Equiibrium of Heterogeneous Congestion Contro Protocos Ao Tang Jiantao Wang Steven H. Low EAS Division, Caifornia Institute of Technoogy Mung Chiang EE Department, Princeton University Abstract When heterogeneous congestion contro protocos that react to different pricing signas share the same network, the resuting equiibrium may no onger be interpreted as a soution to the standard utiity maximization probem. We prove the existence of equiibrium in genera muti-protoco networks under mid assumptions. For amost a networks, the equiibria are ocay unique, and finite and odd in number. They cannot a be ocay stabe uness it is gobay unique. Finay, we show that if the price mapping functions that map ink prices to effective prices observed by the sources are simiar, then goba uniqueness is guaranteed. Numerica exampes are used throughout the paper to iustrate these resuts. Index Terms Congestion contro, Heterogeneous protocos, Mutiprotoco networks, Equiibrium anaysis A. Motivation I. INTRODUCTION Congestion contro protocos have been modeed as distributed agorithms for network utiity maximization, e.g., [0], [4], [22], [33], [], [3]. With the exception of a few imited anaysis on very simpe topoogies [2], [2], [3], [8], existing iterature generay assumes that a sources are homogeneous in that, even though they may contro their rates using different agorithms, they a adapt to the same type of congestion signas, e.g., a react to oss probabiities, as in TCP Reno, or a to queueing deay, as in TCP Vegas or FAST [9]. When sources with heterogeneous protocos that react to different congestion signas share the same network, the current duaity framework is no onger appicabe. With more congestion contro protocos being proposed and ideas of using congestion signas other than packet osses, incuding expicit feedbacks, being deveoped in the networking community, we need a mathematicay rigorous framework to understand the behavior of arge-scae networks with heterogeneous protocos. The purpose of this paper is to propose such a framework. Our emphasis is on genera networks with mutipe sources and inks that use a arge cass of agorithms to adapt their rates and congestion prices. Often, interesting and counterintuitive behaviors arise ony in a network setting where sources interact through shared inks in intricate and surprising ways, e.g., [30]. Such behaviors are absent in singe-ink modes and are usuay hard to discover or expain without a fundamenta understanding of the underying structure. Given Partia and preiminary resuts have appeared in [3]. the scae and heterogeneity of the Internet, it is conceivabe that such behaviors are more common than we reaize, but remain difficut to measure due to the compexity of the infrastructure and our inabiity to monitor it cosey. A mathematica framework thus becomes indispensabe in exporing structures, carifying ideas, and suggesting directions. Some of the theoretica predictions in this paper have aready been demonstrated experimentay in [29]. B. Summary A congestion contro protoco generay takes the form ṗ = g x (t), p (t) () : L() ẋ = f x (t), m (p (t)) (2) L() Here, L() denotes the set of inks used by source, and g ( ) modes a queue management agorithm that updates the price p (t) at ink, often impicity, based on its current vaue and the sum of source rates x (t) that traverse ink. The prices may represent oss probabiities, queueing deays, or quantities expicity cacuated by the inks and fed back to the sources. The function f modes a TCP agorithm that adusts the transmission rate x (t) of source based on its current vaue and the sum of effective prices m (p (t)) in its path. The effective prices m (p (t)) are functions of the ink prices p (t), and the functions m in genera can depend on the inks and sources. When a agorithms use the same pricing signa, i.e., m = m are the same for a sources, the equiibrium properties of () (2) turn out to be very simpe. Indeed, under mid conditions on g and f, the equiibrium of () (2) exists and is unique [3]. This is proved by identifying the equiibrium of () (2) with the unique soution of the utiity maximization probem defined in [0] and its Lagrange dua probem [4]. Here, the equiibrium prices p pay the roe of Lagrange mutipiers, one at each ink. This utiity maximization probem thus provides a simpe and compete characterization of the equiibrium of a singe-protoco network and aso eads to a reativey simpe dynamic behavior. When heterogeneous agorithms that use different pricing signas share the same network, i.e., m are different for
2 different sources, the situation is much more compicated. For instance, when TCP Reno and TCP Vegas or FAST share the same network, neither oss probabiity nor queueing deay can serve as the Lagrange mutipier at the ink, and () (2) can no onger be interpreted as soving the standard network utiity maximization probem. Basic questions, such as the existence and uniqueness of equiibrium, its oca and goba stabiity, need to be re-examined. We focus in this paper on the existence and uniqueness of equiibrium. We prove that equiibrium sti exists, under mid conditions, despite the ack of an underying concave optimization probem (Section III). In contrast to the singeprotoco case, even when the routing matrix has fu row rank, there can be uncountaby many equiibria (Exampe in Section IV) and the set of botteneck inks can be non-unique (Exampe 2 in Section IV). However, we prove that amost a networks have finitey many equiibria and they are necessariy ocay unique (Section IV). We prove the number of equiibria is aways odd, though can be more than one (Section IV). Moreover, these equiibria cannot a be ocay stabe uness the equiibrium is gobay unique (Section IV). Finay, we provide two sufficient conditions for goba uniqueness of network equiibrium (Sections VI and V). The first condition impies that if the price mapping functions that map ink prices to effective prices observed by the sources do not differ too much, then goba uniqueness is guaranteed. The second condition generaizes the fu-rank condition on routing matrix for goba uniqueness from singe-protoco networks to muti-protoco networks. Throughout the paper, we provide numerica exampes to iustrate equiibrium properties or how a theorem can be appied. In [29], we demonstrate experimentay the phenomenon of mutipe equiibria using TCP Reno and TCP Vegas/FAST in ns-2 simuator and Dummynet testbed. C. Reated work Our formuation is cose to the genera equiibrium theory in economics from which we borrow ideas and techniques [8]. See [4], [6], [7], [24], [25], [32], [3], [5] and [7], [] for a fairy compete treatment of reated works in economics iterature. A typica mode of the pure exchange economy consists of L commodities and N consumers. Each consumer i has an initia endowment vector ω i = (ω i 0, =,..., L) and its goa is to choose a consumption vector x i = (x i, =,..., L) to maximize its utiity subect to its weath constraint, i.e., max x i 0 U i(x i ) subect to p T x i p T ω i where p = (p, =,..., L) are unit prices for the goods and T denotes matrix transpose. For each good =,, L, demand and suppy are baanced if N x i = i= N i= A consumption vector x = (x i, i =,..., N) and a price vector p are caed a competitive equiibrium (or Warasian ω i equiibrium) if x i maximizes i s utiity and demand equas suppy for a goods. In genera equiibrium theory, consumers are assumed to be price takers. This aspect is simiar to our mode where sources do not take into account how their decisions affect the ink prices or each other. Both probems are concerned with characterizing fixed points of a continuous mapping, and hence there are consideraby simiarities in terms of the characterizations and the mathematica toos to derive them. The main mathematica toos used in this paper are the Nash theorem in game theory [23], [2], which is an appication of Kakutani s generaized fixed point theorem, and resuts from differentia topoogy, especiay the Poincare-Hopf Index Theorem [20]. They are used to prove existence and study uniqueness of network equiibrium, respectivey. There are however severa differences. First, the effective prices to different sources (consumers) are generay different in our mode, whereas the prices in the economic mode are independent of consumers. Differentia pricing is what makes networks with heterogeneous protocos much more difficut. Second, in the economic mode, there is a concept of initia endowment that defines both the demand-suppy reation and a consumer s consumption possibiity through the weath constraint. In our mode, the weath constraint is repaced by the ink capacity constraint. Third, in the economic mode, consumers maximize their utiities whereas in our mode, sources maximize their utiities minus bandwidth costs. Finay, in our mode, every source consumes exacty the same amount of bandwidth at each ink in its path (x i = x i, for a L(i)), whereas, in the economic mode, consumers can consume different goods at different amounts. This guarantees that the demand for every good is exacty baanced by its suppy in a pure exchange economy, yet in networks, the set of botteneck inks where demand for and suppy of bandwidth is baanced can be nonunique and a strict subset of a inks. The property x i = x i is the key structure that aows us to obtain interesting resuts on goba uniqueness in fairy genera settings. In contrast, goba uniqueness in genera equiibrium anaysis usuay requires very strong conditions and most iterature focuses on oca uniqueness [3], [5], []. A. Notation II. MODEL A network consists of a set of L inks, indexed by =,..., L, with finite capacities c. We often abuse notation and use L to denote both the number of inks and the set L = {,..., L} of inks. Each ink has a price p as its congestion measure. There are J different protocos indexed by superscript, and N sources using protoco, indexed by (, i) where =,..., J and i =,..., N. The tota number of sources is N := N. The L N routing matrix R for type sources is defined by R i = if source (, i) uses ink, and 0 otherwise. The overa routing matrix is denoted by R = [ R R 2 R ] J
3 Every ink has a price p. A type source reacts to the effective price m (p ) in its path, where m is a price mapping function, which can depend on both the ink and the protoco type. By specifying function m, we can et the ink feed back different congestion signas to sources using different protocos, for exampe, Reno with packet osses and Vegas with queueing deay. Let m (p) = (m (p ), =,... L) and m(p) = (m (p ), =,... J). The aggregate prices for source (, i) is defined as q i = R i m (p ) (3) Let q = (q i, i =,..., N ) and q = (q, =..., J) be vectors of aggregate prices. Then q = ( R ) T m (p) and q = R T m(p). Let x be a vector with the rate x i of source (, i) as its ith entry, and x be the vector of x x = [ (x ) T, (x 2 ) T,..., (x J ) T ] T Source (, i) has a utiity function U i (x i ) that is stricty concave increasing in its rate x i. Let U = (U i, i =,..., N, =,..., J). In genera, if z k is defined, then z denotes the (coumn) vector z = (z k, k). Other notations wi be introduced ater when they are encountered. We ca (c, m, R, U) a network. B. Network equiibrium A network is in equiibrium, or the ink prices p and source rates x are in equiibrium, when each source (, i) maximizes its net benefit (utiity minus bandwidth cost), and the demand for and suppy of bandwidth at each botteneck ink are baanced. Formay, a network equiibrium is defined as foows. Given any prices p, we assume in this paper that the source rates x i are uniquey determined by x i ( ) q i = [ ( ) ( ) ] + U i q i ( ) ( ) where U i is the derivative of U i, and U i is its inverse which exists since U i is stricty concave. Here [z]+ = max{z, 0}. This impies that the source rates x i uniquey sove max z 0 U i (z) zq i As ( ) we wi see, under the assumptions in this paper, ( ) U i q i > 0 for a the prices p that we consider, and hence we can ignore the proection [ ] + and assume without oss of generaity that ( ) ( ) ( ) x i q i = U i q i (4) As usua, we use x ( q ) ( ( ) ) = x i q i, i =,..., N and x(q) = ( x ( q ), =,..., J ) to denote the vector-vaued functions composed of x i. Since q = RT m(p), we often abuse notation and write x i (p), x (p), x(p). Define the aggregate source rates y(p) = (y (p), =,..., L) at inks as: y (p) = R x (p), y(p) = Rx(p) (5) In equiibrium, the aggregate rate at each ink is no more than the ink capacity, and they are equa if the ink price is stricty positive. Formay, we ca p an equiibrium price, a network equiibrium, or ust an equiibrium if it satisfies (from (3) (5)) P (y(p) c) = 0, y(p) c, p 0 (6) where P := diag(p ) is a diagona matrix. The goa of this paper is to study the existence and uniqueness properties of network equiibrium specified by (3) (6). Let E be the equiibrium set: E = {p R L + P (y(p) c) = 0, y(p) c} (7) For future use, we now define an active constraint set and the Jacobian for inks that are activey constrained. Fix an equiibrium price p E. Let the active constraint set ˆL = ˆL(p ) L (with respect to p ) be the set of inks at which p > 0. Consider the reduced system that consists ony of inks in ˆL, and denote a variabes in the reduced system by ĉ, ˆp, ŷ, etc. Then, since y (p) = c for every ˆL, we have ŷ(ˆp) = ĉ. Let the Jacobian for the reduced system be Ĵ(ˆp) = ŷ(ˆp)/ ˆp. Then Ĵ(ˆp) = ( ˆR ) x ˆq (ˆp) T ˆm ˆR (ˆp) (8) ˆp where ( ) x ˆq = diag 2 U i (x (9) i )2 ( ) ˆm ˆm = diag (0) ˆp ˆp and a the partia derivatives are evauated at the generic point ˆp. C. Current theory: J = In this subsection, we briefy review the current theory for the case where there is ony one protoco, i.e., J =, and expain why it cannot be directy appied to the case of heterogeneous protocos. When a sources react to the same price, then the equiibrium described by (3) (6) is the unique soution of the foowing utiity maximization probem defined in [0] and its Lagrange dua [4]: max x 0 U i (x i ) () i subect to Rx c (2) Hence we can effectivey modify user utiity functions and infuence rate aocations through the choice of price mapping functions m. In particuar, inear ink-independent m scae user utiity functions ineary; see Theorem 6.
