On Robust Network Planning


 Darleen Perkins
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1 On Robust Network Plannng Al Tzghadam School of Electrcal and Computer Engneerng Unversty of Toronto, Toronto, Canada Emal: Alberto LeonGarca School of Electrcal and Computer Engneerng Unversty of Toronto, Toronto, Canada Emal: Abstract One of the mportant propertes of a relable communcaton network s the robustness to the envronmental changes. Ths paper looks at the desgn of robust networks from a new perspectve. A graphtheoretcal metrc, betweenness, n combnaton wth network weght matrx s used to defne a global quantty, network crtcalty, to characterze the robustness of a network. We show that network crtcalty s a monotone decreasng functon of weght matrx. Furthermore, t s shown that network crtcalty s a strctly convex functon of network weght matrx. Ths leads to a welldefned convex optmzaton problem to fnd the optmal weght matrx assgnment to mnmze network crtcalty. I. INTRODUCTION Robustness to the envronmental changes s an mportant factor n the desgn of relable communcaton networks. Robustness s the capablty of a network to keep tself n a stable mode when envronmental changes take place. Let us begn wth our defnton of robustness. There are three major types of envronmental changes that may affect the performance of the network: Changes n network topology ncludng capacty. 2 Changes n communty of nterest, the set of actve sources and snks for traffc. 3 Changes n traffc demand. Throughout ths paper, we say that a network s robust f ts performance s not senstve to changes n topology, traffc or communty of nterest. In ths sense, a robust network s more relable, snce unantcpated envronmental changes such as traffc shfts cannot sgnfcantly mpact the behavor of network. We am to develop methods to desgn such robust networks wth the help of graphtheoretcal concepts. In ths work our approach s to fnd a metrc to capture the effect of topology, traffc demand, and communty of nterest, and then desgn a network to control the senstvty of the network to the shfts n traffc, or changes n topology by controllng the proposed metrc. Usng ths approach we are able to nvestgate the problem analytcally wth the help of metrcs from graphtheory. We can also dscover some useful aspects of the robustness problem n networks. The paper s organzed as follows. Secton II revews prevous work on network desgn and robustness n networks. In secton III a revew of our man metrc, network crtcalty, s provded. The proposed optmzaton problem and ts attrbutes are dscussed n secton IV, followed by some examples of network plannng for welldefned graphs n secton V. The paper s concluded n secton VI. II. PREVIOUS WORK A wealth of lterature s avalable on network robustness and ts dfferent aspects. [] nvestgates the relatonshp between node smlarty and optmal connectvty, and arrves at the result that a network provdes maxmum resstance to node destructon f t s both nodesmlar and optmally connected. The paper then descrbes a number of ways to desgn robust networks satsfyng these condtons. But ths paper consders only the effect of topology n the robustness of a network. In [2] a way to desgn backbone networks s proposed that s nsenstve to the traffc matrx.e., that works equally well for all vald traffc matrces, and that contnues to provde guaranteed performance under a userdefned number of lnk and router falures. Valant LoadBalancng method s used and the authors argue that t s a promsng way to desgn robust backbone networks. The approach was frst proposed by Valant for processor nterconnecton networks [3], and has receved recent nterest for scalable routers wth performance guarantees [4]. [2] apples Valant method to backbone network desgn problem and provdes approprate capacty allocaton for the lnks of a logcal full mesh topology to support loadbalancng for all possble traffc matrces. In the Valant loadbalancng method, traffc destned for a snk d s forwarded to ntermedate hops wth equal splts to all nodes, and then t s forwarded to the destnaton d. Delay propagaton s one of the shortcomngs of ths method. In [5] we proposed a framework for robust routng n core networks based on the dea of lnk crtcalty and path crtcalty. Further development of the dea of crtcalty s studed n [6], where a mathematcal framework for the defnton of crtcalty s gven wthn the context of Markov chan theory. In ths paper we quantfy the robustness usng the concept of crtcalty from [6]. We study the behavor of crtcalty and derve some useful results for the desgn of robust networks n the face of both topology and traffc varatons. III. NETWORK CRITICALITY In ths secton we summarze the results of our prevous work on robustness [6]. A. Network Model We model a network wth an undrected weghted graph G N, E, W where N s the set of nodes, E s the set of graph lnks, and W s the weght matrx of the graph. Throughout
