Diversity and Degrees of Freedom of Cooperative Wireless Networks

Di v ersity and De grees of Freedom o f Cooperati v e W ireless Networks K. Sreeram Departmen t of E CE Indian I nstitute o f Scien ce Bangalore, India Email: sreeramk annan@ece.iisc.e rnet.in S. Birenjith Departmen t of E CE Indian I nstitute of Science Bangalore, Ind ia Email: b iren@ece.iisc.erne t.in P . V ijay Kumar Departmen t of ECE Indian I nstitute o f Science (on leave from USC) Email: vijayk@usc. edu Abstract — Wireless fading networks with multiple antenn as are t ypically studied infor mation-theoretically fro m two differ - ent perspectives - th e outage characterization and the erg odic capacity characterization. A key parameter in the out age char - acterization of a netwo rk is th e d iv ersity , whereas a first-order indicator for the ergodic capacity is th e degrees of freedom (DOF), which is the pre-log coefficient in the capacity expression. In this paper , w e present max-flo w min-cu t type theorems fo r computing both th e diversity and the degrees of freedom of arbitrary single-source single-sink multi-antenn a networks. W e also show that an amplify-and -for ward protocol is suffi cient to achiev e this. The degrees of freedom characterization is obtained using a con version to a determinist ic wireless n etwork f or which the capacity was recently f ound. W e show that the d iver sity result easily extends to multi-source multi-sin k networks and evaluate the DOF for mult i-casting in single-source mult i-sink networks. I . I N T RO D U C T I O N There has been a re cent interest in determin ing the degree s of f reedom (DOF) of wir eless m ulti-antenn a networks [3 ]. For definitions of diversity and d egrees of f reedom of point- to- point chann els, see [ 6] for example. The DOF fo r the N user interferen ce chan nel was rec ently derived in [4] an d the DOF of single-sourc e single-sink layered network s was obtaine d in [7]. W e characterize the diversity for arbitr ary networks and compute degrees of f reedom ( DOF) for single- source single- sink and multi-cast networks with multiple antennas. W e compute the degrees of freed om using a connection to de- terministic wir eless networks. The capacity o f sing le-source single-sink and multi-cast d eterministic wireless networks were characterized in [1]. Intuition drawn from the d eter- ministic wireless network s were used to identify capa city to within a co nstant for some example network s in [2]. A similar approa ch was used in [ 5] for o btaining DOF for real gaussian interferen ce networks. While the results for wire-line finite-field single-source single-sink network ha ve been known for some time [9], mu lti- cast capacity was found in th e mo re r ecent sem inal work [10]. An alg ebraic ap proach for fin ding the multi-cast c apacity was g iv en in [1 1]. These r esults were extend ed to finite field wireless networks in [1]. In [12] , computation codes were used to study the capacity o f finite field n etworks with interfer ence alone. While it is easy to extend wirelin e finite field n etwork T ABLE I N E T W O R K C O D I N G F O R F I N I T E F I E L D A N D G A U S S I A N N E T W O R K S W irelin e Networks W irele ss Networks Capaci ty of DOF of Capaci ty of DOF of Finite Field Gaussian Finite Field Gaussian Networ ks Networ ks Networ ks Networks Single Min-cut Min-cut Min-cut Min-cut Source [9] (easy rank [1] rank to see) (this paper) Multic ast Minimum Minimum Minimum Mini mum min-cut [10] min-cut min-cut min-cut rank (easy to see) rank [1] (this paper) results to gaussian wireline networks, the extension of wireless finite field network results to the g aussian case is not obvious. In this paper, we a pply these finite field network results to compute the DOF and diversity for gaussian wireless networks. T able I summ arizes th ese developments. The d iv ersity of a family of m ulti-hop networks was ev al- uated in [1 4]. In [13 ], the diversity for two-hop MIMO relay channel with a certain cond ition o n the n umber of antennas. Howe ver the maximum diversity of an arbitr ary network remains an o pen question, which we settle in this paper . A. Repr esentation of a Network W e re present a single-an tenna wireless network by a edge labelled direc ted gr aph G = ( V , E ) , where V is the set o f vertices and E is the set of ed ges. Eac h node is represented by a vertex, each edge represents a transmission link . Let N := |E | be the nu mber of links