4 where we have omitted the superscript =. The strict concavity of U i guarantees the existence and uniqueness of the optima soution of () (2). The basic idea to reate the utiity maximization probem () (2) to the equiibrium equations (3) (6) is to examine the dua of the utiity maximization probem, and interpret the effective price m (p ) as a Lagrange mutipier associated with each ink capacity constraint (see, e.g., [4], [22], [3]). As ong as m (p ) 0 and m (0) = 0, one can repace p in (6) by m (p ). The resuting equation together with (3) (5) provides the necessary and sufficient condition for x i (p) and m (p ) to be prima and dua optima respectivey. This approach breaks down when there are J > types of prices because there cannot be more than one Lagrange mutipier at each ink. In genera, an equiibrium no onger maximizes aggregate utiity, nor is it unique. However, as shown in the next section, existence of equiibrium is sti guaranteed under the foowing assumptions: A: Utiity functions U i are stricty concave increasing, and twice continuousy differentiabe in their domains. Price mapping functions m are continuousy differentiabe in their domains and stricty increasing with m (0) = 0. A2: For any ɛ > 0, there exists a number p max such that if p > p max for ink, then x i (p) < ɛ for a (, i) with R i = These are mid assumptions. Concavity and monotonicity of utiity functions are often assumed in network pricing for eastic traffic. The assumption on m preserves the reative order of prices and maps zero price to zero effective price. Assumption A2 says that when p is high enough, then every source going through ink has a rate ess than ɛ. III. EXISTENCE OF EQUILIBRIUM In this section, we prove the existence of network equiibrium. We start with a emma that bounds the equiibrium prices. Lemma. Suppose A and A2 hod. Given a network (c, m, R, U), there is a scaar p max that upper bounds any equiibrium price p, i.e., p p max for a. Proof. Choose ɛ = min c /N, and et p max be the corresponding scaar in A2. Suppose that there exists an equiibrium price p and a ink, such that p > p max. A2 impies that the aggregate equiibrium rate at ink satisfies R i x i (p) < Nɛ = min c i Therefore, we get a ink with p > 0 but not fuy utiized. It contradicts the equiibrium condition (6). The foowing theorem asserts the existence of equiibrium for a muti-protoco network. Theorem 2. Suppose A and A2 hod. There exists an equiibrium price p for any network (c, m, R, U). Proof. Let p max be the scaar upper bound in Lemma. For any p [0, p max ] L, define a vector function For any ink, et F (p) := Rx(p) c (3) p := (p,..., p, p +...p L ) T Then we may write F (p) as F (p, p ). Define function h as h (p, p ) := F 2 (p, p ) (4) We caim that h (p, p ) is a quasi-concave function in p for any fixed p. By the definition of quasi-concavity in [23], we ony need to check that the set A := { p h (p, p ) a } is convex for a a R. If a > 0, ceary A = by (4). When a 0, the set A can be rewritten as { A = a F (p, p ) } a p Since F (p, p ) is a non-increasing function in p for any fixed p, the set A is convex. Therefore h (p, p ) is quasiconcave in p. Since [0, p max ] is a nonempty compact convex set, by the theorem of Nash [23], the quasi-concavity of h (p, p ) guarantees that there exists a p [0, p max ] L such that for a {, 2...L} p = arg max p [0,p max] h (p, p ) We now argue that, for a, either ) F (p ) = 0, or 2) F (p ) < 0 and we can take p = 0. These conditions impy (6), and hence p is an equiibrium price. Case : F (0, p ) > 0. Since U i is stricty concave, F (p, p ) is non-increasing2 in [0, p max ]. Moreover, the proof of Lemma shows that F (p max, p ) < 0. Therefore, there exists a point p in [0, p max ] where F (p, p ) = 0. This p maximizes h (p, p ). Case 2: F (0, p ) 0. Since F (p, p ) is a non-increasing function in p, we have that F (p, p ) 0 for a p [0, p max ] If c < F (0, p ) 0, then F (p, p ) and h (p, p ) are stricty decreasing in p and hence p = arg max h (p, p ) = 0 p [0,p max ] Otherwise we have F (0, p ) = c from (3). In this situation, a x i going through ink are zero, and hence we can set p = 0 without affecting any other prices. More precisey, a (possiby) new price vector p with p = 0 and p k = p k for k is aso a Nash equiibrium that maximizes h k (p k, p k ) for k =,..., L. Thus we have proved that, for =,..., L, p F (p, p ) = 0, F (p, p ) 0, p 0 which is (6). 2 F (p, p ) is stricty decreasing uness some x i(p) becomes zero.
5 IV. REGULAR NETWORKS Theorem 2 guarantees the existence of network equiibrium. We now study its uniqueness properties. A. Mutipe equiibria: exampes In a singe-protoco network, if the routing matrix R has fu row rank, then there is a unique active constraint set ˆL and a unique equiibrium price p associated with it. If R does not have fu row rank, then equiibrium prices p may be nonunique but the equiibrium rates x(p) are sti unique since the utiity functions are stricty concave. In contrast, the active constraint set in a muti-protoco network can be non-unique even if R has fu row rank (Exampe 2). Ceary, the equiibrium prices associated with different active constraint sets are different. Moreover, there can be mutipe equiibrium prices associated with the same active constraint set (Exampe ). Exampe : unique active constraint set but uncountaby many equiibria In this exampe, we assume a the sources use the same utiity function ( ) 2 U i (x i ) = x i (5) 2 Then the equiibrium rates x of type sources are determined by the equiibrium prices p as x (p) = (R ) T m (p) where is a vector of appropriate dimension whose entries are a s. We use inear price mapping functions: m (p) = K p where K are L L diagona matrices. Then the equiibrium rate vector of type sources can be expressed as x (p) = (R ) T K p When ony inks with stricty positive equiibrium prices are incuded in the mode, we have J y(p) = R x (p) = c = Substituting in x (p) yieds J R (R ) T K p = = J R c = which is a inear equation in p for given R, K, and c. It has a unique soution if the determinant is nonzero, but has no or mutipe soutions if J det R (R ) T K = 0 = When J =, i.e., there is ony one protoco, and R has fu row rank, det(r (R ) T K ) > 0 since both R (R ) T and K are positive definite. In this case, there is a unique equiibrium price vector. When J = 2, there are networks whose determinants are zero that have uncountaby many equiibria. See Appendix VIII-B for an exampe where R does not have fu row rank. We provide here an exampe with J = 3 where R sti has fu row rank. The network is shown in Figure with three unit-capacity inks, c =. There are three different protocos with the Fig.. 2 x x x2 3 x Exampe : uncountaby many equiibria. x3 2 x2 corresponding routing matrices [ R = I, R 2 0 = 0 ] T, R 3 = (,, ) T The inear mapping functions are given by K = I, K 2 = diag(5,, 5), K 3 = diag(, 3, ) It is easy to cacuate that 3 R i (R i ) T K i = i= 7 4 6 6 6 4 7 which has determinant 0. Using the utiity function defined in (5), we can check that the foowing are equiibrium prices for a ɛ [0, /24]: p = p 3 = /8 + ɛ The corresponding rates are x = x 3 = 7/8 ɛ x 2 = x 2 2 = /8 3ɛ p 2 = /4 2ɛ x 2 = 3/4 + 2ɛ x 3 = 4ɛ A capacity constraints are tight with these rates. Since there is an one-ink fow at every ink, the active constraint set is unique and contains every ink. Yet there are uncountaby many equiibria. Exampe 2: mutipe active constraint sets each with a unique equiibrium Consider the symmetric network in Figure 2 with 3 fows. There are two protocos in the network with the foowing routing matrices R = 0 0, R 2 = (,, ) T Fows (, ) and (, 2) have identica utiity function U and source rate x, and fow (2, ) has a utiity function U 2 and source rate x 2.
