Interference alignment-based sum capacity bounds for random dense Gaussian interference networks

In terference alignmen t-based sum capacit y b ounds for random dense Gaussian in terference net w orks Oliver Johnson ∗ Matthew Aldri dg e † Rob ert Piecho cki ‡ August 9, 20 18 Abstract W e consider a dense K user Gaussian inte r ference n et w ork form ed b y paired trans- mitters and receiv ers placed indep endently at random in a fixed s p atial r egio n . Under natural conditions on the no de p osition distributions and signal atten ua- tion, w e p ro v e con verge n ce in p r obabilit y of the a verag e p er-user capacit y C Σ /K to 1 2 E log(1 + 2SNR). The ac hiev abilit y r esult follo ws directly from results based on an in terference alignmen t sc heme presente d in r ecen t work of Naze r et al. Ou r main con tribution comes th r ough an upp er b ound, motiv ated by ideas of ‘b ottlenec k capacit y’ dev elop ed in recent w ork of Jafar. By con trolling the physica l location of transmitter–receiv er p airs, we can matc h a large prop ortion of these pairs to form so-called ǫ -b ottlenec k links, with consequen t cont r ol of the su m capac ity . 1 In tro duction and main res ult 1.1 In terference net w orks and b ottlenec k states Recen t work o f Jafar [9 ] ma de significan t progress tow ards what is referred to as ‘the holy grail of net w ork information theory’, namely the calculation of the capacit y of ∗ Department of Mathematics, Universit y of B ristol, Universit y W alk, Bristol, BS8 1TW, UK. Email: O.John son@bri stol.ac.uk † Department of Mathematics, Universit y of B ristol, Universit y W alk, Bristol, BS8 1TW, UK. Email: m.aldr idge@br istol.ac.uk ‡ Cent r e for Communications Research, Univ ersity of Bristol, Mer c hant V ent ur ers Building, W o od- land Road, Bristol BS8 1 UB, UK. Email: r.j. piechock i@bristol.ac.uk 1 arbitrary Gaussian in terference net works . Jafar pro v es con vergenc e in probabilit y of the av eraged sum capacity of certain dense G aussian in terference netw orks. Although results contained in the pap er [9] made significant progress with this problem, the results w ere describ ed under the constrain t that eac h direct link had the same fading co efficien t √ SNR – a constraint that we relax in this pap er. In Lemma 1 of [9], Ja far show ed t hat a tw o-user Gaussian in terference channel with one of the cross-link strengths INR = SNR has sum capacit y exactly equal to log( 1 + 2SNR). Jafar describ ed suc h a configura tion as a n example of a ‘b ottlenec k state’, in that altering the other cross-link strength do es not affect the capacit y . Jafar w ent on to define the concept of an ǫ -b o t t lenec k link – that is, a cross-link in a t wo-user c hannel with capacity within ǫ of log(1 + 2SNR). He considered a mo del of large netw orks, where eac h v alue of SNR is fixed, and eac h INR is sampled indep enden tly from a fixed distribution. Jafar argues that with probabilty δ , eac h INR lies in the range suc h that the corresp onding t w o -user c hannel b ecomes an ǫ -b ottlenec k. He uses an argument based on Cheb yshev ’s inequalit y t o deduce that P      C Σ K − 1 2 log(1 + 2 SNR)     > ǫ  = O ( K − 2 ) . (1) It is p erhaps surprising that t he existenc e of a p ositive prop ortion of ǫ -b ottlenec k links implies acc ura te pro babilistic b ounds on the s um capacit y C Σ of the whole netw ork. Ho w eve r , w e migh t regard it as analogo us to the so-called ‘birthda y paradox’ – although eac h cross-link has a probabilit y δ of b eing in an ǫ -b ottlenec k, there are K ( K − 1) cross- links that can ha ve this prop ert y , so as K tends to infinit y , the n um b er of cross-links with this prop erty b ecomes muc h larger than K . In this pap er, w e sho w that results suc h as Equation (1) in fact hold more generally , in cases where tra nsmitter no de p ositions T 1 , . . . , T K and corresp onding receiv er no de p ositions R 1 , . . . , R K are chose n independently at random in a region of space D . While exact expres sions for the capacity of net w o r ks with arbitrarily placed no des remain elusiv e, in Theorem 1.5 w e prov e conv ergence in probabilit y for the a v era g e p er-user capacit y C Σ /K of suc h dense net w ork configuratio ns. This ma y p erhaps b e analogous to the fact t ha t Shannon [14] pro ve d that the a verage co de p erformed w ell, while it remains a significantly harder pro blem to establish g o o d p erformance for a particular family of co des. In other words, the randomness w e add to the mo del helps us, rather than making thing s harder. The intuition is that in a dense netw ork of p oin ts, a large prop ortion of no des can b e put together pairwise to form ǫ -b ottlenec k links (see Figure 1). While it would b e p ossible to adjust individual transmitters’ p ow ers to for ce each SNR to b ecome equal ( as a ssumed by [9]), individual p o w er constrain t s mak e this undesirable. F urther, Jafar assumed that the INR are indep enden t and iden ticaly distributed, a 2 ❜ T 5 ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ❜ R 5 ❜ T 1 ❜ R 1 ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❜ T 4 ❜ R 4 ❜ T 2               ❜ R 2 ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ❜ R 3 ❜ T 3 ❜ T 6 ❜ R 6 Figure 1: A dense net work with K = 6 transmitter–receiv er pairs placed on the square [0 , 1 ] 2 , with ǫ -b ottlenec k links emphasised. The distances from T 5 to R 5 , R 5 to T 1 and T 1 to R 1 are all appro ximately equal, similarly for T 4 to R 4 , R 4 to T 2 and T 2 to R 2 . T ransmitter-receiv er pairs ( T 3 , R 3 ) a nd ( T 6 , R 6 ) a r e not matche d in to ǫ -b ottlenec k links. prop ert y that w ould b e lost if user p o w ers w ere scaled in this wa y . 