A Generalized Cut-Set Bound for Deterministic Multi-Flow Networks and its Applications

We present a new outer bound for the sum capacity of general multi-unicast deterministic networks. Intuitively, this bound can be understood as applying the cut-set bound to concatenated copies of the original network with a special restriction on the allowed transmit signal distributions. We first study applications to finite-field networks, where we obtain a general outer-bound expression in terms of ranks of the transfer matrices. We then show that, even though our outer bound is for deterministic networks, a recent result relating the capacity of AWGN KxKxK networks and the capacity of a deterministic counterpart allows us to establish an outer bound to the DoF of KxKxK wireless networks with general connectivity. This bound is tight in the case of the "adjacent-cell interference" topology, and yields graph-theoretic necessary and sufficient conditions for K DoF to be achievable in general topologies.Comment: A shorter version of this paper will appear in the Proceedings of ISIT 201

Capacity Results for Interference Networks and Nested Cut-Set Bound

In this thesis, a full characterization of the sum-rate capacity for degraded interference networks with any number of transmitters, any number of receivers, and any possible distribution of messages among transmitters and receivers is established. It is proved that a successive decoding scheme is sum-rate optimal for these networks. Moreover, it is shown that the transmission of only a certain subset of messages is sufficient to achieve the sum-rate capacity for such networks. Algorithms are presented to determine this subset of messages explicitly. The sum-rate expression for the degraded networks is then used to derive a unified outer bound on the sum-rate capacity of arbitrary (non-degraded) interference networks. Several variations of degraded networks are identified for which the derived outer bound is sum-rate optimal. Specifically, noisy interference regimes are derived for certain classes of multi-user/multi-message large interference networks. Also, network scenarios are identified where the incorporation of both successive decoding and treating interference as noise achieves their sum-rate capacity. Next, by taking insight from the results for degraded networks, an extension to the standard cut-set bound for general communication networks is presented which is referred to as nested cut-set bound. This bound is derived by applying a series of cuts in a nested configuration to the network first and then bounding the information rate that flows through the cuts. The key idea for bounding step is indeed to impose a degraded arrangement among the receivers corresponding to the cuts. Therefore, the bound is in fact a generalization of the outer bound for interference networks: here cooperative relaying nodes are introduced into the problem as well but the proof style for the derivation of the outer bound remains the same. The nested cut-set bound, which uniformly holds for all general communication networks of arbitrary large sizes where any subset of nodes may cooperatively communicate to any other subset of them, is indeed tighter than the cut-set bound for networks with more than one receiver. Moreover, it includes the generalized cut-set bound for deterministic networks reported by Shomorony and Avestimehr which was originally a special case of the outer bound established for the interference networks by the author (2012). Finally, capacity bounds for the two-user interference channel with cooperative receivers via conferencing links of finite capacities are investigated. The capacity results known for this communication scenario are limited to a very few special cases of the one-sided channel. One of the major challenges in analyzing such cooperative networks is how to establish efficient capacity outer iv bounds for them. In this thesis, by applying new techniques, novel capacity outer bounds are presented for the interference channels with conferencing users. Using the outer bounds, several new capacity results are proved for interesting channels with unidirectional cooperation in strong and mixed interference regimes. A fact is that the conferencing link (between receivers) may be utilized to provide one receiver with information about its corresponding signal or its non-corresponding signal (interference signal). As an interesting consequence, it is demonstrated that both strategies can be helpful to achieve the capacity of the channel. Lastly, for the case of Gaussian interference channel with conferencing receivers, it is argued that our outer bound is strictly tighter than the previous one derived by Wang and Tse

Generalized Cut-Set Bounds for Broadcast Networks

A broadcast network is a classical network with all source messages collocated at a single source node. For broadcast networks, the standard cut-set bounds, which are known to be loose in general, are closely related to union as a specific set operation to combine the basic cuts of the network. This paper provides a new set of network coding bounds for general broadcast networks. These bounds combine the basic cuts of the network via a variety of set operations (not just the union) and are established via only the submodularity of Shannon entropy. The tightness of these bounds are demonstrated via applications to combination networks.Comment: 30 pages, 4 figures, submitted to the IEEE Transaction on Information Theor

Linear Network Coding for Two-Unicast-ZZ Networks: A Commutative Algebraic Perspective and Fundamental Limits

We consider a two-unicast-$Z$ network over a directed acyclic graph of unit capacitated edges; the two-unicast-$Z$ network is a special case of two-unicast networks where one of the destinations has apriori side information of the unwanted (interfering) message. In this paper, we settle open questions on the limits of network coding for two-unicast-$Z$ networks by showing that the generalized network sharing bound is not tight, vector linear codes outperform scalar linear codes, and non-linear codes outperform linear codes in general. We also develop a commutative algebraic approach to deriving linear network coding achievability results, and demonstrate our approach by providing an alternate proof to the previous results of C. Wang et. al., I. Wang et. al. and Shenvi et. al. regarding feasibility of rate $(1,1)$ in the network.Comment: A short version of this paper is published in the Proceedings of The IEEE International Symposium on Information Theory (ISIT), June 201

Network error correction with unequal link capacities

This paper studies the capacity of single-source single-sink noiseless networks under adversarial or arbitrary errors on no more than z edges. Unlike prior papers, which assume equal capacities on all links, arbitrary link capacities are considered. Results include new upper bounds, network error correction coding strategies, and examples of network families where our bounds are tight. An example is provided of a network where the capacity is 50% greater than the best rate that can be achieved with linear coding. While coding at the source and sink suffices in networks with equal link capacities, in networks with unequal link capacities, it is shown that intermediate nodes may have to do coding, nonlinear error detection, or error correction in order to achieve the network error correction capacity