Response to DARPA RFI 06-17
Information Theory for Mobile Ad-Hoc Networks (MANETS)
Fundamental Capacity Limits and
Optimized Node Cooperation in MANETs
Stanford
Stephen Boyd
Andrea Goldsmith
Ramesh Johari
Balaji Prabhakar
MIT
Vincent Chan
Robert Gallager
Muriel Medard
Asuman Ozdaglar
Devavrat Shah
Lizhong Zheng
UIUC
Todd Coleman
Ralf Koetter
Sean Meyn
Pierre Moulin
Ada Poon
Caltech
Michelle Effros
Abstract:
While there has been much progress in obtaining the fundamental capacity limits of wireless single and
multiuser channels, there is very limited understanding about the fundamental capacity limits associated
with mobile ad-hoc networks (MANETs), even under fairly simple modeling assumptions. More realistic
system assumptions, such as fading, node mobility, heterogeneous bursty traffic, delay and energy
constraints, limited channel side information, distributed control, robustness, and security may lead to new
definitions of fundamental capacity that differ from the traditional Shannon capacity associated with
infinite complexity and delay. Moreover, even if the fundamental capacity of a network can be obtained, it
is not that valuable in the absence of separation theorems between the source encoding strategy and the
networking strategy. Thus, along with fundamental limits on network capacity, separation theorems
between source coding, channel transmission, and networking must be investigated. When such theorems
fail, we must pursue either bounds to demonstrate that separated strategies yield good performance or
strategies for node cooperation via low complexity non-separated codes.
We have brought together a strong team of interdisciplinary researchers with an extensive history of
collaboration to address these issues. We believe our team is uniquely poised to characterize upper and
lower bounds and scaling laws for the fundamental capacity limits of MANETs, to develop techniques that
approach these fundamental limits given realistic system models and constraints, and to investigate
source/channel/network separation theorems along with optimal node cooperation. The proposed work
along these lines will be described below in response to the questions posed in the RFI.
Question 1: To what extent is a precise characterization of mobile wireless network capacity achievable?
Before one can speculate about a precise characterization, it is necessary to define fundamental network
capacity. For a network with n nodes, its capacity region is of dimension n(n-1), since it defines all rates
simultaneously achievable between any two nodes in the network via single and/or multiple hop
transmission [TouG03]. The shape of this region will depend on the system parameters and constraints such
as energy/power, bandwidth, channel characteristics, node mobility, delay constraints, traffic
characteristics, and robustness to uncertainty and attack. In many settings, most notably delay constrained
situations; the capacity region will be zero if there is no notion of outage or message error. Indeed, defining
capacity only in the case of asymptotically small probability of error, infinite delay, and infinite complexity
has been, at the same time, the enabling but also one of the most limiting aspects of Shannon theory and a
major reason why the marriage between information theory and network theory remains unconsummated
[EphH98]. We believe that a fundamental capacity theory for networks must include notions of nonzero
error probability, channel outage, and lost packets/messages, which give rise to new capacity definitions
such as capacity versus outage and throughput capacity [EffG98a]. Indeed, one of the most challenging
aspects of an information theory for networks is just to find a way that partial decoding, yielding nonvanishing error probabilities, may be incorporated into operational limits. Thus, our characterization of
Capacity
Upper Bound
Delay
network capacity
will include both
traditional Shannon
theoretic capacity
(which will often be
Lower Bound
zero under practical
constraints) as well
as these more
general capacity
definitions, which
Energy
provide more
insightful capacity metrics under the typical operating conditions of practical networks. In the discussion
below on capacity regions, the capacity we compute will encompass this more general metric.
Unfortunately, a precise characterization of the Shannon capacity region for a general ad-hoc network
under the simplest of assumptions - an AWGN channel with no delay or complexity constraints - has been
an open problem for decades. Even simple channels within the broader context of an ad-hoc network, such
as the relay channel or interference channel, have unknown capacity in the general case. However, much
progress has been made on obtaining upper and lower bounds for these component channels in some
settings (e.g. [KraGG05, HosZ05, RezKV05, VisJ04]), as well as in scaling laws for uniform capacity in
some asymptotic settings (e.g. [XieK04, ElG05, GasV05, DigGT05]). Thus, we believe that a tractable
formulation of network capacity for this program must be in terms of upper and lower bounds, which
would be made progressively tighter over the five year program duration via acquired insights to refine the
cut-set upper bounds and more sophisticated networking techniques to increase the lower bounds. Such
bounds will provide tremendous new insights into network performance and practical designs. Specifically,
tight upper bounds would define the best performance that could be expected of the network and how
various constraints and assumptions affect this performance. Tight lower bounds provide much insight into
near-optimal strategies for user cooperation and resource allocation over available degrees of freedom, with
significant potential impact for practical designs. Large gaps between the upper and lower bounds would
indicate that more work is needed to develop new techniques to significantly improve the lower bounds
and/or new insights are necessary to constrain the upper bounds and thereby tighten them. The upper and
lower bounds will meet under certain system parameterizations and/or constraints, as well as in certain
asymptotic regimes associated with single-letter characterizations (such as scaling laws for homogeneous
traffic and node capabilities), in which case a precise capacity characterization would be determined. Part
of the proposed work would be to determine scenarios where the upper and lower capacity bounds meet,
which would help in tightening the bounds over the entire set of system constraints.