1.2 In terference Alignmen t The concept of in terference alignmen t first app eared in [2 ], and represen ts a depar- ture from the recen t paradigm of random (o r r a ther pseudo-random) co de construction. Pseudo-random co des (e.g. T urb o and LDPC co des) hav e rev olutionised p oint-to-p oin t comm unications. How ev er, since m ulti-terminal net w orks are inte rf erence - limited rat her than noise-limited, unstructured (pseudo-random) co des are not suitable. In terference alignmen t advocates a colla bo rativ e solution. Each receiv er divides its sig- nalling space (space/time/frequency/scale resources) in to t w o parts; one for the signal from the in tended transmitter, and the second acts as a w aste bin. The enco ding is structured is suc h a w ay that the transmitted signal from each of the K transmitters is seen in the clear space for the intende d receiv er, and at the same time, it falls into the 3 w aste bin for all other receiv ers. In suc h a scenario, the net work is no longer in terference limited. I n the first suc h sc heme Cadam b e a nd Jafar [2] sho we d tha t spatio/temp oral b eamforming in the SNR → ∞ regime allo ws to achiev e for each user pair a “capacity” equal to half that of the single user c hannel. A differen t idea recen tly app eared in [11], whic h do es not require SNR → ∞ . Consider the follow ing c hannel pair: H a =   1 − 1 1 1 1 − 1 − 1 1 1   , H b =   1 1 − 1 − 1 1 1 1 − 1 1   (2) The c hannels are constructed in suc h a w a y that the simple sum of the receiv ed signals leads to in terference alignmen t since H a + H b = 2 I . In mor e general c hannels, suc h as those with R ayleigh fa ding, one needs to co de o ve r sufficien tly long time in terv als to observ e and matc h complemen tary matrix pairs. 1.3 No de p ositioning mo del W e b eliev e that our tec hniques should w ork in a v ariety of mo dels for the no de p ositions. W e outline one v ery natural scenario here. Definition 1.1 Consid er a fixe d sp atial r e gion D ∈ R D , with two pr ob ability dis tribu- tions P T and P R supp orte d on D . Given an inte ger K , we sa m ple the K tr ansmitter no de p ositions T 1 , . . . , T K indep endently fr om the distribution P T . Sim i l a rly, we sample the K r e c eiver no de R 1 , . . . , R K p ositions indep endently fr om distribution P R . We r efer to such a m o del of n o de plac ement as an ‘II D network’. Equiv alen tly , we could state that transmitter and receiv er p ositions are distributed ac- cording to t wo indep enden t (non-homogeneous) P o isson pro cesses , conditioned suc h that there ar e K p oints of each t yp e in D . W e pair the tra nsmitter and receiv er no des up so that T i wishes to comm unicate with R i for eac h i . W e mak e the following definition: Definition 1.2 We s a y that tr ansmitter and r e c eiver distributions P T and P R ar e ‘sp a- tial ly sep ar ate d ’ if ther e exist c onstants C sep < ∞ and D sep < ∞ such that fo r T ∼ P T and R ∼ P R the Euclide an distanc e d ( T , R ) satisfies P ( d ( T , R ) ≤ s ) ≤ C sep s D sep for al l s. (3) W e argue in Lemma 2.2 b elo w tha t a wide rang e o f no de distributions P T and P R ha v e this spatial separation prop ert y , whic h allow s us to con trol the t a ils of t he distribution of SNR, and hence the maxim um v alue of SNR in L emma 2.3 4 1.4 T ransmission mo d els F or simplicit y , w e first describ e our r esults in the con text of so-called ‘line of sight’ com- m unication mo dels, without multipath interfere nce. That is, we consider a mo del where signal strengths decay deterministically with Euclidean distance d a ccording to some monotonically decreasing con tin uous f unction f ( d ). W e make the followin g definition, whic h complemen ts the definition of spatial separation giv en in Definition 1.2. Definition 1.3 We s a y that the signal is ‘de c aying at r ate α ’ if ther e exists C dec < ∞ such that for al l d f ( d ) ≤ C dec d − α . (4) Standard ph ysical considerations imply that all signals m ust b e decay ing at some rat e α ≥ D , where D is the dimension of the underlying space. Tse and Visw anath [15, Section 2.1 ] discuss a v ariety of mo dels under whic h this condition holds fo r differen t exp o nen t s α . W e define the full action of the Gaussian in terference net w ork: Definition 1.4 Fix tr ans m itter no de p os i tion s { T 1 , . . . , T K } ∈ D and r e c eiver no de p o- sitions { R 1 , . . . , R K } ∈ D , a n d c ons ider Euclide an distanc e d and attenuation function f . F or e ach i and j , define INR ij = f ( d ( T i , R j )) . F or emphasis, for e ach i w e write SNR i for INR ii . We c on s ider the K user Gaussian interfer enc e network define d so that tr an s mitter i sends a message enc o de d as a se quenc e of c omple x numb ers X i = ( X i [1] , . . . , X i [ N ]) to r e c e i v er i , under a p ower c onstr aint 1 N P N n =1 | X i [ n ] | 2 ≤ 1 for e ach i . The n th symb ol r e c e i v e d at r e c eiver j is given as Y j [ n ] = K X i =1 exp( iφ ij [ n ]) p INR ij X i [ n ] + Z j [ n ] , (5) wher e Z j [ n ] ar e indep endent standar d c o m plex Gaussians , and φ ij [ n ] ar e indep enden t U [0 , 2 π ] r andom variables in d ep endent of al l o ther terms. The INR j i r emain fixe d, sinc e the no de p ositions themselves ar e fi xe d, but the phases a r e fast fading. W e write S ij for the random v ar iables 1 2 log(1 + 2INR ij ), whic h are functions of the distance b et w een T i and R j . In particular, since the no des are p ositioned independently in D efinition 1 .1, under this mo del the random v ariables S ii = 1 2 log(1 + 2SNR i ) a re indep enden t and iden tically distributed. In Section 4.2, w e explain ho w our tec hniques can b e extended to apply to more general mo dels, in t he presence o f random fading terms. 