Compressive Sensing over Strongly Connected Digraph and Its Application in Traffic Monitoring

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1 Compressve Sesg over Strogly Coected Dgraph ad Its Applcato Traffc Motorg Xao Q, Yogca Wag, Yuexua Wag, Lwe Xu Isttute for Iterdscplary Iformato Sceces, Tsghua Uversty, Bejg, Cha {qxao3, {wagyuexua, Abstract Compressve sesg over graphs has recetly attracted great research attetos, whch takes lmted umber ed-to-ed measuremets alog paths (walks) to recover sparse vectors represetg lk/ode propertes. Ulke tradtoal compressve sesg, the alog-path measuremets rule out the freedom of radom samplg, whch troduces path costrats to the measuremet matrx. The costrat makes explct aalyss of recovery performace dffcult. Oly for udrected graphs, early results showed that O(klog()) edto-ed measuremets take by radom walks are suffcet to recover k-sparse edge vector. However, the problem becomes more dffcult whe drected graphs are cosdered, because of the easy state absorbg ad the dffculty of evaluatg the statoary dstrbuto. But dgraphs heretly model may etwork systems. I ths paper, partcularly for strogly coected dgraphs wth low ode degrees, we presets bouds for the statoary dstrbuto of radom walks, ad preset delberatve proofs whch put forward that O(klog()) path measuremets are suffcet to recover k-sparse edge vectors. Further more, because urba road etworks are exactly strogly coected, low degree dgraphs, we desged effcet recovery methods to estmate road delays by a small umber of probg cars. Although the road delay vector s actually ot sparse, we leverage the emprcal o-cogested road delays as refereces ad develop a algorthm whch dvde the problem to teratvely recover several k-sparse vectors. Smulato results show that whe less tha % edges are cogested, more tha 9% cogesto states ca be recovered correctly by % measuremets. I. INTRODUCTION Whe etworks become large scale, t wll be costly ad operatoally dffcult for users to motor the propertes of all lks. For effcecy of motorg each lk, users geerally rely o exteral ed-to-ed measuremets, such as measurg path delays from source odes to destato odes to drectly fer the propertes of the teral lks. Ths falls the area of etwork tomography, whch s mportat for locatg fault lks or aalyze performace whose applcato areas rage from teret to trasportato etworks. Because each ed-to-ed exteral measuremet may cur cosderable amout of costs, such as the measuremet take by probg cars equpped wth GPS the trasportato etworks. How to mmze the cost of exteral ed-to-ed measuremets whle accurately recoverg the propertes of all the teral lks has ot oly theoretcal sgfcace but also applcato values. A fact whch softes the problem s that the lk propertes a large graph geerally costtute a sparse vector, because t s geerally true that oly a small porto of lks cur obvous delays or packet loss rates tha others. Ths fact has spred the applcato of compressve sesg (CS) to recover the lk property vector by lmted amout of edto-ed measuremets. Compressve sesg provdes fudametal theores ad effcet recovery algorthms to perfectly recover sgals that are sparse a trasformed doma. However, prevous studes [ 3] showed that the alogpath measuremets graphs pose ew path restrctos to measuremet matrx, whch makes measuremets o loger be free at radom. Ths costrat makes explctly aalyss to the recovery performace dffcult. Oly udrected graphs, [ 3] proved that O(klog()) path measuremets are suffcet to detfy k-sparse lk vector ad proposed l mmzato based recovery algorthms. [4]. The problem becomes more dffcult whe t comes to dgraphs, although dgraphs are more pertet models of may kds of etwork systems, such as trasportato etworks ad sesor etworks. The reaso s that the radom walks (path measuremet) o drected graphs, whe cosdered as Markov cha, coverge towards absorbg states (such as the odes wth out-degree). I other words, the flows the etworks teds to be absorbed by several odes. The statoary vstg probabltes of traset odes (edges) are zero, whch set obstacles for explct aalyss. However, ths paper, we tur our atteto to a specal kd of dgraphs,.e., strogly coected dgraph, whose statoary dstrbutos do exst for all odes (or edges)[5]. Although the statoary dstrbuto caot be formulated explctly as the udrected graph[6], we propose bouds for the statoary dstrbuto ad based o whch we prove that m = O ( c 4 KT 2 ()log ( K)) ed-to-ed measuremets are suffcet to recover k-sparse lk vectors strogly coected dgraphs, where c s a costat, s the umber of the lks ad k K+ 2 ; T () s the mxg tme, whch wll be troduced Secto III. The proof progress troduce a assumpto that the ode/edge degree s O(), whch meas degree s ot large. The strogly coected dgraph wth low degree s a perfect model for the trasportato etworks, where a car at ay road ca fd a path to ay other road, ad the road crosses are aturally low-degree whe the road etwork s modeled by a graph. But the road delay vector s ufortuately ot sparse. To address ths dffculty, we leverage the emprcal o-cogest delays of the roads as a referece. It helps to estmate the cogesto factors of the roads, ad based o whch, we develop algorthms to recover road delay vectors /9/$25. c 24 IEEE