in the network. The label on every ed ge L ( e ) , e ∈ E repre sents the fading coefficient on that tran smission link. By c on vention, we pu t an edge only when the lin k ha s n on-zer o fading co efficient. In th e case o f m ultiple antenna n etworks, we first pass on to an equ iv alen t representation, where every terminal is represented by a super-node and every antenn a attached to the terminal is represen ted by a small node a ssociated with the super-node. Edges repr esenting single-antenn a connections are drawn only between small nodes a nd hence we can still label ed ges b y scalar fading coefficients. For the gaussian network, we assume that co efficients are elem ents of the complex field C . Sin ce we are dealin g with wireless n etworks, we assume that the bro adcast and interference constraints ho ld. W e a ssume throu ghou t the paper that CSIR is p resent. W e also assume fo r the degre es of fre edom part, that CSIT is pre sent. Definition 1 : A cut ω between sou rce S i and destination D j on a multiple-an tenna gau ssian network is defined as a partition of the super-nodes into U and U c such that the source S i is p resent in U and D j is p resent in U c . Let the set of all cuts between source S i and destination D j be d enoted b y Ω ij . Gi ven a cut ω , the matrix o f the c ut, H ω is defined as the tra nsfer matr ix associated with edges crossing the c ut fr om the source side to d estination side. In the single sou rce single sink case, w e will d rop th e unneed ed ij suf fix. Remark 1 : Any wire-line network can be con verted in to a wireless n etwork, by ad ding a sufficient number o f small- nodes at each super-node thereby separating the links so that in effect, the broa dcast a nd interferen ce constraints are nu llified. W e call this the n atural emb edding of a wire-line network in to a wireless network. B. De gr e es of F reedom Definition 2 : Consider a single-sink wir eless n etwork with each node having a symmetr ic transmit p ower co nstraint, ρ . Let the cap acity between the sour ce i an d the sink j be C ij ( ρ ) . The d e gr ees o f fr eedom o f the flow be tween source i and sink j is defined as D ij = lim ρ →∞ C ij ( ρ ) log ρ (1) Remark 2 : The capacity of the flow F ij between can source i and sink j can then be given by: C ij = D ij log( ρ ) + o (log ρ ) (2) Whenever we con sider a single-source s ingle-sink network, we will with out loss of amb iguity , dro p the su ffix from C ij , D ij and simply use C, D instead. Definition 3 : A multi-ca st network is defined as a network with single-source an d mu ltiple-sinks, with the con straint that all the flows need th e same in formatio n from the so urce. Lemma 1 .1: Consider a chann el of the for m Y = HX + W , where H is a N × N ch annel matrix X , Y , W are N length colu mn vecto rs representing the transmitted signal, rec eiv ed signal and the no ise vector distributed as C N (0 , Σ) , where Σ is a full rank corr elation matrix. Th e degrees of freed om of this channel is g iv en by D = rank ( H ) . Remark 3 : W e d efine the DOF o f a matrix H as the DOF of th e channe l Y = H X + W where W is C N (0 , I ) . C. Diversity The Lemma b elow computes the di versity of a channel matrix having a specific structur e. Lemma 1 .2: Consider a chann el of the for m Y = HX + W , where H is a N × N random ch annel matrix, X, Y , W are N length column vecto rs r epresenting the transmitted signal, received sign al and the no ise vector . Let the n oise vecto r W can be representab le as W = Z + Z 0 + P L i =1 G i Z i , where Z i are C N (0 , I ) indepen dent vectors and every entry in the matrices G i are polyno mial f unctions of gaussian r andom variables. If the N 2 entries o f ma trix H co ntain exactly M indepen dent Rayleigh fading coefficients, the n M is the diversity of that matrix. D. Linear Deterministic W ireles s Network In defin ing deterministic 1 wireless network s, we follow [1]. The se networks are similar to multiple-anten na gaussian networks with th e o nly difference being that these n etworks are noise-free and that the complex fading coef ficients are replaced b y finite fields eleme nts. In place of complex vectors, each nod e tr ansmits an q -tuple over the finite field . Cuts are defined as in the gaussian network c ase. In p lace of H ω , we use G ω to de fine the transfe r matrix between nodes on either side of the cu t ω . W e state th e following Theor em from [1]: Theor em 1 .3: [ 1] Gi ven a