5 1.5 Main result: con v ergence in probabilit y of C Σ /K W e no w state the main theorem of this pap er, whic h prov es conv ergence in probabilit y of the a veraged capacit y , under the mo del of no de placemen t described in Definition 1.1 and the model for signal attenuation described in D efinition 1.4. F o r the sak e of clarit y , w e restrict our atten tion to the case where P R and P T are uniform, though w e discuss later to what exten t this assump t io n is neces sary . W e restrict to b ounded regions D with a smo oth b oundary – the sense of this smo othness will b e made precise in the pro of o f Theorem 1.5. Theorem 1.5 will certainly hold for squares and balls D = [0 , 1] D and D = { x : d ( x , 0 ) ≤ 1 } , and indeed for an y con v ex and b ounded p olytop es (with finite surface area). Essen t ia lly we require that the b oundary of D × D has Hausdorff dimension ≤ 2 D − 1 , whic h is v ery natural. Theorem 1.5 Consi d er a Gaussian interfer enc e ne twork forme d by K p airs of no des plac e d in an IID network, with the sig n al de c aying at some r ate α ≥ D . If dis tributions P T and P R ar e b oth uniform on a b ounde d r e gion D with smo oth b ound a ry, then the aver ag e p er-user c a p acity C Σ /K c onver ges in pr ob ability to 1 2 E log (1 + 2SNR) , that is lim K →∞ P      C Σ K − 1 2 E log (1 + 2SNR)     > ǫ  = 0 for al l ǫ > 0 . Pro of W e break the probabilit y in to t w o terms whic h w e deal with separately . That is, writing E = E S ii = 1 2 E log (1 + 2SNR), P      C Σ K − E     > ǫ  = P  C Σ K − E < − ǫ  + P  C Σ K − E > ǫ  . (6) Bounding the first term of (6 ) corresp onds to the ac hiev abilit y part o f the pro of. Bound- ing the second term of (6) corresp onds to the conv erse part, and represen ts our ma jor con tribution. The first term of (6) can b e b o unded relative ly simply , using an ac hiev abilit y argumen t based on an in terference alignmen t sc heme presen ted by Nazer, G astpar, Jafar and Vish w anath [1 1]. Theorem 3 o f [11] implies that the rates R [ i ] = 1 / 2 log(1 + 2 SNR i ) = S ii are achie v able. This implies that C Σ ≥ P K i =1 S ii . This allo ws us to b ound the first term in Equation (6) as P  C Σ K − E < − ǫ  ≤ P P K i =1 ( S ii − E ) K < − ǫ ! . (7) Note that Lemma 2.2 b elo w implies that the spatial separatio n condition holds in the setting of Theorem 1.5. Hence the conditions of Lemma 2.3 hold, implying that S ii has 6 finite v ariance by Equation ( 1 2). This means that Equation (7) can b e b ounded b y V ar ( S ii ) / ( K ǫ 2 ), and tends to zero at rate O (1 /K ). W e consider the prop erties of the second term of Equation ( 6 ) in the remainder of the pap er, completing the pr o of of t he theorem at t he end of Section 3. Theorem 1.5 can b e in terpreted in the same wa y as Theorem 1 of [9], that ‘eac h user is able to ac hiev e the same rate that he would ac hiev e if he had the c hannel to himself with no in terferers, half the time’. The theorem is presen ted under the assumptions of uniform P R and P T with deterministic fading for the sak e of simplicit y of exp osition – we b eliev e that the main result will b e robust to relaxation of these conditions. In Section 4.1 we consider whether the P R and P T need necessarily b e uniform. In Section 4.2 w e introduce a v ariant of the mo del with a random fading term. Although t he theorem is based on probabilistic arguments , in Section 4.3 w e describ e an a sso ciated a lgorithm whic h b ounds the sum capacit y of arbitrary Gaussian in terference net works. 1.6 Relation to pr evious w ork As review ed in more detail b y Ja f ar [9], recen tly progress has b een made in sev eral directions to wards understanding the capacit y of Gaussian in terference net w o r ks. In problems concerning net works with a la rge n umber of no des, w ork of Gupta and Kumar [6] uses tec hniques based on V o r onoi tesselations to establish scaling laws (see also Xue and Kumar [17] for a review of the inf o rmation theoretical tech niques that can b e applied to this problem). Under a similar mo del o f dense random netw ork placemen ts, though using the same p oints as b oth transmitters and receiv ers, ¨ Ozg ¨ ur, L´ ev ˆ eque and Tse [12, 13] use a hierarc hical sc heme, where no des are successiv ely assem bled in to groups of increasing size, eac h gro up collectiv ely acting as a MIMO transmitter or receiv er, and restricting to transmissions at a common rate. Theorems 3.1 and 3.2 o f [13] sho w tha t for an y ǫ > 0 there exists a constan t c ǫ and a fixed constant c 1 suc h that c ǫ K 1 − ǫ ≤ C Σ ≤ c 1 K lo g K . (8) These b ounds are close to stating that C Σ gro ws lik e K , but without the explicit constan t that Jaf ar [9] and Theorem 1 .5 of this pap er ac hiev e. In this pap er, w e pro duce a ve r sion of the upp er b ound of Equation (8) without the logarithmic factor and b eing explicit ab out the constan t c 1 , although note that this result is prov ed under a mo del that differs from that of [13] in the fact that w e hav e a total of 2 K no des rather than K . An alternativ e a pproac h to Gaussian interferenc e netw orks is to consider the limit of the capacit y a s the SNR tends to infinit y , with a fixed n umber o f users. Cadam b e and Jafar 7 [2] used in terference a lignmen t to deduce t he limiting b eha viour within o (log (SNR)). These tec hniques w ere extended b y the same authors [3] to more general mo dels in the presence of feedbac k and other effects. F or small net works , the classical b ounds due to Han and Kobay ashi [7] for the t w o - user Gaussian interference net work hav e recen tly b een extended and refined. F or example Etkin, Tse and W ang [5] hav e pro duced a characterization of capa cit y accurate to within one bit. These results w ere extended by Bresler, P arekh and Tse [1], using insigh ts based on a deterministic c hannel whic h approximates the Gaussian c hannel with suffi- cien t accuracy , to pro ve results for many-to-one a nd one-to-ma ny Gaussian in terference c hannels. W e briefly men tion an alt ernativ e pro of of Theorem 5 of [9], whic h pro vides a fa ster rate of conv ergence than that ac hieve d in Equation ( 1). W e first review some facts from graph theory , concerning random bipartite graphs formed b y tw o sets of v ertices of size N , with edges presen t indep enden tly with proba bility δ . Erd¨ os a nd R´ en yi [4] pro ve that the probabilit y of a complete matc hing failing to exist tends to 0 for a n y δ = δ ( K ) = ( log K + c K ) /K , where c K → ∞ . W e recall the argumen t where δ is fixed, so that w e can b e precise ab out the b ounds, rather than just working asymptotically . As in, for example, W alkup [16], w e sa y that a subset A S ⊂ A o f size k and a subset B S ⊂ B of size N − k + 1 for m a blo ck ing pair of size k if no edge of the graph connects A S to B S . Equation (1) of [16] uses K¨ onig’s theorem to deduce that P (no matc hing A to B ) ≤ N X k =1 X |A S | = k , |B S | = N − k +1 P (( A S , B S ) blo c king pair) (9) = 2 ( N +1) / 2 X k =1  N k  N k − 1  (1 − p ) k ( N − k +1) . By splitting the sum in to terms where k ≤ √ N and k ≥ √ N , so that (1 − p ) k ( N − k +1) is b ounded b y exp( − p ( N + 1) / 2) and exp( − pN 3 / 2 / 2) respectiv ely , a b ound of 2 √ N N 2 √ N exp( − p ( N + 1) / 2) + 2 2 N exp( − pN 3 / 2 / 2) can b e obtained. W e deduce t hat the probability of a complete matc hing failing to exist deca ys at an exp o nen t ia l rate in N . T o pro ve Theorem 5 of [9], w e divide the receiv er- t ransmitter links in to t wo groups A and B of size N = ⌊ K/ 2 ⌋ , and consider complete matc hings o n the bipart it e graph b et w een them. Eac h edge is presen t in the bipar t ite g raph if the corr esponding INR lies in a particular r ange (see Lemma 2.1 b elow for details), whic h o ccurs indep enden tly with probability δ . F or eac h pair tha t is successfully matc hed up, the corresp onding 8 t w o -user c hannel b ecomes an ǫ -b ottlenec k, and con tributes ≤ log( 1 + 2SNR) + ǫ t o the sum capacit y . Hence, the high probabilit y of a complete matc hing implies exp onen tial deca y of P ( | C Σ /K − 1 2 log(1 + 2SNR) | > ǫ ), impro ving the O ( K − 2 ) rate in Equation (1). 2 T ec hnical le mmas W e con tinue to work to wards our pro of of Theorem 1.5, by establishing some tec hnical results. First in Section 2.1, w e iden tify a condition under whic h the sum capacity of the t w o user in terference c hannel can b e b ounded. Next, in Section 2.2, w e sho w that the spatial separation condition of Definition 1.2 holds under a v ar iety of conditio ns. F urther w e show that spatial separation and t he decaying condition D efinition 1.3 together imply that no SNR can b e ‘to o large’. 2.1 Bounds on t w o user c hannel First, we iden tif y a condition on the v alues of SNR and INR under which the capacit y of the tw o user in terference c hannel can b e b ounded. The pro of of the follow ing result is based on Lemma 1 of [9 ], whic h ga v e the k ey definition of a ‘b ottlenec k state’, deducing that in the case SNR i = SNR j = INR j i = SNR, the sum capacity equals log (1 + 2 SNR). W e repro duce t he arg ument used there, to deduce a stability result that allow s us to deduce when an ǫ -b o t t leneck state o ccurs. Lemma 2.1 F or a n y i , j , c onsid e r the two user infe rf e r enc e channe l d efine d by Y i = ex p ( iφ ii ) p SNR i X i + exp( iφ j i ) p INR j i X j + Z i , Y j = ex p ( iφ ij ) p INR ij X i + exp( iφ j j ) p SNR j X j + Z j , wher e Z i and Z j ar e IID stand ar d c omplex Gaussian s, and φ ij ar e indep endent U [0 , 2 π ] r andom variables indep ende n t o f al l o ther terms. If INR j i ≥ SNR j then a ny r eliable tr ansmission r ates satisfy R [ i ] + R [ j ] ≤ log(1 + INR j i + SNR i ) . (10) Pro of W e ada pt the argument of Lemma 1 of [9]. That is, consider reliable transmission rates R [ i ], R [ j ]. Since the transmissions are reliable, then r eceiv er i can determine X i with an arbitrarily lo w probability of error. Again, reliable transmission rates w ould remain reliable if receiv er j w as presen ted with X i b y a genie. In that case, it is easier for receiv er i to determine X j than it is for 9 receiv er j to do so (since the w eighting INR j i is larg er than SNR j ). How ev er, we kno w that receiv er j can reco v er X j , since the rate R [ j ] is reliable, so w e deduce that receiv er i m ust b e able to do the same. Since receiv er i can determine X i and X j reliably , then these messages m ust ha v e b een transmitted at a sum rate lo we r than the sum capacity of a t wo-user m ultiple access c hannel, see for example [15, Equation (6.6)], whic h is log ( 1 + SNR i + INR j i ). 