2 ths paper we assume that G s a connected graph. We assume that SLA Servce Level Agreement parameters are already mapped to the weghts by some approprate method. Some of these methods are dscussed n [7]. Ths permts us to abstract dfferent busness polces and/or SLA s as parts of the weght defnton. In ths paper we are nterested n the study of the weghts and ther effect on robustness. In [8] a probablstc nterpretaton of the betweenness s defned based on random walks. The betweenness of a node lnk k for sourcedestnaton par s d s the expected number of tmes that a random walk passes node k n ts journey from source s to destnaton d. The total betweenness of node k s the sum of ths quantty over all possble s d pars. The randomwalk s defned on a Markov chan M wth transton probablty matrx P accordng to the followng rule: p j { w j k A w k f 0 otherwse j A where A s the set of adjacent nodes of and w k 0 s the weght of lnk, k. B. Defnton of Network Crtcalty We now ntroduce network crtcalty, the metrc that we proposed n [6], to quantfy the robustness of a network. We start by defnng node/lnk crtcalty. Defnton 3.: Node crtcalty s defned as the randomwalk betweenness of a node over ts weght weght of a node s defned as the sum of the weghts of ts ncdent lnks. Lnk crtcalty s also defned as the betweenness of a lnk over ts weght. Let η k be the crtcalty of node k and η j be the crtcalty of lnk l, j. It s shown n [6] that η and η j can be obtaned by the followng expressons: τ sd l ss + + l + dd 2l + sd or τ sd u t sdl + u sd 2 τ τ sd, τ nn τ 3 s d η k b k nn τ τ 4 W k 2 2 η j bj w j τ nn τ 5 where L + s the MoorePenrose nverse of graph Laplacan matrx L [9], n s the number of nodes, and u j [ ] t and are n th and j th poston. Observaton 3.2: Equatons 2 to 5 show that node crtcalty η k and lnk crtcalty η j are ndependent of the node/lnk poston and only depend on τ or τ whch s a global quantty of the network. Defnton 3.3: We refer to τ as the normalzed network crtcalty or smply network crtcalty. One can see that τ s a global quantty on network graph G. Equatons 4 and 5 show that node lnk betweenness conssts of a local parameter weght and a global metrc network crtcalty. τ can capture the effect of topology and communty of nterest va betweenness, and the effect of traffc va weght by approprate defnton of weght. The hgher the betweenness of a node/lnk, the hgher the rsk of usng the node/lnk. Furthermore, one can defne node/lnk capacty as the weght of a node/lnk, then the hgher the weght of a node/lnk, the lower the rsk of usng the node/lnk. Therefore network crtcalty can quantfy the rsk of usng a node/lnk n a network whch n turn ndcates the degree of robustness of the network. Ths motvates the rest of our work n ths paper. We nvestgate network crtcalty as a networkwde metrc to capture and optmze network robustness. In ths paper our goal s to nvestgate τ as a functon of weght matrx W and fnd necessary condtons for ts optmalty. We am to fnd an approprate weght matrx that can optmze the robustness of a network. IV. OPTIMIZATION OF NETWORK CRITICALITY In ths secton we nvestgate the behavor of τ as a functon of the weght matrx. A. Monotoncty and Convexty of τ We start by nvestgatng the convexty of τ. In order to proceed we need the followng lemma. Lemma 4.: Network crtcalty τ s a monotone decreasng functon of weghts. More precsely τ 2 n L+ L + j 2 Furthermore, the second partal dervatve of τ wth respect to the weght w j s nonnegatve and can be found from the followng equaton: Proof: See Appendx A. 2 τ w τ 2 2τj j Theorem 4.2: Network crtcalty τ s a strctly convex functon of weghts. Proof: It s enough to show that the frst dervatve of τ s always negatve and t s second dervatve s always nonnegatve. Consderng the result of lemma 4., these condtons are met n τ. B. Optmzaton of τ Snce τ s a strctly convex functon of the weghts, an optmzaton problem wth lnear constrants has a unque soluton. We consder the mnmzaton of τ under some constrants. We set the followng as a constrant for our optmzaton problem: zjwj C, where zj s the cost of lnk, j. One can consder C as the allowable budget for total network weght. Consder the followng optmzaton problem on graph GN, E, W : Subject to Mnmze τ zjwj C, C s fxed 6 w j 0
