On Outer Bounds to the Capacity Region of Wireless Networks

Transcription

1 On Outer Bounds to the Capacity Region of Wireless Networks Sahand Haji Ali Ahmad Aleksandar Jovičić and Pramod Viswanath August Abstract We study the capacity region of a general wireless network by deriving fundamental upper bounds on a class of linear functionals of the rate tuples at which joint reliable communication can take place. The widely studied transport capacity is a specific linear functional: the coefficient of the rate between a pair of nodes is equal to the Euclidean distance between them. The upper bound on the linear functionals of the capacity region is used to derive upper bounds to scaling laws for generalized transport capacity: the coefficient of the rate between a pair of nodes is equal to some arbitrary function of the Euclidean distance between them for a class of imum distance networks. This upper bound to the scaling law meets that achievable by multihop communication over these networks for a wide class of channel conditions; this shows the optimality in the scaling-law sense of multihop communication when studying generalized transport capacity of wireless networks. Introduction A characterization of the capacity region of wireless networks is one of the long standing open problems in information theory. Progress in this direction has recently come by This research was supported in part by the National Science Foundation under grant CCR-0343 and a grant from Motorola Inc. as part of the Motorola Center for Communication. S. Haji Ali Ahmad A. Jovičić and P. Viswanath are with the department of Electrical and Computer Engineering at the University of Illinois at Urbana-Champaign. {alihajijovicicpramodv}@uiuc.edu

2 relaxing the questions asked: focus has been on the transport capacity [3] a specific linear functional of rate tuples at which jointly reliable communication is possible. The coefficient of the rate between a pair of nodes in this linear functional is equal to the Euclidean distance between the two nodes. Further much emphasis is on the scaling law of the transport capacity as a function of the number of nodes in the network. Recent results have shown that multihop communication is scaling-law optimal for a class of imum distance networks provided the wireless channel decays fast enough [4 7 8]. A study of the sum capacity of the wireless network appears in [5]. In an effort to understand the entire capacity region of the wireless network we shift the focus from transport or sum capacity to studying arbitrary linear functionals of the rate tuples in the capacity region. The capacity region is a compact convex set. Thus understanding the extremum of linear functionals of the rate tuples in the capacity region is an alternative but equivalent characterization (the convex dual in the language of convex analysis; see [6]) of the capacity region. In this paper we derive simple and robust upper bounds to a class of linear functionals: the upper bound simply depends on the distance between the nodes channel attenuation conditions and the coefficients of the linear functional. The key technique is a combination of the results of isometric embeddability of an arbitrary metric space into the l metric space and the familiar information-theoretic cut-set upper bounds to the capacity region. We use this simple upper bound to exactly characterize the scaling law behavior over arbitrary linear and planar networks of generalized transport capacity def C f = max R ij f(r ij ) () (R ij ) in the capacity region where and R ij and r ij denote respectively the rate of reliable communication and the Euclidean distance between a pair of nodes (ij). When f( ) is the identity function i.e. f(x) = x we get back the transport capacity. The scaling law of generalized transport capacity depends crucially on the long range behavior of the function f( ); of particular interest is the parameter α f def = lim x ( log f(x) log x ij= ). () To be able to state the scaling law precisely we briefly describe our wireless channel model adopted from [4]: the received signal at node j at time m is y j [m] = i j h s ij[m] ( + r ij ) δ x i[m] + z j [m]. (3) Here x i [m] is the signal transmitted by node i at time m and z j [m] is i.i.d. white Gaussian noise. Each node j has an average transmit power constraint (denoted by P j ).

