The High, the Low and the Ugly

1. The High, the Low and the Ugly Muriel Médard

2. Collaborators • Nadia Fawaz, Andrea Goldsmith, Minji Kim, Ivana Maric

3. Regimes of SNR • Recent work has considered different SNR regimes • High SNR: • Deterministic models • Analog coding models • Low SNR: • Hypergraph models • Other SNRs: the ugly

4. R1 Wireless Network Y(e1) e1 Model as error free links Y(e3) e3 Y(e2) e2 R2 Y(e3) = β1Y(e1) + β2Y(e2) • Open problem: capacity & code construction for wireless relay networks • Channel noise • Interference • [Avestimehr et al. ‘07]“Deterministic model” (ADT model) • Interference • Does not take into account channel noise • In essence, high SNR regime • High SNR • Noise → 0 • Large gain • Large transmit power

5. ADT Network Background • Min-cut: minimal rank of an incidence matrix of a certain cut between the source and destination [Avestimehr et al. ‘07] • Requires optimization over a large set of matrices • Min-cut Max-flow Theorem holds for unicast/multicast sessions. [Avestimehr et al. ‘07] • Matroidal [Goemans et al. ‘09] • Code construction algorithms [Amaudruz et al. ‘09][Erez et al. ‘10]

6. Our Contributions • Connection to Algebraic Network Coding [Koetter and Médard. ‘03]: • Use of higher field size • Model broadcast constraint with hyper-edges • Capture ADT network problem with a single system matrix M • Prove that min-cut of ADT networks = rank(M) • Prove Min-cut Max-flow for unicast/multicast holds • Extend optimality of linear operations to non-multicast sessions • Incorporate failures and erasures • Incorporate cycles • Show that random linear network coding achieves capacity • Do not prove/disprove ADT network model’s ability to approximate the wireless networks; but show that ADT network problems can be captured by the algebraic network coding framework

7. ADT Network Model • Original ADT model: • Broadcast: multiple edges (bit pipes) from the same node • Interference: additive MAC over binary field Higher SNR:S-V1 Higher SNR: S-V2 interference • Algebraic model: broadcast

8. Algebraic Framework • X(S, i): source process i • Y(e): process at port e • Z(T, i): destination process i • Linear operations • at the source S: α(i, ej) • at the nodes V: β(ej, ej’) • at the destination T: ε(ej, (T, i))

9. System Matrix M= A(I – F )-1BT c a d b f e1 e11 e9 e7 e5 e3 e8 e6 e10 e12 e2 e4 • Linear operations • Encoding at the source S: α(i, ej) • Decoding at the destination T: ε(ej, (T, i))

10. System Matrix M= A(I – F )-1BT c a d b f e9 e7 e11 e3 e1 e5 e12 e10 e8 e6 e4 e2 • Linear operations • Coding at the nodes V: β(ej, ej’) • F represents physical structure of the ADT network • Fk:non-zero entry = path of length kbetween nodes exists • (I-F)-1 = I + F + F2 + F3 + … : connectivity of the network (impulse response of the network) Broadcast constraint (hyperedge) F = MAC constraint(addition) Internal operations(network code)

11. System Matrix M = A(I – F )-1BT c a d b f e1 e5 e3 e7 e11 e9 e8 e10 e12 e4 e2 e6 • Input-output relationship of the network Z = X(S) M Captures rate Captures network code, topology(Field size as well)

12. Theorem: Min-cut of ADT Networks • From the original paper by Avestimehr et al. • Requires optimizing over ALL cuts between S and T • Not constructive: assumes infinite block length, internal node operations not considered • Show that the rank of M is equivalent to optimizing over all cuts • System matrix captures the structure of the network • Constructive: the assignment of variables gives a network code

13. Min-cut Max-flow Theorem • For a unicast/multicast connection from source S to destination T, the following are equivalent: • A unicast/multicast connection of rate R is feasible. • mincut(S,Ti) ≥ R for all destinations Ti. • There exists an assignment of variables such that M is invertible. • Proof idea:1. & 2. equivalent by previous work. 3.→1. If M is invertible, then connection has been established.1.→3. If connection established, M = I. Therefore, M is invertible. • Corollary:Random linear network coding achieves capacity for a unicast/multicast connection.