3 Theorem 4.3: For the optmal weght set, W, n optmzaton problem 6 we have: w jc τ + z j τ 0, j E 7 Proof: We wll prove theorem 4.3 usng the followng lemma. Lemma 4.4: For any weght matrx W of lnks of a graph: V ecw t τ + τ 0 where V ecw s a vector obtaned by concatenatng all the rows of matrx W to get a vector of w j s. Proof: Suppose we scale all the lnk weghts n a graph wth factor t, then usng equaton 5 we have: τtv ecw nn b j tw j Note that the transton probabltes are nvarant to the scalng of the weghts based on ther defnton n equaton. Ths mples the nvarance of lnk betweenness b j. Therefore: τtv ecw τv ecw 8 t By takng the dervatve of τ wth respect to t, we have V ecw t τ τv ecw 9 t2 It s enough to consder equaton 9 at t to obtan V ecw t τ + τ 0. In optmzaton problem 6, τ s a contnuous decreasng functon of lnk weghts and t s strctly convex. Ths mples that n frst constrant of problem 6 the nequalty can be replaced by equalty. Now, to conclude the proof of theorem 4.3 we notce that for the optmal weght matrx W we can wrte V ecz.v ecw τ wjz j τ C τ 0 Combnng lemma 4.4 and equaton 0 one can see C τ.v ecw + V ecz.v ecw τ 0 V ecw.c τ + τv ecz 0 wjc τ + τz j 0, j E Ths completes the proof of theorem 4.3. Equaton 7 gves the necessary condton for the optmalty of convex problem 6. Optmzaton problem 6 can be converted to a semdefnte programmng. In order to fnd the semdefnte programmng representaton of the optmzaton problem 6, we need the followng lemma. Lemma 4.5: τ 2 T n rl+. Proof: Snce τ sd l ss + + l + dd 2l+ sd, we have: τ n l + + n l n τ nn 2nT rl+ 2 n T rl+ l + 2nT rl + Ths completes the proof of lemma 4.5. Suppose we let Γ L+ J n, where J s a nbyn matrx wth all elements equal to, then Γ can be wrtten as a semdefnte Γ I nequalty as follows. We consder matrx Θ. I L + J n The necessary and suffcent condton for postve semdefnteness of Θ s that ts Schur complement [0] be postve semdefnte. In general, the Schur complement of A B a matrx of the form s: A BD C. Hence the C D Schur complement of Θ s Γ L + J n, and Γ I Θ I L + J 0 Γ L + J n n where means postve semdefnte. Snce the optmzaton problem 6 should mnmze T rγ, the equalty n equaton s chosen whch s equvalent to Γ L + J n. Now optmzaton problem 6 can be converted to a semdefnte programmng: 2 Mnmze n T rγ 2 n Subject to z jw j C Γ I,j I L + J n 0 2 Ths new optmzaton problem can be solved wth standard methods of solvng semdefnte programs. In ths paper we solve ths optmzaton problem for some specfc graphs to show how the concept of network crtcalty helps fnd robust network topologes. C. Capacty Plannng In ths secton we consder the capacty plannng as an mportant specal case of network desgn problem. Consder a network GN, E, W where the lnk weghts are equal to the lnk capactes, that s, w j c j, j E c j denotes the capacty of lnk, j. We nvestgate the capacty assgnment problem n whch network topology and traffc load γ j, j E are assumed known and fxed. The goal s to fnd the capacty of the lnks so as to mnmze the network crtcalty under the constrant that the total cost of the plannng s fxed. Let z j be the symmetrc cost of assgnng capacty c j to lnk, j, and suppose that we have a lnear cost functon. The total cost of the capacty assgnment problem s zjcj. We fx ths total cost to C. We can wrte the optmzaton problem for capacty assgnment problem as follows: Subject to Mnmze τ cjzj C, C s fxed 3 c j γ j By applyng the change of varable c j c j + γ j to the optmzaton problem 3, we wll have the followng convex