3 The fading channel { h s ij[m] } is a stationary and ergodic zero-mean stochastic process m that is independent for each pair of nodes (ij) and models the small scale fluctuations of frequency flat fading. For simplicity we assume that E[ h s ij[m] ] = for all ijm. The large scale variations are modeled explicitly through the decay of signal level: a factor of from node i to node j. In free space δ = with a single reflected path along (+r ij ) δ with the line of sight it becomes equal to two and it can be larger than.5 in crowded urban environments. As in [4] we consider two (disparate) models of the wireless fading channel. If the channel variations are slow enough so that coherent communication is possible we model this scenario by allowing full knowledge at each node of the fading channels from all the other nodes. This is also known as the full-csi scenario (channel state information at both the transmitter and receiver nodes). In the scenario where the channel is changing rapidly enough so that coherent communication is not feasible our model is to assume complete ignorance of the channel realizations by all the nodes. This is the no CSI model. Clearly these are two extreme models but they serve to cover the ranges of channel fluctuations and allow us to make simple statements about the network communication problem. Our main results are the following:. The generalized transport capacity of imum distance networks with n nodes on a line is upper bounded by K n for δ > + 0.5α f as long as α f. With no CSI the same bound holds when δ > α f and α f.. The generalized transport capacity of imum distance networks with n nodes on a plane is upper bounded by K n for δ > + 0.5α f as long as α f. With no CSI the same bound holds with δ > + 0.5α f and α f. With multihop communication the generalized transport capacity is lower bounded by K 0 n [8] so the upper bounds to the scaling law we derive are actually tight. This shows the optimality in the sense of the scaling-law of the generalized transport capacity of operating the wireless network in the multihop communication mode. These results also strictly strengthen the scaling law characterization of the transport capacity in [4] and [7] when f(r) = r: the bounds therein require at least δ >.5 for linear networks and δ >.5 for planar networks with full CSI for linear scaling to hold. Further our result is robust to power fluctuations in the network: in particular the result continues The far field signal decay is usually denoted by r δ ij. Here we have written ( + r ij) δ to ensure that our model makes sense when nodes get close; i.e. the average received power is not more than the average transmit power. 3

4 to hold even when only the total power in the network is allowed to grow linearly with the number of nodes. Finally when the function f( ) is super-linear if α f > δ then the generalized transport capacity of imum distance linear networks grows super-linearly with the number of nodes. We have organized the results into two parts. First we derive an upper bound to a class of linear functionals (where the coefficients of the linear functional constitute a metric space) of rate tuples in the capacity region of an arbitrary wireless network; this is done in Section. Second we use this upper bound to derive the optimal scaling law of the generalized transport capacity of imum distance wireless networks; this is done in Section 3. Upper Bounds to Linear Functionals over the Capacity Region We are interested in the extremum of linear functionals over the capacity region: C {aij } def = max (R ij ) capacity region a ij R ij a ij 0 (4) ij= where there are n nodes in the network N def = {...n}. Our interest stems from the fact that the collection of the extremum values C {aij } for all a ij 0 is a precise representation (the convex dual) of the capacity region. In this section we derive upper bounds to C {aij } for a class of linear functionals represented by {a ij }. There are two key ideas that we use:. Given any disjoint partition (or cut) of the network (i.e. C + C N such that C + C = and N = C + C ) we can bound the sum-rate at which the nodes in C + can reliably communicate to the nodes in C by the well-known cut-set bound (Theorem 4.0. in []).. We weight the different cuts appropriately: we want the probability that a pair of nodes (ij) gets cut to be proportional to a ij. Then we can average the bound on the sum-rate over all the possible cuts to arrive at an upper bound to the linear functional of interest. 4

5 The first step can be achieved for any cut of the network. However the second step can be achieved only for a certain specific set of coefficients {a ij }. In particular the set of coefficients needs to satisfy a geometric property: there has to be some dimension m such that n points can be positioned in R m with the property that the set of l distances between every pair of points exactly coincides with the set of coefficients {a ij }. Mathematically this requirement is stated as the following: the set of {a ij } is isometrically embeddable in l. This is an important result from the theory of metric spaces and we state this as a lemma. Lemma. (Prop. 4.. of []). The set of distances {a ij ij N } is isometrically embeddable in l if and only if there exist non-negative numbers { P{C + C } C + N and C = N C +} such that a ij = C + N: i C + j C P {C + C }. One implication of this result is that any nonnegative linear combination of isometrically l -embeddable coefficent sets {a ij }s is also isometrically l -embeddable. Apart from this general observation the embeddability condition imposes some obvious conditions on the coefficients {a ij }; in particular they must be the distances of some n-point metric space. In other words. a ii = 0; this is however not a serious limitation since we do not intend to allow communication from a node to itself and there is no loss in generality in supposing the coefficient of R ii to be zero.. a ij = a ji ; so we are weighing the rate from node i to j equally with the rate from node j to i. 3. a ij + a ik a jk ; the triangle inequality. For n 4 nodes the above three conditions are sufficient as well for the set {a ij } to be isometically embeddable in l. For large n the problem of detering when a set of distances is isometrically embeddable in l is computationally hard: it is NP-complete ([] p. 49). For values of n 7 closed form necessary and sufficient conditions are available in [] pp Our interest however is to use this result only as an intermediary step in deriving an upper bound to the linear functional C {aij }; thus the computational issues are hardly of any concern here. The l distance between a pair of points in R m is the sum of the absolute value of the distances between the pair of points in each of the m dimensions. 5