2.2 Deca y of tails Recall that the no de p o sitioning mo del giv en in Definition 1.1 inv olve s independen t p ositions of no des sampled f r om iden tical distributions P T and P R . W e giv e examples of conditions under whic h the spatial separation prop ert y of Definition 1.2 holds. Lemma 2.2 (i) If either P T or P R has a density with r esp e c t to L eb es g ue me asur e wh ich is b ounde d ab ove on D then P T and P R ar e sp a tial ly sep ar ate d. (ii) If P T and P R ar e supp orte d on sets T an d R that ar e physic al ly sep ar ate d, in that d ∗ = inf { d ( t, r ) : t ∈ T , r ∈ R} > 0 , then P T and P R ar e sp a tial ly sep ar ate d. Pro of (i) If P T has a density with resp ect to Leb esgue measure whic h is b ounded ab ov e b y C , then fo r any ball B s ( y ) of ra dius s cen tred on y , the probability P T ( B s ( y )) ≤ C V D s D , where V D is the v o lume of a Euclidean ball of unit radius in R D . Hence P ( d ( T , R ) ≤ s ) = Z P T ( B s ( y )) d P R ( y ) ≤ C V D s D Z d P R ( y ) = C V D s D , and the result follows , ta king C sep = C V D and D sep = D . The corresp onding result for P R follo ws on exc hanging P R and P T in the displa y ed equation ab ov e. (ii) If s < d ∗ , then P ( d ( T , R ) ≤ s ) = 0. If s ≥ d ∗ , then w e ha ve P ( d ( T , R ) ≤ s ) ≤ 1 ≤ s D /d D ∗ , so that w e can tak e C sep = 1 /d D ∗ and D sep = D . Next we sho w that com bining the spatial separation condition of Definition 1.2 and the deca ying condition Definition 1.3 giv es us go o d control of the maxim um of S ii . The argumen t is similar to that giv en in Theorem 3.1 of [13]. 10 Lemma 2.3 Consi d er an IID network with sp atial ly sep ar ate d P T and P R , with s i gnals de c aying at r ate α . The pr ob ability that the maximum of the K r andom variables S ii is lar ge tends to zer o: lim K →∞ P  max 1 ≤ i ≤ K S ii ≥ α log K D sep  = 0 , (11) wher e D sep is the sep ar ation exp onent fr om Definition 1.2. Pro of W e comb ine Definitions 1.2 a nd 1.3. Since all S ij ha v e the same marg inal distribution, it is enough to deduce that, for any u ≥ 1, P ( S ii ≥ u ) = P (1 / 2 log(1 + 2SNR i ) ≥ u ) = P (SNR i ≥ (exp(2 u ) − 1) / 2) ≤ P (SNR i ≥ exp(2 u ) / 3) = P ( f ( d ( T i , R i )) ≥ exp(2 u ) / 3) ≤ P ( C dec / ( d ( T i , R i )) α ≥ exp(2 u ) / 3) = P ( d ( T i , R i ) ≤ ( C dec / 3) 1 /α exp( − 2 u/α ) ) ≤ C sep ( C dec / 3) D sep /α exp( − 2 uD sep /α )) , (12) where C sep is the separation constan t f rom Equation (3) and C dec is the deca y constan t from Equation (4). The result follo ws from Equation (12) using the union bound: P  max 1 ≤ i ≤ K S ii ≥ α log K D sep  ≤ K P  S ii ≥ α log K D sep  ≤ K C sep ( C dec / 3) D sep /α K 2 , whic h tends to zero as K → ∞ . 3 Pro of of Th e orem 1.5 In this section we complete the pro of o f the upper b ound in Theorem 1.5. W e calculate b ounds on sum capacity using a strategy suggested by the w ork of Jafar [9], who prov es that the presence of a larg e num b er of disjoin t t w o- user c hannels, eac h close to b eing a ‘b ottlenec k state’, allow s go o d con tr o l of the ch a nnel capacit y . First in Section 3.1 w e partition space in to regions B u , v . Lemma 3.2 giv es a n upp er b ound on the sum capacit y o f a t wo-user channe l made up of p oin ts in neigh b ouring regions. F urt her, Lemma 3.3 tells us that w e can con trol the num b er o f links in eac h region. In Section 3.2 w e complete the ar g umen t by mat ching elemen ts of b ox B u , v with elemen t s of B u − e , v + e . This allo ws us to con trol the o ve ra ll sum capacity of the K users, and to complete the pro o f of Theorem 1.5. 11 3.1 Spatial partitioning mo del W e consider each transmitter–receiv er pair a s a p oin t in the joint domain D × D . Each transmitter–receiv er pair ( T i , R i ) ∈ D × D can be placed in well-define d disjoin t r egions B u , v , allo wing us to con trol the p erformance of the corresp onding link. W e write x ( l ) for the l th co ordinate of the v ector x = ( x (1) , . . . , x ( D ) ). Definition 3.1 Given M , we p artition the sp a c e R 2 D and henc e the joint tr ansmitter– r e c e i v er domain D × D by a r e gular grid of sp acing s 1 / M . F or e ach u ∈ Z D and v ∈ Z D , we define b oxes lab el le d by their ‘b ottom-left’ c o rner as B u , v =  ( x , y ) : u ( l ) M ≤ x ( l ) < ( u ( l ) + 1) M , v ( l ) M ≤ y ( l ) < ( v ( l ) + 1) M for al l l  . (13) We w ri te S = { ( u , v ) : B u , v T ( D × D ) 6 = ∅} for the set of p os s i b le lab els ( u , v ) , and split D × D into orthants, index e d by ve ctors E ( u , v ) ∈ { − 1 , 0 , 1 } D , w ith c o-or dinate E ( l ) ( u , v ) =    1 if v ( l ) − u ( l ) > 0 , 0 if v ( l ) − u ( l ) = 0 , − 1 if v ( l ) − u ( l ) < 0 . We intr o duc e two subsets of S that we w i l l no t attempt to match, ac c or din g to the rule that B u , v is matche d with B u − e , v + e , w her e e = E ( u , v ) . (i) First, the ‘spine’ S spine = { ( u , v ) : E ( l ) ( u , v ) = 0 f o r some l } . (ii) Se c ond, the ‘e dge’ S edge = { ( u , v ) : ( B u , v S B u − e , v + e ) 6⊆ D × D } . That is, the r e gions which overlap the b oundary of D × D , or which ar e matche d with a r e gion that overla ps the b oundary. The con tro l of p o sition obtained b y matc hing links in tw o neighbouring regions con v erts in to con trol of the v alues of SNR and INR, allo wing the sum capacit y of the tw o pa ir s to b e b ounded using Lemma 2.1. Figure 2 g ives a sc hematic diagram of a pair of b o xes whic h w e attempt to match using the construction in Lemma 3.2, plotting b oth transmitter and receiv er p ositions singly on D rather than jointly on D × D . Lemma 3.2 Supp ose the r e c eiv er-tr ansmitter p ai r ( R i , T i ) app e ars in r e gion B u , v and the r e c eiver- tr ansmitter p air ( R j , T j ) app e a rs in r e gio n B u − e , v + e , wher e ( u , v ) ∈ S \ S spine and e = E ( u , v ) , then any r eliable r ates for those two l i n ks satisfy R [ i ] + R [ j ] ≤ log (1 + 2 f ( d ( u , v ))) , wher e d ( u , v ) is the minimum tr ansmitter–r e c eiver di stanc e b etwe en ( R i , T i ) ∈ B u , v . 