On Resilience of Multicommodity Dynamical Flow Networks

Transcription

1 On Rsilinc of Multicommodity Dynamical Flow Ntworks Gusta Nilsson, Giacomo Como, and Enrico Loisari bstract Dynamical flow ntworks with htrognous routing ar analyzd in trms of stability and rsilinc to prturbations. Particls flow through th ntwork and, at ach junction, dcid which downstram link to tak on th basis of th local stat of th ntwork. Diffrntly from singlcommodity scnarios, particls blong to diffrnt classs, or commoditis, with diffrnt origins and dstinations, ach racting diffrntly to th obsrd stat of th ntwork. s such, th commoditis compt for th shard rsourc that is th flow capacity of ach link of th ntwork. This implis that, in contrast to th singl-commodity cas, th rsulting dynamical systm is not monoton, hnc hardr to analyz. It is shown that, in an acyclic ntwork, whn a fasibl globally asymptotically stabl aggrgat quilibrium xists, thn ach commodity also admits a uniqu quilibrium. In addition, a sufficint condition for stability is proidd. Finally, it is shown that, diffrntly from th singl-commodity cas, whn this condition is not satisfid, th possibl uniqu quilibrium may b arbitrarily fragil to prturbations of th ntwork. Indx Trms Dynamical flow ntworks, multicommodity flows, rsilinc, distributd routing, htrognous routing. I. INTRODUCTION In a multicommodity ntwork particls of diffrnt classs flow through a ntwork sharing and compting for th channl rsourc. Exampls of multicommodity flow problms ar ubiquitous in nginring scincs. Traffic ntworks, air traffic control, data ntworks, production chains and supply chains can all b intrprtd as multicommodity ntworks, whr th aim is usually to lt th highst possibl olum of particls of th diffrnt classs through th ntwork. Modls for multicommodity flow ntworks basd on PDEs and th clbratd LWR modl ha bn studid [1], [2], but solutions ar usually difficult to obtain n in simpl sttings. In this papr, w propos and analyz a dynamical ODE-basd modl for ntworks with htrognous routing. Th ntwork topology is modld as a dirctd graph in which nods ar junctions and dgs ar links through which particls can flow. Th flow on ach link is boundd from abo by a finit alu calld th link capacity. Particls ntr in th ntwork from origins and la it at dstinations, which possibly ary from commodity to commodity. Whn particls arri at a junction, thy dcid which subsqunt link to tak on th basis of th local stat of th ntwork, Th first two authors ar with th Dpartmnt of utomatic Control, Lund Unirsity, SE Lund, Swdn gusta.nilsson, giacomo.como@control.lth.s. Th third author is with th NCS tam at INRI Grnobl Rhôn-lps and th GIPS-lab, UMR-CNRS 5216, Grnobl, Franc, nrico.loisari@gipsa-lab.fr. Th first two authors ar mmbrs of th xcllnc cntrs ELLIT and LCCC and all th authors wr supportd by th Swdish Rsarch Council through th junior rsarch grant Information Dynamics in Larg-Scal Ntworks. that is, th aggrgat of particls in ach possibl subsqunt link. In othr trms, particls ar unabl to distinguish in th flow th classs of particls. Routing is htrognous in that particls of diffrnt classs ha diffrnt prfrrd paths and ract to th local stat of th ntwork in diffrnt ways. In contrast to singl-commodity scnario, in which all particls blong to th sam class and hnc thr is no comptition among diffrnt classs, multicommodity ntworks show a complx bhaior n in th static stting [3], in which it has bn shown that th maximum throughput in a multicommodity ntwork is boundd away from th alu prdictd by th clbratd max-flow min-cut thorm. Dynamical modls basd on ODEs ha bn proposd in th litratur, but htrognity is usually mbddd in a singl-commodity scnario with fixd turning