2 2 It decomposes the road delay vector to a set of sparse vectors ad recovers them teratvely, whch accomplsh the task of recoverg the o-sparse delay vector. Effcecy of the proposed algorthms s evaluated by tesve smulatos. The rest of the paper s orgazed as follows. Secto II troduces the problem model. Secto III presets the compressve sesg methodology strogly coected dgraphs. Secto IV shows how traffc delay vector s recovered by lmted umber of probg cars. Secto V presets the smulato results. Secto VI revews the related works. Secto VII cocludes the paper wth future works. II. PROBLEM FORMULATION Cosder a drected graph G = (V, E), where V ad E are ode ad edge sets, ad let V = N ad E = be the umbers of odes ad edges respectvely. Let deg( ) ad deg( ) + be fuctos to retur the out-degree ad -degree of, where ca be a edge or a vertex. I ths paper, we use edge degrees by default uless ode degrees are specfed. Cosder a edge e, deg(e) dcates the umber of edges who ed at e s start pot ad deg(e) + dcates the umber of edges who start from e s ed pot. We assume the dgraph s (D, c)-uform dgraph whch satsfes D deg(e),+ cd, where c s a costat. We also cosder G s strogly coected,.e., there must be a path betwee ay two vertexes. Each edge e = (u, v) has a value x e whch s the delay from u to v the etwork cotext. We deote x = [x,, x ] T as the delay vector. A ed-to-ed measuremet alog a path measures the sum value of the edges that partcpate the path. Mathematcally speakg, f m ed-to-ed measuremets are take, the measuremet vector y s a m vector, where: y = Mx, () where M s a m bary matrx, whose elemet M j = f the th path routes through the j th edge ad otherwse. Note that value of edge ca also be statated to lk loss rate, ad formula () ca stll be obtaed easly va calculatg logarthms of lk loss rates[7]. We cosder measuremets are take by radom walks o the dgraph. We defe a edge trasto probablty matrx P E = {p j }, e, e j E where p j = deg(e ) f e + s ed pot equal to e j s start pot, ad otherwse. The statoary dstrbuto, π, of radom walks o G, s defed formally as a dstrbuto that s varat to the trasto probablty, whch s π = πp E. Ulke the statoary dstrbuto of udrected graph, whch s characterzed a closed form, the statoary dstrbuto dgraphs has o closed-form expresso[6]. Eve so, we ca gve a boud for π uder the strogly coected dgraph (detals Secto III). The questo ow s whether we ca estmate the delay vector x va as less the ed-to-ed measuremets as possble,.e., m <. However, dfferece from the tradtoal CS, the measuremet matrx M s costructed by the edto-ed, alog-path measuremet dgraph other tha the freely radom samplg, whch may cotradcts the coheret requremet of CS s observato matrx. Aother problem s x may be ot sparse. But at ths stage, we assume we ca fd a method whch ca covert x to be sparse. Therefore, what s followg, we cosder x s k-sparse,.e., havg at most k o-zero values, ad cocetrate o whether CS ca work, ad f yes, how may ed-to-ed measuremets are suffcet to recover x accurately. III. HOW TO CONSTRUCT THE MEASUREMENT MATRIX Some prevous studes [, 3] have proved that udrected graphs, O(k log()) path measuremets are suffcet to recover k-sparse lk vector. I ths secto, we wll show whether compressve sesg over strogly coected dgraph ca also have theoretcal performace guaratees. If t s possble, how may ed-to-ed measuremets wll suffce to recover the delay vector. Before solvg above two problems, we frstly troduce two useful otatos for cocsely presetg followg sectos. The ull space of matrx M : N (M) = {z : Mz = }. The mathematcal form of k-sparse vector: Γ k = {x : x k}, where x dcates the l orm of x. A. Null Space Codto for Vector Recovery I ths sub-secto, we wll gve a theorem (Theorem ) to characterze what kd codto the measuremet matrx should satsfy such that the k-sparse vector ca be recovered uder the dgraph costrats. Before troducg ths theorem, we gve two deftos for easly presetg the theorem, ad a lemma whch s used to prove t. Defto (K-dsjuct matrx). A m bary matrx M s called K-dsjuct, f for ay o more tha K + colums (dexed by set S {, 2,..., }), we select these K+ colums to costruct a sub-matrx M S, there s at least oe row of M S, whch has oly a sgle that row. We ca uderstad ths defto from the dgraph perspectve, suppose the K-dsjuct matrx s m, each colum represets a drected edge (e j, j ) ad each row dcates a path (W = {e, e,..., e t }) o dgraph G = (V, E), the take m paths ad deote them by Λ = {W, m}, for E E, E =K+, we say Λ ca costruct a K-dsjuct matrx f the followg codto s satsfed W Λ, s.t. W E =. (2) Defto 2 (Matrx Spark [8]). The spark of a gve matrx M m (m ) s the smallest umber of colums of M that are learly depedet. If m = ad rak(m) =, the spark(m)= +. Based o above deftos, we gve a lemma depctg the relatoshp betwee them, whch wll be used the proof of Theorem.

