Minimizing Effective Resistance of a Graph

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1 Proceedings of the 17th Interntionl Symposium on Mthemticl Theory of Networks nd Systems, Kyoto, Jpn, July 24-28, 26 TuP12.2 Minimizing Effective Resistnce of Grph Arpit Ghosh Stephen Boyd Amin Seri Astrct The effective resistnce etween two nodes of weighted grph is the electricl resistnce seen etween the nodes of the corresponding resistor network with rnch conductnces given y the edge weights. The effective resistnce comes up in mny pplictions nd fields in ddition to electricl network nlysis, including, for exmple, Mrkov chins nd continuous-time verging networks. In this pper we study the prolem of llocting edge weights on given grph in order to minimize the totl effective resistnce, i.e., the sum of the effective resistnces etween ll pirs of nodes. We show tht this is convex optimiztion prolem which cn e solved efficiently. We show tht optiml lloction of the edge weights cn reduce the totl effective resistnce of the grph (compred to uniform weights y fctor tht grows unoundedly with the size of the grph. I. INTRODUCTION Let N e network with n nodes nd m edges, i.e., n undirected grph (V, E with V = n, E = m, nd nonnegtive weights on the edges. We cll the weight on edge l its conductnce, nd denote it y g l. The effective resistnce etween pir of nodes i nd j, denoted R ij, is the electricl resistnce mesured cross nodes i nd j, when the network represents n electricl circuit with ech edge (or rnch, in the terminology of electricl circuits resistor with (electricl conductnce g l. In other words, R ij is the potentil difference tht ppers cross terminls i nd j when unit current source is pplied etween them. We will give forml, precise definition of effective resistnce lter; for now we simply note tht it is mesure of how close the nodes i nd j re: R ij is smll when there re mny pths etween nodes i nd j with high conductnce edges, nd R ij is lrge when there re few pths, with lower conductnce, etween nodes i nd j. Indeed, the resistnce R ij is sometimes referred to s the resistnce distnce etween nodes i nd j. We define the totl effective resistnce, R tot, s the sum of the effective resistnce etween ll distinct pirs of nodes, R tot = 1 n R ij = R ij. (1 2 i,j=1 i<j The totl effective resistnce is evidently quntittive sclr mesure of how well connected the network is, or how lrge the network is, in terms of resistnce distnce. The totl effective resistnce comes up in numer of contexts. In n electricl network, R tot is relted to the verge power dissiption of the circuit, with rndom current excittion. The totl effective resistnce rises in Mrkov chins s well: R tot is, up to scle fctor, the verge commute time (or verge hitting time of Mrkov chin on the grph, with weights given y the edge conductnces g l. In this context, network with smll totl effective resistnce corresponds to Mrkov chin with smll hitting or commute times etween nodes, nd lrge totl effective resistnce corresponds to Mrkov chin with lrge hitting or commute times etween t lest some pirs of nodes. The totl effective resistnce comes up in numer of other pplictions s well, including verging networks, experiment design, nd Eucliden distnce emeddings. In this pper we ddress the prolem of llocting fixed totl conductnce mong the edges so s to minimize the totl effective resistnce of the grph. We cn ssume without loss of generlity tht the totl conductnce to e llocted is one, so we hve the optimiztion prolem minimize R tot suject to 1 T g = 1, g. Here, the optimiztion vrile is g R m, the vector of edge conductnces, nd the prolem dt is the grph (topology (V, E. The symol 1 denotes the vector with ll entries one, nd the inequlity symol etween vectors mens componentwise inequlity. We refer to the prolem (2 s the effective resistnce minimiztion prolem (ERMP. We will give severl interprettions of this prolem. In the context of electricl networks, the ERMP corresponds to llocting conductnce to the rnches of circuit so s to chieve low resistnce etween the nodes; in Mrkov chin context, the ERMP is the prolem of selecting the weights on the edges to minimize the verge commute (or hitting time etween nodes. When R ij re interpreted s distnces, the ERMP is the prolem of llocting conductnce to grph to mke the grph smll, in the sense of verge distnce etween nodes. We denote the optiml solution of the ERMP (which we will show lwys exists, nd is unique s g, nd the corresponding optiml totl effective resistnce s R tot. In this pper, we will show tht the prolem (2 is convex optimiztion prolem, which cn e formulted s semidefinite progrm (SDP [BV4]. This hs severl implictions, prcticl nd theoreticl. One prcticl consequence is tht we cn solve the ERMP efficiently. On the theoreticl side, convexity of the ERMP llows us to form necessry nd sufficient optimlity conditions, nd n ssocited dul prolem (with zero dulity gp. Fesile points in the dul prolem gives us lower ounds on R tot; in fct, we otin lower ound on R tot given ny fesile lloction of conductnces. This gives us n (2 1185