linear determin istic single-sou rce single-sink wirele ss network over any fin ite field F , ∀ ǫ > 0 , the ǫ - error c apacity C of su ch a r elay n etwork is given by , C = min ω ∈ Ω rank( G ω ) . (3) where the capacity is specified in terms o f the nu mber o f finite field sym bols per unit time. A strategy utilizin g o nly linear transform ations over F at the relays is sufficient to ac hieve this capacity . Remark 4 : The strategy specified in [1] utilizes m atrix transform ations at eac h relay of the in put vecto r rec eiv ed over a perio d of T time slots. Thu s the achievability shows the existence o f relay m atrices A i at each relay node i ∈ V , each of size q T × q T , that spe cifies the transform ation betwee n the received vector of size q T to the v ector of size q T that needs to be transmitted . I t can be seen using the natur al em beddin g of a wir e-line n etwork into a wireless n etwork, th at this theor em is in deed a generaliza tion of th e max-flow min-cut th eorem. The m ulticast-version of Theore m1.3 appea rs below . Theor em 1 .4: [ 1] Gi ven a linear determin istic single-sou rce D - sink multi-cast wireless network, ∀ ǫ > 0 , the ǫ -err or capacity C of such a network is given b y , C = min j =1 , 2 ,..,D min ω ∈ Ω j rank( G ω ) . (4) where Ω is the set of all cuts betwe en th e sou rce and destination j . A strategy utilizing on ly linea r tran sformation s at the re lays is su fficient to achieve this cap acity . I I . M I N - C U T E Q UA L S M A X D I V E R S I T Y W e begin with a r esult applicable to gaussian n etworks. Definition 4 : W e d efine the value M ω of a cut ω as the number of edges crossing over from the source side to the sink side a cross the cut. W e refer to the value o f the m in-cut as simp ly th e min-cut. 1 By determi nistic netw ork, we will alwa ys mean li near deterministic netw ork. Theor em 2 .1: Consider a multi-term inal fadin g n etwork with nod es ha ving multiple antenna s with each edge having iid Rayleigh-fading coe fficients. T he ma ximum d i versity ach iev- able fo r any flow is eq ual to the min-cut b etween the source and the sin k cor respond ing to the flow . E ach flow can achieve its maximu m div ersity simu ltaneously . Pr oo f: W e will distinguish between two cases. Case I : Network with single a ntenna nodes Choose a sour ce S i and sink D j . Let C ij denote the set of all cuts between S i and D j . From the cu tset b ound [15] o n DM T [16], d (0) ≤ min ω ∈ Ω ij { d ω (0) } = min ω ∈ Ω ij { M ω } ⇒ d (0) ≤ M It is now sufficient to p rove that diversity order of M is achiev able. Let us first consider the case wh en th ere is only one flow . By the Ford-Fulkerson theorem [9], the number of edges in the min-cut is equal to the max imum number of edge disjoint paths between so urce and the d estination. Schedule the network in suc h a way that each ed ge in a g iv en edge disjoint p ath is activ ate d on e by one. Repeat for all the edge disjoint paths. Thus the same data symbol is tran smitted throug h all the edge disjoint path s from S i to D j . Let the number of edges in the i - th edg e disjoin t path be n i . The j th edg e in the the i th edg e disjoint path is denoted by e ij and the associated fading coefficient be h ij . The a ctiv atio n schedule can b e repre sented a s follows: Ac ti vate each of the fo llowing edge individually in successive time instants: e 11 , e 12 , · · · , e 1 n 1 , e 21 , · · · , e 2 n 2 , · · · , e M 1 , e M 2 , · · · , e M n M . Now define h i := Q n i j =1 h ij to be the product fading coefficient on the i -th p ath. Le t the total n umber o f tim e slo ts required be N = Σ M i =1 n i . W ith this protoco l in place, the equi valent chan nel s een from the source to the d estination has chann el matrix H =      h 1 0 . . . 0 0 h 2 . . . 0 . . . . . . . . . . . . 