6 Now we are ready to state and prove the main theorem of this section: an upper bound to C {aij }. Suppose each node i in the network N = {...n} is power limited to P i. Theorem.. Consider an arbitrary configuration N of n nodes in the plane. Assume that the set {a ij } is isometrically l -embeddable. With no channel state information (CSI) at the nodes a ij C {aij } < P i. (5) ( + r ij ) δ With full CSI at the nodes C {aij } < ik= j= ij= { } (a ij a kj ) Pi P k + ( + r ij ) δ ( + r kj ) δ ij= P i a ij. (6) ( + r ij ) δ Remark: Since the second term in (6) is exactly the no-csi bound in (5) the significance of the first term in (6) is that it is a bound on the communication gain of having CSI at all the nodes. The proof of Theorem. is relegated to Appendix A.. Examples of Linear Functionals.. Rate Coefficients as Functions of Distance If we substitute a ij = r ij the Euclidean distance (l distance) between the pair of nodes (ij) we see that expression (6) is a strengthening of expression (6) in Theorem. of [4]: the improvement in the bound comes from using a single fixed distribution of network inputs for all possible cuts of the network (see Appendix A). The bound given in (5) remains the same as expression (5) in Theorem. of [4]. For this special choice of coefficients a ij = r ij we can then use a random line as in [4] to geographically partition the network so that the probability that any pair of nodes is cut by the line is proportional to the Euclidean distance between the nodes. Since a set of distances that are embeddable in the l metric space are isometrically embeddable in the l metric space (Prop []) we can cast the utility of the random line construction in [4] as a special case of the Lemma. above. In particular the probability that a pair 6

7 of nodes (ij) is cut using the random line is (cf. (0) in [4]) P [ i C + j C ] = r ij πd n (7) where d n is the radius of a circle that contains all the n points in the network. To arrive at the statement of Theorem. we choose the probabilities of the cuts explicitly so that (7) is replaced by the implication from Lemma.: P [ i C + j C ] = a ij A (8) for some constant A chosen such that the quantity on the left side of the equation above summed over all possible cuts sums to unity. The proof of expression (6) in Theorem. uses these observations and is given in Appendix A. An important class of coefficients {a ij } that is isometrically l -embeddable is the following ([] p. 4): a ij = r α ij 0 α. (9) In particular α = 0 corresponds to a ij = for each pair of nodes (ij) and thus the corresponding linear functional is simply the sum capacity of the network. Further for any density function g( ) a ij = 0 r α ij g(α) dα is also isometrically embeddable; this is a special case of the earlier observation that nonnegative linear combinations of isometrically l embeddable coefficients also are isometrically l embeddable. We use this result to study the scaling law behavior of generalized transport capacity in Section 3... Extension to an Arbitrary Metric Space Up until now we have required that the finite metric space whose distances are given by {a ij } be isometrically embeddable into an l space i.e. that the distances be exactly preserved. However if we allow embeddings that introduce some relative distortion in the distances we can extend our upper bounds to any linear functional as long as its coefficients are distances in some metric space. Lemma.3 (Bourgain 985 see [] Theorem 0..). Suppose {a ij } is the distance between x i and x j in some n point metric space N. There exists an embedding f : N R m such that f(x i ) f(x j ) a ij c 0 log n f(x i ) f(x j ) 7