4 optmzaton problem. Subject to Mnmze τ c jz j C, C s fxed 4 c j 0 where C C zjγj. The optmzaton problem 4 s now converted to the optmzaton problem 6 wth w j c j and C C, therefore, all the results of ths secton are applcable for the capacty assgnment problem. Later n ths chapter we wll see when the optmzaton problem 4 s equal to the Klenrock s capacty assgnment problem []. V. CASE STUDY In ths secton we nvestgate the optmal network weght allocaton for some welldefned network topologes. In the followng examples f we assume that the weght of a lnk s equal to ts capacty, then the problem s converted to a capacty assgnment problem. A. Complete Graph on n Nodes K n For K n we can obtan the soluton of optmzaton problem 6 analytcally. In ths example we assume z j, j E. In order to fnd the optmal weght set for K n we need the followng lemmas. Lemma 5.: τ can be wrtten as: τ 2 n 2 n 2 λ, where λ λ 2... λ n are egenvalues of graph Laplacan L. Proof: We know from lnear algebra that the sum of egenvalues of a square matrx s equal to the trace of the matrx. On the other hand, the nonzero egenvalues of L + are recprocals of the nonzero egenvalues of L.Consequenty, Lemma 5. wll be a drect result of Lemma 4.5. Lemma 5.2: Consder optmzaton problem 2 and suppose all the lnks have equal costs, that s, Z J I ths s a squarematrx whose entres are all, except the dagonal elements whch are set to zero. If there s an automorphsm on a graph GN,E,W that can map lnk l, j on lnk l, j, then these lnks should have equal optmal weghts. Proof: Let G be the new graph after applyng the automorphsm. Any automorphsm on graph G can be shown by a matrx T so that the Laplacan of transformed graph G can be obtaned from Laplacan of orgnal graph G as LG T LGT t. Ths means that LG and LG have the same egenvalues. As a result accordng to the Lemma 5. crtcalty of graph G and G are the same: τg τg. On the other hand the soluton of optmzaton problem 6 s unque. As a result the weght of lnk l and l are the same. Corollary 5.3: Consder optmzaton problem 2 and assume that the graph of the network s an edgetranstve graph wth equal lnk costs. The optmal weght for a lnk, j E s equal to w j C, where m denotes the number of graph m lnks. Proof: A graph s edgetranstve, f there s an automorphsm that can map any two lnks of the graph. Accordng to lemma 5.2 all the lnk weghts are equal. In addton, suppose w j w, j E, then constrant wj C mples that w C. Ths completes the proof of corollary 5.3. m Complete graph K n s an edgetranstve graph, therefore, accordng to corollary 5.3 the optmal weght of all the lnks of K n are equal. Let denote ths common weght by w. Further, let vector X be the egenvector of Laplacan matrx for egenvalue λ. Then: LK n wni J and LX λx wn x j j x λxj for In addton n x 0 property of the Laplacan matrx for K n. Therefore wn x j x j λx j λ nw for 2... n One can also fnd lnk weght w from corollary 5.3. The total number of lnks n K n s m nn, therefore, accordng to the corollary 5.3 w C. Therefore nn λ n C N 5 It s easy now to calculate network crtcalty for graph K n usng Lemma 5. and equaton 5. τ 2 n n 2 2n λ C 6 Accordng to equaton 6 the optmal network crtcalty n a complete graph s lnearly ncreasng wth the sze of the network. Ths provdes a bass for comparng the normalzed network crtcalty of dfferent networks aganst the fullmesh on n nodes. Case of Unequal Lnk Costs for K n: When the lnk costs z j s are not equal, we can use the semdefnte approach to fnd the best weght assgnment. We use a numercal example to show the effect of changes n lnk costs. We consder the complete graph on 6 nodes K 6, and we assume that the matrx of lnk costs s gven as follows: Z [z j] Further, let C We used the semdefnte form of the optmzaton problem whch s descrbed n equaton 2 and solved semdefnte program for complete graph K 6 usng CVX, a package for specfyng and solvng convex programs [2], [3]. The optmal weght assgnment s gven n the
5 Fg. : Hypercube Topology H, H 2, H 3 followng W The weght matrx shows that the optmal weght assgnment s not unform. The optmal weght of lnk, 6 s w 6 w whch means that lnk, 6 s effectvely down. In other words, the topology of the optmal network s not K 6 any more. B. Hypercube wth 2 n nodes H n Now we consder hypercube of order n H n, another wellstructured graph whose crtcalty can be obtaned analytcally. In ths example we assume z j, j E. Hypercube s an edgetranstve graph, therefore, by corollary 5.3 the optmal soluton of the optmzaton problem 2 for hypercube has equal weghts. Hence, we consder a hypercube wth weght w for all the lnks. Hypercube can be recursvely bult by the use of Cartesan Product of a graph wth K 2 complete graph on 2 nodes: H n+ H n K 2; where denotes the cartesan product. Ths equaton can also be wrtten usng Kronecker Product : H n+ H n K 2; H n I 2 + I 2 n K 2 7 We have used the symbol to denote Kronecker product. Fg. shows hypercube topology for n to 3. We try to obtan egenvalues of adjacency matrx of H n