6 Fig. 2. x 2 x Exampe 2: two active constraint sets. Links and 3 both have capacity c and price mapping functions m (p) = p and m 2 (p) for protocos and 2 respectivey. Link 2 has capacity c 2 and price mapping functions m 2(p) = p and m 2 2(p). In [29], we prove that when assumption A hods, the network shown in Figure 2 has two equiibria provided: ) c < c 2 < 2c ; 2) for =, 2, (U ) (x ) p, possiby, if and ony if x 0. 3) for =, 2, m 2 (p ) p 2 as p p, and satisfy 2m 2 ((U ) (c 2 c )) < (U 2 ) (2c c 2 ) < m 2 2((U ) (c 2 c )) By manipuating buffer sizes and RED parameters, i.e., carefuy designing the price mapping functions m, we have demonstrated experimentay in [29] the phenomenon of mutipe equiibria for this exampe using TCP Reno, which reacts to oss probabiity, and TCP Vegas/FAST, which react to deay. 3 B. Reguar networks Exampes and 2 show that goba uniqueness is generay not guaranteed in a muti-protoco network. We now show, however, that oca uniqueness is basicay a generic property of the equiibrium set. We present our main resuts on the structure of the equiibrium set here, providing conditions for the equiibrium points to be ocay unique, finite and odd in number, and gobay unique. We prove these resuts in the next subsection. Consider an equiibrium price p E. Reca the active constraint set ˆL defined by p. The equiibrium price ˆp for the inks in ˆL is a soution of x2 ŷ( ˆp) = ĉ (6) By the inverse function theorem, the soution of (6), and hence the equiibrium price ˆp, is ocay unique if the Jacobian matrix Ĵ(ˆp ) = ŷ/ ˆp is nonsinguar at ˆp. We ca a network (c, m, R, U) reguar if a its equiibrium prices are ocay unique. The next resut shows that amost a networks are reguar, and that reguar networks have finitey many equiibrium 3 It is pointed out in [27] that there is actuay a third equiibrium for this network where a inks are activey constrained. However, unike the other two equiibria, the third is not ocay stabe and hence did not manifest itsef in the experiments reported in [29]. prices. This ustifies restricting our attention to reguar networks. Theorem 3. Suppose assumptions A and A2 hod. Given any price mapping functions m, any routing matrix R and utiity functions U, ) the set of ink capacities c for which not a equiibrium prices are ocay unique has Lebesgue measure zero in R L +. 2) the number of equiibria for a reguar network (c, m, R, U) is finite. For the rest of this subsection, we narrow our attention to networks that satisfy an additiona assumption: A3: Every ink has a singe-ink fow (, i) with ( U i ) (c ) > 0. Assumption A3 says that when the price of ink is sma enough, the aggregate rate through it wi exceed its capacity. This ensures that the active constraint set contains a inks and faciitates the appication of Poincare-Hopf theorem by avoiding equiibrium on the boundary (some p = 0). 4 Since a the equiibria of a reguar network have nonsinguar Jacobian matrices, we can define the index I(p) of p E as { if det (J(p)) > 0 I(p) = if det (J(p)) < 0 Then, we have Theorem 4. Suppose assumptions A A3 hod. Given any reguar network, we have I(p) = ( ) L p E where L is the number of inks. We give two important consequences of this theorem. Coroary 5. Suppose assumptions A A3 hod. A reguar network has an odd number of equiibria. Proof. Since both I(p) and ( ) L are odd, the number of terms in the summation in Theorem 4 must be odd. Notice that Coroary 5 impies the existence of equiibrium. Athough we proved this in Section III in a more genera setting, this simpe coroary shows the power of Theorem 4. The next resut provides a condition for goba uniqueness. We say that an equiibrium p E is ocay stabe if the 4 It is recenty shown in [27] that A3 is not necessary and one can generaize Theorem 4 to ( ) ˆL(p) I(p) = p E where ˆL(p) is the number of inks of the active constraint set associated with equiibrium p. Ceary, if ˆL(p) = L, it reduces to Theorem 4. This generaized theorem aso aows [27] to concude the number of equiibria is odd (and therefore existence) without A3. In this paper, athough A3 is imposed, a resuts can be viewed as with respect to a fixed active constraint set with appropriate modifications. In particuar, the goba uniqueness resuts in Section V directy appy without A3 since Ĵ has a simiar structure as J except with a smaer dimension.