There are many ways to study the impact of various system assumptions and constraints on the n(n-1)dimensional capacity region bounds. We propose the following characterization. We view energy and delay
as fundamental axes of the capacity region. Hence, for a given set of system assumptions and constraints,
the network capacity upper and lower bounds will be characterized as a 3n(n-1)-dimensional region, of
which a 3-dimensional slice indicating capacity or throughput as a function of energy and delay between
any two nodes in the network is shown in the figure below. This region is then parameterized by various
system assumptions and constraints. For example, multiple antennas at each node will increase the upper
and lower bounds (added degrees of freedom), whereas robustness to attack will decrease these bounds
(reduced degrees of freedom). The precision to which these network capacity bounds can be characterized
will depend on how many network and system parameters one wishes to allow in the characterization. We
will begin our capacity characterization under the most basic assumptions (e.g. heterogeneous nodes,
continuous traffic, no fading or mobility) and, once these bounds have been obtained, consider the most
promising directions in which to generalize them under more complex assumptions. Note that these bounds
will require precise information about the network setup, in particular either the network topology and
channel models or the channel gains between nodes.
Figure 1: Three-dimensional slice of 3n(n-1)-dimensional capacity region
In many circumstances, especially when parameters such as latency, mobility, and heterogeneity are
included, it will be difficult to obtain these multidimensional upper and lower capacity bounds. In these
cases we plan to investigate a rate of growth characterization with upper and lower bounds that match in an
order sense. We have already determined such bounds to characterize a fundamental relationship between
capacity, delay, and energy [GamMPS04a, GamMPS04b, TouG04]. Such single-letter characterizations of
the capacity region coincide with letting one or more of the parameters in the capacity region of Figure 1 go
to infinity. Determining which asymptotics are of value for capacity characterization will be part of the
proposed work.
Question 2: How would one recognize a “complete” mobile wireless network capacity limit formulation if
one were to see it?
A complete mobile wireless network capacity formulation must be sufficiently general to characterize the
capacity region under various system parameterizations, constraints, and assumptions, through either the
exact region when possible or through bounds and scaling laws. However, the region characterization must
be complemented by a characterization of the optimal forms of node cooperation and network resource
allocation that achieve these bounds. Hence, a large part of our efforts will be focused on investigating
optimal forms of node cooperation and resource allocation over the network degrees of freedom. These
optimal strategies will generally depend on the network topology, the channel SNR, and the channel side
information available at the nodes. A complete network capacity formulation must also consider the
fragility of the result to the underlying modeling assumptions: small changes in these assumptions should
not fundamentally change the capacity, and thus any capacity theory for networks requires robustness to the
modeling assumptions, analogous to robust strategies developed by control theorists in the face of modeling
errors.
Novel forms of node cooperation have been the focus of much recent work in MANETs, including network
coding, virtual MIMO, cooperation diversity, conferencing, and relaying with no, full, and partial decoding.
Network coding has proven to be an effective way to increase network capacity over traditional routing
schemes for multicast [KoeM03, LunRKM05, LunMK06, LunMKE05, LunMK05, DebEHKKLMR05].
Virtual MIMO leverages the diversity and multiplexing benefits of multiple antennas by pooling antenna
resources of nodes that are close together on the transmit and/or receive end to increase capacity [SenEA,
LanTW04, Hos04, Hos05, KhoSA04, NgA04, NgA05, NgA06, NgLG06, JinMA04]. The capacity benefits
of conferencing have been characterized for MAC channels and for relay channels [Wil83, NgMGSY06],
and our proposed work will investigate the benefits of conferencing in a more general network setting.
Relaying via Decode-and-Forward (DF) and Amplify-and-Forward (AF) has been well studied and is
capacity achieving in some limited settings [KraGG05, NgA05]. However, more general forms of relaying
need to be developed and explored, especially the notion of partial message decoding at each node.
Network coding is one example where nodes cooperate without decoding at intermediate nodes. Another
form of relaying with partial decoding that we plan to explore is error exponents in list decoding, which is a
natural generalization of DF and AF. . Indeed, we view list decoding as an integral part of expanding the
consideration of issues of distortion measures, which are intrinsically linked to the types of trade-offs we
propose to study [Gur06]. A take-away lesson from all work on node cooperation is that - contrary to
popular belief - interference is not always bad. If coding is done properly and a small amount of
coordination between users is done, low-complexity transmission schemes can be constructed that allow for
an aggregate benefit to the system [ColMEM05].
Part of the complete capacity characterization for MANETs must include the optimal resource allocation
across its many degrees of freedom. We plan to explore dynamic resource allocation over multiple
dimensions including bandwidth, power, rate, antennas, and end-to-end routes, using both centralized and
distributed optimization techniques. Power control impacts many aspects of MANET design and is
therefore a critical parameter to optimize: it can be used at the link layer to compensate for random
variations in the channel, at the MAC layer to maintain tolerable interference between users, and to change
network connectivity, which impacts routing protocols [TouG03]. Another key area of investigation is to
consider how the degrees of freedom associated with multiple antennas – which can be used for
interference cancellation, MIMO, and/or beamforming - should be allocated in conjunction with the upper
layers of a wireless network to optimize network performance [Poo05, PooBT05]. In many resource
allocation problems, sensitivity to certain ‘control parameters’ is extremely low in heavy traffic (operation
near capacity) [ChePM03], which implies a certain robustness of resource allocation techniques in such
settings.