0 0 . . . h M      This matrix is exactly of the struc ture in Lem ma 1.2 except that the re are M prod uct Rayleigh coefficients. Ho wev er , it can be shown that the diversity remain s u nchang ed. Th e noise matrix also obeys the condition s of Lemma 1 .2. Th us the maximum diversity of M can be ac hieved. When there are mu ltiple flo ws in the network , we simply schedule the data o f all th e flows in a time-division ma nner . This will en tail further rate lo ss - howev er , sin ce we ar e interested only in th e diversity , we can still ach iev e each flow’ s maximum diversity simultan eously . Case I I: Network with multiples antenna node s In the multiple anten na case, we regard any link between a n t transmit and n r receive antenna as bein g composed of n t n r links, with one link between each transmit and each receive antenna. Note that it i s possible to selectiv ely acti vate precisely one of the n t n r Tx-anten na-Rx-an tenna pairs b y ap propr iately transmitting f rom just one an tenna an d listenin g at ju st one Rx antenna. The sam e strategy as in the single anten na case ca n then be ap plied to ach iev e th is d iv ersity in the network. I I I . D E G R E E S O F F R E E D O M O F S I N G L E S O U R C E S I N G L E S I N K N E T W O R K S In this section, we pr esent a max -flow min-cut type theo rem for ev aluating the degrees o f freed om in single- source sing le- sink n etworks: Theor em 3 .1: Gi ven a single- source sing le-sink gaussian wireless n etwork, with indep endent fading coefficients h aving an arbitrary density function , the DOF of th e network is given by D = min ω ∈ Ω rank( H ω ) with probab ility one. (5) An amplify- and-fo rward strategy utilizing only linear trans- formation s at the r elays is sufficient to ach iev e this DOF . Remark 5 : If we ass ume that the channel coefficients have a time variation, then we can sho w that the de grees of freedom is identically eq ual to the value shown in the th eorem, by codin g across the time variation. Pr oo f: Th e pr oof pr oceeds as follows: 1) First, a conv erse f or the DOF is pr ovided using simple cutset boun ds. 2) Then, we convert the gaussian network into a determin- istic network with the prop erty that the cutset boun d on DOF for the g aussian network is th e same as th e cutset bound on the c apacity of the d eterministic network. 3) W e then characterize the zero-error-capacity of the linear deterministic wirele ss ne twork. 4) Finally , we conv ert th e achiev ability r esult for the deter- ministic network into an ach iev ability result for gaussian network, which match es the converse. A. The Convers e W e first provide a simple co n verse on the degrees of freedom of a single so urce single sink network. Lemma 3 .2: Gi ven a single-sou rce sing le-sink network, the DOF is u pper bo unded b y the D OF o f every cut: D ≤ min ω ∈ Ω rank( H ω ) where H ω is th e matrix correspo nding to th e cut ω . B. Con version to Linear Deterministic Network In this subsectio n, we co n vert the wireless gau ssian network to a e quiv alent line ar deterministic network . 2 W e use the term ”equiv alent” to signify that the DOF of the gau ssian network and the c apacity of the deter ministic network are the same. In 2 It m ust be noted that the con version to determini stic netw ork used here is dif ferent from that used in [2] and [5]. order to ge t the eq uiv alent d eterministic network, we pro ceed as f ollows: Let the the fading coefficients on the N edges in the gaussian n etwork be h 1 , h 2 , ..., h N . W e first con sider a finite field network with the same graph as the o riginal gau ssian network. W e take q , th e vector length in the deterministic network to be eq ual to the m aximum n umber of an tennas of any nod e in the gaussian network. For nodes with antennas less than q , we leave th e remain ing nodes un- connected . Howev er, we still need to decide the finite field size, p , and a finite field coefficient on each ed ge. G i ven a finite field size, we need N ma ps, ψ i , i = 1 , 2 , ..., N th at co n vert the gaussian fading coefficients into finite field coefficients. Let ξ i := ψ i ( h i ) denote th is m apping. In order to obtain these co efficients and th e finite field size, we require further conditions. In particu lar , we will requ ire the finite field network to h av e at least th e same capac ity as the upper boun d on the gaussian network. W e recognize th e similarity b etween the capacity equation in Theor em 1.3 an d DOF terms in Lemma 3.2 an d r equire th at cut b y cu t, the r ank of th e tran sfer matrix on deter ministic n etwork b e no less tha n the rank on th e gaussian network. Before assignin g values to ξ i , we will treat th em as forma l variables. Consider a cut ω in the gau ssian network. W e want the rank ( G ω ) ≥ rank ( H ω ) for this cut. T o do this, let r ω := rank ( H ω ) be the DOF of the cu t in the gaussian network. Then there exists a r ω × r ω sub-matrix of the H ω which has non-zero determinan t. Let us call this matrix as H ′ ω . Consider the same cut on the deterministic network and find