The Balanced Unicast and Multicast Capacity Regions of Large Wireless Networks

We consider the question of determining the scaling of the $n^2$-dimensional balanced unicast and the $n 2^n$-dimensional balanced multicast capacity regions of a wireless network with $n$ nodes placed uniformly at random in a square region of area $n$ and communicating over Gaussian fading channels. We identify this scaling of both the balanced unicast and multicast capacity regions in terms of $\Theta(n)$, out of $2^n$ total possible, cuts. These cuts only depend on the geometry of the locations of the source nodes and their destination nodes and the traffic demands between them, and thus can be readily evaluated. Our results are constructive and provide optimal (in the scaling sense) communication schemes.Comment: 37 pages, 7 figures, to appear in IEEE Transactions on Information Theor

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

Computation in Multicast Networks: Function Alignment and Converse Theorems

The classical problem in network coding theory considers communication over multicast networks. Multiple transmitters send independent messages to multiple receivers which decode the same set of messages. In this work, computation over multicast networks is considered: each receiver decodes an identical function of the original messages. For a countably infinite class of two-transmitter two-receiver single-hop linear deterministic networks, the computing capacity is characterized for a linear function (modulo-2 sum) of Bernoulli sources. Inspired by the geometric concept of interference alignment in networks, a new achievable coding scheme called function alignment is introduced. A new converse theorem is established that is tighter than cut-set based and genie-aided bounds. Computation (vs. communication) over multicast networks requires additional analysis to account for multiple receivers sharing a network's computational resources. We also develop a network decomposition theorem which identifies elementary parallel subnetworks that can constitute an original network without loss of optimality. The decomposition theorem provides a conceptually-simpler algebraic proof of achievability that generalizes to $L$-transmitter $L$-receiver networks.Comment: to appear in the IEEE Transactions on Information Theor

Multiple Unicast Capacity of 2-Source 2-Sink Networks

We study the sum capacity of multiple unicasts in wired and wireless multihop networks. With 2 source nodes and 2 sink nodes, there are a total of 4 independent unicast sessions (messages), one from each source to each sink node (this setting is also known as an X network). For wired networks with arbitrary connectivity, the sum capacity is achieved simply by routing. For wireless networks, we explore the degrees of freedom (DoF) of multihop X networks with a layered structure, allowing arbitrary number of hops, and arbitrary connectivity within each hop. For the case when there are no more than two relay nodes in each layer, the DoF can only take values 1, 4/3, 3/2 or 2, based on the connectivity of the network, for almost all values of channel coefficients. When there are arbitrary number of relays in each layer, the DoF can also take the value 5/3 . Achievability schemes incorporate linear forwarding, interference alignment and aligned interference neutralization principles. Information theoretic converse arguments specialized for the connectivity of the network are constructed based on the intuition from linear dimension counting arguments.Comment: 6 pages, 7 figures, submitted to IEEE Globecom 201