3 3 Lemma. If the matrx M m (m ) s K-dsjuct, the spark(m) > K + Proof. spark(m)>k + meas ay K ( K +) colums of M are lear depedet, thus for prove ths lemma, we oly eed to prove that ay vector of N (M) has at least (K + 2) o-zeros elemets. We prove t by cotradcto. Assume z N (M), z ad z K +. Suppose ts support set s S (dexes set of o-zero elemets). Select K + colums from M followg S ad costruct a ew matrx M S. Sce there exsts oe row M S wth a sgle, thus M S z must be satsfed, whch cotradcts the assumpto that z N (M). So each vector z N (M) has at least (K + 2) o-zeros elemets. So we ca coclude that ay K colums of M are lear depedet, ad whch fers that spark(m)>k +. Now we gve the frst mportat theorem ths paper: Theorem. Let M m be a K-dsjuct matrx, ad y = Mx. If x s a k-sparse vector, where k K+ 2, the x s the uque k sparsest vector satsfyg y = M x. Proof. We prove ths theorem va cotradcto. Assume Mx = Mx 2 = y, where x, x 2 Γ k ad x x 2. Deote S ad S 2 are the support sets of x ad x 2 respectvely. Sce Mx = Mx 2, we ca get that M(x x 2 ) =. Accordg to the assumpto, x x 2 ad x x 2 2k K +, we ca fer that there exsts oe o-zero vector x = x x 2 such that Mx =, where x K +. However, we kow that there does ot exst such a o-zero vector from Lemma, whch cotradcts the assumpto. From Theorem, we kow that f the measuremet matrx s K-dsjuct, the we ca recovery the k-sparse vector precsely va compressve sesg. So the ext sub-secto, we study the problem that how may path measuremets,.e., radom walks o dgraph, are eeded to costruct a K-dsjuct matrx. B. How May Measuremets are Needed? I ths paper, measuremet matrx s geerated by m depedet radom walks over the dgraph. Our goal s to get the upper boud of the umber of radom walks, whch s suffce to costruct the K-dsjuct measuremet matrx. I other words we eed a theoretcal guaratee that how may radom walks are suffce to costruct Λ whch satsfes the codto (2). I [, 3], a upper boud of the umber of radom walks over the (D, c)-uform udrected graph has bee proved, where D D, D = O(c 2 KT ()). T () s the δ-mxg tme of a radom walk whch wll be troduced soo. I ths paper, we exted the method of [3] to gve a upper boud the (D, c)-uform strogly coected dgraphs, where D e2 c. For dstgush the edge e, we use e as the base of atural logarthm. Theorem 2. Suppose G s a (D, c)-uform strogly coected dgraph wth δ-mxg ( tme) T () (where δ := (/2c) 2 ). For D < e2 c, K = O ed2 ad t = O(/(c 3 KT ())) + (radom walk legth), the measuremet matrx M s K- dsjuct wth probablty o() after m measuremets, where ( ( m = O c 4 KT 2 ()log. K)) We ca uderstad ths theorem from the opposte aspect: f the codtos Theorem 2 are all satsfed, the after m radom walks o G, the probablty that the measuremet matrx M s ot K-dsjuct matrx s o(). I other words, to prove ths theorem, we eed to lower boud the probablty that after m measuremets, o radom walk satsfes codto (2). So the crtcal step s determg the probablty that a radom walk satsfes codto (2) (.e., π e,e, wll be troduced soo), ad the followg part of ths sub-secto, we wll focus o ths problem. For statemet coveece, we frstly troduce a cocept of δ-mxg tme of radom walk ad gve two useful otatos. Accordg to [9] we kow that the radom walk o a dgraph G has a uque statoary dstrbuto f G s strogly coected ad s ot perodc, whch s obvously our cotext. Based o ths fact, we gve the defto of δ-mxg tme uder strogly coected dgraph. Defto 3 (δ-mxg tme[3]). Let G=(V, E) be a (D, c)- uform strogly coected dgraph ad deote by π ts statoary dstrbuto. For e E ad a teger τ, deote by π τ e the dstrbuto that a radom walk of legth τ startg at e. The the δ-mxg tme of G s the smallest teger t such that π τ e π δ, for τ t ad e E. For cocreteess, we defe the quatty T () as the δ-mxg tme of G for δ := (/2c) 2. Cosder a radom walk W := (w, w,..., w t ) of legth t o a dgraph G = (V, E), where the radom varables w E deote the edge vsted by the walk. We gve the followg quattes related to the walk W : π e The probablty that W passes e E π e,e The probablty that W passes e (E\E ) but oe of the edges E, E E. Lower boudg π e,e s exactly the crtcal step the proof of Theorem 2. We gve a lemma about the lower boud of π e,e : Lemma 2. There are two scales K := O ed2 + ad t := O c 3 KT () such that wheever K < K, by settg the path legth t := t the followg holds. let E be a set of at most K edges the graph G, ad let e / E, the π e,e = Ω c 4 KT 2. () For provg Lemma 2, we troduce some basc propostos that wll be used the proof. The frst proposto gves a boud of statoary dstrbuto of strogly coected dgraph.