7 corresponding Jacobian matrix J(p ) defined in (8) is stabe, that is, every eigenvaue of J(p ) = y(p )/ p has negative rea part. For ustification of this definition, oca stabiity of p impies that the gradient agorithm (9) beow converges ocay. Theorem 6. Suppose assumptions A A3 hod. The equiibrium of a reguar network is gobay unique if and ony if every equiibrium point in E has an index ( ) L. In particuar, if a equiibria are ocay stabe, then E contains exacty one point. Proof: The first caim of the theorem directy foows from Theorem 4. We now caim that an equiibrium p E which is ocay stabe has an index I(p ) of ( ) L. To prove the caim, consider a ocay stabe equiibrium price p. A the eigenvaues of J(p ) have negative rea parts. Moreover, since J(p ) has rea entries, compex eigenvaues come in conugate pairs. The determinant of J(p ) is the product of a its eigenvaues. If there are k conugate pairs of compex eigenvaues and L 2k rea eigenvaues, the product of a eigenvaues has the same sign as ( ) L 2k which has the same sign as ( ) L. Hence the index of a ocay stabe equiibrium is ( ) L. This resut may seem surprising at the first sight as it reates the oca stabiity of an agorithm to the uniqueness property of a network. This is because both equiibrium and oca stabiity are defined in terms of the function y(p): an equiibrium p satisfies y(p ) = c and the oca asymptotic stabiity of p is determined by y(p )/ p. The connection between these two properties is made exact by the index theorem. An impication of this resut is that if there are mutipe equiibria, then no agorithm ṗ = f(p(t)), whose inearization around each equiibrium p E satisfies f(p )/ p = y(p )/ p, can be found to ocay stabiize a of the equiibria. Theorem 6 wi be used in Section V to derive a sufficient condition on price mapping functions m for goba uniqueness. We cose this subsection with an exampe that iustrates the appication of Theorem 4 and Coroary 5. Exampe 3: iustration of Theorem 4 and Coroary 5 We revisit Exampe with modified utiity functions. Reca that in Exampe, as ɛ varies from 0 to /24, we trace out a equiibrium points. The components x and q = p of these equiibrium points are shown by the (red) soid ine in Figure 3. Other sources x i and their effective end-to-end prices q i aso ie on simiar straight ines. Since the network has uncountaby many equiibrium points, it is not reguar. To make it reguar, suppose we change the utiity functions of sources (, i) to { U i (x i, α i ) = β i (x i ) α i /( α i ) if α i β i og x i if α i = with appropriatey chosen positive constants α i and β i. These utiity functions can be viewed as a weighted version of the α-fairness utiity functions proposed in [22]. The basic idea of how to choose α i and β i to generate ony finitey many equiibrium points is as foows. First, we pick two points in the equiibrium set of Exampe, say, the points associated with ɛ = 0.0 and ɛ = 0.04. These choices of ɛ provide two distinct equiibrium points (q, x) and ( q, x). For instance, (q, x ) = (0.35, 0.865) corresponds to ɛ = 0.0 and ( q, x ) = (0.65, 0.835) corresponds to ɛ = 0.04, as iustrated in Figure 3. Then, for each source Fig. 3. x 7/8 5/6 =(β /q ) /α x /8 /6 (0.35,0.865) (0.65,0.835) Exampe 3: construction of mutipe isoated equiibria. (, i), find α i and β i such that (4) is satisfied by the two equiibrium points (q i, x i ) and ( q i, x i ) with the new utiity functions. This is iustrated in Figure 3 where reation (4) with the new utiity function is represented by the (bue) curve, and α i, β i are chosen so that the curve passes through the origina equiibrium points (x, q) and ( q, x). More specificay, given two equiibrium points (q i, x i ) and ( q i, x i ), choose α i = og(q i ) og( q i ) og( x i ) og(x i ) β i = q i ( ) α x i i The resuting α i and β i for a fows (, i) are shown in Tabe I. TABLE I EXAMPLE 3: α i AND β i. Fows α i β i x 5.685 0.0592 x 2 4.0285 0.0803 x 3 5.685 0.0592 x 2 0.0322 0.8389 x 2 2 0.0322 0.8389 x 3 0.0963 0.704 By construction, both (p = 0.35, p 2 = 0.230) and (p = 0.65, p 2 = 0.70) are network equiibria. By Coroary 5, there is at east one additiona equiibrium. Numerica search indeed ocated a third equiibrium with (p = 0.42, p 2 = 0.206). We further check the oca stabiity of these three equiibria under the gradient agorithm (9) to be introduced in Section IV-C. The eigenvaues and index for each equiibrium are shown in Tabe II. It turns out that the equiibrium (p = 0.42, p 2 = 0.206) is not stabe and has index, whie the q
8 TABLE II EXAMPLE 3: STABILITY AND INDICES OF EQUILIBRIA. Equiibria (p, p 2, p 3 ) Eigenvaues Index (0.35, 0.23, 0.35) 0.2, 7.43, 26.73 (0.42, 0.206, 0.42) 0.2, 2.32, 22.40 (0.65, 0.7, 0.65) 2.4,.67, 0.67 other two are stabe with index. The dynamics of this network under the gradient agorithm can be iustrated by a vector fied. By symmetry, the equiibrium prices for the first and third ink are aways same. Therefore, we can draw the vector fied restricted on the pane p = p 3 to iustrate the system dynamics. The phase portrait is shown in Figure 4. The (red) dots represent the three equiibria. Note the equiibrium in the midde is a sadde point, and therefore unstabe. The (red) arrows give the direction of this vector fied. Individua traectories are potted with sim (bue) ines. p 2 0.25 0.24 0.23 0.22 0.2 0.2 0.9 0.8 0.7 0.6 0.5 0. 0. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 p Fig. 4. Exampe 3: vector fied of (p, p 2 ). C. Proofs In this subsection we provide proofs for the resuts in Section IV-B. Proof of Theorem 3. The main mathematica too used in our proof is Sard s Theorem [4], [28], of which we quote a version here that is taiored to our probem. Let G be an open subset of R L + and et F be a continuousy differentiabe function from G to R L +. A point y G is a critica point of F if the Jacobian matrix F/ y of F at y is singuar. A point z R L + is a critica vaue of F if there is a critica point y G with z = F (y). A point in R L + is a reguar vaue of F if it is not a critica vaue. Sard s theorem. If F : G R L + is continuousy differentiabe on the open subset G R L +, then the set of critica vaues of F has Lebesgue measure zero in R L +. Fix a routing matrix R and utiity functions U. There are at most 2 L different active constraint sets. Let ˆL L be such a combination with ˆL inks. Consider the set of a possibe ink capacities c = (c, L) under which the active constraint set is ˆL, i.e., with such a capacity vector c, an equiibrium price p has p > 0 if ˆL and p = 0 otherwise. Fix such an equiibrium point p. Again et ˆp denote the price vector ony for inks in ˆL. Then ˆp is not ocay unique if the function ŷ : R ˆL + R ˆL + defined by ŷ(ˆp) = ˆRx(ˆp) has a singuar Jacobian matrix ŷ/ ˆp at ˆp, i.e., if ˆp is a critica point of ŷ. The set of such capacity vectors ĉ R ˆL + under which a inks in ˆL have active constraints in equiibrium satisfy ŷ(ˆp ) = ĉ and hence are critica vaues of ŷ. Since ŷ is continuousy differentiabe by assumption A, we can appy Sard s theorem and concude that the set of such capacity vectors ĉ has zero Lebesgue measure in R ˆL +. The extension to R L + for a ink capacities ceary aso has zero Lebesgue measure in R L +. Since we ony have a finite number of different active constraint sets, the union of ink capacity vectors that give rise to ocay nonunique equiibria sti has zero Lebesgue measure. This proves the first part of the theorem. The equiibrium set E defined in (7) is cosed because y(p) is continuous, and is bounded by Lemma. Hence E is compact. Since (c, m, R, U) is a reguar network, every p E is ocay unique, i.e., for each p E we can find an open neighborhood such that it is the ony equiibrium in that open set. The union of these open sets forms a cover for set E. Since E is compact, it admits a finite subcover [6], i.e., E can be covered by a finite number of open sets each containing a singe equiibrium. Hence, the number of equiibria is finite. Proof of Theorem 4. By assumption A3, we can aways find p min > 0 such that for any price p and ink with p < p min, we have R i x i (p) > c i Let G := [p min, p max ] L where p max is defined in Lemma. Ceary, a equiibria are in the set G. To prove our resut, we wi invoke a version of the Poincare-Hopf Index Theorem taiored to our probem [32], [20]. Poincare-Hopf index Theorem. Let D be an open subset of R and v : D L R L be a smooth vector fied, with nonsinguar Jacobian matrix v/ p at every equiibrium. If there is a G D L such that every traectory moves inward of region G, then the sum of the indices of the equiibria in G is ( ) L. Gradient proect agorithm. To construct the vector fied v required by the index theorem, et D L = G and consider the foowing gradient agorithm from G to G proposed in [4]. The prices are updated at time t according to ṗ(t) = Λ (Rx(t) c) (7) where Λ > 0 is an L L diagona matrix whose eements represent stepsizes. A source updates its rate based on the
9 end-to-end price x(t) = x(p(t)) (8) A consequence of assumption A3 is that p(t) p min > 0 for a t under the gradient agorithm (7) (8). This guarantees a unique active constraint set that is L. Hence the equiibrium set E defined in (7) is equivaent to E = {p R L + y(p) c = 0}. Combining (7) (8) with y(p(t)) = Rx(t) yieds the required vector fied v: ṗ(t) = Λ(y(p(t)) c) =: v(p(t)) (9) whose Jacobian matrix is: v (p) = ΛJ(p) = Λ y (p) (20) p p where J(p) is given by (8). Ceary, p is an equiibrium point of v, i.e., v(p ) = 0, if and ony if p is a network equiibrium, i.e., p E. Since the network (c, m, R, U) is reguar, J(p) is nonsinguar at every network equiibrium p E G. Since Λ is a positive diagona matrix, v(p)/ p is aso nonsinguar by (20) at a its equiibrium points p in G, as the index theorem requires. Consider any point p on the boundary of G. For any, we have one of two cases: ) If p (t) = p max, ink wi be underutiized, y (p(t)) < c, and ṗ < 0 according to (9). 2) If p (t) = p min, the aggregate rate at ink wi exceed c, y (p(t)) > c, and ṗ > 0 according to (9). Therefore, every point p on the boundary of G wi move inward and our resut directy foows from the Poincare-Hopf index theorem. V. GLOBAL UNIQUENESS: MAPPING FUNCTIONS m(p) In this and the next sections, we provide sufficient conditions on the structure of the network for goba uniqueness. We aso provide some important specia cases in Appendix VIII-A where goba uniqueness is set up. In this section, we show that, under assumptions A A3, if the price mapping functions m are simiar, then the equiibrium of a reguar network is gobay unique. A. Main resut To state the resut concisey, we need the notion of permutation. We ca a vector σ = (σ,..., σ L ) a permutation if each σ is distinct and takes vaue in {,..., L}. Treating σ as a mapping σ : {,..., L} {,..., L}, we et σ denote its unique inverse permutation. For any vector a R L, σ(a) denotes the permutation of a under σ, i.e., [σ(a)] = a σ. If a {,..., L} L is a permutation, then σ(a) is aso a permutation and we often write σa instead. Let = (,..., L) denote the identity permutation. Then σ = σ. See [9] for more detais. Finay, denote dm /dp by ṁ. Theorem 7. Suppose assumptions A A3 hod. If, for any vector {,..., J} L and any permutations σ, k, n in {,..., L} L, ṁ [k()] + ṁ [n()] ṁ [σ()] (2) = = = then the equiibrium of a reguar network is gobay unique. Theorem 7 impies that if the (sopes of the) price mapping functions are simiar, then goba uniqueness is guaranteed, as the foowing coroary shows: if m do not differ much across source types at each ink, or they do not differ much aong inks in every source s path, then the equiibrium is unique. Coroary 8. Suppose assumptions A A3 hod. The equiibrium of a reguar network is gobay unique if any one of the foowing conditions hods: ) For each =,..., L, =,..., J ] [a, 2 L a for some a > 0 (22) ṁ 2) For a =,..., J, =,..., L [ ] a, 2 L a for some a > 0 (23) ṁ Proof. If (22) hods, we have for any, ĵ, in {,..., J} = ṁ + = ṁĵ 2 L = a = L = 2 L a = ṁ which impies the sufficient condition in Theorem 7. For the second assertion, fix any in {,..., L} L and any permutations σ, k, n in {,..., L} L. If (23) hods, we have = ṁ [k()] + = ṁ [n()] 2 a = 2 L a = = ṁ [σ()] = which impies the sufficient condition in Theorem 7. Remarks: ) Asymptoticay when L, both conditions (22) and (23) converge to a singe point. Condition (22) reduces to ṁ = a which essentiay says that a protocos are the same (J = ). Condition (23) reduces to ṁ = a, which is the inear ink independent case discussed in Theorem 6. 