rats, i.., in which at ach junction th fraction of hicls turning into ach subsqunt link is fixd [4], [5]. Diffrntly from th lattr approach, w xtnd th framwork proposd in [6], [7] and considr a dynamic rsponsi scnario, in which agnts ha prfrrd paths that thy would follow whn compltly isolatd in th ntwork, but ar also willing to adapt thir bhaior according to th local stat of th ntwork, and aoid prfrrd, but highly congstd, paths. Bsids th analysis of th stability of th ntwork, w study th rsilinc proprtis of multicommodity ntworks with rspct to prturbations, which in this papr ar undrstood as rduction of th link s capacitis. Th main rsults of this papr ar th following: 1) Undr crtain assumptions on th constant inflows in th ntwork, th ntwork admits a globally asymptotically stabl quilibrium for ach commodity, and 2) Whn th ntwork is not singl-commodity, it can b xtrmly fragil with rspct to prturbations. In particular, prturbing a ntwork at quilibrium can triggr a cascad ffct that maks th ntwork unstabl. In addition, xampls show that such a prturbation can b arbitrarily small. Such a bhaior ariss in multicommoditis only, and has no countrpart in singl-commodity ntwork, whr instad, as shown in [8], wll dsignd routing policis can compltly xploit th structur of th ntwork and nsur maximal rsilinc to prturbations. Th papr is organizd as follows: th rst of this sction prsnts th notation. In Sction II w proid a motiing xampl for th fragility of th multicommodity ntwork. In Sction III w propos a modl for dynamical flow ntworks with htrognous routing. Sction IV is dotd to stability analysis of th modl and to a sufficint condition for th stability of th ntwork. Sction V discusss rsilinc and formally pros th claims mad in Sction II. Finally, Sction VI prsnts som futur rsarch dirctions.

2 Lt R b th st of ral numbrs and lt R + := {x 2 R : x 0} dnot th st of non-ngati ral numbrs. For a st, dnots its cardinality and with R (+), w man th (non-ngati) ral ctors indxd by th lmnts in. In th sam mannr, R B (+) ar matrics indxd by th product st of and B. dirctd multi-graph is a pair consisting of a finit st of nods, V, and a finit multi-st, i., a st whr an lmnt is allowd to occur mor than onc, of dirctd links, E, containing ordrd pairs of nods. For a link =( 1, 2 ) 2E w writ = 1 for its tail and = 2 for its had, s Fig. 1a. Th st of outgoing links, E +, for a nod 2Vis dfind as E + := { 2E: = }. In th sam mannr th st of incoming links is dfind as E := { 2 E : = }. Th sts, for a nod, ar illustratd in Fig. 1b. For sak of notation, w put R := R E +. =2 f 1 =1 C 1 =2 f 4 =0.5 C 4 = =1 f3 =0.5 C 2 =2 C 3 =2 f 5 =0.5 C 5 =0.7 Fig. 2: singl-commodity ntwork. Th minimum rsidual capacity 0.2 is achid at nod 3. Hnc, undr any prturbation of magnitud smallr than 0.2 th ntwork is still b abl to transfr th xtrnal inflow 1 to th dstination nod 4. (a) Th prcding and nxt nod for a link Fig. 1: Notation E E + (b) Th sts of incoming and outgoing dgs from a nod II. MOTIVTING EMPLE Lt us considr th ntwork displayd in Fig. 2. First, w focus on singl-commodity dynamical flows, whr th dnsity dynamics on ach link is dscribd by th following consration law = u (t) f ( (t)). Hr, f ( (t)), calld th flow function of link, rprsnts th outflow from, and is gin by f ( (t)) = C (1 (t) ), whr C is th link s maximum flow capacity. On th othr hand, th trm u dscribs how much of th flow through nod should b snt to link. In particular, w st u (t) =G ( (t))( + f j ( j (t))), j2e whr G ( (t)) is a map that dscribs how th fraction of flow though a nod that is routd towards link dpnds on th currnt local stat of th ntwork and 1 =2, i =0 i 6= 1, dnots a static inflow at th origin nod 1. Th routing polics ar