1. Network Coding Beyond Network Coding
Michelle Effros
DIMACS 2015

2. Network Coding is a rich & beautiful fieldBC
MAC
Interference
Multiple Unicast
Multiple
Multicast
Networks
Network
Coding
Index
Coding
with broad implications & applications beyond.
Network Coding Beyond Network Coding
DIMACS Effros

3. These include NC bounds for non-NC networks.
are (lower, upper) bounding models for
(written ) iff
[Koetter, Effros, & Medard 2011,2014] [Jalali & Effros 2015]
Network Coding Beyond Network Coding
DIMACS Effros

4. Example upper and lower bounding networks ...
[Koetter, Effros, & Medard 2009][Wong & Effros 2012] [Jalali & Effros 2015]
Network Coding Beyond Network Coding
DIMACS Effros

5. These models simplify the study of network IT ...
[Koetter, Effros, Medard 2013] + + +
Upper & lower bounds differ by at most 1/2 bit for every P/N,
Prev: Schein + Avestimehr, Diggavi, & Tse indep noise, high SNR
Network Coding Beyond Network Coding
DIMACS Effros

6. ... & enable hierarchical analysis of large networks.
[Ho, Effros, Jalali 2010]
Internet topology [Lun Li, Caltech Ph.D. 2007]
Network Coding Beyond Network Coding
DIMACS Effros

7. Reductive arguments also uncover equivalence relationships between NC & non-NC problems.
Code Equivalence
Linear NC Linear IC [El Rouayheb, Sprintson, Georghiades 2008]
NC IC [Effros, El Rouayheb, Langberg 2013]
k-unicast unicast [Kamath, Tse, Wang, 2014]
Capacity Equivalence
NC IC [Wong, Langberg, Effros 2011][Effros, El Rouayheb, Langberg 2013]
Linear MMIC Linear MUIC [Maleki, Cadambe, Jafar 2012]
MMIC MUIC [Wong, Langberg, Effros 2015]
MMMN MUMN [Wong, Langberg, Effros 2013]
Wireline networks NC networks [Koetter, Effros, Medard 2011]
MN with delays MN without delays [Effros 2012]
...
NC = Network Coding, IC = Index Coding
MM = Multiple Multicast, MU = Multiple unicast
MN = Memoryless network
Simple relationships can be combined to build a rich understanding of the space (“hard” problems, “complete” problems, ...)
special
cases
Network Coding Beyond Network Coding
DIMACS Effros

8. Studying NC distills fundamental questions. For example, ...
The edge removal problem [Jalali, Effros,Ho 2010, 2011] :
Given a pair of networks differing in a single edge
Network Coding Beyond Network Coding
DIMACS Effros

9. Definitions
Network Coding Beyond Network Coding
DIMACS Effros

10. Definitions
Network Coding Beyond Network Coding
DIMACS Effros

11. The question remains unsolved in general.
[Jalali, Effros, Ho 2011, 2012, Langberg, Effros 2012, Lee, Langberg, Effros 2013]
No examples where property fails. No proof that always holds.
The edge removal property holds for some networks.
cut-set bounds are tight (e.g., single- & multi-source multicast)
co-located sources, super-source networks, terminal edges
linear codes, “separable” codes
index coding
...
The edge removal property holds for outer bounds.
Cut-set bound
Generalized network sharing bounds [Kamath, Tse, Anantharam 2011]
Linear Programming (LP) bound [Yeung 1997, Song, Yeung 2003]
Equivalence to other problems (0- vs. -error, dep srcs, NC vs. IC...)
Network Coding Beyond Network Coding
DIMACS Effros

12. Studying NC distills fundamental questions. Intuition:
The edge removal problem [Jalali, Effros,Ho 2010, 2011] :
Given a pair of networks differing in a single edge
Network Coding Beyond Network Coding
DIMACS Effros

13. Studying NC distills fundamental questions. Intuition:
The edge removal problem [Jalali, Effros,Ho 2010, 2011] :
Given a pair of networks differing in a single edge
Network Coding Beyond Network Coding
DIMACS Effros

14. Studying NC distills fundamental questions. Intuition:
The edge removal problem [Jalali, Effros,Ho 2010, 2011] :
Given a pair of networks differing in a single edge
Network Coding Beyond Network Coding
DIMACS Effros

15. The question remains unsolved in general.
[Jalali, Effros, Ho 2011, 2012, Langberg, Effros 2012, Lee, Langberg, Effros 2013]
No examples where property fails. No proof that always holds.
The edge removal property holds for some networks.
cut-set bounds are tight (e.g., single- & multi-source multicast)
co-located sources, super-source networks, terminal edges
linear codes, “separable” codes
index coding
...
The edge removal property holds for outer bounds.
Cut-set bound
Generalized network sharing bounds [Kamath, Tse, Anantharam 2011]
Linear Programming (LP) bound [Yeung 1997, Song, Yeung 2003]
Equivalence to other problems (0- vs. -error, dep srcs, NC vs. IC...)
Network Coding Beyond Network Coding
DIMACS Effros