Even in the cases where significant progress has been made in determining network capacity, these results
are often highly fragile relative to the assumptions used in the problem formulation. In this context a central
step towards an information theory for networks that is relevant to practical designs must include capacity
characterizations that are relatively insensitive to small variation in the investigated scenario, e.g. when the
network topology, the channel state information (CSI), or the characteristics of interfering signals are not
perfectly known or cover a range of parameters. For example, the capacity regions obtained through
network coding are demonstrated in [DouFZ05] to be highly dependent in methodology and final result on
the network in question. Similarly, it is known that capacity of a broadcast channel with perfect transmit
and receive CSI is achieved with dirty paper coding [WigSS05]. However, it was shown in [LapSW05] that
the sum rate capacity of this channel is greatly reduced if the fading in the channel is not known precisely,
and therefore there is no graceful transition from a perfectly known channel to approximate knowledge in
terms of channel capacity. The development of robust capacity characterizations that are relatively
insensitive to small variations in the investigated scenario is essential for practically useful results.
A companion question to how a complete capacity characterization would be recognized is how such a
characterization would be used. Shannon’s landmark source-channel separation theorem decoupled the
design of source encoders and channel transmission techniques in theory and (mostly) in practice as well.
In other words, when separation is optimal on a point-to-point link, channel capacity becomes the only
metric of importance to source code design: the underlying capacity-achieving techniques are irrelevant to
this design. While separation greatly simplifies source and channel code design, it does entail some
performance penalty in practice, especially under delay and complexity constraints [EffG98b]. However, it
is not at all clear that the optimality of source-channel separation in point-to-point links can be extended to
optimality in separating the source coding, channel modulation and coding, multiple access, and
routing/relaying in MANETs [EffKGM04]. In fact, it is known that separation of source and channel
coding does not hold even for simple networks like a multiple access channel [CovT91]. Likewise,
separation between source and network coding and channel and network coding can also fail in simple
networks [EffMHRKK03]. Mirror site selection is another example that separation between source coding
and networks is suboptimal. Indeed, if such separation were optimal, then the deliberate duplication of
information at the ingress of the network would never be of use. In effect, the presence of duplicated
information allows connections to occur even in the event of congestion, which may throttle information
sufficiently to preclude certain connections from taking place. There is in effect a lack of source/network
coding separation [RamKCE04].
We plan to investigate under what conditions separation of source-channel-network coding is optimal,
where coding in this context is generalized to encompass all transmission strategies associated with the
source, channel, and network, respectively. When such separation is not optimal, it is critical to understand
the performance cost of separation, as design modularity is clearly desirable and often essential in complex
systems. Performance cost must also be clearly defined, and this cost definition will depend on the nature
of the sources, channels, and networks. When the cost of separation is high, then we will explore joint
source-channel-network coding to close the capacity gap while maintaining feasibility and scalability of the
overall network design. For example, it has recently been shown [ColMO06, RKCE04] that when
correlated sources are broadcast across a network, separation of the distributed data compression from the
channel coding at the encoders and decoders does not in general achieve capacity. However, separate
distributed compression and channel encoding with joint source-channel decoding does achieve capacity.
We plan to explore such ideas for more general source-channel-network designs. An interesting open
question here is whether partial joint designs are feasible while providing significant performance gains
and, if so, which parts should be jointly designed: the source and channel codes, the channel and network
codes, or the source and network codes (source coding for multihop networks). These results could have
great practical relevance because source-channel separation at the encoders (which are usually the most
energy and resource-constrained - for instance sensor networks) is a very pleasing, modular and robust
engineering solution that many network designers would embrace.
Joint source-network coding also naturally exploits arbitrary correlation of sources. Such correlation may
be aided by the natural correlation of data in the system (for instance spatially-induced correlation in a
sensor setting) or introduced deliberately in a way similar to that in which mirror sites are created. We plan
to explore such joint source-network coding based on random code constructions that generalize those of
the Slepian-Wolf settings. While the decoding of such codes using traditional methods such as minimumentropy approaches is NP-complete, our recent results [ColME05] have shown that an asymptotically errorfree approach can be implemented using relaxation methods that yield polynomial-time complexity.
Regardless of the optimality of separation, the capacity region of a MANET, once characterized, can be
used to optimize end-to-end performance using network-aware application design. In particular, the
network capacity region captures the tradeoffs between capacity, energy, and delay for various network
parameterizations. The performance metric associated with a given application can then be optimized by
finding the best operating point on this tradeoff curve. Thus, network-aware application design entails
finding this optimal operating point, which is a nontrivial optimization problem given the dimensions of the
capacity region. Our team includes significant expertise in optimization of complex systems, and this
expertise will be applied to defining appropriate metrics for simultaneous heterogeneous applications with
different QoS requirements as well as optimizing the network operation with respect to these metrics.
Question 3: What is the largest number of problem dimensions (latency, mobility, heterogeneity, etc) for
which the responder believes an upper bound on capacity can be understood within the timeframe of a 5year program?
As stated in the response to Question 1, we believe that the fundamental capacity limits for MANETs must
encompass the tradeoffs between capacity, energy, and delay under parameterizations of bandwidth,
channel characteristics and CSI assumptions, node mobility, delay constraints, traffic characteristics, and
robustness to uncertainty and attack. Our team has developed preliminary results characterizing the impact
on capacity under these parameterizations in some network settings, and hence we are optimistic that these
preliminary results can be generalized to wider range of network models and constraints over the five-year
program duration.
In particular, as discussed above our team has made significant contributions to the characterization of
scaling laws and capacity regions as a function of delay and energy in MANETs, as well as the impact of
node cooperation on these tradeoffs. We plan to expand this work to characterize the capacity region
depicted in Figure 1 under general wireless network models. This characterization will likely build upon
the recent progress in capacity scaling laws in conjunction with the topological properties of network (such
as spectral graph properties).