constructd as 1 3 G 1 ( 1, 2 )= 1 +, G 3( 3, 4 ) = 2 3 +, 4 2 G 2 ( 1, 2 )= 1 +, G 4 4( 3, 4 ) = 2 3 +, 4 and G 5 ( 5 ) 1. With ths routing policis, it can b rifid that th ntwork dynamics admits an quilibrium with corrsponding flow ctor f whos ntris ar spcifid in Fig. 2. Such quilibrium is globally asymptotically stabl [6]. W want to study how th limit flow changs whn th ntwork is prturbd, namly, whn th flow capacity is rducd from C to C <C on som links. Dfin th margin of rsilinc to b th infimum aggrgat flow capacity rduction P 2E (C C ), or prturbation magnitud, such that th prturbd systm =ũ (t) f ( (t)), ũ (t) = fj ( j (t)) G ( (t)). j2e is unstabl, i.., th dnsity ctor (t) blows up in th limit of larg t. For a singl-commodity ntwork, it was shown that th margin of rsilinc quals th minimun nod rsidual capacity [6], [7]. This implis that th ntwork in Fig. 2, with th gin routing policis, can absorb any prturbation of magnitud smallr than 0.2. Now, lt us mo to a multicommodity scnario, whr th particl dnsity on ach link is mixtur of particls of two diffrnt classs, and B, such that = + B. W assum that th particls ar fully mixd, so that th dynamics for particls of class k =, B ar whr k = u k (t) u k (t) =G k ( (t))( k + k (t) (t) f ( (t)), j2e k j (t) j (t) f j( j (t))), and 1 = B 1 =1, i = B i =0for i 6= 1ar static inflows. W lt th particls ha diffrnt routing policis, i., th two commodity flows and B ha diffrnt path prfrncs. In particular, w considr routing polics of th

3 form G k 1( 1, 2 )=1 G k 2( 1, 2 )= G k 3( 3, 4 )=1 G k 4( 3, 4 )= f1 k k 1 1 f1 k k k k 2 2 f3 k k 3 3 f3 k k f4 k k 4 4 and G 5 ( 5 ) G B 5 ( 5 ) 1. Hr, f k is th limit flow for commodity k on link as gin in Fig. 3a. Obsr that th aggrgat limit flows coincid with thos in th singl-commodity cas. On th othr hand, k > 0 ar paramtrs which do not ffct th limit flows. Howr, ths paramtrs do affct how th rspons to prturbations. In ordr to illustrat th fragility of th multicommodity stting, w lt 1 = B 2 = 1000 and B 2 = B 1 = 1 and k 3 = k 4 =0.01, and considr now a prturbation of magnitud 0.01 which rducs C 1 =2to C 1 =1.99. Th limit flows for th prturbd dynamics ar shown in Fig. 3b. Th prturbation causs th limit flow on link 3 to incras and xcd th capacity of th subsqunt link 5. Consquntly, th dnsity on link 5 grows unboundd. This implis that th margin of rsilinc in th multicommodity cas is not largr than This xampl thn indicats that a dynamical multicommodity ntwork can b much mor fragil than a singl-commodity on with th sam topology and aggrgat quilibrium flow. III. MODEL FOR DYNMICL FLOW NETWORKS WITH HETEROGENOUS ROUTING W modl a dynamical multicommodity flow ntwork as a dirctd multigraph M =(V, E), V bing th st of nods and E bing th st of links, that is shard by a finit st K of diffrnt commoditis. For ry k 2K, s k and d k in V will dnot, rspctily, th sourc and dstination nods of commodity k, and k 0 will stand for th inflow of such commodity at nod s k from outsid th ntwork. In ordr to account for th fact that, in crtain applications, not all commoditis can accss ry link, for ry nod w dnot by E k E + th st of accssibl (out-)links of for commodity k. Th st of all accssibl links for commodity k 2 K is thn dnotd by E k := [ E k.w mak th following stady assumption, which nsurs that particls of ach commodity can rach thir dstination. ssumption 1 (Existnc of origin-dstination paths): For k 2Kwith k > 0 and ry 2E k thr xists a dirctd path in th sub-multigraph (V, E k ) from to d k. Particls flow through th ntwork quuing up on th links. W dnot by k 2 R + th dnsity of particls of commodity k 2Kon link 2E, and w dnot by := P k2k k th aggrgat dnsity of particls on. ll particl dnsitis in