16. Intuition
Network Coding Beyond Network Coding
DIMACS Effros

17. The question remains unsolved in general.
[Jalali, Effros, Ho 2011, 2012, Langberg, Effros 2012, Lee, Langberg, Effros 2013]
No examples where property fails. No proof that always holds.
The edge removal property holds for some networks.
cut-set bounds are tight (e.g., single- & multi-source multicast)
co-located sources, super-source networks, terminal edges
linear codes, “separable” codes
index coding
...
The edge removal property holds for outer bounds.
Cut-set bound
Generalized network sharing bounds [Kamath, Tse, Anantharam 2011]
Linear Programming (LP) bound [Yeung 1997, Song, Yeung 2003]
Equivalence to other problems (0- vs. -error, dep srcs, NC vs. IC...)
Network Coding Beyond Network Coding
DIMACS Effros

18. The question remains unsolved in general.
[Jalali, Effros, Ho 2011, 2012, Langberg, Effros 2012, Lee, Langberg, Effros 2013]
No examples where property fails. No proof that always holds.
The edge removal property holds for some networks.
cut-set bounds are tight (e.g., single- & multi-source multicast)
co-located sources, super-source networks, terminal edges
linear codes, “separable” codes
index coding
...
The edge removal property holds for outer bounds.
Cut-set bound
Generalized network sharing bounds [Kamath, Tse, Anantharam 2011]
Linear Programming (LP) bound [Yeung 1997, Song, Yeung 2003]
Equivalence to other problems (0- vs. -error, dep srcs, NC vs. IC...)
Network Coding Beyond Network Coding
DIMACS Effros

19. The same question arises in hybrid networks.
Network Coding Beyond Network Coding
DIMACS Effros

20. Definitions. Lossless links Noisy channel
Network Coding Beyond Network Coding
DIMACS Effros

21. [Noorzad, Effros, Langberg, Ho 2014]
The edge removal question for hybrid networks.
[Noorzad, Effros, Langberg, Ho 2014]
Given a pair of networks differing in a single edge,
does
Network Coding Beyond Network Coding
DIMACS Effros

22. From prior literature, it seemed that the impact of an edge is bounded by its capacity.
[Willems 1983]
Network Coding Beyond Network Coding
DIMACS Effros

23. It is possible to construct examples where the power of an edge exceeds its capacity.
[Noorzad, Effros, Langberg, Ho 2014] X
Network Coding Beyond Network Coding
DIMACS Effros

24. The impact of a single edge can be large.
[Noorzad, Effros, Langberg, Ho 2014] X
Statement fails for any polynomial f() (“almost exponential”)!
Network Coding Beyond Network Coding
DIMACS Effros

25. Intuition ... [Noorzad, Effros, Langberg, Ho 2014]
Network Coding Beyond Network Coding
DIMACS Effros

26. Intuition ... [Noorzad, Effros, Langberg, Ho 2014]
Network Coding Beyond Network Coding
DIMACS Effros

27. The impact of a single edge can be large.
[Noorzad, Effros, Langberg, Ho 2014] X
Statement fails for any polynomial f() (“almost exponential”)!
Network Coding Beyond Network Coding
DIMACS Effros

28. We use this observation to study the
the potential power of cooperation.
Cooperation
facilitator.
[Noorzad, Effros, Langberg, Ho 2014]
When edge removal fails,
the “benefit” of cooperation can exceed its “cost”
Network Coding Beyond Network Coding
DIMACS Effros

29. Can the benefit of cooperation exceed its cost
in more realistic networks?
Cooperation
facilitator.
[Noorzad, Effros, Langberg 2015]
Network Coding Beyond Network Coding
DIMACS Effros

30. Can the benefit of cooperation exceed its cost
in more realistic networks?
[Noorzad, Effros, Langberg 2015]
Network Coding Beyond Network Coding
DIMACS Effros

31. Can the benefit of cooperation exceed its cost
in more realistic networks?
[Noorzad, Effros, Langberg 2015]
Network Coding Beyond Network Coding
DIMACS Effros

32. Conclusions
The implications and applications of network coding extend far beyond traditional network coding networks
Bounding general network performance
Lending insight through equivalences (and their failure)
Highlighting and distilling fundamental problems
Providing tools for understanding hybrid networks
Network Coding Beyond Network Coding
DIMACS Effros