8 for all xy N where denotes the l norm and c 0 < 9 is a constant. This theorem allows us to state the following corollary to Theorem.: Corollary.. Consider an arbitrary configuration N of n nodes in the plane. Assume that the set {a ij } represents distances in some metric space. With no channel state information (CSI) at the nodes With full CSI at the nodes C {aij } < c 0 log n ik= C {aij } < c 0 log n j= ij= P i a ij ( + r ij ) δ. { } (a ij a kj ) Pi P k + c ( + r ij ) δ ( + r kj ) δ 0 log n ij= P i a ij ( + r ij ) δ. The proof of this corollary follows the proof of Theorem. (Appendix A) along with the following observation: by Lemma.3 ij R ija ij c 0 log n ij R ijã ij where ã ij := f(x i ) f(x j ). Now Lemma. guarantees that there exists a distribution on the cut-sets (C + C ) for which P [i C + j C ] is proportional to ã ij. Averaging over this distribution and using the left-hand inequality in Lemma.3 we obtain the desired result...3 Metric Spaces Defined on Planar Graphs Lemma. states that it is possible to embed any metric space into l with distortion at most O(log n). However if the metric space in question is defined on a certain type of planar graph there exist l embeddings with distortion that is independent of n. In particular a graph metric 3 defined on any treewidth- planar graph can be embedded into l with distortion of at most 4 (see Theorem 4. of [9]). Hence for such metric spaces we can strengthen the statement of Corrollary. by replacing the log n factor with the constant 4. The long-standing conjecture is that every metric defined on a planar graph can be embedded into l with constant distortion. Moreover the set of all planar graphs whose metrics are isometrically l embeddable (i.e. embeddable with unit distortion) is exactly the set of planar graphs that do not 3 The distance between any two nodes is the smallest sum of (non-negative) weights on any path between them. 8

9 contain K 3 as a or (see Proposition 3. of [9]). For such metric spaces we can directly apply Theorem.. weight= weight= Figure : It can be shown (see [] Example 6.3.) that the metric space generated by this complete bipartite graph (also known as K 3 ) is not isometrically embeddable in l. 3 Scaling Laws for Generalized Transport Capacity Our upper bound to the extremum of linear functionals of the capacity region was for arbitrary network topologies. Here we focus on a specific network topology imum distance networks to study the scaling law behavior of generalized transport capacity. We consider both linear and planar networks with imum distance between the nodes denoted by r > 0. Suppose the average power constraint of each radio is uniformly upper bounded by P. Our main result is the following: the generalized transport capacity scales linearly with the number of nodes n provided:. the function f(r) grows sublinearly with the Euclidean distance r i.e. α f (cf. ()) and. the channel decays fast enough relative to the growth of the function f(r). Theorem 3.. Suppose the nodes lie on a line at a distance of at least r > 0 from each other.. With no CSI at the nodes the generalized transport capacity is upper bounded by for all δ > +α f and α f. C f < Pζ (δ α f) n r δ α f 9

10 . With full CSI at the nodes the generalized transport capacity is upper bounded by C f < P r δ α f as long as δ > + α f 4 and α f. {G(α f δ) + ζ (δ α f )} n Here we have defined the constant def G(α f δ) = ( {ζ (δ α f ) + K (α f δ) ζ δ α f ) [ ( + δ α f 4 ζ δ α ) ( ( f + K (α f δ) c (α f δ) ζ δ α ) ( f ζ δ α f ))]} 4 4 where ζ (δ) def = i= i δ is the Riemann-Zeta function (a finite number for δ > ) ( c (α f δ) def ) ( = 3 α f δ+ 3 4 α f δ+ ) and K (α f δ) def = (δ 3α f ). Theorem 3.. For planar networks with a distance of at least r > 0 between any two nodes the scaling law is similar:. With no CSI at the nodes the generalized transport capacity is upper bounded by for δ > + α f and α f. C f < (π + )Pζ (δ α f ) n r δ α f. With full CSI at the nodes the generalized transport capacity is upper bounded by C f < P r δ α f for δ > + α f 4 and α f. Here we have defined the constant H(α f δ) + δ α f 4 def [ ζ {H(α f δ) + (π + )ζ (δ α f )} n { ( ζ δ α f 3 ) = (π + ) 3 ( δ α ) f 4 + K (α f δ) c (α f δ)ζ (δ α f ) + K (α f δ) ζ (δ αf ) ]} where ζ(δ) is as defined in Theorem 3. K (α f δ) def = ( δ 3α f ) and c (α f δ) def = ( ) ( ) 3 α. f δ+ 3 4 α f δ+ 0

