Response to DARPA RFI 06-17Information Theory for Mobile Ad-Hoc Networks (MANETS)

Fundamental Capacity Limits and Optimized Node Cooperation in MANETs

Stanford MIT UIUC Caltech

Stephen Boyd Vincent Chan Todd Coleman Michelle Effros

Andrea Goldsmith Robert Gallager Ralf Koetter

Ramesh Johari Muriel Medard Sean Meyn

Balaji Prabhakar Asuman Ozdaglar Pierre Moulin

Devavrat Shah Ada Poon

Lizhong Zheng

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 non-vanishing error probabilities, may be incorporated into operational limits. Thus, our characterization of network capacity will include both traditional Shannon theoretic capacity (which will often be zero under practical constraints) as well as these more general capacity definitions, which 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

Capacity

Delay

Energy

Upper Bound

Lower Bound

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 minimum-entropy approaches is NP-complete, our recent results [ColME05] have shown that an asymptotically error-free 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 5-year 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 diversity-multiplexing 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 Shannon-type 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? Whatintermediate 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 ``utility-maximization'' 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)

[AceDMK05] S. Acedanski, S. Deb, M. Médard, and R. Koetter, “How Good is Random Linear Coding Based Distributed Networked Storage?”, invited paper, First Workshop on Network Coding, Theory, and Applications, (6 pages), April 2005 [AceJO06] D. Acemoglu, R. Johari, A. Ozdaglar, “Paradoxes of traffic engineering with partially optimal routing”.  Proc. Conf. Inform. Scien. Syst (CISS). To appear, 2006.[AceOS04] D. Acemoglu, A. Ozdaglar and R. Srikant, “The Marginal User Principle for Resource Allocation in Wireless Networks,” Proc. Conf. Dec. Cont., Dec. 2004.[BerNO03] Bertsekas, D. P., Nedic, A., and Ozdaglar A., Convex Analysis and Optimization, Athena Scientific, 2003.[BoyL05] S. Boyd and L. Vandenberghe, Convex Optimization, , Cambridge University Press, 2005. [ChenPM03] M. Chen, C. Pandit, and S. P. Meyn. “In search of sensitivity in network optimization”. Queueing Syst. Theory Appl., pp. 313–363, 2003.[ColME05] T. Coleman, M. Médard, and M. Effros, , “Practical Universal Decoding for Combined Routing and Compression in Network Coding” First Workshop on Network Coding, Theory, and Applications [ColMO06] T. Coleman, E. Martinian, E. Ordentlich. “Joint Source-Channel Decoding for Transmitting Correlated Sources over Broadcast Networks”. Submitted to 2006 IEEE Intern. Symp. Inform. Theory (ISIT), January 2006[ColMEM05] T. Coleman, E. Martinian, M. Effros, M. Medard. “Interference Management via Capacity-Achieving Codes