4 4 Proposto. Let G = (V, E) be a (D, c)-uform strogly coected dgraph, ad deote by π the statoary dstrbuto of G. The for each e E, /c π(e) c/ The proof ca be foud Secto VIII-A. Actually, [6], the authors proposed a approxmated method to measure the statoary dstrbuto uder strogly coected dgraph va P E. But our paper, we do t eed to get the exact statoary dstrbuto, boudg t s eough. Proposto 2. Suppose a radom walk W :=(w, w,..., w t ) o dgraph G startg from a arbtrary edge ad set j +T (). Let ε deote ay evet that oly depeds o the frst edges vsted by W. The for every µ, ν E, P r[w =µ w j =ν, ε] P r[w = µ ε] 2 3c. The proof detals of ths proposto are Secto VIII-B. The followg proposto gves a boud of π e, whch s proved Secto VIII-C. Proposto 3. For the probablty that W passes e E,.e., π e, we have t π e = Ω, ct () where t s the walk legth. Now we beg to prove Lemma 2, whch s the techcal core of ths paper. Proof of Lemma 2. Let B deote the evet that W = {w,..., w t } vsts some edges E. Now π e,e = P r[ B, e W ] = P r[e W ]P r[ B e W ] = π e ( P r[b e W ]). Next we eed to upper boud P r[b e W ]. Frstly we fx > 2T (), ad assume that w = e. The fx some edge e / E ad assume that w = e. Now we try to upper boud P r[b w =e, w =e ]. Let β := T () ad γ := +T (), ad assume that T ()+ < β <γ <t. Partto W to four segmets [3]: W := (w, w,..., w T () ), W 2 := (w T ()+, w T ()+2,..., w β ), W 3 := (w β, w β+,..., w γ ), W 4 := (w γ+, w γ+2,..., w t ). For j =, 2, 3, 4, defe ρ j := P r[w j passes E w = e, w = e]. Now we upper boud each ρ j. I some degeerate stuato, W j may be empty, thus the correspodg ρ j wll be. W 2 ad W 4 are oblvous of the codtog o w ad w sce they are suffcetly far from both. I partcular, the dstrbuto of each edge o W 4 s pot-wse close to statoary dstrbuto π. Therefore, uder our codtog the probablty that each such edge belogs to E s at most E (c/ + δ) < 2cK/. Smlarly, each edge o W 2 has a almost-statoary dstrbuto wthout the codtog o w. However, by Proposto 2, the codtog o w chages ths dstrbuto by up to δ := 2/(3c) at each edge. Altogether, for each j {T () +,..., β }, we have P r[e j E w =e, w =e ] E (c/+δ+δ ) 2cK/. Usg a uo boud o the umber of steps, we coclude ρ 2 + ρ 4 tk(c/ + δ + δ ) 2ctK/. For boudg ρ 3, we eed to dvde W 3 /w to two segmets ad boud them respectvely. The detals are show Secto VIII-D, here we just lst the results: for j = +,..., γ, for j =β,...,, P r[w j E w =e, w =e ] K. (3) P r[w j E w =e, w =e ] K + Kδ. (4) Altogether, usg a uo boud ad by combg (3) ad (4), we get that 4T ()K ρ 3 K +KT 5KT () ()δ K. Usg the same reasog, ρ ca be bouded as Fally, we obta P r[b w =e, w =e ] 3KT () ρ K. 4 j= 8KT () ρ j K + 2ctK. The ext step s to relax the codtog o the startg edge of the walk. The probablty that the tal edge s E s at most K, ad by Proposto 2, codtog o w chages ths probablty by at most Kδ K. Now we wrte P r[b w =e] P r[w E ]+P r[b w =e, w / E ] 4 + ρ j j= 8KT () K (ct + ) +. Now by takg t = O(/c 3 KT ()), we kow that 8KT () P r[b w =e] K + (co(/c3 KT ()) + ) T ()K =O, K ad usg Proposto 3 we ca get the cocluso: ) T ()K π e,e π e ( O K t = Ω = Ω ct () c 4 KT 2. ()

5 (b) Correspodg Drected Graph Fg.. Example of Cty Roads Networks ay car ca reach ay posto from ay startg pot o (a) A Porto of Cty Road Network the graph. The dgraph model also has small ode ad edge degree, because a cross s geerally tersecto of two or three roads. So that followg part of ths secto, we study how to recover the delays of all roads by m probg cars walkg radomly o the dgraph to measure ed-to-ed path delays. 6 5 y =y Mx r = Mx. (5) Lemma 2 gve a lower boud of π e,e whch exactly satsfes our requremet. Now we are ready to prove Theorem 2. The Proof of Theorem 2. Wthout loss of geeralty, take a arbtrary set of edges E E wth cardalty E = K +. Deote by π E the probablty that a radom walk vsts oe ad oly oe elemet from E. I fact, K + π E =(K +)π e,e /e =Ω c 4 KT 2. () Deote by P f the falure probablty, amely that the resultg matrx M s ot K-dsjuct. By a uo boud we get P f ( π E ) m K + K+ ( ) m e K + Ω K + c 4 KT 2 () ( ) ( ) m e exp (K + ) log Ω K + c 4 T 2. () If ( ) ( ) e (K +) log +m log Ω K + c 4 T 2 <. () Thus by choosg m = O ( ( c 4 KT 2 ()log K)) we ca esure that P f = o(), ad hece, M s K-dsjuct wth overwhelmg probablty. IV. COMPRESSIVE TRANSPORTATION STATE MONITORING Compressve sesg over dgraphs ca beeft may applcatos. As a stace studed ths paper, we show that the abstract models of trasportato etworks are exactly strogly coected dgraphs wth low ode/edge degrees. Therefore applyg compressve sesg o trasportato etworks to motor road delays becomes promsg, whch reduce the cost of takg measuremets by probg cars. However the delays o roads,.e, the edge delay vector are o loger sparse. Eve though, ths secto we preset a ew method to recover the edge delay vector eve t s ot sparse. Partcularly, we utlze the emprcal o-cogested delays of roads to help to estmate the cogesto factors of edges ad paths at frst, whch ultmately leads to a effcet algorthm to recover the edge delays. To avod ambguty, ths secto road ad edge are both used to represet road segmets ad path meas the route traveled by cars. A. Cty Road Network Model Gve a dgraph G = (V, E), where V = N ad E =. Each vertex v V dcates a road crossg, ad each drected edge (u, v) E represets a road from crossg u to v. Fg. shows a example, where Fg.