2) The sufficient condition in Theorem 7 can be conservative because many r π may be zero (no source of type takes path π). 3) These ink-based uniqueness resuts hod for a network whenever no fow uses more than L inks. B. Proof We now prove Theorem 7. By Theorem 6, we ony need to prove that I(p) = ( ) L for any equiibrium p E. Since det(j(p)) = ( ) L det( J(p)), the condition reduces to det( J(p)) > 0. Now J(p) = = R D (p) ( R ) T m p (p) B M
0 where M = M (p) = m p (p) is a diagona matrix, and B = B (p) is defined by its eements B k = ) R ki R i ( 2 U i i (x (24) i )2 Hence det( J(p)) = det = k sgnk r π = r π(p) = i I π B M = = J [ B M ] k 2 U i (x i )2 (25) Here, the summation over k = (k,..., k L ) {,..., L} L is over a L! permutations of the L items {,..., L}. The function sgnk is if the minimum number of pairwise interchanges necessary to achieve the permutation k starting from (, 2,..., L) is even and if it is odd. Let π denote an L-bit binary sequence that represents the path consisting of exacty those inks k for which the kth entries of π are, i.e., π k =. Let Π(k, ) := {π π k = π = } be the set of paths that contain both inks k and. Let Iπ = {i R i = if and ony if π = } be the set of type sources on path π, possiby empty. Let ( ) (26) where rπ is zero if Iπ is empty. Since a utiity functions are assumed concave, rπ 0. Then we have from (24) and (26) B k = rπ (27) π Π(k,) This together with (25) impies det( J(p)) = k sgnk = J ṁ rπ = π Π(k,) (28) Consider any sequence a i, J i, i =,..., I, where J i is a finite index set that depends on i. We have I i= J i a i = I i= a ii (29) where denotes the vector index = (,..., J I ) and the summation is over a vaues in J... J I. Using (29) to change the order of product over and summation over in (28), we have det( J(p)) = k sgnk = ṁ r π π Π(k,) where the vector index = (,..., L ) ranges over {,..., J} L. Appying (29) again to change the order of product over and summation over the index π, we have det( J(p)) = sgnk µ() ρ(, π)(30) k π Π(k,) where µ() := ρ(, π) := = = ṁ (3) r π (32) The ast summation in (30) is over the vector index π = (π,..., π L ) that takes vaue in the set { a L-bit binary sequences } L. As mentioned above, = (,..., L) denotes the identity permutation, and π Π(k, ) is a shorthand for π Π(k, ), =,..., L. Denote by (a) the indicator function that is if the assertion a is true and 0 otherwise. Then (30) becomes det( J(p)) = C(, π) ρ(, π) (33) π where C(, π) := k (π Π(k, )) sgnk µ() (34) Hence det( J(p)) is a summation, over the index (, π), of terms ρ(, π) with coefficients C(, π). We now show that ony those terms for which the constituent r π in the product ρ(, π) are a distinct have nonzero coefficients. Lemma 9. Consider a term in the summation in (33) indexed by (, π). If there are integers a, b {,..., L} such that a = b and π a = π b, then C(, π) = 0. Proof. Fix any (, π). Suppose without oss of generaity that = and π = π 2 and ρ(, π) 0. We now show that its coefficient C(, π) = 0. Consider any permutation k in (34) that gives a nonzero coefficient in C(, π): This means that (π Π(k, )) sgnk µ() = sgnkµ() (35) π Π(k, ) and π 2 Π(k 2, 2) Hence, since π = π 2, the path π goes through a inks, 2, k, k 2. In particuar π Π(k 2, ) and π 2 Π(k, 2) Therefore there is a permutation ˆk in (34) with ˆk = k 2, ˆk 2 = k, and ˆk = k for 3 for which (π Π(ˆk, )) = but sgnˆk = sgnk. This yieds a term sgnk µ() in C(, π) which exacty cances the term in (35). Since the argument appies to any k in (34), C(, π) = 0. In view of Lemma 9, we wi restrict the summation over the index (, π) in (33) to the argest subset of {,..., J} L where the constituent rπ in ρ(, π) are a distinct. Let Θ denote this subset. We abuse notation and define permutation σ {,..., L} L on Θ by σ(, π) = (σ(), σ(π)) Then et Θ 0 be the argest subset of Θ that is permutationay distinct, i.e., no vector in Θ 0 is a permutation of another
vector in Θ 0. The set of permutations σ {,..., L} L is in one-one correspondence with the set of (, π ) that are permutations of a given (, π) in Θ 0. 5 This aows us to carry out the summation over (, π) in (33) first over (, π) that are permutationay distinct and then over a their permutations. Notice that, given any (, π) and any permutation σ, we have from (32) ρ(σ(), σ(π)) = ρ(, π) i.e., ρ is invariant to permutations. Hence, we can rewrite (33) (34) as det( J(p)) = D(, π) ρ(, π) (36) (,π) Θ 0 where D(, π) = σ k (σ(π) Π(k, )) sgnk µ(σ()) (37) In the above, L-vectors σ and k are permutations. The next emma converts a condition on σ(π) into one on π. It foows directy from the definition of permutation. Lemma 0. For any π and any permutations σ, k, we have σ(π) Π(k, ) π Π(σ k, σ ) i.e., [σ(π)] Π(k, ) for a if and ony if π Π(k σ, σ ) for a. Appying Lemma 0 to (37), we have D(, π) = (π Π(σ k, σ )) sgnk µ(σ()) σ k Since k, and hence σ k, range over a possibe permutations, we can repace the index variabe σ k by k to get D(, π) = (π Π(k, σ )) sgn(kσ) µ(σ()) σ k (38) We now use (38) to derive a sufficient condition under which D(, π) are nonnegative for a permutationay distinct (, π). The main idea is to show that for every negative term in the summation in (38), either it can be exacty canceed by a positive term, or we can find two positive terms whose sum has a arger or equa magnitude under the given condition. This emma directy impies Theorem 7. Lemma. Suppose assumptions A A3 hod. Suppose for any {,..., J} L and any permutations σ, k, n in {,..., L} L, we have for a reguar network µ(k()) + µ(n()) µ(σ()) Then, for a (, π) Θ 0, D(, π) 0. Proof. Fix any (, π) Θ 0. Each term in (38) is indexed by a pair (σ, k). 