12 ❜ R j ❜ R i ❜ T i ❜ T j   ✒ u − e   ✒ u   ✒ v   ✒ v + e Figure 2: Sch ematic plot of matc hed b o xes, where b o xes are lab elled according to their b ottom-left corner. W e consider the case where D = [0 , 1] 2 , a nd plot t he receiv er and transmitter p ositions on the same square, with e = E ( u , v ) = ( − 1 , 1). The k ey prop ert y is that we can observ e that d ( T j , R j ) ≥ d ( T j , R i ) ≥ d ( T i , R i ). W e prov e this rigorously in order to pro v e Lemma 3.2. 13 Pro of The rates are o nly improv ed b y b eing presen ted with the messages of all the other users, reducing the situation to t hat of Lemma 2.1. F or each l such that co-ordinate e ( l ) = 1 , by construction (a) T ( l ) i − R ( l ) i > 0, ( b) R ( l ) j < u ( l ) ≤ R ( l ) i , (c) T ( l ) j ≥ v ( l ) > T ( l ) i . Hence, b y (b), ( T ( l ) j − R ( l ) i ) = ( T ( l ) j − R ( l ) j ) + ( R ( l ) j − R ( l ) i ) < ( T ( l ) j − R ( l ) j ) , and b y (c), ( T ( l ) j − R ( l ) i ) = ( T ( l ) i − R ( l ) i ) + ( T ( l ) j − T ( l ) i ) > ( T ( l ) i − R ( l ) i ) . Ov erall then, in the case e ( l ) = 1, 0 < ( T ( l ) i − R ( l ) i ) < ( T ( l ) j − R ( l ) i ) < ( T ( l ) j − R ( l ) j ) . A similar argumen t applies for eac h l with e ( l ) = − 1, with the order of the signs rev ersed. Ov erall, we deduce that d ( T j , R j ) ≥ d ( T j , R i ) ≥ d ( T i , R i ) ≥ d ( u , v ), or that f ( d ( u , v ) ) ≥ SNR i ≥ INR j i ≥ SNR j , so that Lemma 2.1 applies, allowing us to b ound R [ i ] + R [ j ] ≤ log(1 + INR j i + SNR i ) ≤ log (1 + 2 f ( d ( u , v ))) , as required. Eac h of the K transmitter–receiv er links are placed in the regions B u , v where ( u , v ) ∈ S . Our no de p ositioning rule implies that eac h link is placed in region B u , v with some probabilit y p u , v indep enden tly of the others. W e write N u , v for the total n umber of links placed in region B u , v , noting that the marginal distribution of each N u , v is Bin( K, p u , v ). Lemma 3.3 We c an b ound the pr ob ability that any of the r e g i o ns c ontain a s i g n ific antly differ en t numb er of links to that exp e cte d at r andom: P  max ( u , v ) ∈S | N u , v − K p u , v | ≥ K η  ≤ K 1 − 2 η . Pro of A standard a r g umen t using the union b o und and Cheb yshev g iv es P  max ( u , v ) ∈S | N u , v − K p u , v | ≥ K η  = P   [ ( u , v ) ∈S {| N u , v − K p u , v | ≥ K η }   ≤ X ( u , v ) ∈S P ( | N u , v − K p u , v | ≥ K η ) ≤ X ( u , v ) ∈S V ar ( N u , v ) K 2 η ≤ K 1 − 2 η , 14 since V a r ( N u , v ) = K p u , v (1 − p u , v ) ≤ K p u , v , so tha t P ( u , v ) ∈S V ar ( N u , v ) ≤ K . 3.2 Matc hing links W e no w complete the pro of of Theorem 1 .5 – recall t hat we consider uniform no de distributions P T and P R on a b o unded domain D with smo oth b oundary . Pro of of Theorem 1.5 The total sum capa city C Σ ≤ I M + J M , where I M is the con tribution from matched pairs of links and J M is the contribution from unmatc hed links. W e will consider M gro wing as a p ow er of K , but for now, it is enough to regard M as fixed. W e pair up the remaining edges in S \ ( S spine ∪ S edge ), w or king or t han t b y orthant. In particular, the matching b et w een u , v and u − e , v + e is one-to- one, and each region is coun ted at most once. F or eac h e ∈ { − 1 , 1 } D , w e can define the function Π e b y Π e ( w ) = w · e . The k ey observ ation is that f or each ( u , v ) / ∈ S spine , if e = E ( u , v ) then the inner pro duct 0 ≤ Π e ( v − u ) ≤ Π e (( v + e ) − ( u − e ))Π e ( u − v ) + 2 D , so that ( u − e , v + e ) / ∈ S spine . W e can sort the regions B u , v b y v alue o f Π e ( u − v ), a t eac h stage adding some ( u , v ) with the lo w est v alue of Π e ( u − v ) that has not ye t b een matc hed to the set S bo dy . W e depict this matc hing in Figure 3. S S S S S S B B B B B B B B B B ❅ ❅ ❘ ❅ ❅ ❘ ❅ ❅ ❘ ❅ ❅ ❘ ❅ ❅ ❘ ❅ ❅ ■ ❅ ❅ ■ ❅ ❅ ■ ❅ ❅ ■ ❅ ❅ ■ E E E E E E E E E E Figure 3: P a rtition of regions in to S spine , S edge and S bo dy . This illustrates the case where D = [0 , 1] and M = 6, and we plot the regions as squares on D × D = [0 , 1 ] 2 . W e lab el regions in S spine b y S , regions in S edge b y E and regions in S bo dy b y B , with an a r r o w in to the region they are matc hed with. 15 This means that w e can select a set of regions S bo dy suc h that for u , v ∈ S bo dy , the B u , v and B u − e , v + e b et w een them cov er S \ ( S spine ∪ S edge ), that is [ u , v ∈S b ody ( B u , v ∪ B u − e , v + e ) = S \ ( S spine ∪ S edge ) . F or D with v olume V and eac h ( u , v ) ∈ S bo dy the probabilities p u , v = p u − e , v + e = 1 / ( V 2 M 2 D ), since t he t wo b o xes do not intersec t the b oundary of D × D . By assumption there are a t least K / ( V 2 M 2 D ) − K η links in eac h of the regions B u , v and B u − e , v + e , since Lemma 3.3 shows that the proba bility that this do es not o ccur is ≤ K 1 − 2 η . W e choo se K/ ( V 2 M 2 D ) − K η links randomly from eac h pair of sets and matc h them, with Lemma 3.2 implying that each pa ir of matc hed links contributes at most log ( 1 + 2 f ( d ( u , v ))) to the sum capacit y . Ov erall, the total contribution to the sum capacit y from all the matc hed links satisfies I M ≤ X u , v ∈S b ody K V 2 M 2 D log(1 + 2 f ( d ( u , v ))) . (14) By the definition of R iemann in tegratio n, b y pick ing M sufficien tly large, the term I M /K ≤ 1 2 E log (1 + 2SNR) + ǫ/ 2, for ǫ arbit r a rily small. Next w e control J M , the con tributio n f r o m the unmatche d links. Sp ecifically , without loss of generality , if D is b ounded with v olume V , w e can assume D × D ⊆ [0 , L ] 2 D for some L . (i) W e do not attempt to match some regio ns b ecause they b elong in S edge or S spine . (a) W e assume that the b o undar y of D is sufficien tly smo oth that there exists a finite A (‘surface area’) suc h that |S edge | ≤ AM 2 D − 1 for a ll M . (F or