(a) s a porto of the cty road etworks ad Fg.(b) s the correspodg dgraph model. We otce that there s oe apparet ad mportat fact that the dgraph model of road etworks s strogly coected, because B. Problem formulato for trasportato motorg For coveece, TABLE I lsts some otatos whch wll be used the followg parts. The real-tme road delay TABLE I SUMMARY OF KEY NOTATIONS Notato Meag x The real-tme road delay vector, x dcates the curret delay (travel tme) o road x r The emprcal, o-cogested road delays vector x The estmated (recovered) road delays vector r The road cogesto rak vector α, α2 The factor of sem-cogesto ad cogesto, where <α <α2 < p, p The path ad road cogesto factor vectors x The guessed approxmate traffc codto vector, where x=xr + p x r MD, M D Two bary dagoal matrxes, the former cotrols whch roads wll be recovered; the latter dcates whch roads determed the road cogesto factor. vector s represeted by x, whose elemet dcates the curret delay of each road. Vector x r represets the emprcal, ocogested road delays, whose elemets represet the ormal travel tme o the correspodg roads wthout presece of cogesto. Let vector y represet ed-to-ed delays measured by m probg cars the cty, ad M be a m bary matrx, whose elemet M,j = f the th car passes through the jth road ad otherwse. Apparetly, y = Mx provdes a approprate formulato about the trasportato motorg. Ufortuately, x,.e., the road delay vector s ot sparse. Nevertheless, wth kowledge of the emprcal o-cogesto delay x r, the dfferece vector x = x xr ca rule out some o-cogested roads because ther dffereces from emprcal o-cogested delays are almost zero. Therefore, we cosder the trasformed problem as followg:

6 6 We hope x s sparse, but most tmes, t ca ot meet our expectato. The reaso s that the rages of x r are dfferet, whch meas that eve some road wth bg ormal delay s umpeded, the dfferece value may stll be greater tha some cogested road wth small ormal delay values. Although we ca ot recover x drectly most tmes, we stll ca fer some roads codto va Formula (5). For example, f y () =, the the real-tme traffc codto of roads vsted by the th measuremet path are equal to the correspodg elemets values x r ad we ca make sure that these roads are umpeded. Further more, f we ca fer whch roads are most possble cogested, the we ca recover these roads wth hgh prorty. Suppose we select k most possble cogested roads by our speculato, ad dex them by the set S = {s, k}. Gve a bary dagoal matrx M D = {d j }, where d sjs j =, s j S or otherwse. The the vector x = x M D x must be k-sparse. The formulato becomes a ew form: y =y MM D x = Mx. (6) Actually, y ca ot be gotte sce x s ukow whch meas that MM D x ca ot be calculated. Nevertheless, we ca try to costruct a vector x, a substtute of x such that MM D x MM D x. Assume x 2x r, we eed to fd a factor vector p, ad based o ths factor vector, we ca costruct a vector x = x r + p x r. Algorthm shows a heurstc terato method to get p. The fudametal dea s that frstly utlzg M D we delete the roads whose cogesto state have bee determed ad mus ther cotrbutos from y, the we cosder the left roads average the cotrbuto to ew y. We wll gve prorty to the rows where oly oe or two roads are left whe calculate p. The the fal verso of the measuremet formula become as the followg: y =y MM D x Mx. (7) Sce every tme oly k elemets are recovered, thus the recovery algorthm should be executed utl x totally be recovered. C. The Recovery Algorthm The algorthm ca roughly dvded to two steps: step Calculate road cogesto factor vector. step 2 Recover x by the descedg order of the road cogesto factor utl x totally be recovered. For step 2, we ca utlze some commo CS recovery algorthm, for example, l -orm mmzato. The crtcal step s calculatg the road cogesto factor. So pror to troducg the algorthm detals, we wll show how to estmate the road cogesto factor vector p. Specfcally we eed frstly calculate the path cogesto factor vector p, the based o p, we ca guess some road cogesto factors, ad these estmated factors wll tur fluece the path cogesto factors. Ths process wll cotue utl all road cogesto factor are determed. Algorthm shows the detals about calculato of p ad p. Note that M( ) dcate the th row ad M T represets the trasform of M. Algorthm Calculate Path Cogesto Factors Iput: Measuremet matrx M m = {m j }, x r, y. Output: The estmated road cogesto factors vector p. : M D = I ;// I s the detty matrx. 