason, the observ ation of heter o geneous patterns in R TT time series has b een often pr op osed as evidence of per i- o ds of congestion [3]. Apart from these indirect measures, congestion ev ents are difficult to monitor and s tudy , so that a clear pheno menologica l pictur e is still missing. In spite o f the wide interest in developing o ptimal routing algorithms, muc h less atten tion is devoted to explore the- oretically the dynamical mec ha nisms resp onsible of con- gestion. It is poss ible that a better comprehension of these mec hanisms could help in understanding experi- men ta l data, in pr edicting cong estion even ts and design- ing better routing proto cols. The top olo gical and dynamical prop erties of dis- tributed informa tion systems, suc h as the In ternet [4], po se theoretical challenges of a similar nature of those addressed in statistical physics. Therefor e, unders ta nd- ing netw or k congestion phenomena has b e c ome a sub ject of int ens e rese a rch in this field [5], in particular after the works by T ak ay asu and collab ora to rs [6], in which the evidence o f a pha s e tra nsition fro m a fr e e-flow regime to a cong ested phase dep ending o n the load level w as rep orted. In a recen t work, E chenique et al. [7] ha ve shown us ing nu mer ical simulations that the natur e o f the co ngestion trans itio n depends on the t y p e of ro uting rules. They ha ve adopted a routing scheme that co uld be considered a first a pproximation of r e alistic transmis- sion c ontrol proto co l (TCP) routing: pa ck ets follow the shortest path b etw een their source and destination, but small detours are a dmitted in order to a void cong ested no des. In c ase of purely top olo gical routing (e.g. along the shortest paths) they found that the congested phase app ears contin uously , wher eas the transition is discon- tin uo us if so me traffic- aware scheme is c o nsidered. The effect o f ro uting rules on net work p erforma nce has also bee n addressed in [8]. In this Letter, w e put fo r ward a minimal mo del of traf- fic that preserves all interesting features previously ob- served in s imu la tions but is simple enough to be studied analytically . Both contin uous and discon tinuous pha s e transitions observed in [7] a re repro duced, and their re- lation to micro scopic pac kets dynamics is clarified. Let us consider a net work o f N nodes a nd let v ( i ) de- note the set of neighbors of node i . W e describ e tra ffic dy- namics as a con tinuous time sto chastic pro cess, in w hich pack ets are gener ated at each no de i with a rate p i . Each no de is endow ed with a first-in first-out (FIFO) queue in which pack ets are store d waiting to be pro cesse d. Let n i be the num ber of pac kets in the queue of node i . If n i > 0, no de i attempts to tra ns mit packets at a rate r i , which represents bandwidth, to one of the neighbo r s j ∈ v ( i ). W e assume the following pro babilistic routing proto col. Fir st, the no de j is c ho sen at random among the neighbo rs v ( i ) of i . Sec o nd, the fate of the packet being transmitted dep ends i) on whether j is the desti- nation no de for tha t pack et and ii) on the state of conges- tion of no de j . W e mo del b oth as probabilistic ev ents: i) we call µ j the probability that node j is the destinatio n of the pack et b eing s ent , mea ning that with pr obabil- it y µ j the pack et is “absorb ed” in the tra ns fer; a nd ii) we assume that the transfer is refused b y node j with a probability η ( n j ), which is