for the Deterministic Broadcast Channel”. Proc. 2005 IEEE Inform. Theory Wshop. (ITW) , Rotorua, New Zealand, August 29 - Sep 1, 2005[ColM04] T.P. Coleman, M. Médard, “A Distributed Scheme for Achieving Energy-Delay Tradeoffs with Multiple Service Classes over a Dynamically Varying Network,” IEEE Journal on Selected Areas in Communications Special Issue on Advanced Mobility Management and QoS Protocols for Wireless Internet, Volume 22, Issue 5, June 2004, pp. 929-941 [ColM03] T. Coleman, M. Médard, "The Impact of User Information on Power-Delay Tradeoffs Between in Bursty Packetized Systems,” IEEE International Symposium on Information Theory (ISIT), June 2003, pg. 440 [ColM01] T.P. Coleman, M. Médard, “Trade-off Between Power Consumption and Delay in Wireless Packetized Systems,” invited paper, 39th Annual Allerton Conference on Communication, Control, and Computing, October 2001, pp. 501-512 [ColLME04b] T.P. C oleman, A.H. Lee, M. Médard, M. Effros, “A New Source-Splitting Approach to the Slepian-Wolf Problem,” IEEE International Symposium on Information Theory, June 2004, pg. 332 [ColME05] T.P. Coleman, M. Médard, M. Effros, “Towards Bridging the Gap Between Theory and Practice for the Slepian-Wolf Problem,” invited paper, IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), (4 pages, proceedings pending) March 2005 [ColLME04a] T. P. Coleman, A. H. Lee, M. Médard, M. Effros, "On Some New Approaches to Practical Slepian-Wolf Compression Inspired by Channel Coding,” 2004 IEEE Data Compression (DCC) Conference, Snowbird Utah, March 2004, pp. 282-291 [Cov91] T. M. Cover and J. A. Thomas, Elements of Information Theory. Wiley-Interscience, 1991.[CuiGB04] S. Cui, A. J. Goldsmith and A. Bahai, “Energy-efficiency of MIMO and Cooperative MIMO in Sensor Networks ”, IEEE J. Select. Areas Commun., Aug. 2004. [DanGHEM00] A.F. Dana, R. Gowaikar, B. Hassibi, M. Effros, M. Médard, “Should we Break a Wireless Network into Subnetworks?,” 41st Allerton Annual Conference on Communication, Control and Computing, October 2000, pp. 50 – 59, Vol. 1 [DaiM95] J. G. Dai and S. P. Meyn. Stability and convergence of moments for multiclass queueing networks via fluid limit models. IEEE Trans. Automat. Control, pp. 1889–1904, Nov. 1995.[DebEHKKLMR05] S. Deb, M. Effros, T. Ho, D. R. Karger, R. Koetter, D. S. Lun, M. Médard, and N. Ratnakar. “Network coding for wireless applications: A brief tutorial.” Proc. International Workshop on Wireless Ad-hoc Networks (IWWAN) 2005, May 2005. Invited paper. [DigGT05] S. Diggavi, M. Grossglauser, and D. Tse, “Even One-Dimensional Mobility Increases the Capacity of Wireless Networks,” IEEE Trans. Inform. Theory, pp. 3947-3954, Nov. 2005.[EffG98a] M. Effros and A. Goldsmith, “Capacity definitions and coding strategies for general channels with receiver side information” Proc. IEEE Int. Symp. Inform. Theory, pg. 39, Aug. 1998.