usng equaton 7. We fnd the egenvalues for w, then we multply these normalzed egenvalues by w to fnd the egenvalues for the general case. H n+ H n I 2 + I 2 n K 2 H n 0 0 I 2 n + 0 H n I 2 n 0 H n+ H n I 2 n I 2 n H n For smplcty of notaton, we drop the subscrpt from I 2 n and use I nstead, whch means the dentty matrx of approprate order. Now we try to buld the determnant of characterstc matrx H n+ λi: H n λi I H n+ λi I H n λi H n λi I d n+ H n+ λi det I H n λi I H n λi det H n λi I Now we multply the frst row by H n λi, and then subtract the frst row from the second row. We have: I H n λi d n+λ det H n λi H n λi I H n λi 2 I H n λi det 0 I H n λi 2 I H n λi 2 H n λ I H n λ + I d n+λ d nλ d nλ + Usng ths recursve formula for determnant of hypercube, one can fnd wth nducton that the egenvalues of H n are n! 2k n, k 0,,..., n wth multplcty Cn, k. k!n k! Ths result s true when all the weghts are set to. In general case where we have a weght w for each lnk, the egenvalue s also multpled by ths weght. We notce that hypercube s a regular graph degree of all nodes are n. Ths means that we can fnd egenvalues of Laplacan usng egenvalues of the adjacency matrx of H n: L n ni H n λ k n 2k n wth multplcty Cn, k λ k 2n k wth multplcty Cn, k Now one can easly fnd the network crtcalty for H n usng lemma 5.. τ 2 2 n k n 2 2 n k0 λ k Cn, k 2n kw n Cn, k 2 n w n k k0 8 On the other hand, by consderng the fact that the number of lnks n H n s m n2 n, by corollary 5.3, we have: w C 9 n2 n The fnal expresson for network crtcalty of H n can be found
6 0. By usng ths relaton n equaton 23 we get: τ nj w 2 j zjτ C 24 w j njc z jτ 2 25 Fg. 2: The Rato of Normalzed Network Crtcalty of Hypercube and Complete Graph by applyng equaton 9 n 8: τ τ n2 n n 2 n C k0 n n 2 n C Cn, k n k Cn, 20 To obtan the last equaton we appled the change of varable n k and used the fact that Cn, n Cn,. Equaton 20 shows the behavor of normalzed network crtcalty when the sze of hypercube ncreases. We can compare the normalzed crtcalty of a hypercube H n wth a complete graph K 2 n to see how the robustness s decreased by changng a complete graph to a hypercube wth the same number of nodes. τh n τk 2 n n n Cn, 2 n C 22 n C n 2 n+ n Cn, 2 Fg. 2 shows the graphcal behavor of equaton 2 for dfferent values of n. It can be seen that for hgher values of n, fracton τhn τk 2 approaches. Note that even for hgh n values of n the dfference between the normalzed crtcalty of H n and K 2 n s consderable, although the rato s decreasng. C. Optmal Network Crtcalty for a Tree We note that a tree s an acyclc smple graph, whch means that there s exactly one path between every two nodes of a tree. It follows that network crtcalty of a tree can be found from the followng equaton. τ τ nj w 2 j n j w j where n j denotes the number of tmes that lnk, j s n the path connectng any source to any destnaton. But from equaton 7 we know that for optmal weghts C τ +τz j From the constrant of the optmzaton problem we have zjwj C. Hence: njzjc τ τ njzj 2 C 26 C Now t s enough to substtute τ from equaton 27 n equaton 25 to have optmal weght for tree. Fnally w j njc z j 2 n j z j C 2 w j C z j n jz j 2 njzj 2 28 Equaton 28 shows that the optmal weght of a lnk n a tree s proportonal to the square root of n j. Capacty Plannng for a Tree: The capacty assgnment problem for a tree can be solved analytcally usng the gudelnes provded n secton IVC. It s enough to apply the followng changes n equaton 28: w j c j γ j C C z jγ j The optmal capacty assgnment for a tree would be: c j γ j + C zjγj z j n jz j 2 njzj 2 29 There s a close analogy between our result and Klenrock s result for capacty assgnment. In [] Klenrock showed that under the ndependence assumpton the optmal capacty to mnmze average delay of the network of a lnk s proportonal to the square root of the lnk rate. Note that for a tree n j s proportonal to the lnk load γ j snce there s only one path between every sourcedestnaton par. As a result, equaton 29 s smlar to the Klenrock s equaton for optmal capacty [], 5.7, equaton Ths result s not surprsng because the network crtcalty of a tree accordng to equaton 22 s equal to τ n j c j γ j consderng w j c j. Ths s the same expresson that s used n [] to fnd the average delay of a network [], 5.6, equaton 5.9, therefore, the mnmzaton of network crtcalty s equal to the mnmzaton of the average network delay when the network s a tree.