Error exponents will also be a critical tool in optimizing capacity versus delay tradeoffs. A properly
designed code for a given rate can give a very large error exponent, which translates to low latency since
code words are short. This new viewpoint on coding leads to remarkable performance improvements in
point-to-point communications [HuaSM05]. We plan to generalize these results to networks, as short-time
decoding seems even more important in delay-constrained wireless networks with multihop routing than in
point-to-point links. In a sense, the diversity-multiplexing tradeoff is one call for designs of error exponent
optimal codes. Such codes can be widely used in wireless mobile networks, when the environment is
dynamic, and nodes would like to start cooperation quickly rather than waiting to receive a long codeword.
This also becomes important in node cooperation that employs relaying or conferencing under partial
decoding. Different error exponents are associated with different points on the capacity region. This
problem has already been explored by our team in [LunMKE05], and we plan to generalize this analysis to
investigate error exponents in heterogeneous networks.
In MANETs there is a fundamental tradeoff between energy and delay, since the minimum energy per bit
required for successful transmission entails infinite delay. The optimal trade-off between energy-per-bit and
delay scaling as well as the optimal throughput-delay scaling tradeoff at minimum energy-per-bit was
obtained in [ElGMPD04a, ElGMPD04b], and we plan to extend this tradeoff beyond scaling laws to the
entire capacity region. Node cooperation also plays a significant role in energy-delay tradeoffs, and our
team has demonstrated that cooperative transmission and reception can simultaneously achieve both energy
savings and delay reduction [CuiGB04]. These preliminary results were for a given network topology and
cooperative strategy – we plan to extend these ideas to more general topologies and cooperative techniques.
Cooperative strategies between multiple nodes naturally incur higher overhead in terms of communication
and computation, which can increase energy consumption and delay. However, such strategies may have
benefits for the overall capacity of the network. Thus the "degree of cooperation" is an important
dimension along which capacity needs to be evaluated; this will be particularly valuable if studied in
conjunction with the value of additional on-board computational resources. We plan to use foundational
methods from distributed control and game theory to provide quantitative insight into characterizing
bounds on the capacity/delay/energy tradeoffs associated with cooperation. Joint source-channel-network
coding plays a role in the energy-delay tradeoff as well, since we can achieve great savings in power when
a receiver wants not the data itself but merely a function of the data received at a collection of nodes in the
network, as is typical of sensor network applications.
Node mobility and the channel fading that results are generally considered to be detrimental to wireless
systems, but our team has shown that fading can actually increase network capacity when the fading gains
are known and can be exploited by network nodes [TouG04]. We plan to extend this work to determine the
impact of mobility and fading under different constraints, such as delay. Note that the benefits of mobility
and fading depend crucially on the availability of channel CSI. Indeed, it was shown recently that without
such CSI, there is no benefit to node cooperation in a MANET, since a given node has no knowledge of
which nodes to partner up with to cooperate [Jaf05]. The dynamic nature of MANETs will make it difficult
for the channel gain between pairs of nodes to be perfectly estimated at the transmitter or the receiver.
Therefore, it is important to characterize network capacity under imperfect or unknown channel CSI. This
is quite a challenging problem even for simple networks. For example, lack of transmitter CSI makes the
broadcast channel nondegraded, and hence its Shannon capacity region is unknown. The best way to
analyze capacity under unknown CSI, which mirrors techniques used in practice, is to introduce the notion
of capacity versus outage. In this setting the transmitter can optimize its transmission strategy given the
statistics of the CSI, and in unfavorable channel conditions that transmission may be corrupted, which is
precluded in traditional Shannon analysis. Capacity versus outage is commonly used to characterize MIMO
channels without transmitter CSI – we plan to extend these ideas to multihop networks. More generally, as
much of the theoretical study of network capacity is based on simplifying assumptions, such as a fully
scattered fading environment, a static or stationary fading process, perfect channel state information, no
peak power constraints and/or no signal processing cost, etc., the sensitivity of the capacity results with
respect to the variation of these modeling assumptions forms a new dimension of the problem. With the
sensitivity analysis, we plan to develop performance limits and optimal operations over networks that are
robust and agile.
MANETs must also be designed to deal with potential adversarial conditions, where information gathering
and transmission are not only limited by the network capacity, but also by external threats and other
handicaps inherent in the environments where these networks operate. These may arise from a multitude of
reasons, such as rogue users attempting to prioritize their traffic, intentional jamming, or attacks to the
network infrastructure. We plan to characterize the impact of security on network capacity in two ways: the
capacity reduction that results from assigning network degrees of freedom to ensure security, and in a
game-theoretic context as described in more detail in the response to Question 7 below.
Denial-of-service (DoS) attacks – whereby the attacker attempts to flood a network to prevent legitimate
traffic or attempts to disrupt the connection of a given user – are prevalent in wireless networks due to the
broadcast nature of wireless channels. Any information that is transmitted from the sender to the intended
receiver can be intercepted by attackers. Thus, for example, in CDMA systems, if the attacker listens the
channel long enough, it can estimate the spreading code used by sender- receiver pair and use this
knowledge to disrupt the received data. Note, however, that the detected signal is the convolution of the
receive waveform, channel response, and transmit waveform. Thus, the channel response between the
sender and the intended receiver can be used as a secret code between them, especially the directional part
of the channel response when MIMO systems are deployed. We will investigate the possibility of using
channel reciprocity and the directional channel response to deal with the DoS attacks in MANETs.