increas ing with n j ; in this 2 case the pack et do es not lea ve no de i . This mo del relates in a s t y lized manner 1) the structural featur es, enco ded in the netw or k a nd in no des’ character is tics ( p i , µ i and r i ), 2) the proto c o l r esp onse to traffic, detailed in the function η ( n ) and 3) the ensuing tra ffic pr o cess. Our model is a simplifyied version o f more elabo rate mo dels[7, 9]. In these, packets ar e created at each no de with a g iven destination no de, ra ndomly chosen. Then they a re dispatched b y a given routing proto col, minimiz- ing some cost along the path. It is r easona ble to assume that the statistical natur e of the collective b ehavior do es not depend crucially on the details of the routing proto- col, and that it can be ca ptured b y the muc h simpler ran- dom diffusion-annihila tion pro ces s ass umed b y our mo de. In particular , the probabilistic traffic-aw a re routing as - sumed ab ov e is a simple analytica lly tra c table w ay to in- tro duce a tr affic dep endence in pack ets’ dyna mics. Also , in shortest- path r outing a node o f degree k i = | v ( i ) | is visited with a probability ∝ k β i , with β ≈ 2, wher eas in the random walk proto co l assumed by our mo de l the probability of visiting no de i is prop ortio nal to the de- gree k i . In both, high degree nodes are more expos ed to even ts of congestion, which is wh y star-like top olo gical structures are particularly vulnerable to congestio n and per form o ptimally only a t low traffic [9]. Our mo del, can easily acc ommo date for this statistical features with a sp ecific degree dependence of r i or b y consider ing degree- biased random w alks, lik e in [10]. The phase tra nsition from free-flow to a jammed phas e, is c hara cterized [7 ] by the order par a meter ρ = lim t →∞ N ( t + τ ) − N ( t ) τ P (1) i.e. the per cent a ge o f not a dsorb ed pa ck ets for unit time, where N ( t ) = P i n i ( t ) is the total num b er of pac kets in the system at time t , P = P i p i is the rate of crea tion of pack ets and τ is the observ a tion time. Note that a lo cal order par ameter, replacing N ( t ) by n i ( t ) and P by p i , can be defined in the sa me way . The mo del discussed ab ov e can efficien tly be analyzed, in the asymptotic time limit, within a mean field a pprox- imation P ( n 1 , . . . , n N ) = Q i P i ( n i ). The corr esp onding master equations ca n b e solved by messa g e pas s ing-type algorithm [11]. Rather than inv estig a ting sp ecific examples, w e prefer here to fo cus on the mec ha nisms inducing the change from a contin uo us to a discontin uous phase trans ition when traffic-aw a re routing is considered. A ma jor insight is obtained rephrasing the pr oblem in terms of ense m bles of graphs. W e co nsider uncorrelated random gra phs with degree distribution P ( k ), so that n k represents now the av erag e queue length of no des in classes of degree k . W e fo cus on the simple case µ i = µ , p i = p and r i = 1 for all i , and the routing proto co l η ( n ) = 0 for n < n ∗ and η ( n ) = ¯ η for n ≥ n ∗ . The mea n-field transition rates for no des with degree k are w k ( n → n + 1) = p + (1 − µ )(1 − 0 0.5 1 1.5 2 p 0 0.2 0.4 0.6 0.8 ρ 0 0.5 1 η 0 1 2 p /µ η = 0.75 η = 0.25 congested free /µ FIG. 1: ρ ( p /µ ) for an homogeneous graph from theoretical predictions for η = 0 . 25, 0 . 