th whol ntwork can thn b dscribd by th matrix 2 R E K +. Th rst of th sction is dotd to dscribing th dynamics of th dnsitis k. To this aim, for ry link, w dnot by f th total outflow from, and w assum that it is a function of th aggrgat dnsity of th link, namly, f = f ( ). Th quantity C := sup 0 f ( ) rprsnts th (maximum f 1 = C 1 =2 f 4 = C 4 =2 =1, B = = f3 = C 2 =2 C 3 =2 f 5 = C 5 =0.7 (a) Equilibrium flows for a simpl ntwork. ll flows ar lss than dgs capacitis, so that th ntwork is abl to fully transfr th xtrnal inflows, B to thir dstinations, nod 4. f 1 = C 1 =1.99 f 4 = C 4 =2 =1, B = = f3 = C 2 =2 C 3 =2 f 5 = C 5 =0.7 (b) Th sam ntwork whn dg 1 s capacity is slightly dcrasd. Now th inflow to link 5 is largr than its capacity, and hnc th ntwork is not abl to handl th flow dmands. Fig. 3: simpl ntwork, with two xtrnal inflows, B at nod 1. Nod 4 is th dstination for both flows. Th flows shown in figur ar gin in trms of th sum of flows of th two classs on ach link, i.., f = f + f B. flow) capacity of link. Throughout th papr, w shall rfr to a ntwork N as th pair of a topology M =(V, E) and a st of flow functions {f } 2E satisfying th following: ssumption 2 (Flow function): For ach link 2Eth flow function f : R +! R + is a strictly incrasing continuously diffrntiabl function with boundd driati, with f (0) = 0 and C =sup 0 f ( ) < +1. W furthr mak th simplifying assumption that particls of diffrnt commoditis ar always homognously mixd in ach link. s a consqunc, th outflow f k of particls of class k from link is proportional to th fraction of particls of class k on link, i.., f k = k f ( ). For ry non-dstination nod 2V\{d k } k2k, dnot by k th total inflow of commodity k into, gin by 8 < Pj2E k j (t) k := f j(t) j( j (t)) + k if = s k : Pj2E k j (t). (1) f j(t) j( j (t)) othrwis For ry link 2E k, w dnot by u k th inflow of particls of commodity k into link. s alrady mntiond, particls only quu up on links, i., th nods ha no buffr capacitis, thrfor inflows must satisfy u k = k, 8 2V\d k. (2) 2E k

4 Sinc th family of signals {u k } 2E k,k2k dscribs how particls split at nods, w rfr to it as a routing control. With th prious dfinitions, th dynamics of dnsity of commodity k on link is gin by th following massconsration law k = u k (t) k (t) (t) f ( (t)). (3) Th inflow u k can b in principl any signal that satisfis (2). In this papr w assum u k = k G k ( ). (4) namly, th routing control is a function of th stat of th ntwork. In particular, w considr distributd policis as pr th following dfinition: Dfinition 1 (Distributd routing policy): distributd routing policy is a family of diffrntiabl functions G := {G k : R!R + } 2E,k2K satisfying, for all k 2K, a) G k ( ) 1 for all 2V\{d k } 2E + b) G k ( ) 0 for all 2V\{d k }, /2E k c) ( ) 0 for all 2V\{d k }, 2E k, j 62 E k d) ( ) 0 for all 2V\{d k },, j 2E k, 6= j ) For ry 2V\{d k } and ry propr subst I ( E k thr xists a continuously diffrntiabl family of functions, Ḡ, Ḡ k : R!R + such that P 2E k Ḡ k ( ) 1 and such that if thn!1, 8 2E k \I, j! I j, 8j 2I, G k ( )! 0, 8 2E k \I, G k j ( )! Ḡk j ( I ), 8j 2I. W can now gi th dfinition of a Dynamical Multicommodity Ntwork. Dfinition 2 (Dynamical multicommodity ntwork): dynamical multicommodity ntwork is a ntwork N associatd with a family of distributd routing policis G and a st of commodity dmands {s k,d k, k} 8k2K, whr th dynamics is gin by (3) and controlld by (4). Rmark 1: In Dfinition 1, proprty a) nsurs mass consration at ach nod, whil proprty b) nsurs that particls of commodity k ar routd to links on which commodity k is allowd only. For ach nod 2Vand 2E +, proprty c) nsurs that ach G k ( ) only dpnds on dnsitis of links in E +, hnc it is