2: p =, p = ; 3: y = y MM D xr ; 4: Choose the edges where y () = ad m j =. Idex them by S ={s, t} ad set d ss =, t; y () j= mjxr j 5: p =, for all ; djj 6: whle # of udetermed roads > do 7: M =MM D ; 8: [mv, mi]=m(m ( )) for all m; 9: Choose the edges where m mi,j = ad dex them by S ={s, t}, ad set d ss = for all t; : p s = y (mi) M (mi )x, for all t; r : y = y M M ( ) T p x r. 2: p = 3: ed whle y (mi) j= mjxr j djj, for all m; Actually, there are stll some roads ca ot be assged cogesto factor sce they are ever be vsted by ay path. I fact, we do ot eed to calculate these roads cogesto factors ad just eed estmate ther road codto whe we fshed the recovery algorthm. The detals wll be troduced the followg part. Algorthm 2 Traffc Codto Vector Recovery Algorthm Iput: Measuremet matrx M m = {m j }, x r, y. Sparsty level k. Output: The estmated traffc codto vector x for x. : x = x r, M D = I ; 2: Calculate p va Algorthm ; 3: y =y MM D x=y MM D (x r + p x r ); 4: whle # of urecovered roads > do 5: Select k roads followg descedg order of p s elemets, whch are dexed by S = {s, k}; 6: d s s =, for all k; 7: x = M D x + (I M D ) l m(y, M); 8: M D = I; 9: ed whle The detals of recovery algorthm s show Algorthm 2. There are two matters eed atteto: I le 7, l m(y, M) dcates the l orm mmzato fucto, whch s a commo algorthm compressve sesg, so we omt the detals. At last terato, the umber of the roads watg for

7 7 recovery may be less tha k. But for cocse statemet, we assume the value of k ca chage at last terato. We have metoed that there may be some roads ever be vsted by ay measuremet path. So we eed fd a method to estmate these roads traffc codto. we propose a smple method based o the assumpto: oe road ad ts eghbour roads has hghly smlar traffc codto. For example road e = (u, v) s ever be vsted measuremet process, deote by E = {e j = (u, u), u V/v} ad E + = {e j = (v, v ), v V/u} the backward ad forward eghbor roads set respectvely, the the calculato formula s lke followg: x = x r + e j E x ej x r e j deg(e j ) + + e j E + x ej x r e j deg(e j ). (8) Based o the recovered traffc codto vector x, we ca easly rak the roads cogesto levels. As the wdely used red, yellow, gree represetato of road codtos, by comparg the real-tme road delay wth the emprcal ocogesto delay, we preset r r whch ca take oly three values to dcate the real-tme road codtos. If the curret traffc codto vector s kow, we ca rak the road codto lke followg: r = f x r x <(+α )x r f (+α )x r 2 f (+α 2 )x r x <(+α 2 )x r x 2x r V. SIMULATION RESULTS // gree // yellow // red Smulatos are carred out to evaluate the traffc delay motorg algorthms. We pay specal attetos to the detecto rate of cogesto lks ad the tradeoffs betwee the measuremet cost ad detecto accuracy. A. Groud Truth Data For evaluatg the performace of Algorthm 2, we eed to geerate the groud truth data,.e., smulate the traffc codto of the cty road etworks ad take measuremets o t. For ths purpose, we bult a abstracted but reasoable traffc etwork model Matlab to verfy the coceptoal correctess of proposed algorthms. The model omts the dyamc behavors of cars ad effects of traffc lghts, but grasps the essetal spatal correlato of traffc delays coected ad eghborg roads. The desg methodology of the model s that the cogesto at road e wll propagate slowly to ts comg lks ad ts sblgs. The releasg from cogesto of ths road wll also reduce the cogesto of comg lks ad sblgs. We have verfed dfferet fluecg coeffcets to comg lks ad sblgs ad foud that the proposed algorthm worked well dfferet correlato scearos. (9) B. Smulato Results We evaluate the accuracy of recovered vector x va calculatg the SNR value betwee x ad x followg the formula x 2 2 SNR db = log x x 2 2 Fg.2 was draw for measuremet umber ad SNR values amog 9 dfferet road umbers ad 3 dfferet cogesto ratos ( k ). We ca fd that as the road umber ad cogesto rato creasg, the measuremet umber must crease to keep the accuracy of x. Although the SNR values dcate that the accuracy are ot perfect, but almost all them are less tha 5dBm, whch s also a acceptable results compressve sesg. Measuremet Number Rato=% (a) Measuremet Number for D- fferet ad Rato SNR Rato=% (b) SNR betwee x ad x Fg. 2. Measuremet Number ad SNR betwee x ad x Based o the recovered vector x, we ca further rak the roads cogesto levels followg (9), because t geerally true that oly the cogesto levels are cared by users. TABLE II shows a example whe the road umber s fxed 72. We ca fd that for cogested roads, ot oly the detecto rate ( Commo Commo Orgal ) but also the accuracy rate ( Recovered ) are greater tha 9%; for sem-cogested roads, the detecto rate s also hgher tha 9%. TABLE II FIXED ROAD NUMBER = 72, m = 73 Orgal Recovered Commo Gree Yellow Red Further more, we have also compared the detecto ad accuracy rate regardg dfferet road umber ad cogesto rato. From Fg 3, we fd that although the accuracy of x s ot perfect, we stll ca detect the cogested ad sem-cogested roads wth hgh probablty. For example, the detecto rate for cogested ad sem-cogested roads (Fg.3(a) ad 3(c)), we ca easly fd that almost all the red ad yellow roads be checked out ad the lower cogesto rato the better performace. Although the accuracy rate (Fg.3(b) ad 3(d)) for cogested ad sem-cogested roads are ot very well, we ca cosder these results as pessmstc estmato, other words, f some road s raked as a gree road based o our recovery vector, the bascally ths road ca be cosdered as a umpeded road.