75. Inset: phase diagram for the same graph. ¯ q ) k z [1 − ¯ ηθ ( n − n ∗ )] and w k ( n → n − 1 ) = θ ( n )(1 − ¯ χ ). Here z is the a verage deg r ee, q k is the pr obability that a no de o f degree k has empty q ueue and χ k = ¯ η P { n i ≥ n ∗ | k i = k } is the pr obability that a node of degree k refuses pack ets. Likewise ¯ q = P k q k P ( k ) and ¯ χ = P k k z χ k P ( k ). The average queue length n k follows the rate e q uation ˙ n k = p + (1 − µ )(1 − ¯ q ) k z (1 − χ k ) − (1 − q k )(1 − ¯ χ ) (2) Note tha t summing ov er k and dividing by p w e obtain a measure o f the o r der par ameter ρ ( p ). Since ˙ n k depe nds linearly on k , high de g ree no de s are more likely to b e con- gested. Therefor e , in the stationary state for a given p , there exis ts a thr e shold re a l v alue k ∗ such that all nodes with k > k ∗ are congested whereas nodes with degre e less than k ∗ are no t conges ted. Congested no des ( k > k ∗ ) hav e q k = 0 and χ k = ¯ η . F or k < k ∗ , the ge nerating func- tion G k ( s ) = P n P ( n k ) s n of the pa ckets distribution can be computed fro m detailed ba lance. This takes the fo r m G k ( s ) = q k n 1 − ( a k s ) n ∗ 1 − a k s + ( a k s ) n ∗ 1 − ( a k − b k ) s o corres p o nding to a double exp onential, where a k = [ p +(1 − µ ) k z (1 − q )] / [1 − ¯ χ ] and b k = ¯ η [(1 − µ ) k z (1 − q )] / [1 − ¯ χ ]. F rom the no rmal- ization G k (1) = 1 and the condition ˙ n k = 0, we get expressions for q k , χ k and, finally , for ¯ q , ¯ χ . The v a lue k ∗ is self-consistently determined imp osing that no des with k = k ∗ hav e q k ∗ = 0, χ k ∗ = ¯ η and ˙ n k ∗ = 0 that translates in to the equation k ∗ = [1 − p − ¯ χ ] / [(1 − µ )(1 − ¯ η )(1 − ¯ q )] (3) The a bove set of clo sed equatio ns ca n b e solved nu- merically for any degr ee distr ibution P ( k ) and ρ ( p ) can be according ly c o mputed. The solution is particular ly simple for regular gra phs: k i = z , ∀ i in the limit n ∗ ≫ 1. Then the congestion free solution with ρ = 0 has q k = ¯ q = 1 − p /µ and χ k = ¯ χ = 0 and it exists for p ≤ µ . In the congested phase, instea d, all no des hav e n i → ∞ , i.e. ¯ χ = ¯ η and ¯ q = 0. This so lution has ρ = ˙ n / p = 1 − (1 − ¯ η ) µ/p and exists for p ≥ (1 − ¯ η ) µ . Therefore, in the in terv al p ∈ [(1 − ¯ η ) µ, µ ] b oth a co n- gested and a free phase c o exist, as shown in the inset of 3 0 0.1 0.2 0.3 0.4 0.5 p 0 0.2 0.4 0.6 0.8 1 ρ 0 0.1 0.2 p 0 0.5 1 ρ FIG. 2: ρ ( p ) for an uncorrelated scale-free graph ( P ( k ) ∝ k − 3 , k min = 3, k max = 110, N = 3000), µ = 0 . 2, n ∗ = 10 and ¯ η = 0 . 05 (b elow, green) and ¯ η = 0 . 95 (ab ov e, red), from b oth sim ulations and theoretical predictions. Inset: H ysteresis ci- cle for th e same graph for ¯ η = 0 . 95. Fig. 1. The b ehavior o f ρ a s a function o f p exhibits hysteresis: The s ystem turns from a free to a congested phase discontin uously as p increases at p = µ and it re- verts back to the free phase from a c ongested pha se at p = (1 − ¯ η ) µ as p decreases. This simple case also shows that tra ffic control is useless in homogeneous graphs, a s it does not enlarge the sta bilit y region of the free phase, while making a conge s ted pha se stable for p ∈ [(1 − ¯ η ) µ, µ ] (see inset o f Fig. 1). The c a se of he ter ogeneous g raphs instead is muc h richer. In Fig. 2, we display ρ ( p ) for a scale- fr ee net- work from bo th simulations (p oints) and n umer ical cal- culations (full line). The ag reement is very go o d and the behavior o f the curves repro duces the scenario o bserved in [7]. The fig ure is obtained for µ = 0 . 2 and n ∗ = 1 0, but the b ehavior do es not qualitativ ely c ha nge for dif- ferent v alues of these parameter s . The dependence on ¯ η brings instead qualitative changes. Increa sing ¯ η from 0 . 