Network coding can also be exploited in DoS attacks, since it involves packets being linearly combined
together. Thus, it is conceivable that even a single malicious node could subvert the entire network, since
its fake packets could be mixed with and contaminate the information being generated by all the other
nodes in the network. The challenges for reliable communication using network codes are even greater in a
wireless environment, where it is possible that the malicious nodes can hear almost everything, packets can
be probabilistically erased, and there is no fixed channel topology. We plan to investigate a low-complexity
decontamination scheme for network coding that operates in a rateless manner. The information generated
by the adversary is treated as a new source in the network, and the task is to decode not only the source's
true information, but also the fake information being generated by the malicious node(s). To do this
requires us to first estimate the channel, i.e., the random decentralized network code used by the network.
We propose to show that when enough degrees of freedom get through to each receiver, as long as there is
a guarantee that even a small amount of provably uncontaminated information gets through to each
receiver, each receiver can use this information to boot-strap itself into estimating the channel, and the true
and fake information. Our team has some preliminary results in the wireline case, which points to the fact
that network coding can not only detect Byzantine attacks [HoLKMEK04], but can also, through the use of
MDS codes, effectively parry such attacks [JagLHE05] up to a fundamental limit.
Jamming is another mechanism for DoS that can be dealt with in an information theoretic context. In recent
work our team has explored jamming in wideband fading channels, and uncovered some surprising
findings: under some technical assumptions, a jammer is unable to significantly affect capacity or even
probability of error exponents for suitably randomized transmission schemes. We would like to explore
extensions of these results in a network setting, and see what type of randomization would be appropriate to
obtain high resilience to jamming.
Question 4: What enabling insights have recently emerged that give grounds for optimism? Are there any
theoretical breakthroughs or experimental results that suggest promising research directions for the
proposed program?
Many enabling insights on capacity of MANETs have been developed over the last few years and our
proposed work builds on many of these preliminary findings. In particular, scaling laws and the impact of
energy and delay on these laws, as described above, provide a simple characterization of network capacity
that is tractable and informative. There has also been much progress on new forms of node cooperation and
the capacity improvements that result (see above discussion). Network coding has taken the networking
community by storm, with a flurry of research over the last few years on its capacity and performance in
general network settings. Capacity definitions have also been expanded to include outage and diversitymultiplexing tradeoffs, especially in MIMO systems [ZheT03, TseVZ04]. The results and insights
associated with many of these techniques, along with our plan for building on them to characterize network
capacity in general settings and parameterizations, have been discussed in detail above.
In effect, the use of network coding promises to shed the artificial constraints posed on information
handling in the network. By removing such constraints, we may finally be able to consider, in a unified
manner, the use of all resources – be they energy or degrees of freedom afforded by bandwidth, time or
antennas. Conversely, all limitations to systems may be considered in terms of costs. Our preliminary work
in this area has already established a clean equivalence between channel uncertainty and energy cost in
multiple access channels [YouMZ04].
There has also be some recent development in coding techniques and applications that extend the
conventional block-coding framework. The systematic studies of variable length codes offer new insights
of dynamic information processing. Such dynamic processing is fundamentally different from the Shannontype capacity formulation, originally proposed to study static channels. The use of such techniques in
cooperation over networks has been particularly fruitful, including the dynamic decode-and-forward
operation at the relay node.
A further research direction that has been fairly active in recent years and that gives rise to some optimism
is centered around the notion of side information in communication scenarios. In fact, properly dealing
with side information at both the transmitter and the receiver appears to be a crucial step towards
decomposing network problems. The recent breakthrough results for the capacity region of Gaussian
MIMO broadcast channels [WeinSS04] is an excellent example of how a network problem can be
decomposed into a standard communication problem that is being solved in conjunction with side
information at the transmitter.
Question 5a: What are the principal obstacles to developing the desired theory? What
intermediate questions need to be answered to achieve the result?
There are many obstacles to developing the desired theory in the most general context. The problem of
network capacity under even relatively simple topologies and assumptions has been open for decades,
which indicates the challenges associated with this line of research and the fact that new perspectives and
insights are needed to make progress. This progress requires intermediate analysis for relatively basic
network assumptions and constraints. Once the scaling laws and/or capacity upper and lower bounds have
been obtained for these settings, it will be more apparent how to extend these results to more general
settings, as well as which of these extensions may appear intractable.
A major obstacle in obtaining capacity bounds and/or scaling laws is the complexity of large networks
under various system assumptions. To address this issue, we plan to investigate if the model reduction
techniques of [Mey04, ChePM03] can be used for complexity reduction. If so, then the next question is
how a given policy for relaxation can be adapted to the original network. Our team has also developed
extensive results on optimization techniques for very large systems, including MANETs [XiaJHBG03,
AceJO06, AceOS04, BerNO03, BoyL05, JohT05, JohT06]. We plan to leverage these preliminary results
to optimize capacity, node cooperation, joint source-channel-network coding, and network-aware
application design in MANETs.
Another obstacle to developing the desired theory is the tradeoff between accurate models versus problem
tractability relative to the assumptions we make about the wireless channel. For instance, the assumption
that the fading between different antennas in a MIMO setting is i.i.d. has allowed much progress in the field
of MIMO capacity. However, this assumption breaks down for closely packed antennas: if it held then
capacity could be increased indefinitely by packing more antennas onto a given device. The block-fading
model has also been used extensively in capacity analysis due to its tractability in obtaining results.
However, the ensuing capacity results can change dramatically under slight variations in the model
[KocL05]. Using assumptions that are physically plausible is required to develop capacity characterizations
that will significantly impact the engineering of real-world wireless networks. The intermediate question
that needs to be answered here is what wireless channel models allow for tractability of network capacity
calculations while still capturing the realities of the underlying physical channel model.