distributd in th sns that dcisions ar takn on th basis of local information only. Proprty d) dscribs th following monoton bhaior: incrasing th dnsity of a link rducs th fraction of flow 1 3 Fig. 4: local ntwork with E + = { 1, 2, 3 }. routd towards that link, and icrsa. Such a stting can b intrprtd as an attmpt to aoid congstd links, and will b instrumntal in th proof of our main rsult. Finally, proprty ) stats that whn a link is compltly congstd, i.., its dnsity is infinit, it cannot b usd. Rmark 2: Th dfinition of distributd routing policy follows th dfinition in [6], [7]. Th ky nolty is that in th prsnt stting diffrnt commoditis ar allowd to ha diffrnt routing policis, namly, to ha diffrnt routing prfrncs and to rspond in a diffrnt way to congstion. On th othr sid, all commoditis compt for th sam shard rsourc, which is th flow capacity on ach of th links of th ntwork. Rmark 3 (Loss of monotonicity): Sinc th controllrs u k and th flows ar dtrmind by th aggrgat dnsitis, th monotonicity proprty, that is a cntral proprty for th rsults in th singl-commodity cas, [6], [7], is no longr guarantd. Finally, a ntwork that can fulfil all flow dmands is calld fully transfrring, as pr th following dfinition: Dfinition 3 (Fully transfrring): dynamical multicommodity ntwork is said to b fully transfrring if lim inf t!1 2 2E dk f k (t) = k, 8 k 2K. IV. STBILITY NLYSIS In this sction w will stat a sufficint condition for an acyclic dynamical multicommodity ntwork to ha finit limit dnsitis and a uniqu limit flow. First of all, w analyz a local ntwork, s Fig. 4, namly a ntwork with a singl nod. For a local ntwork, th dynamics is gin by k = k (t)g k ( (t)) k (t) (t) f ( (t)), 8 2E +, 8k 2K. From now on, w shall rfr to a dnsity which is an quilibrium for (5) as a (dnsity) quilibrium, and to th corrsponding flows {f ( )} 2E as a flow quilibrium. Th nxt rsult offrs a ncssary and sufficint condition for th ntwork to admit a globally asymptotically quilibrium. Thorm 1: Considr a local dynamical multicommodity ntwork N. ssum moror that th inflows ar conrging, namly lim t!+1 k (t) = k, 8k 2K. Thn it holds that (5)

5 a) if P j2j j < P 2E C J for ry nonmpty J K, thn thr xists a finit uniqu such that lim t!1 k (t) = k for ry 2E + and k 2K. b) if thr xists a nonmpty J Ksuch that P P j2j j 2E C, thn thr xists at last on k 2J such J that lim t!+1 (t) =+1 for all 2E k. Thorm 1 dals with stability of a local ntwork. In th rst of this sction w shall addrss th stability of an acyclic ntwork with a singl origin by intrprting it as a cascad of local ntworks. To this nd, lt J K and V J := { 2V E J 6= ;}. Moror, lt U J V J J := { =(a, b) 2E J a 2 U,b /2U}. Dfin th minimum cut capacity btwn two nods o, s 2V, Co!s k as C J o!s := min U J V J s.t o2u J,s/2U J 2E C. For sak of simplicity, considr now an acyclic ntwork with th sam origin o 2Vfor all th commoditis, i.., s k = o for all k 2K. Th following proposition offrs a sufficint condition for such a ntwork to admit uniqu limit flow and dnsity. Proposition 1: Considr an acyclic dynamical multicommodity ntwork with singl origin. Thn a sufficint condition for it to admit a uniqu limit dnsity and a uniqu limit flow is that for ry k 2Kand for ry 2V k min Co!, J < C. (6) J o 2E J V. RESILIENCE In this sction w instigat how th dynamic multicommodity ntwork rsponds to prturbations. In this papr a prturbation of a flow ntwork corrsponds to th rduction of th flow function as a function of th dnsity, on possibly mor than on link. Formally, following [6], [7], a prturbation is modld as a family of prturbd flow functions, { f ( )} 2E such that f( ) appl f ( ), 8 2 E and f satisfis ssumption 2. Th magnitud of th prturbation on on link 2Eis