05 to 0 . 95 , the tra nsition becomes discontin uous and p c increases. Fig.2 (inset) a lso shows that in case of discontin uous tr ansition, the system exhibits hysteresis phenomena: a cong ested system does not immediately decongest if the creation rate p is decrea sed under the threshold v alue p c . When the whole system is cong ested, simple ar guments of queueing theory show that ρ ( p ) follows the curve 1 − (1 − ¯ η ) µ p ; how ever the most interesting situatio n o ccurs when the netw or k is only par tially cong e s ted. This ca se can be better understo o d considering the limit n ∗ → ∞ , in whic h the ca lculations simplify considerably without mo difying the o verall qualitative b ehavior. In this limit, unconges ted no des hav e a k < 1, hence χ k → 0 and q k = 1 − a k , meaning that these no des hav e short queues and do not re ject arr iving pac kets. If a k > 1, then q k → 0 and χ k = ¯ η a k − 1 b k ; more precisely we have χ k = 1 − k ∗ k (1 − ¯ η ) for k < k ∗ and χ k = ¯ η for k ≥ k ∗ . The latter cla ss iden tifies c ongeste d no des, while we call fickle those with k ∗ (1 − ¯ η ) ≤ k < k ∗ . The unc ongeste d nodes exis t up to k F = k ∗ (1 − ¯ η ). Using this classification, we get a first express ion for ¯ χ , i.e. ¯ χ 1 = P k ∗ k = k F h 1 − k ∗ (1 − ¯ η ) k i k z P ( k ) + ¯ η P k max k = k ∗ k z P ( k ). Eq. (3) provides a further rela tion be tw ee n ¯ q , ¯ χ and k ∗ . ¯ q can be eliminated using its definition wich leav es us with an implicit equation for ¯ χ , whose solution w e call ¯ χ 2 . In Fig . 3 we plot the difference ∆ χ = ¯ χ 1 − ¯ χ 2 vs. k ∗ , for ¯ η = 0 . 05 (left) a nd 0 . 95 (right) and differ ent v alues of p . The zer os ∆ χ ( k ∗ ) corresp ond to the o nly p o ssible v a lues assumed b y k ∗ . In both cases, ∆ χ decreases a s p . F or small rejection probability ( ¯ η = 0 . 05 in Fig. 3), there is only one s olution k ∗ ( p ), which decreases fro m + ∞ when increasing p fr o m 0. The v alue p c at which k ∗ ( p c ) = k max is the c r itical creation ra te a t which hig h- est degree no des beco me congested. A t larger p , k ∗ ( p ) de- creases monotonously un til all no des ar e congested when k ∗ ( p ) = k min . Hence for low v a lues of ¯ η , the transition from free- flow to co ngested phas e o ccurs contin uous ly at the v alue of p for which k ∗ ( p ) = k max . A t larg e ¯ η ( ¯ η = 0 . 95 in Fig . 3), the scenar io is muc h more co mplex. Dep ending on p , the equa tio n ca n hav e up to three solutions, k ∗ 1 ( p ) ≤ k ∗ 2 ( p ) ≤ k ∗ 3 ( p ). It is ea s y to chec k that only k ∗ 1 and k ∗ 3 are stable solutions. F or p ≪ 1 there is only one solution at k ∗ 3 ( p ) ≫ k max , cor- resp onding to the free phase. As p increases, t wo cases are pos sible: i) k ∗ 3 ( p ) reaches k max befo re the loca l min- im um cr osses z ero, in which cas e the congested phase emerges contin uously; ii) if ins tea d the solution k ∗ 1 ( p ) app ears when still k ∗ 3 ( p ) > k max , a co ngested pha se a p- pea rs abruptly . F or sufficiently large v alues of p , o nly k ∗ 1 ( p ) sur vives, a nd the netw ork b ecomes fully cong ested when k ∗ 1 ( p ) ≤ k min . Therefo re, the ex is tence of a purely discontin uous transitio n depends strongly on the tail of the degree distributio n. In the latter case ii) the hysteresis pheno menon b ecomes evident upo n v ar ying p in oppos ite directions: Star ting from the free phas e at low p , the system selects the solu- tion k ∗ 