Asymptotic Transport Capacity of Wireless Erasure Networks

1. Asymptotic Transport Capacity of Wireless Erasure Networks Brian Smith and Sriram Vishwanath University of Texas at Austin Allerton Conference on Communication, Control, and Computing September 28th, 2006 September 28th, 2006

2. Overview • Introduction • Motivation • Wireless Erasure Networks • Erasure Model • Upper Bounds • Dense Networks • Summary September 28th, 2006

3. Introduction • Capacity of Multiple-Source Multiple-Destination Networks (Multiple Unicast) • Hard Problem • Transport Capacity: Convenient Scalar Description • Distance Weighted Rate-Sum (bit-meters) • Work by Gupta & Kumar[2000] and Xie & Kumar[2004] • Gaussian Interference Channel Model • Information Theoretic Linear Bound on Transport Capacity Growth • Under high attenuation model September 28th, 2006

4. Wireless Erasure Networks: Background • Dana, Gowaikar, Hassibi & Effros [2006] • Packetized Wireless Network • Model • GF2 Source Alphabet • Broadcast requirement on directed graph • Links are independent erasure channels • No receiver interference – Receiver gets vector • Result • Multicasting from single source to multiple receivers can be performed at generalized min-cut max-flow rate Modified Cut-Set Bound Example Bound Evaluation September 28th, 2006

5. Motivation • Desire to investigate transport capacity from a network layer point of view for a wireless network • Thus, Erasure Network • Packet is either correctly decoded or nothing is known about the contents • Wireless Networking • Broadcast Requirement • Fully connected graph • Interference - physical layer phenomenon • Main Result: Linear Bound on Transport Capacity Growth September 28th, 2006

6. Connecting Transport Capacity with Wireless Erasure Networks • Erasure Probability as Function of Geographic Distance • Minimum Node Separation Constraint: d>dmin • Threshold Model • (d)=0, dd*; 1, d>d* • Exponential Model • (d)=1-e-d/d* • Polynomial Decay Model • (d)=1-1/(1+d), >3 • In all cases • (0)=0, (∞)=1 x1 x2 x3 x4 y6 x5 y7 y8 y9 September 28th, 2006

7. Aside: Cycles • Dana, Gowaikar, Hassibi, Effros 2006 • Includes a ‘directed acyclic graph’ requirement • Our model: • Completely connected, many back cycles • ΣR ≤ I(XS;YSC|XSC) • Use data processing inequality (as in XK2004) to show that only symbols from outside the cut matter • Back cycles do not increase sum-rate bound S SC September 28th, 2006

8. Source-Destination Rate Pairs Bounding Transport Capacity • Examine one-dimensional case for intuition • Every source-destination path crosses at least one cut dmin Rate Cut m September 28th, 2006

9. Threshold Model Converse • Rate Cut-set: One bit for every node “within range” • Each node within range of less than d*/dmin cuts to its right • 2-D: Different Constants dmin Length d* represents range of transmission Rate Cut m September 28th, 2006

10. Source-Destination Rate Pairs Bounding Transport Capacity (2) • Index nodes in order of increasing position • Place one cut between consecutive nodes • Every source-destination path crosses at least one cut dm Node m Rate Cut m September 28th, 2006

11. 1-D Exponential Converse • Model: (d)=1-e-d/d* • Transport Capacity ≤κn • Constant depends only on dmin, d* September 28th, 2006

12. Squeeze dm dm Geometric Sequences Bounded by Kn September 28th, 2006

13. Polynomial Case Differences • Model: (d)=1-(1+d), >3 • Instead of a geometric summation, bound the summation with integrals • Per-node Transport Capacity Result: k dm3- • As long as >3, upper-bounded • dm>dmin September 28th, 2006

14. 2-D: Different Approach Needed • For 1-D, bounded the transport capacity across every cut • Doesn’t work for 2-D, because some cuts can diverge with n: • Attempt: Place one horizontal cut in between each node. Allow overlapping cuts • Lots of zeros-valued distances, though Rate cut-set for this cut increases (n) Multiple overlapping cuts add zero to TC September 28th, 2006

15. Explanation of General Transport Capacity Bound • Notation d1h d2h dij≥dmin d34 Node 1 Cut 1 September 28th, 2006

16. Squish dih di+1h dm-1h Move all nodes with index greater than ‘m’ in as close as possible, within minimum distance constraint, to bound summation Node i Node m September 28th, 2006

17. Proof Sketch • Replace the product term with a summation • Bound the summation with integrals to get transport capacity in terms of horizontal components only • Rearrange the summations • Bound each of the terms in the new summation by a constant September 28th, 2006

18. Dense Networks • Drop the minimum distance requirement • Place n nodes equally spaced on unit square • Without Interference: • Upper bound and achievablity both (n) September 28th, 2006

19. d* ln n Super-linear Growth • Place n nodes in two groups, spaced d*ln n apart • n ln n growth • But, specifying erasure locations requires ln n extra information September 28th, 2006

20. Current Work:Additive Finite-Field Interference • Bhadra, Gupta & Shakkottai 2006 • Capacity Bounds - Asymptotically tight in field size q for uniform fading over the field • Received signal: Yj=ijXi • ij is a 0-1 Bernoulli r.v. with P[ij=0]=ij • Cut-Set Bound • Write the vector of transmitted and received symbols on each side of the cut as Y=HX • H is a 0-1 random matrix x1 x2 Modified Cut-Set Bound x3 x4 y6 E[rank(H)] lg(q) x5 y7 y8 y9 September 28th, 2006

21. Summary • Transport capacity linearly bounded in number of nodes with minimum distance requirement • Threshold Model • Exponential Model • Polynomial Model with >3 for 1-D • Results correspond well to Xie-Kumar results for Gaussian Network case • What properties of network models cause linear growth? • Analog to CRIS in low-attenuation case? • Dense Networks • Remove minimum distance requirement? • Achievability with interference? September 28th, 2006