thn dfind as := sup f 0 ( ) f ( ) and th total magnitud of th prturbation is thn gin by := P 2E. Gin a family of prturbd flow functions { f ( )} 2E, a prturbd ntwork Ñ is a ntwork with th sam graph, commoditis, origin, dstinations and routing policy as N, and with flow functions f. Th rsilinc of a dynamical flow ntwork associatd to a ntwork N and routing policis G is thn dfind as th infimum total magnitud of prturbations making th prturbd dynamical flow ntwork Ñ not fully transfrring. It was pron in [6], [7] that, in th singl-commodity cas, th rsilinc of an acyclic dynamical flow ntwork coincids with th minimum rsidual capacity, dfind as min { C f }, 6=d 2E + whr f is th limit flow of th unprturbd dynamical flow ntwork. t th cor of th rsult is a diffusiity proprty of singl-commodity local dynamical flow ntworks (cf. [7, Lmma 1]) guaranting that a prturbation of total magnitud in ithr som of th outlinks, and/or an incras of th inflow, dos not incras th limit flow of any outlink by mor than th sum of and of th inflow incras. In othr words, th ntwork dos not orract to prturbations. Th goal of this sction is to show that, whn mor than on commodity ar prsnt, dynamical flow ntworks can b instad arbitrarily fragil. In particular, w will construct a family of simpl xampls of multicommodity dynamical flow ntworks (with topology illustratd in Fig. 3) that, irrspcti of thir minimal rsidual capacity, can los thir fully transfrring proprty n by mans of arbitrarily small prturbations. This will show that thir rsilinc quals 0. W will procd by first stating som proprtis of local multicommodity dynamical flow ntworks that ha th fully accssibl proprtis. Th first on can b considrd as a wakr rsion of th aformntiond diffusiity proprty for multicommodity dynamical ntworks. Lmma 1: Considr a fully accssibl local dynamical multicommodity ntwork N, with inflow such that k < C. k2k 2E + Lt f dnot th limit flow for this ntwork. Moror, lt Ñ b an admissibl prturbd ntwork with inflow such that k < C. k2k 2E + Lt f ( ) dnot th limit flow of th prturbd ntwork, with th inflows. Thn for ry I E + it holds that f i ( ) fi appl i h k k +. + i2i k2k 2E + Lmma 1 proids a bound on th diffrnc btwn aggrgat limit flows bfor and aftr th prturbation in trms of its magnitud and of th diffrnc btwn th inflows. Obsr that, whn thr is only on commodity, i.., K = 1, Lmma 1 rducs to Lmma 1 in [7]. On th othr hand, th following two rsults show that, whn mor than on commodity is prsnt, ach commodity flow can chang in an arbitrary way as long as th bound on th aggrgat flow proidd by Lmma 1 is satisfid. Lmma 2: Considr a local dynamical ntwork with two outgoing links 1, 2 and two commodity inflows, B. Lt f k b a fasibl quilibrium flow. Thn, for >0 small nough, thr xist distributd routing policis G and G B such that a) f k is th quilibrium flow of th dynamical local ntwork, b) thr xits a prturbation of magnitud such that th prturbd limit flow, for on commodity k and for on link, satisfis f k > min( k,f ),

6 whr >0 can b chosn arbitrary small. Notic in particular that th prturbation considrd in Lmma 2 dos not chang th inflows and B, and hnc by Lmma 1 f appl f + for = 1, 2. lso notic that k triially f appl min{ k, f }applmin{ k,f + }. Lmma 2 nsurs thn that aftr prturbation w gt min( k,f ) appl f k appl min( k,f + ). Sinc and ar arbitrary, w can str arbitrarily clos to min( k,f ). Lmma 3: Considr a local dynamical ntwork, with two B outgoing links 1, 2 and two commodity inflows,. Lt f b a fasibl limit flow. Thn, if th commodity inflow changs to and th nw limit flows satisfy C 1 > f 1 >f 1 and f 2 <f 2, thr xist routing policis G,G B, such that for a gin >0 f 1 > f 1 f k + f B 1 B. B W ar now