11 The first step in the proofs of the above two theorems an upper bound to the linear functional of interest has been generalized for a class of coefficients {a ij } in Theorem.. Since the choice of coefficients a ij = r α ij for α is isometrically l -embeddable (see (9)) the bound in Theorem. can be used to evaluate an upper bound to the generalized transport capacity for functions of the form f(r) = r α for α. The proof of the full-csi cases is given in Appendix B and C. The no-csi case is treated very similarly to Corollaries 3. and 3. in [4] except that the conditions on the signal decay parameter δ now depend α f. Remark : For the special case of a ij = r ij the result of Theorem. is a strict improvement over the previous results on the scaling behavior of transport capacity [4 7]: the bounds therein require at least δ >.5 for linear networks and δ >.5 for planar networks with full CSI for linear scaling to hold. Remark : When a ij = for all ij we establish the optimality of multihop with respect to sum-capacity of the network for δ > in a linear and δ > in a planar network. Sum capacity bounds for a particular traffic pattern (across a single cut of the network) have been studied in [5]. Our results hold regardless of the assumptions on the traffic pattern. Remark 3: Exaing the origin of the constant factors in the bounds of Theorems 3. and 3. we observe that it is the communication between near-by nodes that deteres the requirement on the path-loss parameter for linear scaling to hold. This vicinity is quantified in the proofs. So far we have only considered functions f( ) that are sub-linear in n. What happens when the function is super-linear in n? It turns out that in this case the generalized transport capacity can grow faster than the number of nodes in the network. The communication strategy that achieves this is coherent multi-stage relaying and interference cancellation described in Theorem 4.3 of [7]. The following statement based directly on Theorem 3.6 of [7] makes this precise: Proposition 3.. The generalized transport capacity for linear imum-distance networks can grow super-linearly with the number of nodes n according to Ω(n α f/δ ) as long as α f > δ. This scaling law is also optimal when < δ < and α f > δ. A Proof of the full-csi case in Theorem. The cut-set bound (Theorem 4.0. of []) applied to a network N with the channel model given in (3) and with full CSI at all the nodes can be stated in the following way:

12 There exists a n n covariance matrix Q (with (Q) ii P i for all i N) such that the sum-rate at which the nodes in C + can reliably communicate to the nodes in C satisfies i C + j C R ij < log det(i + H C Q C H C ) for every disjoint partition C = (C + C ) of N where the C + C + matrix Q C is obtained by retaining only the elements lying in the rows and columns of Q that are indexed by i C + (i.e. it is a principal submatrix) and where the C C + matrix H C is obtained by retaining only the rows and columns of H that are indexed by j C and i C + respectively. The entries (H) ij are independent zero mean random variables with variance (+r ij. Note that the principal submatrix Q ) δ C is always positive semi-definite. We can then further upper-bound the sum-rate by using the Hadamard bound in the log-det expression followed by the inequality log( + x) x: log det(i + H C Q C H C ) ) ( + h j (C)Q C h j (C) j C log = log + h ji (Q) ik h jk j C ik C + (Q) ik h ji h jk {i C + k C + j C }. ik= Here we have used h j (C) to denote the j-th row of the submatrix H C and { } to denote the indicator function. Thus we obtain an upper-bound on C {aij }: [ [ ]] C {aij } < A E H max E C (Q) ik h ji h jk {i C + k C + j C } Q 0: (Q) ii P i ik= j= [ ] = A E H max (Q) ik h ji h (a ji a jk ) jk (0) Q 0: (Q) ii P i A ik= j= [ ] E H Pi P k h ji h jk (a ji a jk ) () ik= j= [ = Pi P k E H h ji h ] jk (a ji a jk ) + P i E H h ji a ji () j= j= ik= i k { Pi P k ik= i k j= j= } (a ji a jk ) + ( + r ji ) δ ( + r jk ) δ i= ij= P i a ji (3) ( + r ji ) δ