2 esily computle upper ound on the suoptimlity, i.e., dulity gp, given conductnce lloction g. We use this dulity gp in simple interior point lgorithm for solving the ERMP. We show tht for the rell grph, the rtio of Rtot to R tot otined with uniform edge weights converges to zero s the size of the grph increses. Thus, the totl effective resistnce of grph, with optimized edge weights, cn e unoundedly etter (i.e., smller thn the totl effective resistnce of grph with uniform lloction of weights to the edges. This pper is orgnized s follows. In II, we give forml definition of the effective resistnce, derive numer of formuls nd expressions for R ij, R tot, nd the first nd second derivtives of R tot, nd estlish severl importnt properties, such s convexity of R ij nd R tot s function of the edge conductnces. In III, we give severl interprettions of R ij, R tot, nd the ERMP. We study the ERMP in IV, giving the SDP formultion, (necessry nd sufficient optimlity conditions, two dul prolems, nd simple ut effective custom interior point method for solving it. In V, we nlyticlly solve the ERMP for the rell. We descrie some extensions in VII. A. Relted prolems The ERMP is relted to severl other convex optimiztion prolems tht involve choice of some weights on the edges of grph. One such prolem (lredy mentioned ove is to ssign nonnegtive weights, tht dd to one, to the edges of grph so s to mximize the second smllest eigenvlue of the Lplcin: mximize λ 2 (L suject to 1 T g = 1, g. Here L denotes the Lplcin of the weighted grph. This prolem hs een studied in different contexts. The eigenvlue λ 2 (L is relted to the mixing rte of the Mrkov process with edge trnsition rtes given y the edge weights. In [SBXD], the weights g re optimized to otin the fstest mixing Mrkov process on the given grph. The prolem (3 hs lso een studied in the context of lgeric connectivity [Fie73]. The lgeric connectivity is the second smllest eigenvlue of the Lplcin mtrix L of grph (with unit edge weights, nd is mesure of how well connected the grph is. Fiedler defines the solute lgeric connectivity of grph s the mximum vlue of λ 2 (L over ll nonnegtive edge weights tht dd up to m, i.e., m times the optiml vlue of (3. The prolem of finding the solute lgeric connectivity of grph is discussed in [Fie9], [Fie93], nd n nlyticl solution is presented for tree grphs. Other convex prolems involving edge weights on grphs include the prolem of finding the fstest mixing Mrkov chin on given grph [BDX4], [GHW5], [SBXD], the prolem of finding the edge weights (which cn e negtive tht give the fstest convergence in n verging network [XB4], nd the prolem of finding (3 edge weights tht give the smllest lest men-squre (LMS consensus error [XBK5]. Convex optimiztion cn lso e used to otin ounds on vrious quntities over fmily of grphs; see [GB]. For n overview of such prolems, see [Boy6]. In [BVGY1], Boyd et l consider the sizing of the wires in the power supply network of n integrted circuit, with unknown lod currents modeled stochsticlly. This turns out to e closely relted to our ERMP, with the wire segment widths proportionl to the edge weights. Some ppers on vrious spects of resistnce distnce include [Kle2], [XG3], [Bp99], [KR93]. A. Definition II. THE EFFECTIVE RESISTANCE Suppose edge l connects nodes i nd j. We define l R n s ( l i = 1, ( l j = 1, nd ll other entries. The conductnce mtrix (or weighted Lplcin of the network is defined s m G = g l l T l = A dig(ga T, where dig(g R m is the digonl mtrix formed from g, nd A R n m is the incidence mtrix of the grph: A = [ 1 m ]. Since g l, G is positive semidefinite, which we write s G. (The symol denotes denotes mtrix inequlity, etween symmetric mtrices. The mtrix G stisfies G1 =, since T l 1 = for ech edge l. Thus, G hs smllest eigenvlue, corresponding to the eigenvector 1. Throughout this pper we mke the following ssumption out the edge weights: The sugrph of edges with positive edge weights is connected. (If this is not the cse, the effective resistnce etween ny pir of nodes not connected y pth of edges with positive conductnce is infinite, nd mny of our formuls re no longer vlid. With this ssumption, ll other eigenvlues of G re positive. We denote the eigenvlues of G s < λ 2 λ n. The nullspce of G is one-dimensionl, the line long 1; its rnge hs co-dimension one, nd is given y 1 (i.e., ll vectors v with 1 T v =. Let G (k e the sumtrix otined y deleting the kth row nd column of G. Our ssumption (4 implies tht ech G (k is nonsingulr (see, e.g., [DK69]. We will refer to G (k s the reduced conductnce mtrix (otined y grounding node k. Now we cn define the effective resistnce R ij etween pir of nodes i nd j. Let v e solution to the eqution Gv = e i e j, (4 1186

3 where e i denotes the ith unit vector, with 1 in the ith position, nd elsewhere. This eqution hs solution since e i e j is in the rnge of G. We define R ij s R ij = v i v j. This is well defined; ll solutions of Gv = e i e j give the sme vlue of v i v j. (This follows since the difference of ny two solutions hs the form α1, for some α R. We define the effective resistnce mtrix R R n n s the mtrix with i, j entry R ij. The effective resistnce mtrix is evidently symmetric, nd hs digonl entries zero, since R ii =. B. Effective resistnce in n electricl network The term effective resistnce (s well s severl other terms used here comes from electricl network nlysis. We consider n electricl network, with conductnce g l on rnch (or edge l. Let v R n denote the vector of node potentils, nd suppose current J i is injected into node i. The sum of the currents injected into the network must e zero, in order for Kirchhoff s current lw to hold, i.e., we must hve 1 T J =. The injected currents nd node potentils re relted y Gv = J. There re mny solutions of this eqution, ut ll differ y constnt vector. Thus, the potentil difference etween pir of nodes is lwys well defined. One wy to fix the node potentils is to ssign potentil zero to some node, sy the kth node. This corresponds to grounding the kth node. When this is done, the circuit equtions re given y G (k v (k = J (k, where G (k is the reduced conductnce mtrix, v (k is the reduced potentil vector, otined y deleting the kth entry of v (which is zero, nd J (k is the reduced current vector, otined y deleting the kth entry of J. In this formultion, J (k hs no restrictions; lterntively, we cn sy tht J k