7 Method Average Network Delay Network Crtcalty Klenrock Mester Crtcalty Method TABLE II: Average Network Delay and Network Crtcalty usng Dfferent Methods D. Klenrock s Network Fg. 3: Klenrock s Network In ths followng the proposed optmal weght assgnment method s compared wth Klenrock s method for capacty assgnment [4], [] and Mester s extenson [5] usng the example of telegraph network n Klenrock llustrated n Fg. 3see [4], pp In ths example the lnk cost factors z j are all consdered equal to unty. Klenrock s method fnds capactes of the lnks n such a way to mnmze the average delay of the network under the ndependence assumpton and when the lnk loads are known. One problem wth Klenrock s approach s that t assgns very long delays to the lnks wth small loads. Mester s method s an alternatve approach whch assgns equal delays to all the lnks, of course at the expense of a large devaton from optmal average network delay whch can be acheved by Klenrock s soluton. The proposed soluton n ths paper assgns capacty of the lnks n a way to balance the ndvdual lnk delays so as to have acceptable lnk delays whle stll we have a good average network delay. Table I shows the capacty assgned to the lnks usng all the methods. The second column of table I shows the ndvdual Lnk Load Klenrock Mester Crtcalty Method TABLE I: Capacty Assgnment usng 3 Dfferent Methods lnk loads. Columns 3, 4, and 5 show the optmal capacty assgnment usng Klenrock s method, Mester s method, and our proposed method whch we call t crtcalty method respectvely. The mnmum average network delay for these methods are gven n second column of table II. The thrd column also shows the value of network crtcalty. In the crtcalty method we actually optmze the robustness not the average delay as t s the case n Klenrock and Mester, therefore t s not surprsng to see that the average delay obtaned by crtcalty method s between two extremes of Klenrock to mnmze the average network delay and Mester to mnmze the maxmum lnk delay. Table III shows Lnk Klenrock Mester Crtcalty Method TABLE III: Indvdual Lnk Delays usng 3 Dfferent Methods ndvdual lnk values for three methods. Klenrock s method assgns very large delay to lnk 3 because the demand on lnk 3 s much less than other lnks. Mester s method assgns equal delays for all the lnks. Ths resolves the ssue wth Klenrock s method, but ntroduces a farness problem. In our proposed method, the lnk delays are nt equal to allow for farness based on the demand for each lnk, and at the same tme the ndvdual lnk delay are kept n a reasonable range. VI. CONCLUSION In ths paper we proposed an approach to the network desgn problem and network plannng usng graphtheoretcal concepts. We used network crtcalty metrc to quantfy the robustness of a network and nvestgated the propertes of network crtcalty. We showed that network crtcalty s a strctly convex functon of lnk weghts and nvestgated the convex optmzaton problem of mnmzng the network crtcalty under some constrants on the weght matrx. We also found a semdefnte programmng representaton of ths problem whch permts us to use avalable lterature on semdefnte programmng to solve the optmzaton problem and fnd the optmal weghts. Capacty assgnment problem can be consdered as a specal case of ths general problem where the weght of a lnk s equal to ts capacty. APPENDIX A PROOF OF LEMMA 4. We can calculate the frst dervatve of τ sd wth respect to the weght of a typcal lnk l, j usng equaton 2: τ sd u t sd L+ u sd 30 Now, we use the followng fact from graphtheory about generalzed nverse Laplacan of a graph [6]. L + L + J n J n 3