[EffG98b] M. Effros and A. Goldsmith, “Joint design of fixed-rate source codes and multiresolution channel codes”, IEEE Trans. Commun.. pp. 1301-1312, Oct. 1998. [EffMRKK03] M. Effros, M. Médard, T. Ho, S. Ray, D. Karger, R. Koetter, “Linear Network Codes: A Unified Framework for Source Channel, and Network Coding,” invited paper to the DIMACS Workshop on Network Information Theory, 2003, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, volume 66, American Mathematical Society, pp. 197 - 216 [EffKGM04] M. Effros, R. Koetter, A. Goldsmith, and M. Medard, “On source and channel codes for multiple inputs and outputs: does multiple description beat space time?” Proc IEEE Inform. Theory Wshop, pp. 324-329, Oct. 2004. [EffMHRKK03] M. Effros, M. Medard, T. Ho, S. Ray, D. Karger and R. Koetter, “Linear network codes:  a unified framework for source, channel, and network coding”, Proc. DIMACS Workshop on Network Information Theory, 2003. [ElGMPD04a] “Throughput-delay trade-off in wireless networks”, A. El Gamal, J. Mammen, B. Prabhakar, and D. Shah, Proc. INFOCOM, March 2004.[ElGMPD04b] A. El Gamal, J. Mammen, B. Prabhakar, and D. Shah, Throughput-delay trade-off in energy constrained wireless networks, Proc. Intl. Symp. Inform. Theory, pg. 439, June-July 2004[ElG05] H. El Gamal, “On the scaling laws of dense wireless sensor networks: the data gathering channel,” IEEE Trans. Inform. Theory, pp. 1229-1234, March 2005. [EphH98] A. Ephremides and B. Hajek, “Information theory and communication networks: an unconsummated union” IEEE Trans. Inform. Theory, pp. 2416-2434, Oct. 1998.[EryOM06] A. Eryilmaz, A.Ozdaglar, M. Médard, “On Delay Performance Gains from Network Coding”, invited paper, Proceedings of the Conference on Information Sciences and Systems, Princeton, 2006.[FenES04] H. Feng, M. Effros, and S. A. Savari, “Functional source coding for networks with receiver side information,” Proc. Aller. Conf. Commun. Cont. Comp. Pp. 1419-1427, Sept. 2004. [MedGb97] R.G. Gallager, M. Médard, “Bandwidth Scaling for Fading Channels,” IEEE International Symposium on Information Theory (ISIT), July 1997. pg. 471[GalMBF98] R.G. Gallager, M. Médard, R.A. Barry, S.G. Finn, “Multicast Automatic Protection Switching in Arbitrary Redundant Graphs,” IEEE International Conference on Communications (ICC), June 1998, pp. 640 – 644, Vol.1[GasV05] M. Gastpar and M. Vetterli, “On the capacity of large Gaussian relay networks,” IEEE Trans. Inform. Theory, pp. 765-779, March 2005[GasV04] M. Gastpar and M. Vetterli, “Power-bandwidth-distortion scaling laws for sensor networks,” Intl. Symp. Inf. Proc. Sens. Net., pp. 320-329, April 2004[Gur06] V. Guruswami. “ List Decoding from Erasures: Bounds and Code Constructions.” IEEE Trans. Inform. Theory, to appear.[HoLKMEK04] T. Ho, B. Leong, R. Koetter, M. Médard, M. Effros and D. R. Karger, “Byzantine Modification Detection in Multicast Networks Using Randomized Network Coding”, International Symp. Inform. Theory (ISIT), 2004[HoMK02a] T. Ho, M. Médard, R. Koetter , “A Coding View of Network Capacity, Recovery and Management,” International Symposium on Information Theory, July 2002, pg. 137[HoMK02b] T. Ho, M. Médard, R. Koetter, “A Coding View of Network Recovery and Management for Single Receiver Communications,” Conference on Information Sciences and Systems (CISS), Princeton, April 2002, pp. 590- 597 [Hos05] A. Host-Madsen, “The multiplexing gain of wireless networks,” in Proc. IEEE Int. Symp. Inform. Theory, pp. 2065–2069, Sept. 2005[Hos04] A. Host-Madsen, “On the achievable rate for receiver cooperation inad-hoc networks,” Proc. IEEE Int. Symp. Inform. Theory, p. 272, June 2004.[HosZ05] A. Host-Madsen and J. Zhang, “Capacity bounds and power allocation for wireless relay 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. [HuaSM05] J. Huang, S. Meyn, and M. Medard. Error exponents for channel coding and signal constel-lation design. Journal of Selected Areas in Comm. Special Issue on Nonlinear Optimization of Communication Systems (submitted), 2005.[Jaf05] S. Jafar, “Too much mobility limits the capacity of wireless ad hoc networks,” IEEE Trans. Inform. Theory, pp. 3954-3965, Nov. 2005.