the same r ω × r ω sub-matrix G ′ ω correspo nding to the transfer matrix G ω . Now consider the determina nt of th e m atrix G ′ ω . T he determinant is a poly nomial in se veral variables ξ i , i = 1 , 2 , .., N with rational integer coefficients. Let us call th is polyno mial as f ω ( ξ 1 , ξ 2 , .., ξ N ) . This p olynom ial is not iden tically zero as a po lynomial over Q , since in that case even the substitution ξ i = h i will lead t o a zero value, mak ing the determina nt zer o even for th e gau ssian case, which is clear ly a contrad iction. T herefor e we have that f ω is a non -zero polyn omial. W e also have an ob servation that the degree of f ω in each of the variable ξ i is at-most one. W e want a field F p and an a ssignment to ξ i that ma kes the f ω non-ze ro over the ch osen field. For any gi ven cut, this can be easily done. Ho wever we want to do this simultaneously for all cu ts. T o d o so, we will employ the fo llowing lem ma, proven easily u sing e lementary algebra: Lemma 3 .3: Gi ven a polyno mial f ( ξ 1 , ξ 2 , ..., ξ N ) with in- teger coefficients, which is n ot identically z ero, ther e exists a prime field F p with p large en ough , su ch th at the poly nomial ev alu ates to a non -zero value at least for one assignment o f field values to th e formal variables. Now consider the polyno mial f ( ξ 1 , ξ 2 , .., ξ N ) := Y ω ∈ Ω f ω ( ξ 1 , ξ 2 , .., ξ N ) (6) Now , the po lynom ial f is non-zero since it is a product of non-ze ro po lynom ials f ω and the degree of f in any of the variables is at-m ost | Ω | . W e want a field F p and an assignment for ξ i from th e field su ch th at f is non zero. Using Lem ma 3.3, we have that such an assignment exists. L et us choose that p and the assignment that makes f no n-zero. T hus we h av e a deterministic wireless network who se capacity is g uarantee d to be greater tha n or equal to γ of the converse. C. Zer o Err or Capacity of Deterministic Networks W e establish the zero error capacity of deterministic wireless networks. W e have the fo llowing definition Definition 5 : [8 ] The zero error capacity is defined as th e supremum of all achievable ra tes such tha t the pr obability of error is exactly zero. Theor em 3 .4: The z ero er ror capacity of a single sourc e single sink deterministic wire less n etwork is equ al to C Z E = min ω ∈ Ω rank( G ω ) This cap acity can b e ach iev ed using a linear code and linear transform ations in all relays. Pr oo f: W e will prove this th eorem using the ǫ er ror capacity result from Theorem 1.3. W e will assum e the field F appearing in the theorem to be the finite field F p of size p where p is the prim e previously identified. From th e achiev ability result in the proo f of Theo rem 1 .3, we have that giv en any ǫ > 0 an d r ate r < C , ther e exists a blo ck-leng th T , linear transform ations A j , j = 1 , 2 , ..., M of size q T × q T u sed by all relays an d a code b ook C for the so urce, such that the probab ility of er ror is lesser th an or equal to ǫ . E ach cod ew ord X i ∈ C is a q T × 1 vector that specifies the en tire tr ansmission from th e source. Le t X 1 , ..., X |C | be the codewords. Let us assume that the sink listens for a duratio n T ′ ≥ T in general to acco unt for th e presence of path s of un equal lengths in the network between sou rce and sink, (for large T , we would have T ′ T → 1 , so this d oes no t affect rate calcula tions). The transfer equation between the sou rce and the destination vectors are specified by: Y = GX since all tran sformation s in the n etwork ar e ind eed linear . Here G is a q T ′ × q T matrix, X is a T length transmitted vector, and Y is a T ′ length vector . Now , gi ven that a vector X i is tr ansmitted, either the decoder always makes an error or n ev er makes erro r because the channel is a determ inistic map Y i = GX i . Let P i e be the p robability of erro r co nditioned on transmitting the i -th codeword. Then P i e ∈ { 0 , 1 } and the average codeword err or probab ility P e = 1 |C | |C | X i =1 P i e ≤ ǫ ⇒ |C | X i =1 P i e ≤ ǫ |C | This means that at least (1 − ǫ ) |C | cod ew ords have z ero p rob- ability of error . Ther efore if we choo se only these (1 − ǫ ) |C | codewords as an expu rgated cod e-book C ′ , then the code-bo ok has zero pr obability o f er ror under the same relay matrices and decod ing r ule. The rate of the codeb ook is however ¯ r = r − log(1 − ǫ ) T . Let δ = log(1 − ǫ ) T be the rate loss and therefor