While a complete characterization of mobile wireless network capacity is a worthy goal, it is not the only
worthy goal. While we set development of such bounds as a critical long-term goal, we recognize the need
for good intermediate solutions to practical problems in this domain. A central tenet of our approach will
be the development of systematic techniques for deriving simple, calculable capacity bounds, source coding
limits, and, when separation fails, joint coding bounds for general mobile wireless networks. These
intermediate steps will not only be important in their own right, but will also provide insight into obtaining
the complete capacity characterization.
Question 5b: Is it possible to lay out a research agenda that marches inexorably toward the proposed
objective of capacity understanding, and what might the agenda involve?
The discussion above identifies a research agenda to address the open problem of network capacity
consisting of several main thrusts:
-
Definitions of capacity that extend Shannon’s definition to incorporate packet loss and
message error, and perhaps generalize the capacity region to a region of user utility
Upper and lower bounds on capacity/delay/energy tradeoffs regions under various
parameterizations
Scaling laws for capacity/delay/energy tradeoffs under various parameterizations
Robustness of capacity results to system assumptions
Optimized node cooperation in MANETs, including network coding, virtual MIMO,
generalized relaying with full/partial/no decoding, and conferencing
Optimized resource allocation across network degrees of freedom, including bandwidth,
power, and multiple antennas
Error exponents to address delay constraints
Separation theorems and joint source-channel-network coding
Network-aware application design and centralized/distributed optimization
Impact of network security on capacity
Our team has developed preliminary results in each of these thrust areas, and we have mapped out a
research agenda for extending these results with concrete intermediate milestones every eighteen months
over the five-year program duration. These intermediate results involve pursuing progressively more
general techniques and system assumptions that build on our early results developed for more limited
settings.
Question 6: Apart from MANET capacity understanding, what is the most important set of related tractable
network-information-theoretic questions that could be addressed within the timespan of the envisioned
program? What would be the practical uses of answers to those secondary questions?
For the capacity characterization of a MANET to be useful, the insights and techniques drawn from the
capacity-achieving strategies must be transferable to practical network designs. One of the key insights we
expect to obtain from the capacity analysis is optimal forms of node cooperation. We expect that the
optimal cooperation strategy will depend on various network assumptions and constraints, and the insights
obtained from capacity analysis can then be used to develop a unified form of node cooperation that adapts
to given operating conditions. Another important question for practical network designs is how the degrees
of freedom associated with multiple antennas should be used to optimize network performance: should they
be used for multiplexing, diversity, beamforming, or interference cancellation? We expect this question to
be answered from a capacity perspective in our proposed work, and this will provide insight into practical
designs as well.
Another key open problem is optimizing dynamic control of MANETs to achieve good performance. This
requires the development of a systematic approach for the analysis and synthesis of dynamic control
policies for wireless networks. We plan to explore the performance and robustness trade-offs inherent to
the resulting policies and to devise efficient distributed algorithms for their implementation. Among the
performance characteristics that our formulation will tackle are network delay and rate of convergence of
the obtained policies. Computational ideas will play a significant role in our developments, where
convexity properties will be exploited through the use of approximate dynamic programming techniques.
Capacity results often require centralized information and control, while in practice MANETs require
distributed local optimization. If the nodes of a MANET are considered to be distributed local optimizers,
then high performance protocols must determine how local optimization by the nodes yields good global
behavior. Traditional analysis techniques are ill-equipped to deal with this issue, because they do not
include any elements of the communication and computation constraints that are binding in MANET
environments. We are currently studying a family of models where information theoretic bounds are
placed on the communication ability of nodes, with a view towards finding game environments that have
provably good performance properties (e.g., high throughput or low delay). This research will yield design
recommendations for MANETs without centralized control where local optimization by many nodes is a
required feature. Game theory will also be used to address issues of network security. For example,
freeriding can be solved by introducing incentive mechanisms under the umbrella of game theory.
Spectrum allocation - how should spectrum be shared among competing users - is another major design
issue to be addressed in our proposed work [RayMZ06]. Opportunistic radios can find and use "white
spaces" in frequency, space, and time, but the techniques for doing so in an optimal distributed fashion are
unknown. Our capacity region analysis will yield great insight into how spectrum should be allocated in
MANETs, and we plan to use these insights to investigate dynamic distributed spectrum allocation
considering models that incorporate both technical aspects of wireless device management, as well as
economic aspects of resource sharing and incentive alignment. It is expected that this effort will give
specific protocol recommendations for dynamic spectrum allocation.
Network source coding bounds and techniques provide another critical element in the investigation
considered here. Understanding network capacity is futile without the source codes required to efficiently
gather and distribute the information acquired in MANETs. Generalizing capacity beyond asymptotically
negligible error probabilities is useless without source codes that can deliver meaningful information in the
resulting unreliable communication environment. Recent advances in source coding bounds and practical
code constructions. We plan to build on our preliminary work in this area (e.g. [ColME05, FenES05] in
developing network source coding bounds to complement the channel capacity regions.
Question 7: To what extent do results from other fields (e.g. physics) seem to contribute to the desired
theory and in what way?
The fields of electromagnetics, distributed optimization and control, economics and game theory, statistical
physics, and robust control contribute to various aspects of the desired theory. In particular, as mentioned
above, models used in capacity analysis often ignore the reality of signal propagation in wireless channels.
Capacity results for artificial channels will yield little insight into capacity limits of real networks. Hence
work that combines information theory with electromagnetics is needed to develop network capacity results
based on realistic underlying channel assumptions. The combination of electromagnetics and information
theory has been applied to MIMO channels in [Poo05, PooBT05], and we plan to extend these ideas to
wireless networks with multiple antenna nodes.