[JagEHM04] S. Jaggi, M Effros, T. Ho, M. Médard, “On Linear Network Coding.”, invitedpaper, 42nd Allerton Annual Conference on Communication, Control and Computing,October 2004, paper 42-263[JagLHE05] S. Jaggi, M. Langberg, T. Ho and M. Effros, Correction of Adversarial Errors in Networks, Int. Symp. Inform. Theory (ISIT), 2005[JinMA04]N. Jindal, U. Mitra, and A. Goldsmith, “Capacity of ad-hoc networks with node cooperation,” Proc. IEEE Int. Symp. Inform. Theory, p. 271, June 2004,[JinZM06] S. Jing, L. Zheng, M. Médard, “On the use of sounding in wideband channels”, invited paper, IEEE International Zurich Seminar on Communications, February 2006[JohT06] R. Johari and J.N. Tsitsiklis, “A scalable network resource allocation mechanism with bounded efficiency loss,”.  To appear, J. Select. Areas Commun.[JohT05] R. Johari and J.N. Tsitsiklis, “Communication requirements of VCG-likemechanisms in convex environments,” Allerton Conference on Communication,Control, and Computing, 2005.[KhoSA04] M. A. Khojastepour, A. Sabharwal, and B. Aazhang, “Improved achievable rates for user cooperation and relay channels,” Proc. IEEE Int. Symp. Inform. Theory, p. 4. June 2004.[KimWME06] M. Kim, C. Wook Ahn, M. Médard, M. Effros, “On Minimizing Network Coding Resources: An Evolutionary Approach”, Netcod 2006 (Second Workshop on Network Coding, Theory, and Applications), April 2006[KocL05] T. Koch and A. Lapidoth, “The fading number and degrees of freedom in non-coherent MIMO fading channels: a peace pipe,” Proc. IEEE Int. Symp. Inform. Theory, pp. 661-665, Sept. 2005. [KoeM01] R. Koetter, M. Médard, “An algebraic approach to network coding and robust networks,” IEEE International Symposium on Information Theory (ISIT), June 2001, pg. 104[KoeM03] R. Koetter, M. Médard. “Beyond Routing: An Algebraic Approach to Network Coding,” IEEE/ACM Transactions on Networking, pp. 782-796, Oct. 2003 (selected as one of the outstanding papers from INFOCOM for transfer to IEEE/ACM Transactions on Networking).[KoeEHM04] R. Koetter, M. Effros, T. Ho, M. Médard, “Network Codes as Codes on Graphs,” invited paper, 38th Annual Conference on Information Sciences and Systems, Princeton, March 2004, paper 632 (6 pages, proceedings to appear) [KraGG05] G. Kramer, M. Gastpar, and P. Gupta, “Cooperative strategies and capacity theorems for relay networks,” IEEE Trans. Inform. Theory, pp. 3037-3063, Sept. 2005. [LanTW04] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversityin wireless networks: Efficient protocols and outage behavior,” IEEE Trans. Inform. Theory, pp. 3062–3080, Dec. 2004. [LapSW05] A. Lapidoth, S. Shamai (Shitz), M. A. Wigger, “On the Capacity of Fading MIMO Broadcast Channels with Imperfect Transmitter Side-Information,” Proceedings Allerton Conference on Communication, Control, and Computing, Sept. 28–30, 2005. (Invited Paper ).[LiaK] X. Liang-Liang and P.R. Kumar, “A network information theory for wireless communication: scaling laws and optimal operation,” IEEE Trans. Inform. Theory, pp. 748-767, May 2004. [LuoMZ06] Luo, C.,  M. Médard, L. Zheng, "On Achieving Wideband Capacity Using Multitone FSK,"To appear, J. Select Areas Commun. Special Issue on Differential and Noncoherent Wireless Communications.[LuoMZL05] C. Luo, M. Médard, L. Zheng, D.S. Lun, “Multi-tone FSK with Feedback”, International Symposium on Information Theory, September 2005 [LunRKMAL05] D. S. Lun, N. Ratnakar, R. Koetter, M. Médard, E. Ahmed, and H. Lee. “Achieving Minimum Cost Multicast: A Decentralized Approach Based on Network Coding.” Proc. IEEE Infocom, pp. 1607-1617, March 2005[LunMK06] D. S. Lun, M. Médard, and R. Koetter. “Network Coding for Efficient Wireless Unicast.” In Proc. 2006 International Zurich Seminar on Communications (IZS 2006), February 2006. Invited paper. [LunMKE05] D. S. Lun, M. Médard, R. Koetter, and M. Effros. “Further results on coding for reliable communication over packet networks”. In Proc. 2005 IEEE International Symposium on Information Theory (ISIT 2005), pages 1848-1852, Sept. 2005. [LunMK05] D. S. Lun, M. Médard, and R. Koetter. “Efficient operation of wireless packet networks using network coding.” In Proc. Intn. Wshop Conv. Tech. (IWCT) 2005, June 2005. Invited paper. [LunRMKKHAZ] D. S. Lun, N. Ratnakar, M. Médard, R. Koetter, D. R. Karger, T. Ho, E. Ahmed, F. Zhao, “Minimum-Cost Multicast over Coded Packet Networks”, accepted to the IEEE Transactions on