13 where we have used E H to denote expectation with respect to the channel statistics and E C to denote the expectation with respect to the distribution of the cuts. Observe that (0) follows from (8) and () follows from the fact that (Q) ii (Q) jj (Q) ij since every principal or of a positive semidefinite matrix is non-negative. Equality () is obtained by sumg over i k and i = k separately. Inequality (3) is obtained by applying Jensen s inequality to the concave function x x / and making use of the independence of the {h s ij} as well as their zero-mean property. This concludes the proof of (6) in Theorem.. B Proof of the full-csi case in Theorem 3. Let P = max i N P i. We focus on the case where a ij = r α ij for α. Then we can express the first term in (6) as CSI gain C{a ij } P { ik= j= Pn k= + (r α jir α jk ) ( + r ji ) δ ( + r jk ) δ j: r ji >r jk r jk >r ki k= } ( + r ji ) δ ( + r jk ) δ α j: r ji >r jk r jk <r ki ( + r ji ) δ ( + r jk ) δ α (4) where i = i := n yields the largest inner sum (over k and j) in (). 4. Since the summations over j are symmetric around the mid-point between node k and node i there is a factor of multiplying both of the inner sums (over j) above. In additon to splitting the above summations we have also used the inequality (a + b) a + b. Henceforth we will use B and B to denote the first and second terms respectively in expression (4). We first focus on the term B above and observe that r ji > r jk and r jk > r ki that r ji > r ki so imply ( + r ji ) δ ( + r jk ) α ( + rki ) δ α (5) 4 Since we are deriving upper bounds we can always add another node to make n odd 3

14 as long as δ > α. Substituting this calculation and using the fact that δα 0 we get 4 that B < (6) < k= k= j: r ji >r jk r jk >r ki ( + r ki ) δ α 4 ( + r ki ) δ α ( + r jk ) δ α+ α j: r jk >r ki ( + r jk ) δ α+ α Now let l ki := r ki /r and l jk := r jk /r. Clearly for a given i l ki for all k N and for a given k l jk for all j N. The largest number of nodes k which have a common value l ki is simply since no two nodes can be closer than r and they all lie on a line. Thus B < l ki = ( + r l ki ) δ α 4 l jk =l ki ( + r l jk ) δ α+ α Note that the inner summation (over l jk ) in the above expression is simply a lower Riemann sum. Thus we can further upper bound it as follows: l jk =l ki ( + r l jk ) δ 3 α < = < r δ 3 4 α r δ 3 4 α r δ 3 4 α { dx + l δ 3 α ki l ki { K (αδ) + l δ 3 α ki l δ 3 α ki x δ 3 α } { } K (αδ) + l δ 3 4 α ki l δ 3 4 α ki } where we have used the notation K (αδ) := (δ 3 α ). Substituting this calculation in (8) we obtain the bound B < ( {ζ (δ α) + K r δ α (αδ) ζ δ α )} where ζ(δ) def = i= i δ is the Riemann-Zeta function and is finite for δ >. Thus the above bound on B is valid for δ > 3 + α. 4 (7) (8) 4

15 Now turning to term B in (4) we see that r ji > r jk and r jk < r ki imply that so inequality (5) becomes r ji > r ki and the bound on B is given by B < < ( + r ji ) δ ( + r jk ) α ( k= l ki = + r ki ) δ α 4 +δ α 4 ( + r l ki ) δ α 4 ( + rki j: r jk <r ki l ki l jk = ) δ α (9) ( + r jk ) δ α+ α ( + r l jk ) δ 3 α Again upper bounding the lower Riemann sum over l jk by its integral we get l ki l jk = ( + r r jk ) δ 3 α r δ 3 4 α { lki + dx x δ 3 α }. (0) () = r δ 3 4 α r δ 3 4 α { + l 3 α δ+ ki 3 + } α δ + ( l 3 α δ+ ) ki ( δ 3 α ) () where the last line holds when δ > + 3α. Now we would like to find a number c 4 (αδ) ) ( ) independent of l such that ( l 3 α δ+ c (αδ) l 3 4 α δ+ for all l =... Observe that for l = the inequality is satisfied with c (αδ) =. Also the function ) ( ( ) l 3 α δ+ l 3 4 α δ+ is monotonically decreasing with l. Thus we can choose c (αδ) def = B < 3 +δ α 4 r δ α ) ( ( ) 3 α δ α δ+ Substituting this calculation in () we hence upper-bound term B in (0) as follows: { ( ζ δ α ) + K (αδ) c (αδ) 4 [ ( ζ δ α ) ( ζ δ α )]} 4 We note that the above bound on B is valid only for δ > + α. Observe that if δ > + α 4 4 then δ > + 3 α thus () is well defined in this regime. 4 5