is implicitly defined s J k = 1 T J (k. From our ssumption (4, G (k is nonsingulr, so there is unique reduced potentil vector v (k for ny vector of injected currents J (k. Now consider the specific cse when the externl current is J = e i e j, which corresponds to one mpere current source connected from node j to node i. Any solution v of Gv = e i e j is vlid vector of node potentils; ll of these differ y constnt. The difference v i v j is the sme for ll vlid node potentils, nd is the voltge developed cross terminls i nd j. This voltge is R ij, the effective resistnce etween nodes i nd j. (The effective resistnce etween two nodes of circuit is defined s the rtio of voltge cross the nodes to the current flow injected into them. The effective resistnce R ij is the totl power dissipted in the resistor network when J = e i e j, i.e., one mpere current source is pplied etween nodes i nd j. This cn e shown directly, or y power conservtion rgument. The voltge developed cross nodes i nd j is R ij (y definition, so the power supplied y the current source, which is current times voltge, is R ij. The power supplied y the externl current source must equl the totl power dissipted in the resistors of the network, so the ltter is lso R ij. C. Some formuls for effective resistnce In this section we derive severl formuls for the effective resistnce etween pir of nodes nd the totl effective resistnce of generl grph. Our first expressions involve the reduced conductnce mtrix, which we write here s G (since the prticulr node tht is grounded will not mtter. We form the reduced conductnce mtrix G y removing, sy, the kth row nd column of G. Let ṽ, ẽ i, nd ẽ j e, respectively, the vectors v, e i nd e j, ech with the kth component removed. If Gv = e i e j, then we hve Gṽ = ẽ i ẽ j. This eqution hs unique solution, ṽ = G 1 (ẽ i ẽ j. The effective resistnce etween nodes i nd j is given y v i v j = ṽ i ṽ j, i.e., R ij = (ẽ i ẽ j T G 1 (ẽ i ẽ j. (5 (This is independent of the choice of node grounded, i.e., which row nd column is removed. When neither i nor j is k, the node tht is grounded, we cn write (5 s R ij = ( G 1 ii + ( G 1 jj 2( G 1 ij. If j is k, the node tht is grounded, then ẽ j =, so (5 ecomes R ij = ( G 1 ii. We cn lso write the effective resistnce R ij in terms of the pseudo-inverse G of G. We hve G G = I 11 T /n, which is the projection mtrix onto the rnge of G. (Here we use the simpler nottion 11 T /n to men (1/n11 T. Using this it cn verified tht G = (G + 11 T /n 1 11 T /n. (6 The following formul gives R ij in terms of G (see, e.g., [KR93]: R ij = (e i e j T G (e i e j (7 = (G ii + (G jj 2(G ij. (8 To see this, multiply Gv = e i e j on the left y G to get (I 11 T /nv = G (e i e j, so (e i e j T G (e i e j = (e i e j T v = v i v j (since e i e j 1. From (6, we get nother formul for the effective resistnce, R ij = (e i e j T (G + 11 T /n 1 (e i e j. (9 We cn derive severl formuls for the effective resistnce mtrix R, using (5 nd (7. From (7, we see tht R = 1 dig(g T + dig(g 1 T 2G, (1 where dig(g R n is the vector consisting of the digonl entries of G. 1187

4 Using (6, this cn e rewritten s R = 1 dig((g + 11 T /n 1 T (11 + dig((g + 11 T /n 1 1 T 2(G + 11 T /n 1. We cn lso derive mtrix expression for R in terms of the reduced conductnce mtrix G. Suppose G is formed y removing the kth row nd column from G. Form mtrix H R n n from G 1 y dding kth row nd column, with ll entries zero. Then, using (5, R cn e written s R = 1 dig(h T + dig(h1 T 2H. (12 D. Some formuls for totl effective resistnce In this section we give severl generl formuls for the totl effective resistnce, From (1 we get R tot = i<j R ij = (1/21 T R1. R tot = (1/21 T 1 dig(g T 1 + (13 (1/21 T dig(g 1 T 1 1 T G 1 = n Tr G (14 = n Tr(G + 11 T /n 1 n, (15 using G 1 = to get the second line, nd (6 to get the third line. (Tr Z denotes the trce of squre mtrix Z. We cn use (14 to get formul for R tot in terms of the eigenvlues of G. The eigenvlues of G re 1/λ i, for i = 2,..., n, nd. So we cn rewrite (14 s R tot = n n i=2 1 λ i. (16 This expression for the totl effective resistnce cn e found in [AF3, 3.4]. The totl effective resistnce cn lso e expressed in terms of the reduced conductnce mtrix G. Multiplying (12 on the left nd right y 1 T nd 1 nd dividing y 2, we hve R tot = n Tr G 1 1 T G 1 1 = n Tr(I 11 T /n G 1. (17 (Note tht G R (n 1 (n 1, so the vectors denoted 1 in this formul hve dimension n 1. The totl effective resistnce cn lso e written in terms of n integrl: R tot = n Tr (e tg 11 T /n dt. (18 This cn e seen s follows. Let the eigenvectors of G e n 1/2 1, v 2,..., v n, corresponding to the eigenvlues λ 1 = < λ 2 λ n. The mtrix e tg hs the sme eigenvectors, with corresponding eigenvlues 1, nd e λit, for i = 2,..., n. Therefore we hve using Tr v i v T i n Tr = n Tr = n = n E. Bsic properties n i=2 n i=2 ( e tg 11 T /n dt i=2 1 λ i, n e λit v i vi T e λit dt dt = v i 2 = 1 to get the second line. The effective resistnce R ij, nd the totl effective resistnce R tot, re rtionl functions of g. This cn e seen from (5, since the inverse of mtrix is rtionl function of the mtrix, nd R ij is liner function of G 1. They re lso homogeneous with degree 1: if ĝ = cg, where c >, then ˆR ij = R ij /c, nd ˆR tot = R tot /c. The effective resistnce R ij, with i j, is lwys positive: the mtrix G 1 is positive definite (since G, so from (5, R ij > when i j. Nonnegtivity of R ij cn lso e seen y noting tht G is n M-mtrix. The inverse of n M-mtrix is elementwise nonnegtive [HJ91], nd since R ij is the (i, ith entry of (G (j 1, it is nonnegtive s well. The effective resistnce lso stisfies the tringle inequlity (see, e.g., [KR93]: R ik R ij + R jk. (19 Therefore, the effective resistnce defines metric on the grph, clled the resistnce distnce [KR93]. The effective resistnce R ij is monotone decresing function of g, i.e., if g ĝ, then R ij ˆR ij. To show this, suppose g ĝ, nd let G nd G denote the ssocited conductnce mtrices. Evidently we hve G + 11 T /n Ĝ + 11 T /n, so (G + 11 T /n 1 (Ĝ + 11T /n 1. From (9, R ij = (e i e j T (G + 11 T /n 1 (e i e j (e i e j T (Ĝ + 11T /n 1 (e i e j = ˆR ij. F. Convexity of effective resistnce The effective resistnce R ij is convex function of g: for g, ĝ (oth stisfying the sic ssumption (4, nd ny θ [, 1], we hve R ij (θg + (1 θĝ θr ij (g + (1 θr ij (ĝ. To show this, we first oserve tht f(x = c T Y 1 c, where Y = Y T R n n nd c R n, is convex function of Y, for Y (see, e.g., [BV4, 3.1.7]. Since G + 11 T /n is n ffine function of g, R ij is convex function of g. 1188