e, the expurgated code -book has negligible rate loss as T b ecomes large. No w , we hav e established a zero erro r codebo ok of rate r − δ . By choo sing r arbitrarily close to C and T large, we g et that in deed C Z E = C . Howe ver, the c ode C ′ like the code C used in [1], is a non - linear code. W e obtain a linear c ode by utilizing the following technique : Since there is a zero erro r co de for rate ¯ r , it mea ns that the tra nsfer m atrix G ha s rank at least ¯ r T an d th erefore that ther e is a sub -matrix G ′ of size ¯ rT × ¯ r T , which is full rank. I f we commu nicate on ly o n th ese ¯ r T dimension s we can obtain the transfer matrix G ′ . Th us we get a linear zero error code of rate ¯ r . D. Achievable DOF in Ga ussian Networks In this sub-section, we will lift the zero-erro r-capacity achiev ability result fro m dete rministic networks to determin e an ach iev able DOF for gaussian networks. In th e achievability for cap acity of determin istic n etworks, the relays perform ed m atrix operation s A i on receiv ed vectors for T time duratio ns. Since e ach receiv ed vector is of size q , the matr ix A i is of size q T × q T . No w we use the same strategy for the gaussian network, i.e., all relays use the same matrices A i that they used in th e deter ministic n etwork. T his makes sense, since in a prime finite field F p , all field elements are in tegers modulo p . There fore the m atrices A i can also be interpreted as ma trices over C . This strategy yields a e ffecti ve channel m atrix H , i.e ., Y = H X + W . It is su fficient to pr ove that H has rank ( H ) ≥ ¯ rT since DOF is equal to rank ( H ) . T o d o so, we first establish that there exists an assignmen t of h i such th at rank ( H ) ≥ ¯ r T . Let us consider the same ¯ r T × ¯ r T sub-matrix H ′ by deleting rows and colum ns in the same way that G ′ was o btained from G . W e have that det( H ′ ) is a multi-variate po lynomial in h i , i = 1 , 2 , ..., N , if we treat h i as formal variables. Now this poly nomial has integer co efficients and therefore can be treated as a polynom ial over a ny finite field, in particular over the finite field F p . Over F p , we kn ow that this polyno mial is a no n-zero polyno mial, since the assign ment of h i = ξ i giv es a n on-zer o value. It follows th at this polynomial is nonzero, ev en wh en viewed as a polyn omial over the integers. Since C is algeb raically closed, we hav e that any non- zero polyno mial must h av e a assign ment of variables in C that gives non -zero value to the po lynomial. Using this assignment for h i giv es us that det ( H ′ ) 6 = 0 and thereb y H h as rank ( H ) ≥ ¯ r T . W e h av e the following lemm a: Lemma 3 .5: Consider a multi-variate polyno mial f in sev- eral variables h i , i = 1 , 2 , .., N . Let h i be in depend ent random variables in C g enerated accordin g to any pr obability density function . If th e polynom ial has a no n-zero assignment, then the polyn omial is n on-zer o with probab ility one. Now using the Lemma above along with the fact that we have an a ssignment fo r h i such that r ank ( H ) ≥ ¯ r T with probab ility o ne. T herefo re, for chann els with fr equency (tim e) selecti vity an d co ding over multiple f requen cy (time) slots, we have that the achievable d egrees of freedom is eq ual to γ − δ . Since DOF is defined as the supremum over all achiev a ble DOF values, we have that DOF = γ or D OF = min { ω ∈ Ω } rank( H ω ) E. Multi-casting over Gaussian Networks W e state the following Th eorem witho ut p roof: Theor em 3 .6: Gi ven a single-source D - sink multi-cast gaussian wireless network, with indepen dent fading co effi- cients ha ving a n arbitrary d ensity fu nction, the DOF of the network is given by D = min { j =1 , 2 ,..., D } min ω ∈ Ω j rank( H ω ) with probab ility one (7) An am plify-an d-for ward stra tegy utilizing o nly linear transfor- mations at the r elays is su fficient to achieve this DOF . I V . C O N C L U S I O N This pap er presen ted two max- flow min-cut typ e theo rems for computing d iv ersity a nd DOF of multi-antenn a wireless gaussian n etworks. I n addition, a conn ection was established between DOF of gaussian networks an d capacity of deter- ministic networks [1] f or the sin gle-source single-sink and the multi-cast case. Along the way , we proved that the zero error capacity of deterministic networks is the same as th e ǫ -error cap acity . 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