As mentioned above, capacity results often require centralized information and control, while in practice
MANETs require distributed local optimization Unfortunately, the best numerical methods for optimization
mostly rely on centralized control. The challenge of harnessing distributed, heterogeneous computing
resources to solve optimization problems has yet to be fully addressed, and requires expertise in the field of
distributed optimization. Fortunately, we have already seen that the use of network coding in multicast
settings replaces routing-based NP-complete problems with polynomial complexity problems that can be
solved in a distributed fashion [LunRKMAL05]. Thus, our taking full advantage of all degrees of freedom
may actually simplify, rather than complicate, optimization issues. We plan to develop this basic
framework for large-scale optimization, including new algorithms, methods, and protocols for convex
optimization in a distributed, networked environment. A well-designed framework for this distributed
network optimization will provide a number of benefits, including verifiability, plug and play operation,
self-optimization, and fault tolerance.
Economics and game theory provide a number of techniques that are needed to support users with diverse
bandwidth and QoS requirements (typically represented in terms of data transmission rate and associated
bit error rate). In the presence of heterogeneity in QoS requests, the resource allocation problem becomes
nonstandard and traditional network optimization techniques cannot be applied directly to come up with
efficient distributed methods that would yield optimal allocations. We plan to use the ``utilitymaximization'' framework of market economics, which is increasingly attracting attention in the
networking community, to provide different access privileges to users with different QoS requirements in a
distributed manner [Ozda06]. In this framework, each user (or equivalently traffic type) is represented by a
utility function that is a measure of his preferences over transmission rates. The overall goal is to devise
distributed algorithms for flow (congestion) control that maximizes the sum of the utilities of all the agents
in the system (i.e., maximize the social/system objective function). Economics can also be used in
optimization of network coding by determining a minimum-cost subgraph for a single multicast session
given a fixed (i.e., inelastic) rate demand, where the cost is defined as the sum of the link costs that may
represent monetary costs (which may be set by third parties, possibly for different objectives).
The fluid-model approach has proven to be a valuable tool to address stability [DaiM95], approximate
optimality in complex networks [Mey97, Mey01a], and as a means to model reduction in complex systems
[Mey01b, ChePS03]. This viewpoint is analogous to the O.D.E. method for recursive algorithms, or the
mean-field models in physics. Finally, robust control has developed advanced techniques for maintaining
controller stability in the face of modeling errors – such techniques may well be used to avoid fragility in
our capacity results due to modeling errors.
Team Collaborations:
The research team responding to this RFI has a long and extensive history of collaborations to address the
problems described in the RFI. These collaborations are illustrated below.
The joint publications resulting from these collaborations include [AceDMK05,
AceJO06, ColM01, ColM03, ColME05, ColMO06, ColLME04 , ColMEM05, DebEHKKLMR05,
EffG98a, EffG98a, EffKGM04, EffMHRKK03, ElGMPD04a, ElGMPD04a, [EryOM06], GalMBF98,
HoLKMEK04, HoMK02, HuaSM05, KimWME06, KimWME06, KoeM03, [JagLHE05], JinZM06,
LunMK06, LunMHK04, LunMKE05, LunRMKKHAZ, LuoMZ06, LunRKMAL05, LunMK06,
LunMKE05, LunMK05, LunMK05, MasMZ04, MedBFG99, MedEHK03, MedG95, MedG02, MedG99,
MedG97, MedHGMC04, PanHMMV05, PerMG03, RayMM06a, RayMM06b, RayMZ06, RayMZ06,
RayZM06, SunMKE06, TraRKLM06, WienSS04, WeiCM03, WeiCM04, XiaJHBG03, YouMZ04,
ZheM03, ZheT03,ZheTM04 ] .
References (those highlighted in magenta have web links)
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channels,” IEEE Trans. Inform. Theory, pp. 2020-2040, June 2005.
[HuaO06] X. Huang and A. Ozdaglar, “Power Control and Network Design in Mobile Sensor Networks,”
Proc. WIOPT, 2006.
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[PooBT05] A. S. Y. Poon, R. W. Brodersen, and D. N. C. Tse, “Degrees of freedom in multiple-antenna
channels: a signal space approach,” IEEE Trans. Inform. Theory, vol. 51, no. 2, pp. 523-536, Feb. 2005.
[RayMZ06] S. Ray, M. Médard and L. Zheng, "On Non-coherent MIMO Channels in the Wideband
Regime: Capacity and Reliability", submitted to IEEE Trans. Inform. Theory.
[RayMZA04] S. Ray, M. Médard, L. Zheng, J.Abounadi, “On the Sublinear Behavior of MIMO Channel
Capacity at low SNR,” International Symposium on Information Theory and its Applications (ISITA 2004),
October 2004, pp. 1031-1034
[RayMZ05a] S. Ray, M. Médard and L. Zheng, "Fiber Aided Wireless Network Architecture: A SISO
wireless-optical channel, Allerton Conference on Comunication, Control and Computing, Sep 2005
[RayMM06a] S. Ray, P. Moulin and M. Medard, ``On Jamming in the Wideband
Regime,'' submitted to IEEE Int. Symp. Inform. Theory, Jan. 2006.
[RayMM06b] S. Ray, P. Moulin and M. Medard, ``On Optimal Signaling and Jamming
Strategies in Wideband Fading Channels,'' submitted to IEEE Conf. ob Signal Processing Applications for
Wireless Communications (SPAWC), Jan.2006.