5 It follows tht R tot is lso convex, since it is sum of convex functions. The totl effective resistnce is, in fct, strictly convex function of g: for g, ĝ (oth stisfying the sic ssumption (4, with g ĝ, nd ny θ (, 1, we hve R tot (θg + (1 θĝ < θr tot (g + (1 θr tot (ĝ. To estlish this, we first show tht Tr X 1 is strictly convex function of X, for X symmetric nd positive definite. Its second order Tylor pproximtion is Tr(X + 1 Tr X 1 Tr X 1 X 1 + Tr X 1 X 1 X 1. The second order term cn e expressed s Tr X 1 X 1 X 1 = X 1 X 1/2 2 F, where F denotes the Froenius norm. This second order term vnishes only if = (since X 1 nd X 1/2 re oth invertile, i.e., it is positive definite qudrtic function of. This shows tht Tr X 1 is strictly convex function of X = X T. Since the ffine mpping from g to G + 11 T /n is one-to-one, we conclude tht R tot = n Tr(G + 11 T /n 1 n is strictly convex function of g. G. Grdient nd Hessin In this section we work out some formuls for the grdient nd Hessin of R tot with respect to g. (A similr pproch cn e used to find the derivtives of R ij with respect to g, ut we will not need these in the sequel. We will use the following fct. Suppose the invertile symmetric mtrix X(t is differentile function of the prmeter t R. Then we hve ([BV4, A.4.1] X 1 t 1 X = X t X 1. Using this formul, nd R tot = n Tr(G+11 T /n 1 n, we hve R tot = n Tr(G + 11 T 1 G /n (G + 11 T /n 1 g l g l = n Tr(G + 11 T /n 1 l T l (G + 11 T /n 1 = n (G + 11 T /n 1 l 2. (2 We cn express the grdient s R tot = n dig(a T (G + 11 T /n 2 A. The grdient cn lso e expressed in terms of reduced conductnce mtrix: R tot = n dig(ãt G 1 (I 11 T /n G 1 Ã. For future reference, we note the formul R T totg = R tot, (21 which holds since R tot is homogeneous function of g of degree 1. It is esily verified y tking the derivtive with respect to α of R tot (αg = R tot (g/α, evluted t α = 1. We now derive the second derivtive or Hessin mtrix of R tot. From (2, we hve 2 g l g k R tot = n g k (G + 11 T /n 1 l 2 = 2n T l (G + 11 T /n 2 k T k (G + 11 T /n 1 l. (22 We cn express the Hessin of R tot s 2 R tot = 2n ( A T (G + 11 T /n 2 A ( A T (G + 11 T /n 1 A, where denotes the Hdmrd (elementwise product. A similr expression cn e derived using reduced mtrices: 2 R tot = 2n(ÃT G 1 (I 11 T /n G 1 Ã (ÃT G 1 Ã. (23 A. Averge commute time III. INTERPRETATIONS The effective resistnce etween pir of nodes i nd j is relted to the commute time etween i nd j for the Mrkov chin defined y the conductnces g [CRR + 89]. Let M e Mrkov chin on the grph N, with trnsition proilities determined y the conductnces: P ij = g ij l (i,k g, l where g ij is the conductnce cross edge (i, j, nd l (i, k mens tht edge l lies etween nodes i nd k. This Mrkov chin is reversile, with sttionry distriution l (i,j π i = g l m g. l The hitting time H ij is the expected time tken y the rndom wlk to rech node j for the first time strting from node i. The commute time C ij is the expected time the rndom wlk tkes to return to node i for the first time fter strting from i nd pssing through node j. The following well known result reltes commute times nd effective resistnce (see, for exmple, [AF3, 3.3]: C ij = (1 T gr ij. Tht is, the effective resistnce etween i nd j is proportionl to the commute time etween i nd j. Therefore, the totl effective resistnce is proportionl to C, the commute time verged over ll pirs of nodes: C = 2(1T g n(n 1 R tot. Since C ij = H ij + H ji, R tot is lso proportionl to H, the hitting time verged over ll pirs of nodes: H = 1T g n(n 1 R tot. In the context of Mrkov chins, the ERMP (2 is the prolem of choosing edge weights on grph so s to minimize its verge commute time or hitting time. 1189

6 B. Power dissiption in resistor network The totl effective resistnce is relted to the verge power dissipted in resistor network with rndom injected currents. Suppose rndom current J R n is injected into the network. The current must stisfy 1 T J =, since the totl current entering the network must e zero. We ssume tht E J =, E JJ T = I 11 T /n. Roughly speking, this mens J is rndom current vector, with covrince mtrix I on 1. The power dissipted in the resistor network with injected current vector J is J T G J. The expected dissipted power is E J T G J = Tr G E JJ T = Tr G = 1 n R tot, where the second equlity follows from G 1 =. Thus, the totl effective resistnce is proportionl to the verge power dissipted in the network when the injected current is rndom, with men nd covrince I 11 T /n. A network with smll R tot is one which dissiptes little power, under rndom current excittion; lrge R tot mens the verge power dissiption is lrge. The ERMP (2 is the prolem of llocting unit conductnce mong the rnches of resistor network, so s to minimize the verge power dissipted under rndom current excittion. (See, e.g., [BVGY1]. We cn lso give n interprettion of the grdient R tot in the context of resistor network. With rndom current excittion