rady to construct an xampl showing that rsilinc can b arbitrarily low. To this aim, considr th ntwork in Fig. 3. Start from a gin fasibl limit flow f such that and assum that 1 = f 3 > 2 = f 3 B f B 2 min(, ) > C W claim that w can construct routing policis such that th ntwork will not b fully transfrring aftr an arbitrarily small prturbation. Considr first th local ntwork around nod 1. Using Lmma 2, w know that can construct routing policis such that aftr a small prturbation on link 1 th flow of commodity on link 2 is strd clos to th alu f 2 min(, ) > C 5 2., 1 2 In nod 3, w construct thn th routing policis according to Lmma 3. In this way, whn, aftr prturbation, approachs f 2 th prturbd limit flow on link 3 conrgs to f 3 = f 3 f 2 + f 3 B f B B 2 >C 5. Sinc th prturbd limit flow on link 3 is gratr than th capacity of link 5, th ntwork loss th fully transfrring proprty, and th claim is prod. To illustrat this bhaior numrically, rcall th motiating xampl in Sction II. Sinc C 1! 3 = 2 > C 5 th sufficint condition statd in Proposition 1 is iolatd. Howr, as th xampl shows, th systm conrgs to finit limit dnsitis. But aftr th prturbation on link 1, th systm s prturbd limit flow f 3 =0.8 > 0.7 and th systm is not fully transfrring anymor. In Fig. 5 w show how th flows on link 2 and 3 ol, starting from zro initial stat. Th prturbation occurs at t = t p t p Tim Fig. 5: Th tim olution of flows on link 2 and 3. t tim t = t p a prturbation occurs and causs th flow f 3 on link 3 to xcd th capacity of link 5. s a consqunc, th dnsity on link 5 grows unboundd and th ntwork loss th proprty of bing fully transfrring. VI. CONCLUSIONS In this papr, a modl for a dynamical multicommodity ntworks has bn proposd and studid. sufficint condition for th stability of th ntwork has bn proidd. If th condition is iolatd, th ntwork can b ry fragil to prturbations, and n a small prturbation can modify th limit flows drastically and in such a way that th ntwork bcoms unstabl. Futur rsarch dirctions includ and ar not limitd to analysis of cyclic ntworks, study of th rsilinc undr constraind routing policis, and dsign of robust controllrs. REFERENCES [1] J. Lbacqu and M. Khoshyaran, First ordr macroscopic traffic flow modls for ntworks in th contxt of dynamic assignmnt, in Transportation Planning, sr. pplid Optimization, M. Patriksson and M. Labbé, Eds. Springr US, 2002, ol. 64, pp [2] M. Hrty, C. Kirchnr, S. Moutari, and M. Rascl, Multicommodity flows on road ntworks, Communications in Mathmatical Scincs, ol. 6, no. 1, pp , [3] T. Lighton and S. Rao, Multicommodity max-flow min-cut thorms and thir us in dsigning approximation algorithms, J. CM, ol. 46, no. 6, pp , No [4] P. Varaiya, Th max-prssur controllr for arbitrary ntworks of signalizd intrsctions, in dancs in Dynamic Ntwork Modling in Complx Transportation Systms. Springr, 2013, pp [5] K. boudolas, M. Papagorgiou,. Koulas, and E. Kosmatopoulos, rolling-horizon quadratic-programming approach to th signal control problm in larg-scal congstd urban road ntworks, Transportation Rsarch Part C: Emrging Tchnologis, ol. 18, no. 5, pp , [6] G. Como, K. Sala, D. cmoglu, M.. Dahlh, and E. Frazzoli, Robust distributd routing in dynamical ntworks Part I: Locally rsponsi policis and wak rsilinc, IEEE Transactions on utomatic Control, ol. 58, no. 2, pp , Dc [7], Robust distributd routing in dynamical ntworks Part II: strong rsilinc, quilibrium slction and cascadd failurs, IEEE Transactions on utomatic Control, ol. 58, no. 2, pp , Dc [8] G. Como, E. Loisari, and K. Sala, Throughput optimality and orload bhaior of dynamical flow ntworks undr monoton distributd routing, IEEE Transactions on Control of Ntwork Systms, in prss, B f3 f3 B