5 The one-one correspondence fais to hod for permutations not in Θ. Fix aso a permutation σ in (38). Suppose there is ony one permutation k for which the term indexed by (σ, k) has a negative sign given by (π Π(k, σ ))sgn(kσ) =. This term is then µ(σ()) < 0. Since the summation over k ranges over a permutations, we can find a positive term, indexed by (σ, ˆk) with ˆk = σ, that exacty cances this negative term. This is because (π Π(ˆk, σ )) is aways and sgn(ˆkσ) = sgn =, yieding the term µ(σ()). Hence we have shown that, given σ, if there is ony one k that yieds a negative term, then it is aways canceed by another positive term indexed by (σ, ˆk) with ˆk = σ. Given a σ, suppose now there are two permutations k, n for which π Π(k, σ ) and π Π(n, σ ) (39) and sgn(kσ) = sgn(nσ) =. Each of (σ, k) and (σ, n) yieds a negative term µ(σ()) in the summation in (38). Notice that (39) says that, for a =,..., L, paths π contains ink pairs (k, σ ) and (n, σ ). Hence π aso pass through ink pairs (σ, σ ), (k, n ) and (n, k ), i.e., π Π(σ, σ )) (40) π Π(k, n), π Π(n, k) (4) (40) impies that there is a positive term in the summation in (38) indexed by (σ, ˆk) with ˆk = σ : sgn(σ σ)µ(σ()) = µ(σ()) > 0 It cances the negative term µ(σ()) in the summation indexed by (σ, k). To dea with the negative term µ(σ()) indexed by (σ, n), note that (4) impies that there are two nonzero terms in the summation, indexed by (n, k) and (k, n), that we now argue are positive. Indeed the term indexed by (n, k) is sgn(kn ) µ(n ()) = sgn(kσ(nσ) ) µ(n ()) = sgn(kσ) sgn(nσ) µ(n ()) = µ(n ()) > 0 where we have used the hypothesis that sgn(kσ) = and sgn(nσ) = sgn(nσ) =. Simiary, the term with index (k, n) is µ(k ()). The hypothesis of the emma impies that µ(n ()) + µ(k ()) µ(σ()) 0 Hence, we have shown that, given σ, if there are two negative terms in the summation in (38) indexed by (σ, k) and (σ, n), then we can aways find three positive terms, indexed by, (σ, σ ), (n, k) and (k, n), so that the sum of these five terms are nonnegative. If there are more than two negative terms, take any additiona negative term, indexed by, say, (σ, ˆn). The same argument shows that it wi be compensated by the two (unique) positive terms indexed by (ˆn, k) and (k, ˆn). This competes the proof. Since the network is reguar, det( J(p)) 0. Lemma, together with (36), impy that det( J(p)) > 0, or equivaenty, I(p) = ( ) L for any p E, under the condition of the emma. Theorem 7 then foows from Theorem 6.
2 C. Specia case: L = 3 and J = 2 We now speciaize our uniqueness resut to the case of L = 3, J = 2, and iustrate with an exampe the proofs of Theorem 7 and Lemma. This case is of both theoretica and practica interest. Theoreticay, it represents the smaest network that can exhibit non-unique equiibrium points if A A3 are satisfied. Practicay, empirica study shows that very few paths in the Internet (about 3%) experience more than three botteneck inks [26]. λ 2 0 9 8 7 6 5 4 From Theorem 2 λ =λ 2 =λ 3 From Coroary 3 Theorem 2. Suppose assumptions A A3 hod for a 3-inks reguar network with 2 protocos. If the foowing 6 inequaities hod, the network has a unique equiibrium: 3 2 λ 2 + λ 3 λ, λ + λ 3 λ 2, λ + λ 2 λ 3 +, λ 2 λ 3 λ λ + λ 3 λ 2, + λ λ 2 λ 3 where λ := ṁ (p)/ṁ2 (p). Proof. It is straightforward to check that ony the foowing six ρ(, π) in (36) can have negative coefficients D(, π): (λ 2 + λ 3 λ )ṁ 2 ṁ 2 2ṁ 2 3r r 2 0r 2 0 (42) (λ + λ 3 λ 2 )ṁ 2 ṁ 2 2ṁ 2 3r r 2 0r 2 0 (λ + λ 2 λ 3 )ṁ 2 ṁ 2 2ṁ 2 3r r 2 0r 2 0 ( λ 2 + λ 3 λ )ṁ ṁ 2ṁ 3r 2 r 0r 0 ( λ + λ 3 λ 2 )ṁ ṁ 2ṁ 3r 2 r 0r 0 ( λ + λ 2 λ 3 )ṁ ṁ 2ṁ 3r 2 r 0r 0 Fig. 5. λ 3 0 0 2 3 4 5 6 7 8 9 0 λ Region of λ for goba uniqueness: proection to λ λ 2 pane..2. 0.9 0.8 0.7 0.6 0.5 From Coroary 3 From Theorem 2 The condition in the theorem guarantees that these terms are a nonnegative. By (36), det( J(p)) 0. Since the network is reguar, we have det( J(p)) > 0 for a equiibria p. Hence the equiibrium is gobay unique. A straightforward coroary is the foowing Coroary 3. Suppose assumptions A A3 hod. For a 3-inks reguar network with 2 protocos, if, for a, λ [a, 2a] for some constant a > 0, the network admits a gobay unique equiibrium. Remark: If ṁ = k are ink independent, then λ = k /k 2 [a, 2a] for any k /2k 2 a k /k 2. Hence goba uniqueness is guaranteed, which agrees with Theorem 6. We iustrate in Figures 5 and 6 the regions of λ in Theorem 2 and Coroary 3. They are both cones. The first one is the proection to λ λ 2 pane and the second one is the crosssection cut by pane λ + λ 2 =. We cose this subsection by iustrating how we determine the coefficient D(, π) in the proof of Lemma. Consider the term for ρ(, π) = r r 2 0r 2 0 in (42). Here = (, 2, 2) and π = ((), (0), (0)). By (37), we need to ook at the sum over σ and k. First, ook at σ = (3,, 2), the ony k such that (σ(π) Π(k, )) = and sgnk = is k = (2,, 3). By the argument in the proof of Lemma, if we et k = = (, 2, 3), we have (σ(π) Π(k, )) = and sgnk = and the sum of these two terms in (37) is zero. 0.4 0.3 λ =λ 2 =λ 3 0.2 0.4 0.3 0.2 0. 0 0. 0.2 0.3 0.4 0.707tg(arctg(λ λ 2 ) 0.25π) Fig. 6. Region of λ for goba uniqueness: cross-section cut by pane λ + λ 2 =. We can visuaize this operation as foow. Each entry of J(p) is a sum of ṁ r π with appropriate signs. When we expand its determinant, we obtain, from (36) (37), a sum, over a set of source types, paths π and permutations σ, k, of terms ρ(, π) which are products of rπ. Hence we can identify each term in (36) (37), indexed by (, π, σ, k), with the origina position in J(p) of each constituent rπ in ρ(, π). This is iustrated beow: the negative term r 2 0 r r 2 0 is canceed exacty by the positive term r2 0 r r 2 0 (σ = (3,, 2), k = (2,, 3), sgnk = ) (σ = (3,, 2), k = (, 2, 3), sgnk = ) Simiary, we have the foowing two terms that cance one