J, with E J =, E JJ T = I 11 T /n, the prtil derivtive R tot / g l is proportionl to the men squre voltge cross edge l. This cn e seen from (2 s follows. The voltge v l cross edge l, with current excittion J is T l G J = T l (G + 11T /n 1 J, since 1 =. The expected vlue of the squred voltge is T l E( T l (G + 11 T /n 1 J 2 = T l (G + 11 T /n 1 E JJ T (G + 11 T /n 1 l = T l (G + 11 T /n 1 (I 11 T /n(g + 11 T /n 1 l = T l (G + 11 T /n 2 l, where the lst equlity follows since (G+11 T /n 1 1 = 1, nd 1 T l =. Compring this with (2, we see tht R tot g l = n E v 2 l. (24 The grdient R tot is equl to n times the vector of men squre voltge ppering cross the edges. C. Elmore dely in n RC circuit We consider gin resistor network, with rnch (electricl conductnces given y g l. To this network we dd seprte ground node, nd unit cpcitnce etween every other node nd the ground node. The vector of node voltges (with respect to the ground node in this RC (resistor-cpcitor circuit evolves ccording to v = Gv. This hs solution v(t = e tg v(. Since e tg hs lrgest eigenvlue 1, ssocited with the eigenvector 1, with other eigenvlues e λit, for i = 2,..., n, we see tht v(t converges to the vector 11 T v(/n. In other words, the voltge (or equivlently, chrge equilirtes itself cross the nodes in the circuit. Suppose we strt with the initil voltge v( = e k, i.e., one volt on node k, with zero voltge on ll other nodes. It cn e shown tht the voltge t node k monotoniclly decreses to the verge vlue, 1/n. The Elmore dely t node k is defined s T k = (v k (t 1/n dt (see, for exmple, [Elm48], [WH4]. The Elmore dely T k gives mesure of the speed t which chrge strting t node k equilirtes. The verge Elmore dely, over ll nodes, is 1 n n T k = 1 n k=1 n k=1 e T k = 1 n Tr ( = 1 n 2 R tot, (e tg 11 T /ne k dt (e tg 11 T /n dt where the lst equlity follows from (18. The totl effective resistnce of the network is thus equl to the sum of the Elmore dely to ech node in the RC circuit. The ERMP (2 is the prolem of llocting totl conductnce of one to the resistor rnches of n RC circuit, so s to minimize the verge Elmore dely of the nodes. D. Totl time constnt of n verging network We cn interpret R tot in terms of the time constnts in n verging network. We consider the dynmicl system ẋ = Gx, where G is the conductnce mtrix. This system crries out (symptotic verging: e tg is douly stochstic mtrix, which converges to 11 T /n s t, so x(t = e tg x( converges to 11 T x(/n. The eigenvlues λ 2,..., λ n of G determine the rte t which the verging tkes plce. The eigenvectors v 2,..., v n re the modes of the system, nd the ssocited time constnts re given y T k = 1 λ k. (This gives the time for mode k to decy y fctor e. Therefore, R tot = n n k=2 T k is proportionl to the sum of the time constnts of the verging system. E. A-optiml experiment design The ERMP cn e interpreted s certin type of optiml experiment design prolem. The gol is to estimte prmeter vector x R n from noisy liner mesurements y i = v T i x + w i, i = 1,..., K, 119

7 where ech v i cn e ny of the vectors 1,..., m, nd w i re independent rndom (noise vriles with zero men nd unit vrince. Thus, ech mesurement consists of mesuring difference etween two components of x, corresponding to some edge of our grph, with some dditive noise. With these mesurements of differences of components, we cn only estimte x up to some dditive constnt; the prmeter x nd x + α1, for ny α R, produce exctly the sme mesurements. We will therefore ssume tht the prmeter to e estimted stisfies 1 T x =. Now suppose totl of k l mesurements re mde using l, for l = 1,..., m, so we hve m k l = K. The minimum vrince unised estimte of x, given the mesurements, is ( K ( K ˆx = v i vi T vi T y i = i=1 i=1 ( m ( m k l l T l k l T l y. The ssocited estimtion error e = ˆx x hs zero men nd covrince mtrix ( m Σ err = k l l T l. (There is no estimtion error in the direction 1, since we hve ssumed tht 1 T x =, nd we lwys hve 1 T ˆx =. The gol of experiment design is to choose the integers k 1,..., k l, suject to l k l = K, to mke the estimtion error covrince mtrix Σ err smll. There re severl wys to define smll, which yield different experiment design prolems. In A-optiml experiment design, the ojective is the trce of Σ err. This is proportionl to the sum of the squres of the semi-xis lengths of the confidence ellipsoid ssocited with the estimte ˆx. We now chnge vriles to θ l = k l /K, which is the frction of the totl numer of experiments (i.e., K tht re crried out using v = l. The vriles θ l re nonnegtive nd dd to one, nd must e integer multiples of 1/K. If K is lrge, we cn ignore the lst requirement, nd tke the vriles θ l to e rel. This yields the (relxed A-optiml experiment design prolem [Puk93]: minimize suject to (1/K Tr( m θ l l T l m θ l = 1, θ l. This is convex optimiztion prolem, with vrile θ R m. (See [BV4, 7.5] nd its references for more on experiment design prolems. Identifying θ l with g l, we see tht the A-optiml experiment design prolem ove is the sme s our ERMP (up to scle fctor in the ojective. Thus, we cn interpret the ERMP s follows. We hve rel numers, x 1,..., x n t the nodes of our grph, which hve zero sum. Ech edge in our grph corresponds to possile mesurement we cn mke, which gives the difference in its djcent node vlues, plus noise. We re going to mke lrge numer of these mesurements, in order to estimte x. The prolem is to choose the frction of the experiments tht should e devoted to ech edge mesurement. Using the trce of the error covrince mtrix s our mesure of estimtion qulity, the optiml frctions re exctly the optiml conductnces in the ERMP. IV. MINIMIZING TOTAL EFFECTIVE RESISTANCE In this section we study the ERMP, minimize R tot suject to 1 T g = 