1. March 18, 2005 Network Coding in Interference Networks Brian Smith and Sriram Vishwanath University of Texas at Austin March 18 th, 2005 Conference on Information Sciences and Systems

2. March 18, 2005 Overview  Introduction  Interference vs. Non-Interference Networks  Network Coding in Non-Interference Networks  Gaussian Interference Networks  Classic Network Coding Example on a Gaussian Network  Network Coding on “Bow-Tie” Network  Network Coding with Node Cooperation  Summary

3. March 18, 2005 Introduction  Show by series of examples  Network-coding in interference network can act same as network coding in equivalent non-interference network  Can provide a benefit in interference network when it does not in the equivalent non-interference network  Node-cooperation and network-coding used together can additionally increase throughput

4. March 18, 2005 Non-Interference vs. Interference Networks  Arbitrary network configuration with conditional probability distributions across links Source Node #2 Terminal Node #1 Terminal Node #2 Terminal Node #3 Source Node #1

5. March 18, 2005 Non-Interference vs. Interference Networks X1X1 X2X2 X3X3 Y X Y1Y1 Y2Y2 Y3Y3 Interference NetworksNon-Interference Networks Sender is constrained to send the same signal to each receiver Sender not constrained to send the same signal to each receiver X=(X 1,X 2,X 3 ) Signals from different senders arriving at the same receiver interfere with each other Signals from different senders arriving at the same receiver are received independently Y=(Y 1,Y 2,Y 3 ) Multiple Access Model: p(y|x 1,x 2,x 3 ) Multiple Access Model: p(y|x 1,x 2,x 3 )=p(y 1 |x 1 )p(y 2 |x 2 )p(y 3 |x 3 ) Broadcast Model: p(y i |x) Broadcast Model: p(y i |x)=p(y i |x i ) We use additive Gaussian noise channels in examples In most network coding work, channels assumed perfect

6. March 18, 2005  Ahlswede, Cai, Li, Yeung, 2000  Model  Links transmit single symbol with no errors  Nodes can perform linear operations on received symbols  Result  Multicasting from single source to multiple receivers can be performed at min-cut max-flow rate [ACLY 2000]  Example  Ubiquitous routing vs. network coding example Classic Network Coding

7. March 18, 2005 The Network Coding Example S B D1D1 D2D2 A+B A BA BA S B D1D1 D2D2 A BA BA B BB Routing Network Coding 1.5 bits to each receiver2 bits to each receiver

8. March 18, 2005  Use additive complex white Gaussian noise channels in the illustrative examples  Each node has transmit power constraint  Received signal: Y=  X i +   ~ N (0,N)  Point-to-point channel  Capacity: R=lg(1+P/N)  Multiple access channel with independent messages  Sum Rate Capacity: R 1 +R 2 ≤lg(1 +P 1 /N+ P 2 /N) Gaussian Interference Networks X1X1 XY P1P1 X2X2 Y P2P2 P

9. March 18, 2005  Broadcast Channel with independent messages  Sum Rate Capacity:  R 1 +R 2 ≤ lg (1+P/N)  Multicasting over Broadcast Channel  Same message can be sent to both receivers at point-to-point rate  Broadcast channel with common message  First example: Classic network-coding network configuration with Gaussian channels  Use P=3/2 and N=1 at all nodes  To compare with bit-pipe model Gaussian Interference Networks X P Y1Y1 Y2Y2

10. March 18, 2005 Gaussian Network Coding Example B D1D1 D2D2 A X Y C=lg(1+3/2+3/2)=2 C=lg(1+3/2) C1C1 C2C2 P P P P

11. March 18, 2005  Strategies for the classic configuration in interference model  Ignore center nodes X,Y completely  Receive lg(5/2) bits/transmission each  Routing strategy  Constraints if multicasting at broadcast nodes  MACs: 2 bits/transmission  Direct links: lg(5/2) bits/transmission  R A =R B =R XY =R T =1 is viable  Route so that X-Y and Y-D links carry A’s information  1 bit to D 1, 2 bits to D 2, 1.5 bits on average  Can show with linear programming that no better pure routing strategy exists Interference Network Example: Routing B D1D1 D2D2 A X Y P P P P RARA RARA RBRB RBRB RTRT RTRT R XY