8 8 Detecto Rate Rato=% (a) Detecto Rate for Cogested Roads Detecto Rate Rato=% (c) Detecto Rate for Sem- Cogested Roads Accuracy Rate Rato=% (b) Accuracy Rate for Cogested Roads Accuracy Rate Rato=% (d) Accuracy Rate for Sem- Cogested Roads Fg. 3. Detecto ad Accuracy Rate VI. RELATED WORK We revew related work ths secto. Compressve sesg s a ew paradgm sgal processg area, whch offers fudametal theores ad effcet recovery algorthm to recover sparse sgal from reduced dmeso measuremets[4, 8, ]. A promsg applcato area of compressve sesg s etwork tomography [ 3], whch vestgate the problem of motorg lk propertes by lmted umber of ed-to-ed measuremets. Prevous studes about ths problem [ 3] showed that the ed-to-ed, alog-path measuremets graphs troduced ew path restrctos to the measuremet matrx, so that the measuremets could o loger be depedet ad freely radom. Ths some exted may cotradct the coheret requremet of observato matrx CS. But by delberate proofs, [ 3] proved that, despte the path costrats, udrected graphs, O(klog()) path measuremets are suffcet to detfy k-sparse lk vector ad they proposed l mmzato algorthm that has theoretcal guaratee recoverg the k-sparse lk vector. Some researchers have explored to apply the dea of compressve sesg oto trasportato etwork motorg[4, 5]. Other tha the state of the art estmatg traffc utlzg statc sesors such as loop detectors or traffc cameras, some recet studes have bega to vestgate the use of GPS devces as dyamc traffc probes for ferrg traffc volume usg exstg mathematcal models[6, 7], or they estmate traffc speed from GPS[8, 9]. The utlzato of dyamc prob sesors provdes a great support for the trasportato motorg va compressve sesg. However, most of these studes are based o a data mg type of approach. They ether coverge the trasportato flow matrx by sgular value decomposto or dscovered the sparseess feature by processg the traffc data statstcally. Dfferet from these exstg approaches, we at frst vestgate theoretcally the applcato of CS o strogly coected dgraphs, whch the frst tme shows the effectveess of CS dgraphs to the best of our kowledge. We the preset effcet recovery algorthms essetal model of traffc etworks, whch provdes algorthms to recover road delays whch are eve ot sparse. VII. CONCLUSION AND FUTURE WORKS I ths paper, for strogly coected dgraphs wth low ode (or edge) degrees, we prove that O(klog()) path measuremets are suffcet to recover k-sparse edge vectors. It s derved by gvg bouds to the statoary dstrbuto of radom walk whch s dfferet from the aalyss udrected graphs. Further more, because the urba road etworks are exactly strogly coected ad wth low ode degrees, we desged a effcet, teratve recovery algorthm, whch recovers the road delays ad the cogesto states of the traffc etworks by small umber of probg cars. I future work we wll study the traffc delay recovery algorthm o the real data set. Secodly, based o the curret cogesto state, predctg the future chage of the etwork also falls our research terests. ACKNOWLEDGMENT Ths work was supported part by the Natoal Basc Research Program of Cha Grat 2CBA3, 2C- BA3, the Natoal Natural Scece Foudato of Cha Grat 62236, 633, 67374, REFERENCES [] W. Xu ad A. Tag, Compressve sesg over graphs: How may measuremets are eeded? Commucato, Cotrol, ad Computg (Allerto), 2 48th Aual Allerto Coferece o. IEEE, 2, pp [2] M. Wag, W. Xu et al., Sparse recovery wth graph costrats: Fudametal lmts ad measuremet costructo, INFO- COM, 22 IEEE. IEEE, 22, pp [3] M. Cheraghch, A. Karbas et al., Graph-costraed group testg, Iformato Theory Proceedgs (ISIT), 2 IEEE Iteratoal Symposum o. IEEE, 2, pp [4] E. J. Cades ad T. Tao, Decodg by lear programmg, Iformato Theory, IEEE Trasactos o, vol. 5, o. 2, pp , 25. [5] C. Cooper ad A. Freze, Statoary dstrbuto ad cover tme of radom walks o radom dgraphs, Joural of Combatoral Theory, Seres B, vol. 2, o. 2, pp , 22. [6] A. Mohase, H. Tra et al., O the mxg tme of drected socal graphs ad securty mplcatos, Proceedgs of the 7th ACM Symposum o Iformato, Computer ad Commucatos Securty, ser. ASIACCS 2. New York, NY, USA: ACM, 22, pp [7] H. X. Nguye ad P. Thra, The boolea soluto to the cogested p lk locato problem: Theory ad practce, INFOCOM, 27 IEEE. IEEE, 27, pp [8] M. A. Daveport, M. F. Duarte et al., Itroducto to compressed sesg, Preprt, vol. 93, 2. [9] K. Costello, Radom walks o drected graphs, 25. [] E. J. Cadès, Compressve samplg, Proceedgs oh the Iteratoal Cogress of Mathematcas: Madrd, August 22-3, 26: vted lectures, 26, pp