1, g, (25 in detil. This is convex optimiztion prolem, since the ojective is convex function of g, nd the constrint functions re liner. The prolem is clerly fesile, since g = (1/m1, the uniform lloction of conductnce to edges, is fesile. Since the ojective function is strictly convex, the solution to (25 is unique. We denote the unique optiml point s g, nd the ssocited vlue of the ojective s R tot. A. SDP formultion The ERMP (25 cn e formulted s semidefinite progrm (SDP, minimize n Tr Y suject to 1 T g = 1, g, [ ] G + 11 T /n I, I Y (26 where G = m g l l T l. The vriles re the conductnces g R m, nd the slck symmetric mtrix Y R n n. To see the equivlence, we note tht whenever G + 11 T /n (which is gurnteed whenever the sic ssumption (4 holds, [ ] G + 11 T /n I Y (G + 11 T /n 1. I Y To minimize the SDP ojective n Tr Y, suject to this constrint, with G fixed, we simply tke Y = (G+11 T /n 1, so the ojective of the SDP ecomes R tot + n. B. Optimlity conditions The optiml conductnce g stisfies 1 T g = 1, g, R tot + R tot 1, (27 where R tot is the totl effective resistnce with g. Conversely, if g is ny vector of conductnces tht stisfies (27, then it is optiml, i.e., g = g. The first two conditions in (27 require tht g e fesile. These optimlity conditions cn e derived s follows. Since the ERMP is convex prolem with differentile ojective, necessry nd sufficient condition for optimlity of fesile g is R T tot(ĝ g for ll ĝ with 1 T ĝ = 1, ĝ 1191

8 (see, e.g., [BV4, 4.2.3]. This is the sme s R T tot(e l g, l = 1,..., m. Since R tot is homogeneous function of g of degree 1, we hve Rtotg T = R tot (see (21, so the condition ove cn e written s R tot g l + R tot, l = 1,..., m, (28 which is precisely the third condition in (27. From the optimlity conditions (27 we cn derive complementry slckness condition: g l ( Rtot g l + R tot =, l = 1,..., m. (29 This mens tht for ech edge, we hve either g l = or R tot / g l + R tot =. To estlish the complementrity condition, we note tht g T ( R tot + R tot 1 =, since g T R tot = R tot nd g T R tot 1 = R tot. If g stisfies (27, then this sttes tht the inner product of two nonnegtive vectors, g nd R tot + R tot 1, is zero; it follows tht the products of the corresponding entries re zero. This is exctly the complementrity condition ove. We cn give the optimlity conditions simple interprettion in the context of circuit driven y rndom current, s descried in III-B. We suppose the circuit is driven y rndom current excittion J with zero men nd covrince E JJ T = I 11 T /n. By (24, we hve R tot / g l = n E vl 2, where v l is the (rndom voltge ppering cross edge l. The optimlity condition is tht g is fesile, nd we hve E v 2 l (1/nR tot, l = 1,..., m. Thus, the conductnces re optiml when the men squre voltge cross ech edge is less thn or equl to (1/nR tot. Using the complementrity condition (29, we cn e it more specific: ech edge tht hs positive conductnce llocted to it must hve men squre voltge equl to (1/nR tot ; ny edge with zero conductnce must hve men squre voltge no more thn (1/nR tot. C. The dul prolem In this section we derive the Lgrnge dul prolem for the ERMP (25, s well s some interesting vritions on it. We strt y writing the ERMP s minimize suject to n Tr X 1 n X = m g l l T l + 11 T /n, 1 T g = 1, g, (3 with vriles g R m, nd X = X T R n n. Associting dul vriles Z = Z T R n n, ν R with the equlity constrints, nd λ R m with the nonnegtivity constrint g, the Lgrngin is L(X, g, Z, ν, λ = n Tr X 1 n + ν(1 T g 1 λ T g ( m + Tr Z X g l l T l 11 T /n. The dul function is h(z, ν, λ = inf L(X, g, Z, ν, λ X, g = inf X Tr(nX 1 + ZX + ( m inf g l ( T l Z l + ν λ l n g = ν (1/n1 T Z1 T ν (1/n1 T Z1 + 2 Tr(nZ 1/2 n if T l Z l + ν = λ l, l = 1,..., m, Z ; otherwise. To justify the lst line, we note tht Tr(nX 1 +ZX is unounded elow, s function of X, unless Z ; when Z, the unique X tht minimizes it is X = (Z/n 1/2, so it hs the vlue Tr(nX 1 + ZX = Tr(n(Z/n 1/2 + Z(Z/n 1/2 = 2 Tr(nZ 1/2. When Z is positive semidefinite, ut not positive definite, we get the sme miniml vlue, ut it is not chieved y ny X. (This clcultion is equivlent to working out the conjugte of the function Tr U 1, for U, which is 2 Tr( V 1/2, with domin V ; see, e.g., [BV4, Ex.3.37]. The Lgrnge dul prolem is mximize h(z, ν, λ suject to λ. Using the explicit formul for g derived ove, nd eliminting λ, which serves s slck vrile, we otin the dul prolem mximize ν (1/n1 T Z1 + 2 Tr(nZ 1/2 n suject to T l Z l ν, l = 1,..., m, Z. (31 This prolem is nother convex optimiztion prolem, with vriles Z = Z T R n n nd ν R. The sclr vrile ν could e eliminted, since its optiml vlue is evidently ν = mx l T l Z l. Since the ERMP is convex, hs only liner equlity nd inequlity constrints, nd Slter s condition is stisfied (for exmple y g = (1/m1, we know tht the optiml dulity gp for the ERMP (3 nd the dul prolem (31 is zero. In other words, the optiml vlue of the dul (31 is equl to Rtot, the optiml vlue of the ERMP. In fct, 1192