[RayMZ05b] S. Ray, M. Médard, L. Zheng, “Fiber Aided Wireless Network Architecture: A SISO
wireless-optical channel”, invited paper, Allerton Annual Conference on Communication, Control and
Computing, October 2005
[RayZM06] S. Ray, L. Zheng, M. Médard, “On Error Probability for Non-coherent MIMO Channels in the
Wideband Regime”, accepted to ISIT, July 2006
[RayMZ05c] S. Ray, M. Médard, L. Zheng, “On Error Probability for Wideband MIMO Channels”,
accepted to 2005 Conference on Information Sciences and Systems (CISS), Johns Hopkins, (5 pages),
March 2005
[RayMZ05d] S. Ray, M. Médard, L. Zheng, “Wideband Non-coherent MIMO Capacity”, International
Symposium on Information Theory, September 2005
[RezKV05] A. Reznik, S.R. Kulkarni, S. Verdu, Proc. IEEE Intl. Symp. Inform. Theory, pp. 820-824,
Sept. 2005
[RamKCE04] A. Ramamoorthy, K. Jain, P. A. Chou, and M. Effros, Separating Distributed Source Coding
from Network Coding, in Proc. Allerton Conference on Communication, Control and Computing, 2004
[SendEA] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity—Part I: System
description,” IEEE Trans. Commun., pp. 1927–1938, and Part II: Performance analysis, pp. 1939-1948,
Nov. 2003.
[SunMKE06] J.K. Sundararajan, M. Médard, R. Koetter, E. Erez, “A Systematic Approach to Network
Coding Problems using Conflict Graphs”, invited paper, UCSD Information Theory and Applications
Inaugural Workshop, February 2006
[TouG04] S. Toumpis and A. Goldsmith, “Large wireless networks under fading, mobility, and delay
constraints” Proc. INFOCOM, March 2004.
[TouG03] S. Toumpis and A. J. Goldsmith, “Capacity regions for wireless ad hoc networks”, IEEE Trans.
on Wireless Communications., pp. 736-748, July 2003.
[TraRKLM06] D. Traskov, N. Ratnakar, R. Koetter, D.S. Lun, M. Médard, “Network Coding for Multiple
Unicasts: An Approach based on Linear Optimization”, accepted to ISIT, July 2006
[TseVZ04] D. Tse, P. Viswanath, and L. Zheng, "Diversity-Multiplexing Tradeoff in
Multiple Access Channels", IEEE Transactions on Information Theory, Vol. 50,
September 2004, pp. 1859-74
[VisJ04] S. Vishwanath and S. Jafar, “On the capacity of vector Gaussian interference channels,”
IEEE Inf. Theory Wshop, pp. 365-369, Oct. 2004
[WienSS04] H. Weingarten, Y. Steinberg, and Shamai (Shitz) Shlomo, “The Capacity Region of the
Gaussian MIMO Broadcast Channel”, 38th Annual Conference on Information Sciences and Systems
(CISS 2004), March 17-19, 2004.
[Will83] F. M. J. Willems, “The discrete memoryless multiple access channel with
partially cooperating encoders,” IEEE Trans. Inform. Theory, vol. 29,
no. 3, pp. 441–445, May 1983.
[XiaJHBG03] L. Xiao, M. Hohansson, H. Hindi, S. Boyd, and A. Goldsmith, “Joint optimization of
communication rates and linear systems,” IEEE Trans. Auto. Cont. pp. 148-153, Jan. 2003.
[YouMZ04] P.G. Youssef-Massaad, M. Médard, L. Zheng, “On the Capacity of Multiple-access Channels
with Processing Power,” invited paper, Proc. Allerton Conference on Communication, Control and
Computing, pp. 1342-1351, Oct. 2004.
[WeiCMb04] G. Weichenberg, M. Médard, V. Chan, “Designing for Resilience to Multiple Failures,”
invited paper, Optical Fiber Communication Conference (OFC), sponsored by the Optical Society of
America, March 2004, pp. 224 – 227, Vol. 2
[WeiCMa04] G. Weichenberg, V. Chan, M. Médard, “High-Reliability Architectures for Networks Under
Stress,” Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM)
2004, March 2004, pp. 131-141
[WeiCM03] G. Weichenberg, V. Chan, M. Médard, “Reliable Architectures for Networks Under Stress,”
Fourth International Workshop on the Design of Reliable Communication Networks (DRCN), October
2003 , pp. 263 – 271, (WINNER OF THE BEST PAPER AWARD)
[ZheM03] C. Zheng, M. Médard, “How Far Should We Spread Using DS-CDMA in Time and Frequency
Selective Fading Channels?,” IEEE Global Telecommunications Conference (Globecom) Communication
Theory Workshop, December 2003, pp. 1563 - 1567 , Vol. 3
[ZheT03] L. Zheng and D. Tse, “Diversity and multiplexing: a fundamental tradeoff in multiple-antenna
channels,” IEEE Trans. Inform. Theory, pp. 1073-1096, May 2003.
[ZheTMa04] L. Zheng, D. Tse, M. Médard , “Channel Coherence in the Low SNR Regime,” invited paper,
38th Annual Conference on Information Sciences and Systems, Princeton, March 2004, paper 604 632 (6
pages, proceedings to appear)
[ZheTMb04] L. Zheng, D. Tse, M. Médard, “On the Costs of Channel State Information,” invited paper,
Information Theory Workshop, (6 pages) October 2004
[ZheMTL03] L. Zheng, M. Médard, D. Tse, C. Luo, “On the Interplay Between SNR and Coherence in
Wideband Channels,” invited paper, 41st Allerton Annual Conference on Communication, Control and
Computing, October 2003, pp. 420, 429, Vol. 1