9 9 [] T. Bu, N. Duffeld et al., Network tomography o geeral topologes, ACM SIGMETRICS Performace Evaluato Revew, vol. 3, o.. ACM, 22, pp [2] Y. Che, D. Bdel et al., Algebra-based scalable overlay etwork motorg: algorthms, evaluato, ad applcatos, Networkg, IEEE/ACM Trasactos o, vol. 5, o. 5, pp , 27. [3] A. Gopala ad S. Ramasubramaa, O detfyg addtve lk metrcs usg learly depedet cycles ad paths, IEEE/ACM Trasactos o Networkg (TON), vol. 2, o. 3, pp , 22. [4] H. Zhu, Y. Zhu et al., Seer: Metropolta-scale traffc percepto based o lossy sesory data, INFOCOM 29, IEEE. IEEE, 29, pp [5] Z. L, Y. Zhu et al., Compressve sesg approach to urba traffc sesg, ICDCS, 2. IEEE, 2, pp [6] M. C. Gozalez, C. A. Hdalgo, ad A.-L. Barabas, Uderstadg dvdual huma moblty patters, Nature, vol. 453, o. 796, pp , Jue 28. [7] J. Aslam, S. Lm et al., Cty-scale traffc estmato from a rovg sesor etwork, Proceedgs of the th ACM Coferece o Embedded Network Sesor Systems. ACM, 22, pp [8] D. B. Work, O.-P. Tossavae et al., A esemble kalma flterg approach to hghway traffc estmato usg gps eabled moble devces, Decso ad Cotrol, 28. CDC th IEEE Coferece o. IEEE, 28, pp [9] D. B. Work, S. Blad et al., A traffc model for velocty data assmlato, Appled Mathematcs Research express, vol. 2, o., pp. 35, 2. A. Proof of Proposto VIII. APPENDIX I [5], there are two lemmas (Lemma 6 ad 8) about the boud of statoary dstrbuto o dgraph, whch s ( o()) deg(e) π(e) ( + o()) deg(e), where s the edge umber of a ew dgraph G duced from G whose edges become the vertexes of G. Suppose two edges e = (u, v ), e 2 = (u 2, v 2) of G, f v = u 2, the there exsts a drected edge from e to e 2 G. Sce G s (D, c)-uform dgraph, whch meas D deg(e) cd ad D cd, thus we coclude that c = D cd B. The Proof of Proposto 2 We kow that P r[w =µ w j =ν, ε]= π(e) cd D = c P r[wj =ν w =µ, ε] P r[w =µ ε]. P r[w j =ν ε] Now, accordg to the defto of δ-mxg tme, we get that P r[w j =ν w =µ, ε] P r[w j =ν ε] 2δ, because regardless of the kowledge of w = µ, the dstrbuto of w j must be δ-close to the statoary dstrbuto. Therefore, P r[w = µ w j = ν, ε] P r[w = µ ε] 2δ/P r[w j = ν ε] 2δ/(/c δ) 8δc/3 C. The Proof of Proposto 3 Let t := t/t (), for each {,,..., t }, w := w T (). Deote by W :={w,..., w t } a subset of t + edges vsted by W. Obvously, π e s at least the probablty that e W. By the defto of δ-mxg tme, regardless of the choce of w, the dstrbuto of w s δ-close to the statoary dstrbuto π, whch assgs a probablty betwee /c ad c/ (by Proposto ) to e. Therefore, P r[w e w ] /c+δ 2c. Smlarly, P r[w 2 e w, w ] /2c, ad so o. Altogether, ths meas that ( P r[w e, w e,..., w t e] ) t/t () 2c ( t/2ct ()) e Ω(t/cT ()). I the last equalty we used the fact that e x x/2 for x. Thus the complemet probablty s lower bouded by Ω(t/cT ()). D. The upper boud of ρ 3 Deote D = cd, for boudg ρ 3, we frstly calculate P r[w + E w = e, w = e ] D (K ) j D j D j K = D j j= = D K D D j K j D j K j= K + D K D ( D K j K D j K K + ( K K D K + ( K K D =2 ) D D ( ed j j j=2 ) D D ) j ) j ( K K K K K where the secod equalty s due to ( D ) ( ed ) j j j for D e 2, ( ed ) j ( j K ad the thrd equalty because of j j K) < for K <K. Smlarly, j=2 P r[w +2 E w = e, w = e, w +] K, regardless of w + whch meas P r[w +2 E w = e, w = e ] K, ad geeral, for j = +,..., γ, P r[w j E w = e, w = e ]=P r[w j E w = e] K. Smlarly j = β,...,, we have P r[w j E w = e] K, ad by Proposto 2, codtog o w chages ths probablty by at most Kδ. Therefore, P r[w j E w = e, w = e ] K +Kδ. ) j