14. Extensions to Non-multicast Connections • [Multiple multicast] Multiple sources S1 S2 … Skwants to transmit to all destinations T1 T2… TN • Connection feasible if and only if mincut({S1 S2 … Sk}, Tj) ≥ sum of rate from sources S1 S2 … Sk • Proof idea: Introduce a super-source S, and apply the multicast min-cut max-flow theorem.

15. Extensions to Non-multicast Connections • [Disjoint Multicast]The connection is feasible if and only if: • Proof idea:Introduce a super-destination T, and apply the multicast min-cut max-flow theorem.

16. Extensions to Non-multicast Connections • [Two-level Multicast] A set of destinations, Tm, participate in a multicast connection; rest of the destinations, Td, in a disjoint multicast. The connection is feasible if and only if: • Tm: Satisfy single multicast connection requirement. • Td: Satisfy disjoint multicast connection requirement. • Random linear network coding at intermediate nodes; but a carefully chosen encoding matrix at source achieves capacity Single multicast Disjoint multicast

17. Incorporating Erasures in ADT network • Wireless networks: stochastic in nature • Random erasures occur • ADT network • Models wireless deterministically with parallel bit-pipes • Min-cut as well as previous code construction algorithm needs to be recomputed every time the network changes • Algebraic framework: • Robust against some set of link failures (network code will remain successful regardless of these failures) • The time average of rank(M) gives the true min-cut of the network • Applies to the connections described previously

18. Incorporating cycles in ADT network • Wireless networks intrinsically have bi-directional links; therefore, cycles exists • ADT network mode • A directed network without cycles: links from the source to the destinations • Algebraic framework • To incorporate cycles, need a notion of time (causal); therefore, introduce delay on links D • Express network processes in power series in D • The same theorems as for delay-less network apply

19. Network Coding and ADT • ADT network can be expressed with Algebraic Network Coding Formulation [Koetter and Médard ‘03]. • Use of higher field size • Model broadcast constraint with hyper-edge • Capture ADT network problem with a single system matrix M • Prove an algebraic definition of min-cut = rank(M) • Prove Min-cut Max-flow for unicast/multicast holds • Extend optimality of linear operations to non-multicast sessions • Disjoint multicast, Two-level multicast, multiple source multicast, generalized min-cut max-flow theorem • Show that random linear network coding achieves capacity • Incorporate delay and failures (allows cycles within the network) • BUT IS IT THE RIGHT MODEL?

20. Different Types of SNR • Diamond network [Schein] • As a increases: the gap between analog network coding and cut set increases [Avestimehr, Diggavi & Tse] • In networks, increasing the gain and the transmit power are not equivalent, unlike in point-to-point links

21. Let SNR Increase with Input Power

22. Analog Network Coding is Optimal at High SNR

23. What about Low SNR? • Consider again hyperedges • At high SNR, interference was the main issue and analog network coding turned it into a code • At low SNR, it is noise

24. What about Low SNR? • Consider again hyperedges • At high SNR, interference was the main issue and analog network coding turned it into a code • At low SNR, it is noise

25. Peaky Binning Signal • Non-coherence is not bothersome, unlike the high-SNR regime

26. What Min-cut? • Open question: Can the gap to the cut-set upper-bound be closed? • An ∞ capacity on the link R-D would be sufficient to achieve the cut like in SIMO • Because of power limit at relay, it cannot make its observation fully available to destination. • Conjecture: cannot reach the cut-set upper-bound

27. What About Other Regimes? • The use of hyperedges is important to take into account dependencies • In general, it is difficult to determine how to proceed (see the difficulties with the relay channel) – the ugly • Equivalence leads to certain bounds for multiple access and broadcast channels, but these bounds may be loose