16 Combining the two bounds on B and B we establish the scaling of the first term in expression (6). The second term exactly corresponds to the no-csi bound of (5) whose scaling law is directly obtained from [4] Corollary 3.. Hence we obtain the scaling law of C f when f(r) = r α α for linear imum distance networks. To extend this to generalized transport capacity we make the following observation: For any general f( ) with α f < we have f(r) r α f+ɛ for all r sufficiently large for every ɛ > 0. Choosing ɛ small enough such that α f + ɛ < we can upper bound the generalized transport capacity C f of a imum distance network by C r α f +ɛ multiplied by a constant that does not depend on the number of nodes n. Since α f + ɛ we can now use the scaling law result for polynomial functions of the Euclidean distance derived above to upper bound the scaling law behavior of C f : the scaling law behavior of C f is the same as that when f(r) = r α f+ɛ. Since ɛ > 0 can be chosen arbitrarily small this completes the proof of Theorem 3.. C Proof of the full-csi case in Theorem 3. As in the proof of Theorem 3. we first focus on the case where a ij = rij α for α but we consider arbitrary imum distance networks on the plane. We focus on bounding the first term in (6) and follow exactly the same steps up until inequality (8). Here we observe that (see [4] Corollary 3.) since we are dealing with networks on a plane there are at most (π + )l jk nodes j that share a common value of l jk for fixed k and similarly there are at most (π + )l ki nodes k that share a common value of l ki for fixed i. 5 Thus expression (8) now becomes B < r δ α l ki = (π + )l ki ( + r l ki ) δ α 4 l jk =l ki (π + )l jk ( + r l jk ) δ 3 α Following the same steps as in the proof of Theorem 3. we obtain 3 { ( (π + ) B < ζ δ α 3 ) + K (αδ) ζ (δ α ) }. (3) where K (αδ) := ( δ 3α ). The above bound on B is valid for δ > 3 + α. We can make similar observations for bounding the term B to obtain: B < δ α 4 + (π + ) 3 { ζ (δ α ) } r δ α 4 + K (αδ) c (αδ)ζ (δ α ) 5 Similarly to the linear case i is here set to be the center node. We can always add more nodes to the configuration to make any node we choose the center node. 6

17 where c (αδ) def ) = ( ( ) 3 α δ+ 3 4 α δ+ and the bound is valid for δ > + α. 4 Combining the two bounds on B and B we establish the scaling of the first term in expression (6). The second term exactly corresponds to the no-csi bound of (5) whose scaling law is directly obtained from [4] Corollary 3.. Hence we obtain the scaling law of C f when f(r) = r α α for planar imum distance networks. To extend this to generalized transport capacity we make the same observation as in the proof of Theorem 3.. Hence we establish Theorem 3. D Acknowledgements The authors thank Devavrat Shah for an interesting discussion; in particular this led to the development in Section... References [] T. M. Cover and J. Thomas Elements of Information Theory John Wiley 99 [] M. M. Deza and M. Laurent Geometry of Cuts and Metrics Encyclopedia of Mathematics and its applications Reading MA: Addison Wesley 976 Volume. [3] P. Gupta and P. R. Kumar Capacity of wireless networks IEEE Trans. on Information Theory Vol. 46() pp Mar 000. [4] A. Jovičić P. Viswanath and S. R. Kulkarni Upper Bounds to Transport Capacity of Wireless Networks IEEE Trans. on Information Theory Vol. 50 No. pp November 004. [5] O. Lévêque and E. Telatar Information theoretic upper bounds on the capacity of large extended ad hoc wireless networks IEEE Transactions on Information Theory Vol. 5 No. 3 pp March 005. [6] R. T. Rockafellar Convex Analysis Princeton University Press 97. [7] L.-L Xie and P. R. Kumar A Network Information Theory for Wireless Communication: Scaling Laws and Optimal Operation IEEE Transactions on Information Theory Vo. 50(5) pp May

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