12. March 18, 2005 Interference Network Example: Network Coding  Instead, use the same strategy as in the non-interference network  Set all links to rate 1 bit/transmission  Send A B on links X-Y, Y-D 1, and Y-D 2  Each terminal receives 2 bits/transmission  Rate of 2 bits/transmission is maximum with independent codebooks  cut C 1 across multiple access channel  Exceeding the independent codebooks rate is the subject of the node-cooperation section  Essentially, taking the same actions as the non-interference network B D1D1 D2D2 A X Y P P P P C1C1

13. March 18, 2005 Aside: Multicasting  In the previous example, all broadcast nodes operated by multicasting  The same amount of information crosses cut C 1 in either mode  But, greater amount crosses each of C 2 and C 3 in the multicasting mode C1C1 C3C3 C2C2 R1R1 R2R2 P C1C1 C3C3 C2C2 R1R1 R2R2 PP (1-  )P R 1 =R 2 =lg(1+P) Multicasting R 1 +R 2 =lg(1+P) Broadcasting

14. March 18, 2005 Example: Multicasting Outperforms Broadcasting  In the following example, multicasting is superior to broadcasting  Three independent sources  Achievable rate across cut C 1 when center node multicasts:  Final term removes rate of common message  When center node broadcasts R1R1 R2R2 PP’ S1S1 S0S0 S2S2 C1C1

15. March 18, 2005 Effect of Multicasting  Intuition: Broadcast requirement of the interference network acts like a “T” configuration in the non-interference network  Adding a “bottleneck” link of equal capacity onto the node in the non- interference case constrains the two downlinks to send the same signal  Otherwise, not operating at full capacity  Analogy useful for next example P Multicasting in Interference Network “T” Configuration in Non- Interference Network “Bottleneck” link

16. March 18, 2005 Bow-Tie Network  Non-interference network: routing performs optimally  Interference Network Strategies  Ignore center node X  Allows lg(1+3/2) bits/transmission to each terminal node  Routing – Center node X alternately chooses A or B to forward  Rate of 1.5 bits/transmission to each terminal node  Network Coding  Node X multicasts A B to both destination nodes  Rate of 2 bits/transmission to each terminal node is optimal  Network Coding is useful in some interference networks when it provides no benefit in the non-interference network with the same configuration A B X D1D1 D2D2

17. March 18, 2005 Node Cooperation  Network coding is one method of inducing correlations on a network  Other methods:  Using correlated codebooks at separate nodes  Block-Markov coding for relay channel  For non-interference networks, independent coding shown optimal  Demonstrate interference network counter-example  Example: Network coding and node-cooperation in tandem increasing performance P0P0 U U,V V PP D1D1 D2D2

18. March 18, 2005 Node Cooperation  Nodes T 1 and T 2 have access to data sources U and V, respectively, while node T 0 has access to both U and V  Nodes T 1 and T 2 have power constraint P, while node T 0 has constraint P 0  When P=P 0 =3/2  Routing: 1.5 bits/transmission  Network coding with independent codebooks: 2 bits/transmission P0P0 U U,V V PP D1D1 D2D2 T1T1 T0T0 T2T2

19. March 18, 2005 Node Cooperation  Network coding with node cooperation scheme:  Three codebooks – for U message, V message, and U V message  Equal timesharing over two modes of operation  In first mode,  Node T 1 transmits U codeword scaled to power P+   Node T 2 transmits V codeword scaled to power P-   Node T 0 splits its power to transmit both U and U V  Second mode: Reverse of first mode P0P0 U U,V V PP D1D1 D2D2 T1T1 T0T0 T2T2

20. March 18, 2005 Node Cooperation  Network coding with node cooperation scheme:  T 1 and T 0 cooperate to send the codeword for U to D 1  T 0 sends codeword for U V to both D 1 and D 2  T 2 sends codeword for V to D 2  Destinations receive their own sources directly  Destinations can decode the opposite source with U V message  With appropriate choice of parameters  Average rate of 2.06 is achievable at each node P 0 -  0 U U,V V P+  P-  D1D1 D2D2 T1T1 T0T0 T2T2 00

21. March 18, 2005 Summary  “Bit-pipe” non-interference model is not complete  Neglects interactions between channels  In wireless scenario, imposes orthogonality constraint on channels  Any gains due to node-cooperation necessarily lost  Noisy channel model captures these effects  More difficult to handle  Results for non-interference networks can be duplicated  Configurations exists where bit-pipe model has no network coding gain, while interference model benefits from network coding  Node cooperation strategies can increase the throughput beyond that rate achieved by using independent codebooks