Information Theory [LunMHK04] D.S. Lun, M. Médard , T. Ho, R. Koetter, “Network Coding with a Cost Criterion,” International Symposium on Information Theory and its Applications (ISITA 2004), October 2004, pp. 1232-1237 [LunMKE05] D.S. Lun, M. Médard, R. Koetter, M. Effros, “Further Results on Coding for Reliable Communication over Packet Networks”, International Symposium on Information Theory, September 2005 [LunMK06] D.S. Lun, M. Médard, R. Koetter, “Network Coding for Efficient Wireless Unicast”, invited paper, IEEE International Zurich Seminar on Communications, February 2006 [LunPFMK06] D.S. Lun, P. Pakzad, C. Fragouli, M. Médard, R. Koetter, “An Analysis of Finite-Memory Random Linear Coding on Packet Streams”, Netcod 2006 (Second Workshop on Network Coding, Theory, and Applications), April 2006 [MasMZ04] P.G. Massaad, M. Médard, L. Zheng, “Impact of Processing Energy on the Capacity of Wireless Channels,” International Symposium on Information Theory and its Applications (ISITA 2004), October 2004, pp. 1580-1585 [MedG97] M. Médard, A.J. Goldsmith, “Capacity of Time-Varying Channels with Channel Side Information,” IEEE International Symposium on Information Theory (ISIT), July 1997, pg. 372[MedG99] M. Médard, A.J. Goldsmith, “Capacity of Time-Varying Channels with Side Information at the Sender and the Receiver,” Miniconference on Information Theory, IEEE International Conference on Communications (ICC), June 1999. pp. 16-20[MedEHK03] M. Médard, M. Effros, T. Ho, D. Karger, “On Coding for Non-Multicast Networks,” invited paper, 41st Allerton Annual Conference on Communication, Control and Computing, October 2003, pp. 11 – 20, Vol. 1[MedG02] M. Médard, R.G. Gallager, “Bandwidth Scaling for Fading Multipath Channels,” IEEE Transactions on Information Theory, Volume 48, Issue 4 , April 2002 , pp. 840 -852[MedG97a] M. Médard, R.G. Gallager, “The Effect of Channel Variations upon Capacity,” IEEE Vehicular Technology Conference (VTC), April 1996, pp. 1781 – 1785, Vol.3[MedG95a] M. Médard, R.G. Gallager, “The Effect of a Randomly Time-varying Channel on Mutual Information,” IEEE International Symposium on Information Theory (ISIT), September 1995, pg. 139[MedG95b] M. Médard, R.G. Gallager, “The Issue of Spreading in Multipath Time-Varying Channels,” IEEE Vehicular Technology Conference (VTC), July 1995, pp. 1 – 5, Vol.1[MedHGMC04] M. Medard, J. Huang; A. Goldsmith, S.P. Meyn, and T.P. Coleman, “Capacity of time-slotted ALOHA packetized multiple-access systems over the AWGN channel,” IEEE Trans. Wireless Commun., pp. 486-499, March 2004[MedBFG98] M. Médard, R.A. Barry, S.G. Finn, R.G. Gallager, “Automatic Protection Switching for Multicasting in Optical Mesh Networks,” in Trends in Optics and Photonics (TOPS) – Volume 20 - Optical Networks and their Applications, R.A. Barry, editor, published by the Optical Society of America, 1998, pp. 216-221[MedBFG99] M. Médard, S.G. Finn, R.A. Barry, R.G. Gallager, “Redundant Trees for Preplanned Recovery in Arbitrary Vertex-Redundant or Edge-Redundant Graphs,” IEEE/ACM Transactions on Networks, Volume 7, Issue 5, October 1999, pp. 641 -652 [Mey04] S. P. Meyn. Control Techniques for Complex Networks (new monograph and graduatecourse at UIUC). http://decision.csl.uiuc.edu/~meyn/pages/497SM.html, 2004.[Mey97] S. P. Meyn. Stability and optimization of queueing networks and their fluid models. Math. Stoc. Man. Sys. (Williamsburg, VA, 1996), pages 175–199. Amer. Math. Soc., Providence, RI, 1997.[Mey01a] S. P. Meyn. Sequencing and routing in multiclass queueing networks. Part I: Feedback regulation. SIAM J. Control Optim., 40(3):741–776, 2001.[Mey01b] S. P. Meyn. Sequencing and routing in multiclass queueing networks. Part II: Workloadrelaxations. SIAM J. Control Optim., 42(1):178–217, 2003.[NgA04] C. T. K. Ng and A. J. Goldsmith, “Transmitter cooperation in ad-hocwireless networks: Does dirty-payer coding beat relaying?” in Proc.IEEE Inform. Theory Workshop, pp. 277–282, June 2004.[NgA05] C. T. K. Ng and A. J. Goldsmith,, “Capacity gain from transmitter and receiver cooperation,” Proc. IEEE Int. Symp. Inform. Theory, pp. 397–401, Sept. 2005.[NgA06] C. T. K. Ng and A. J. Goldsmith,, “Capacity and power allocation for transmitter and receivercooperation in fading channels,” to appear, Proc. IEEE Int. Conf. Commun., 2006.[NgLG06] C. T. K. Ng, J. N. Laneman, and A. J. Goldsmith, “The role of SNR inachieving MIMO rates in cooperative systems,” in Proc. IEEE Inform.Theory Workshop, 2006