9 we cn e very explicit: if X is the optiml solution of the priml ERMP (3, then Z = n(x 2, ν = mx T l Z l l re optiml for the dul ERMP (31. Conversely, if Z is optiml for the dul ERMP (31, then X = (Z /n 1/2 is the optiml point for the priml ERMP (3. We cn use the dul prolem (31 to derive useful ound on the suoptimlity of ny fesile conductnce vector g, y constructing dul fesile point from g. With G = A dig(ga T, we define Z = n(g + 11 T /n 2, with ν = mx l T l Z l. The pir (Z, ν is evidently fesile for the dul prolem, so its dul ojective vlue gives lower ound R on Rtot: Rtot ν (1/n1 T Z1 + 2 Tr(nZ 1/2 n = mx n (G + 11 T /n 1 l 2 l + 2n Tr(G + 11 T /n 1 2n = R, where we use (G + 11 T /n 1 1 = 1 in the second line. Let η denote the difference etween this lower ound R nd the vlue of R tot chieved y the conductnce vector g. This is dulity gp ssocited with g, i.e., n upper ound on the suoptimlity of g. Using R tot = n Tr(G + 11 T /n 1 n, we cn express this dulity gp s η = R tot R = n Tr(G + 11 T /n 1 + n + mx n (G + 11 T /n 1 l 2 l ( = R tot + mx R tot l g ( l Rtot = min + R tot. l g l In summry, we hve the following inequlity: given ny fesile g, its ssocited totl effective resistnce R tot stisfies ( R tot Rtot Rtot min + R tot. (32 l g l When g = g, the righthnd side is zero, y our optimlity condition (27. This shows tht the dulity gp converges to zero s g converges to g. A second formultion of the dul, which leds to tighter dulity gp, cn e found in longer version of this pper [GBS5]. D. An interior-point lgorithm The ERMP cn e solved numericlly using severl methods, for exmple vi the SDP formultion (26, using stndrd solver such s SeDuMi [Stu99] or DSDP [BY4], or y implementing stndrd rrier method [BV4, 11.3], using the grdient nd Hessin formuls given in II-G. In this section we descrie simple custom interior-point lgorithm for the ERMP, tht uses the dulity gp ˆη derived in IV-C. This interior-point method is sustntilly fster thn n SDP formultion, or more generic method. The logrithmic rrier for the nonnegtivity constrint g is m Φ(g = log g l, defined for g >. In priml interior-point method, we minimize tr tot + Φ, suject to 1 T g = 1, using Newton s method, where t > is prmeter; the solution of this suprolem is gurnteed to e t most m/t suoptiml. Our formul (32 gives us nice ound on suoptimlity, η = ( R tot min + R tot l g l, given ny fesile g. We cn turn this round, nd use this ound to updte the prmeter t in ech step of n interiorpoint method, y tking t = βm/η, where β is some constnt. (If η =, we cn stop ecuse g is optiml. This yields the following lgorithm. Given reltive tolernce ɛ (, 1, β 1. Set g := (1/m1. while ˆη > ɛr tot repet 1. Set t = βm/η. 2. Compute Newton step δg for tr tot + Φ y solving, with H = t 2 R tot + 2 Φ, nd f = t R tot + Φ, [ H 1 1 T ] [ δg ν ] = [ f ]. 3. Find step length s y cktrcking line serch ([BV4, 9.2]. 4. Set g := g + sδg. When the lgorithm exits, we hve R tot R tot η ɛr tot, which implies tht R tot R tot R tot ɛ 1 ɛ. Thus, the lgorithm computes conductnce vector gurnteed to e no more thn ɛ/(1 ɛ suoptiml. Using β = 1 nd reltive tolernce ɛ =.1, we found the lgorithm to e very effective, never requiring more thn 2 or so steps to converge for the mny grphs we tried. The min computtionl effort is in computing the Newton step (i.e., step 2, which requires O(m 3 rithmetic opertions, if no structure in the equtions is exploited. For grphs with no more thn round m = 2 edges, the lgorithm is quite fst. V. THE BARBELL GRAPH Consider the rell grph K n K n on 2n nodes, which consists of two fully connected components of size n, joined y single edge etween nodes n nd n+1, shown in figure 1. For this grph, we will show tht the rtio 1193

10 g replcements Fig. 1. c A rell grph on 8 nodes. of R tot with uniform weights to Rtot grows unoundedly with n. Using symmetry nd convexity of the ERMP, it cn e shown tht the optiml g hs exctly three distinct weights: the weights on edges neither of whose endpoints is n or n + 1, ; the weights on edges with exctly one endpoint n or n+1, ; nd the weight on the edge etween n nd n+1, c. The conductnce mtrix G for these weights is G = αi 11 T 1 1 γ c c γ 1 1 αi 11 T, where 1 R n 1, α = (n 1+, nd γ = (n 1+2c. The 2n eigenvlues of the conductnce mtrix with these weights re shown in the longer version of this pper [GBS5] to e, (n 1 + with multiplicity 2n 4, n, (1/2(n + 2c ± ( (2c + n 2 8c 1/2. Therefore, we hve reduced the ERMP prolem (which hs m = n(n vriles to the prolem minimize (2n 4/((n /n + (n + 2c/2c suject to (n 1(n 2 + 2(n 1 + c 1,,, c, (33 which hs three vriles:,, nd c. This prolem hs n nlyticl solution: the optiml weights re = µ ( 2 n + 1 n 1 n n c = µ 2, n + 1, = µ, n where µ is normlizing constnt, ( 2 n + 1 1/µ = (n 2 n n + 1 n + 2(n 1 + n 2. PSfrg replcements 1194 c x Optiml Uniform Fig. 2. The optiml vlue Rtot, nd Rtot with uniform weights, s function of n, for the rell grph. Clerly, Rtot scles s n 3 for the rell grph K n K n. For the sme grph, the totl effective resistnce otined with uniform weights g l = 1/m scles s n 4. We conclude tht the suoptimlity of the uniform weights grows unoundedly with n, i.e., the optiml conductnces re unoundedly etter thn uniform conductnces. In figure 2, the optiml Rtot is plotted s function of n for the rell grph, long with the totl effective resistnce with uniform weights. VI. EXAMPLES In this section we show some exmples of optiml conductnce lloctions on grphs. In ech exmple, we drw the edges with width nd color sturtion proportionl to the optiml edge conductnce. Our first two exmples re pth on 11 nodes, nd tree on 25 nodes, shown in figure 3. As expected, the conductnce is lrger on edges with more pths pssing through them thn edges ner the leves, which hve fewer pths pssing through them. Our next exmple is n 8 8 mesh, shown in figure 4. We plot the optiml conductnces for the 8 8 mesh, nd for grph tht is formed y removing some edges from the mesh. In figure 5, we plot the optiml conductnces for rell. Finlly, in figure 6, we plot the optiml conductnces for rndomly generted grph with 25 nodes nd 88 edges. Here too we see lrge conductnce llocted to edges cross sprse cuts. VII. EXTENSIONS We conclude y listing some vritions on the ERMP tht re convex optimiztion prolems, nd cn e hndled using similr methods. Since ech R ij is convex function of the conductnces, we cn minimize ny nondecresing convex function of the R ij (which is convex. Some interesting (convex ojectives, in ddition to R tot, include the following.