example if D = [0 , 1] D then |S edge | ≤ 4 D M 2 D − 1 , since there are 2 D co-ordinates that can ta k e v alues 0 or M − 1, a nd then M 2 D − 1 v alues for the remaining co- ordinates). (b) The n um b er of regio ns in |S spine | ≤ D ( LM ) 2 D − 1 , since there are D co- ordinates whic h can agree, and at most LM p ossible v alues the remaining co-ordinates can tak e. Ov erall, there ar e at most ( A + D L 2 D − 1 ) M 2 D − 1 regions w e do not attempt to matc h. Eac h region w e do not att empt to matc h contains at most K/ ( V 2 M 2 D ) + K η links. (ii) W e attempt to perfo rm matc hing b et w een at most ( LM ) 2 D regions, with at most 2 K η unmatc hed links remaining from each. 16 In total w e deduce there are ( A + D L 2 D − 1 )( K / ( V 2 M ) + K η M 2 D − 1 ) + 2( LM ) 2 D K η un- matc hed links. W riting β = 1 / (3(2 D + 1)), and choo sing M = K 3 β (1 − η ) , and for example taking η = 2 / 3, there are O  K 1 − β  unmatc hed links. The single user capacity b ound (see for example [1 5, Equation (6 .4 )]) tells us that reliable rates satisfy: R [ i ] ≤ log(1 + SNR i ) ≤ log (1 + 2SNR i ) = 2 S ii ≤ 2 ma x i S ii , (15) so that J M = c 1 K 1 − β max 1 ≤ i ≤ K S ii for some c 1 . Hence ov erall, the probabilit y P ( J M /K ≥ ǫ/ 2) ≤ P  max 1 ≤ i ≤ K S ii ≥ ǫK β 2 c 1  , whic h tends to zero b y Equation (11). Note that the upp er b ound in Equation (8) obtained b y ¨ Ozg ¨ ur, L´ ev ˆ eque, and Tse [13] essen tially has the extra factor of log K since the b ound is only made up on the term J M . It is precisely the matc hing argumen t that giv es rise to the I M term whic h has reduced the order of the b ound, as w e tak e adv antage of the extra randomness prov ided b y placing transmitter and receiv er no des separately . Note that Equations (11) and (15) together giv e pro babilistic b ounds on ma x 1 ≤ i ≤ K R [ i ]. Sp ecifically , since they prov e that max 1 ≤ i ≤ K R [ i ] = O P (log K ), they con trol the exten t to whic h a larg e sum capa cit y can b e ac hiev ed by a small num b er of links that op erate at particularly high capacit y . This suggests that in this case the sum capacit y is not to o unfair a measure of netw ork p erformance. 4 F uture work and exten s ions W e briefly commen t on some extensions of Theorem 1 .5 to mor e general mo dels. 4.1 Non-uniform n o de d istributions W e w ould lik e to consider more general distributions P R and P T , rather than simply assuming that these distributions a re unifo r m. Ind eed, w e migh t wish to extend to a situation where P R,T ha v e a joint distribution from whic h w e sample indep enden tly to find receiv er and transmitter distributions. The main issue tha t arises is whether w e can quan tize the jo int distribution into regions B u , v whic h can b e uniquely paired off with other regions B u ′ , v ′ , with the paired regions 17 ha ving equal probability , and with b ounds on the sum of reliable rates b eing p ossible according to results suc h as Lemma 3 .2 . One case that certainly w orks is that where the co-ordinates of P R and P T are indep en- den t. In this case w e can extend Equation (13) to obtain B u , v =  ( x , y ) : q R ,l ( u/ M ) ≤ x ( l ) < q R ,l (( u + 1) / M ) , q T ,l ( v / M ) ≤ y ( l ) < q T ,l (( v + 1) / M ) , for all l  , where q R ,l ( x ) is the x th quantile of the distribution of the l th comp onen t o f the re- ceiv ers, and q T ,l ( x ) is the x th quantile of the distribution of the l th comp onen t of the transmitters. The pro of o f Theorem 1.5 follow s exactly as b efore in this case. 4.2 Random fading mo del W e briefly remark on an adaption of Definition 1.4 that w ould ha v e the same indep en- dence structure, while in tro ducing random fading into the mo del. That is, we could set INR ij = M ij f ( d ( t i , r j )) , where the M ij are I ID random v ariables with a densit y . Under our no de pla cemen t mo del, the SNR i will ag ain b e IID , making t his mo del t ractable in m uc h the same wa y . In pa r t icular w e can extend t he ta il b ehav iour b ounds giv en in Lemma 2.3 to this case, adjusting the constan t to tak e account of the random fading term. Lemma 4.1 Consi d er an IID network with sp atial ly sep ar ate d P T and P R , with sig- nals de c aying a t r ate α . If the fading r andom v a riables M ij have finite m e an then the pr ob ability that the maximum of the K r andom variables S ii is lar ge tends to zer o: lim K →∞ P  max 1 ≤ i ≤ K S ii ≥ max  2 α D sep , 1  log K  = 0 . Pro of An equiv alent of Equation (12) holds, since as b efore, for an y u ≥ 1, P ( S ii ≥ u ) ≤ P ( M ii f ( d ( T i , R i )) ≥ exp(2 u ) / 3) ≤ P  { M ii ≥ exp( u ) / √ 3 } ∪ { f ( d ( T i , R i )) ≥ exp( u ) / √ 3 }  ≤ P ( M ii ≥ exp( u ) / √ 3) + P ( C dec /d ( T i , R i ) α ≥ exp( u ) / √ 3) = √ 3 E M ii exp( − u ) + P ( d ( T i , R i ) ≤ ( C dec √ 3) 1 /α exp( − u/α )) ≤ √ 3 E M ii exp( − u ) + C sep ( C dec √ 3) D sep /α exp( − uD sep /α )) . 