[NgMGSY06] C. T. K. Ng, I. Maric, A. J. Goldsmith, S. Shamai, and R. D. Yates, “Iterative and one-shot conferencing in relay channels,” to appear, Proc. IEEE Inform. Theory Workshop, 2006.[Ozda06] A. Ozdaglar, “Price Competition with Elastic Traffic,” submitted to Networks, special issue on Games, Interdiction, and Human Interaction Problems in Networks, 2006.[OzdaT06] A. Ozdaglar and P. Tseng, “Existence of Global Minima for Constrained Optimization,” Journal of Optimization Theory and Applications, vol. 128, no. 3, 2006.[PanHMMV05] C. Pandit, J. Huang, S. Meyn, M. Medard, and V. Veeravalli, “Entropy, Inference, and Channel Coding”. Proceedings of the Institute for Mathematics and its Applications Summer Workshop on Wireless Communications, (25 pages), July 2005.[PerMG03] N. Peranginangin, M. Médard, R. Gallager, “Capacity of a Multi Output Channel with Distributed Processing,” IEEE International Symposium on Information Theory (ISIT), Yokohoma, Japan, June 2003, pg. 97[Poo05] A. S. Y. Poon, “Angular-domain processing for MIMO systems,” submitted to IEEE Trans. Signal Processing, Dec. 2005.[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 WidebandRegime,'' submitted to IEEE Int. Symp. Inform. Theory, Jan. 2006.[RayMM06b] S. Ray, P. Moulin and M. Medard, ``On Optimal Signaling and JammingStrategies in Wideband Fading Channels,'' submitted to IEEE Conf. ob Signal Processing Applications forWireless 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 inMultiple 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 withpartially 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