11 Fig. 6. Optiml conductnce lloction on rndomly generted grph with 25 nodes nd 88 edges. Fig. 3. Left: Optiml conductnce lloction on pth on 11 nodes. Right: Optiml conductnce lloction on tree with 25 nodes. Fig. 4. Left: Optiml conductnce lloction on n 8 8 mesh. Right: Optiml lloction for modified 8 8 mesh. Minimizing effective resistnce etween specific pir of nodes. We llocte conductnce to minimize the effective resistnce etween specific pir of nodes, i nd j. This prolem hs the following simple solution: Allocte the conductnce eqully to the edges lying on shortest pth etween i nd j. This conductnce lloction violtes our ssumption (4; the effective resistnce etween ny pir of nodes not on the pth is undefined. Minimizing sum of effective resistnces to specific node. This prolem cn e formulted s n SDP, nd solved y modifictions to the methods discussed in this pper. Exmples show tht the optiml lloction of weights need not e tree. Minimizing mximum effective resistnce. The mximum effective resistnce over ll pirs of nodes, mx i,j R ij, is the pointwise mximum of the convex functions R ij, nd is therefore lso convex function of the conductnces g [BV4, 3.2.3]. The prolem of minimizing the mximum effective resistnce is convex optimiztion prolem, cn e formulted s n SDP, nd solved using stndrd interior-point methods. We mention one more extension: minimizing R tot without the nonnegtivity constrints on the edge conductnces: minimize R tot suject to 1 T g = 1. (34 Fig. 5. Optiml conductnce lloction on rell grph with 16 nodes. In this prolem the conductnces cn e negtive, ut we restrict the domin of the ojective R tot to {g G + 11 T /n }. This prolem cn e solved using Newton s method, using the derivtives found in II-G. The optimlity conditions for this prolem re simply tht 1 T g = 1 (fesiility, nd tht ll components of the grdient, R tot, re equl, specificlly, R tot (g = R tot

12 ACKNOWLEDGMENTS We thnk Ali Jdie for ringing to our ttention the literture on resistnce distnce. REFERENCES [AF3] D. Aldous nd J. Fill. Reversile Mrkov Chins nd Rndom Wlks on Grphs. 23. Book in preprtion. Aville t stt- [Bp99] R. Bpt. Resistnce distnce in grphs. Mthemtics Student, 68:87 98, [BDX4] S. Boyd, P. Diconis, nd L. Xio. Fstest mixing Mrkov chin on grph. SIAM Review, prolems nd techniques section, 46(4: , 24. [Boy6] S. Boyd. Convex optimiztion of grph lplcin eigenvlues. volume 3, 26. Aville t www/ oyd/cvx_opt_grph_lpl_eigs. [BV4] S. Boyd nd L. Vndenerghe. Convex Optimiztion. Cmridge University Press, 24. Aville t www/ oyd/cvxook. [BVGY1] S. Boyd, L. Vndenerghe, A. El Gml, nd S. Yun. Design of roust glol power nd ground networks. In Proc. ACM/SIGDA Int. Symposium on Physicl Design (ISPD, pges ACM, April 21. [BY4] S. Benson nd Y. Ye. DSDP5 user guide softwre for semidefinite progrmming. Technicl Report ANL/MCS- TM-277, Mthemtics nd Computer Science Division, Argonne Ntionl Lortory, Argonne, IL, July 24. Aville online t: enson/dsdp. [CRR + 89] A. Chndr, P. Rghvn, W. Ruzzo, R. Smolensky, nd P. Tiwri. The electricl resistnce of grph cptures [DK69] its commute nd cover times. In Proceedings of the 21st Annul Symposium on Foundtions of Computer Science. ACM, C. Desoer nd E. Kuh. Bsic Circuit Theory. McGrw-Hill, Inc., [Elm48] W. Elmore. The trnsient response of dmped liner networks. Journl of Applied Physics, 19:55 63, Jnury [Fie73] M. Fiedler. Algeric connectivity of grphs. Czechoslovk Mthemtics Journl, 23:298 35, [Fie9] M. Fiedler. Asolute lgeric connectivity of trees. Liner nd Multiliner Alger, 26:85 16, 199. [Fie93] M. Fiedler. Some minimx prolems for grphs. Discrete Mthemtics, 121:65 74, [GB] A. Ghosh nd S. Boyd. Upper ounds on lgeric connectivity vi convex optimiztion. To pper in Liner Alger Appl. [GBS5] A. Ghosh, S. Boyd, nd A. Seri. Optimizing effective resistnce of grph. Sumitted to SIAM Review, Prolems nd Techniques, 25. Aville t www/ oyd/eff_res. [GHW5] F. Göring, C. Helmerg, nd M. Wppler. Emedded in the shdow of the seprtor. Preprint 25-12, Fkultät für Mthemtik, Technische Universität Chemnitz, Chemnitz, Germny, Septemer 25. [HJ91] R. Horn nd C. Johnson. Topics in Mtrix Anlysis. Cmridge University Press, Cmridge, UK, [Kle2] D. Klein. Resistnce-distnce sum rules. Croti Chem. [KR93] [Puk93] [SBXD] Act, 73: , 22. D. Klein nd M. Rndic. Resistnce distnce. Journl of Mthemticl Chemistry, 12:81 85, F. Pukelsheim. Optiml Design of Experiments. Wiley nd Sons, J. Sun, S. Boyd, L. Xio, nd P. Diconis. The fstest mixing Mrkov process on grph nd connection to mximum vrince unfolding prolem. [Stu99] J. Sturm. Using SeDuMi, Mtl toolox for optimiztion over symmetric cones, sedumi.mcmster.c. [WH4] N. Weste nd D. Hrris. CMOS VLSI Design. Addison Wiley, 24. [XB4] L. Xio nd S. Boyd. Fst liner itertions for distriuted verging. In Systems nd Control Letters, volume 53, pges 65 78, 24. [XBK5] L. Xio, S. Boyd, nd S.-J. Kim. Distriuted verge [XG3] consensus with lest-men-squre devition. Sumitted to Journl of Prllel nd Distriuted Computing, My 25. Aville t www/ oyd/lms_consensus. W. Xio nd I. Gutmn. Resistnce distnce nd Lplcin spectrum. Theor. Chem. Acc., 11: ,

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