18 so the result fo llo ws exactly as befo re, using the union b ound. The key to proving conv ergence in probability of C Σ /K is to sho w that matc hing is p ossible b et w een elemen ts of B u , v and B u − e , v + e , as b efore. In the case of deterministic fading, any links ( R i , T i ) ∈ B u , v and ( R j , T j ) ∈ B u − e , v + e could b e matc hed, since the pro of of Lemma 3 .2 sho wed that in this case d ( T j , R j ) ≥ d ( T j , R i ) ≥ d ( T i , R i ) . (16) In the case of random fa ding, this is not enough to control the relev an t v alues of INR. Ho w eve r , Lemma 4.2 sho ws that w e can match a high prop ortion of links, by lo oking for ( i, j ) suc h that M j j ≤ M j i ≤ M ii , (17) whic h can b e com bined with Equation (16) to deduce that SNR j ≤ INR j i ≤ SNR i , so that again t he sum capacit y of the relev ant t w o user channe l ≤ log(1 + 2SNR i ). Note that it is enough for our purp oses to consider the case of uniform M ij with densities, since only the ordering b et wee n random v ar iables matters in Equation (17). W e g ive a tec hnical lemma that will imply the con trol that w e require. Lemma 4.2 Consi d er a bip a rtite gr aph with n vertic es in e ac h p art which we r efer to as A = ( A 1 , . . . , A n ) and B = ( B 1 , . . . , B n ) r esp e ctively. Al l the vertic es ar e lab el le d with indep endent U [0 , 1 ] r andom variables, with A i lab el le d by U i and B j lab el le d by V j . The bip artite gr aph has an e dge fr om A i to B j iff U i ≤ W ij ≤ V j , wher e W ij ar e U [0 , 1] , indep endent of ( U , V ) and e ach other. F or any γ ≥ 2 / 3 , ther e exists a matching of al l but O ( n γ ) vertic e s , with pr ob ability ≥ 1 − 5 exp( − 2 n 2 γ − 1 ) . Pro of W e thro w aw ay the 3 n γ v ertices with the biggest v alues of U i and the 3 n γ v ertices with the lo w est v alues of V j . This leav es new sets A and B each of size N = N ( n ) = n − 3 n γ . W e will show that there exists a match ing betw een A a nd B with high probabilit y , again using Equation (9) and con trolling the probability of blo ckin g pairs. W e condition on the ev ent  sup t     # { i : U i ≤ t } n − t     ≤ n γ − 1  [  sup t     # { i : V i ≤ t } n − t     ≤ n γ − 1  , (18) since Massart’s form o f the D v oretzky–Kiefer–W olf o witz theorem [10] tells us that this do es not ta ke place with probability ≤ 4 exp( − 2 n 2 γ − 1 ). Conditional o n the ev en t (18), for any k w e kno w there are more than n − k − 2 n γ v alues of U i whic h are less than 1 − k /n − n γ − 1 . Equiv alen tly , there are few er than k + 2 n γ 19 v alues of U i larger than 1 − k /n − n γ − 1 . Since w e thro w a wa y the la rgest 3 n γ v alues of A , an y subset of A S ⊂ A of size k has at least n γ v ertices with U i v alues less than 1 − k /n − n γ − 1 (‘small v ertices’). By a similar argumen t, any subset B S ⊂ B o f size N − k + 1 has at least n γ v ertices with V j v alues greater than 1 − k /n + n γ +1 (‘large ve rtices’). See Figure 4 for a depiction of these ev en ts. U V 1 − k /n 1 − k /n − n γ − 1 ✲ 1 − k /n + n γ − 1 ✛ x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x Figure 4 : P osition of v ertices in subsets A S and B S forced b y throw ing a w ay 3 n γ largest v alues of U and 3 n γ smallest v alues of V . There is an edge b et we en eac h of these small v ertices in A S and large ve rtices in B S indep enden tly with probabilit y at least 2 n γ − 1 . Hence, t he probabilit y that a particular A S and B S form a blo cking pair is less t ha n (1 − 2 n γ − 1 ) n 2 γ ≤ exp( − 2 n 3 γ − 1 ) . Substituting in Equation (9) the proba bilit y of no matc hing is ≤ N X k =1  N k  N N − k + 1  exp( − 2 n 3 γ − 1 ) =  2 N N + 1  exp( − 2 n 3 γ − 1 ) ≤ 2 n exp( − 2 n 3 γ − 1 ) , and the result fo llo ws since 3 γ − 1 ≥ 1, combining with the probability of (18) failing to o ccur. The remainder of the pro of of Theorem 1.5 carries o v er as b efore. W e need only alter Section 3.2, and this can b e done the increase in num b ers of unmatc hed vertice s remains sublinear in K . 20 4.3 Constructiv e algorithm Although Theorem 1.5 only giv es a result concerning a ve ra g e p erformance of la r g e net- w orks, it do es suggest some tec hniques that can b e used to approximate the sum capa city of any particular Gaussian in terference netw ork. If the net work is created via a spatial mo del, then w e can attempt to matc h cross-links into ǫ -b ottlenec k channels using the constrain ts on spatial p osition described in the pro of of Theorem 1.5, deducing b ounds as a result. Ho w eve r , eve n if w e are only presen ted with the v a lues of SNR i and INR ij , it may b e p ossible to find b ounds on sum capacit y using t he insigh ts given b y Lemma 2.1. Using the in terference alignmen t sc heme of [11], we kno w that a low er b ound on C Σ is giv en b y P i 1 2 log(1 + 2 SNR i ). One p ossible algorithm to find an upp er b ound w orks as follows: (i) Sort the indices b y v alue of SNR i , and for some M , partition the t ransmitter– receiv er links in to 2 M categories B r of appro ximately equal size K / (2 M ). (ii) F or each M , w e attempt to matc h links b etw een B 2 M − 1 and B 2 M , conside r ing t he bipartite graph b et w een t hem. (a) W e add an edge t o the bipartite graph b et w een j ∈ B 2 M − 1 and i ∈ B 2 M if SNR j ≤ INR j i ≤ SNR i . (b) W e lo ok f or a maximal matching o n the bipartite graph, using (for example) the Hop croft–Karp a lgorithm [8 ], whic h has complexit y √ V E , where V is the n um b er of v ertices and E the n um b er o f edges. (iii) By L emma 2.1 eac h edge ( j, i ) in each maximal matc hing con tributes log ( 1 + INR j i + SNR i ) as an upper b ound on the sum capacity , and eac h unmatc hed v ertex i simply con tributes the single user upp er b ound of lo g(1 + SNR i ) By v arying the size o f M , this algorithm will find a ra nge of upp er b ounds, of whic h w e can c ho ose the tightes t. W e w an t M lar g e enough that categories B r eac h con tain a narrow range of SNR v alues, but M small enough that there are plent y of p oints in eac h range B r to ensure a larg e maximal matching. 5 Conclus ions In this pap er, w e hav e deduced sharp b ounds for the sum capacit y of a Gaussian inter- ference net work. Our main contribution comes thro ug h the upp er b ound, whic h uses 21 argumen ts ba sed on controlling the p o sition o f pairs of v ertices to matc h them into b ot- tlenec k states. 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