3 ( p ) and follows it up on increasing p unt il the so- lution k ∗ 3 ( p ) disapp ear s . On the contrary , s ta rting from the co ngested phase (large p ) the system lo cks into the congested phase k ∗ 1 ( p ) and remains congested until the solution k ∗ 1 ( p ) disapp ear s, with a disco nt inuous transi- tion (see inset of Fig. 2). Hyster e sis also o ccurs in cas e i) for the s a me rea sons. A detailed account of this rich phenomenology will be rep orted elsewhere [1 1]. W e computed the phase dia g ram (see Fig. 4) in the plane ( p, ¯ η ) for the same uncorr elated scale-free ra ndom net works with γ = 3 co ns idered in Fig . 2, in the case n ∗ → ∞ . The dashed line represents the contin uous phase transition, sepa rating fr ee-flow regime from con- gestion. A t the p oint C , the critical line splits in tw o branches that define a co existence r egion. The upp er full line r epresents the discontin uous tr a nsition from the fre e - phase to the jammed state, whereas the lower indicates the opp osite tra nsition from conges tion ba ck to free- flow. 4 1 10 100 k* -2 -1.5 -1 -0.5 0 0.5 ∆ χ p = 0.01 p = 0.1 1 10 100 k* -0.2 -0.1 0 0.1 0.2 p = 0.01 p = 0.1 p = 0.2 η = 0.05 η = 0.95 [!h] FIG. 3: The zeros of ∆ χ ( p ) vs. k ∗ define the threshold degree for th e onset of congestion in a netw ork. The picture refers to a scale-free random netw ork with γ = 3 . 0 ( k max = 110), and differen t v alues ¯ η = 0 . 05 , 0 . 95 and p . The solution k ∗ 1 ( p ) in the right panel falls outside the plot. 0.2 0.4 0.6 0.8 1 η 0 0.05 0.1 p 100 10000 N 0 0.1 0.2 p c congested free c FIG. 4: ( η , p ) p hase diagram for an uncorrelated scale-free graph ( P ( k ) ∝ k − 3 , k min = 3, k max = 110, N = 3000), µ = 0 . 2, n ∗ = ∞ from theoretical p red ictions. The inset sho ws p c ( N ) for η = 1 (red line) an d η = 0 (black line). The dotted line decrea s ing from the maxim um of the crit- ical line, is a n unphysical branch of the solutio n obtained from calcula tions. Indeed if a free flow phas e is stable for a given v alue of ¯ η , this will p ersist for lar ger v alues of ¯ η be c ause traffic control only affects conges ted no des in the limit n ∗ → ∞ (i.e. a free stationar y state ca nnot bec ome congested if we increa se ¯ η ). This phenomenolo gy crucially dep ends on the tail of the degree distr ibution. Since k max depe nds on the s ys- tem’s size, we expect that p c depe nds on N as well; the inset of Fig.4 sho w that fo r the same sca le-free netw or k of Fig. 3 the critical rate of pa ckets’ creation g o es to zero as p c ( N ) ∝ 1 / √ N , fo r η = 0, but it go es to a constant for η = 1, in the limit of large N . Hence the p oint C separates tw o regions in ¯ η with distinct behavior of finite size effects. The mechanisms trigger ing the emer gence of conges- tion is somewhat reminiscent of jamming or bo otstrap per colation, where a node is o cc upied if the n umber of o ccupied neighbors exceeds a given threshold. In these mo dels, as the threshold incr e ases, the transition turns from co ntin uo us to discontin uous [1 2]. Similar ly , as traf- fic control in the routing pr oto col is enha nc e d, the con- gestion transition may turn from contin uous to disco n- tin uo us, as a res ult of a co op erative pr o cess in which the more co ngested neighbors a no de has, the mo re it is lik ely that it w ill b e congested. In co nclusion, we hav e pro p osed a minimal mo del to study the emergence of congestio n in informa tion net- works. 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