8 Matrx J n ths equaton s an n n matrx wth all elements L equal to. Usng equaton 3 we have: + L+ n J. On the other hand: L + J n L + J n I L + J n L + J n + L + J n L + J n 0 L + L + J n L L + J n 32 Replacng equaton 32 n equaton 30 wll result n: τ sd u t sd L+ u sd u t sd L + J n L L + J n u sd 33 To obtan L we notce that n four elements of matrx L the weght w j appears: l j,, l j,, l,, l jj,. Based on the defnton of Laplacan L wjujut j [9] we have: L u j u t j 34 Combnng equatons 34 and 33 we have τ sd u t sd L + J n One can also easly verfy that: u j u t j L + J n u sd 35 L + J n u j L + J n J n uj L+ u j L + L + j 36 where L + s the th column of L +. Here we used the fact that Ju j u t jj 0. Usng equaton 36 n 35 gves: τ sd u t sdl + L + j L + L + j t u sd L + L + j t u sd t L + L + j t u sd l + s l + d l + js l + jd 2 l + s l + js 2 + l + d l + jd 2 2l + s l + jsl + d l + jd Now we are ready to obtan the dervatve of τ. τ d s τ sd l + s l js d s d 2 l + s l jsl + + d l jd + d s n L + u j 2 + n L + u j 2 2 d l + d l + jd s l + s l + js l + d l jd + 2 s Equaton 37 shows that the dervatve of τ s always nonpostve. It s also nonzero snce f t would be zero for some weghts, t would mean that two columns of L + are equal. Another way to put ths s to say that the rank of L + s n 2 whle the rank of L + has to be n n order to guarantee connectvty of the graph. 2 τ 2 u t w 2 jl + L + u j j n 2 L + n ut j L + u j + u t jl + L + u j 2 n 2τjL+ u ju t jl + 2 τ τ 2τ w 2 j 38 j Equaton 38 clearly shows that second dervatve of τ s nonnegatve snce ts frst dervatve s always negatve accordng to lemma 4. and τ j s by defnton a nonnegatve functon of weghts. Ths completes the proof of lemma 4.. REFERENCES [] A. H. Dekker and B. D. Colbert. Network Robustness and Graph Topology. Australasan Computer Scence Conference, 26: , Jan [2] R. ZhangShen and N. McKeown. Desgnng a Predctable Internet Backbone wth Valant LoadBalancng. In Thrteenth Internatonal Workshop on Qualty of Servce IWQoS, Passau, Germany, June [3] L. Valant and G. Brebner. Unversal Schemes for Parallel Communcaton. In 3th Annual Symposum on Theory of Computng, May 98. [4] C. S. Chang, D. S. Lee, and Y. S. Jou. Load Balanced Brkhoffvon Neumann Swtches, Part I: OneStage Bufferng. In HPSR 0, Dallas, May 200. IEEE. [5] A. Tzghadam and A. LeonGarca. A Robust Routng Plan to Optmze Throughput n Core Networks. ITC20, Elsver, pages 7 28, [6] A. Tzghadam and A. LeonGarca. On Robust Traffc Engneerng n Core Networks. In IEEE GLOBECOM, December [7] P. Van Meghem and F. A. Kupers. Concepts of Exact QoS Routng Algorthms. IEEE/ACM TRANSACTIONS ON NETWORKING, 25:85 864, October [8] M. Newman. A Measure of Betweenness Centralty Based on Random Walks. arxv condmat/ , [9] C. Godsl and G. Royle. Algebrac Graph Theory. SprngerVerlag, 200. [0] Denns S. Bernsten. Matrx Mathematcs. Prnston Unversty Press, [] L. Klenrock. Queueng Systems, volume II. John Wley & Sons, 975. [2] M. Grant and S. Boyd. CVX: Matlab Software for Dscplned Convex Programmng Web Page and Software. http : //stanford.edu/ boyd/cvx. September [3] M. Grant and S. Boyd. Graph Implementatons for Nonsmooth Convex Programs, Recent Advances n Learnng and Control a trbute to M. Vdyasagar, V. Blondel, S. Boyd, and H. Kmura, edtors, http : //stanford.edu/ boyd/graph d cp.html. Lecture Notes n Control and Informaton Scences, Sprnger, [4] L. Klenrock. Communcaton Nets, Stochastc Message Flow and Delay. McGrawHll, New York, 964. [5] B. Mester, H. R. Muller, and H. R. Rudn. New optmzaton crtera for message swtchng networks. IEEE Transactons on Communcaton Technology, 93: , June 97. [6] C. R. Rao and S. K. Mtra. Generalzed Inverse of Matrces and ts Applcatons. John Wely and Sons Inc., 97. 2n L + u j 2 τ 2n L + L + j 2 37
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