We consider the problem of network coding across three unicast sessions over a directed acyclic graph, where the sender and receiver of each unicast session are both connected to the network via a single edge of unit capacity. We consider a network model in which the middle of the network can only perform random linear network coding, and restrict our approaches to precoding-based linear schemes, where the senders use precoding matrices to encode source symbols. We adapt a precoding-based interference alignment technique, originally developed for the wireless interference channel, to construct a precoding-based linear scheme, which we refer to as precoding-based network alignment scheme (PBNA). A primary difference between this setting and the wireless interference channel is that the network topology can introduce dependencies among the elements of the transfer matrix, which we refer to as coupling relations, and can potentially affect the achievable rate of PBNA. We identify all these coupling relations and interpret them in terms of network topology. We then present polynomial-time algorithms to check the presence of these coupling relations in a particular network. Finally, we show that, depending on the coupling relations present in the network, the optimal symmetric rate achieved by precoding-based linear scheme can take only three possible values, all of which can be achieved by PBNA.

We consider the problem of learning the underlying graph structure of discrete Markov networks based on power-law graphs, generated using the configuration model. We translate the learning problem into an equivalent channel coding problem and obtain necessary conditions for solvability in terms of problem parameters. In particular, we relate the exponent of the power-law graph to the hardness of the learning problem, and show that more number of samples are required for exact recovery of discrete power-law Markov graphs with small exponent values. We develop an efficient learning algorithm for accurate reconstruction of graph structure of Ising model on power-law graphs. Finally, we show that order-wise optimal number of samples suffice for recovering the exact graph under certain constraints on Ising model parameters and scalings of node degrees.

We study epidemic spreading processes in large networks, when the spread is assisted by a small number of external agents: infection sources with bounded spreading power, but whose movement is unrestricted vis-à-vis the underlying network topology. For networks, which are spatially constrained, we show that the spread of infection can be significantly speeded up even by a few such external agents infecting randomly. Moreover, for general networks, we derive upper bounds on the order of the spreading time achieved by certain simple (random/greedy) external-spreading policies. Conversely, for certain common classes of networks such as line graphs, grids, and random geometric graphs, we also derive lower bounds on the order of the spreading time over all (potentially network-state aware and adversarial) external-spreading policies; these adversarial lower bounds match (up to logarithmic factors) the spreading time achieved by an external agent with a random spreading policy. This demonstrates that random, state-oblivious infection-spreading by an external agent is in fact order-wise optimal for spreading in such spatially constrained networks.

This paper considers the problem of compressive sensing over a finite alphabet, where the finite alphabet may be inherent to the nature of the data or a result of quantization. We show that there are significant benefits to analyzing the problem while incorporating its finite alphabet nature, versus ignoring it and employing a conventional real alphabet based toolbox. Specifically, when the alphabet is finite, our techniques have a lower sample complexity compared to real-valued compressive sensing for low levels of sparsity, facilitate constructive designs of sensing matrices based on coding-theoretic techniques, and allow for lesser amount of data storage.

We consider the problem of linear network coding over communication networks, representable by directed acyclic graphs, with multiple groupcast sessions: the network comprises of multiple destination nodes, each desiring messages from multiple sources. We adopt an interference alignment perspective, providing new insights into designing practical network coding schemes as well as the impact of network topology on the complexity of the alignment scheme. In particular, we show that under certain (polynomial-time checkable) constraints on networks with K sources, it is possible to achieve a rate of 1/(L+d+1) per source using linear network coding coupled with interference alignment, where each destination receives messages from L sources (L <; K), and d is a parameter, solely dependent on the network topology, that satisfies 0 ≤ d <; K - L.

This paper presents a rate distortion approach to Markov graph learning. It provides lower bounds on the number of samples required for any algorithm to learn the Markov graph structure of a probability distribution, up to edit distance. We first prove a general result for any probability distribution, and then specialize it for Ising and Gaussian models. In particular, for both Ising and Gaussian models on p variables with degree at most d, we show that at least Ω((d - s/p)log p) samples are required for any algorithm to learn the graph structure up to edit distance s. Our bounds represent a strong converse; i.e., we show that for a lower number of samples, the probability of error goes to 1 as the problem size increases. These results show that substantial gains in sample complexity may not be possible without paying a significant price in edit distance error.

We study infection spreading on large static networks when the spread is assisted by a small number of additional virtually mobile agents. For networks which are “spatially constrained”, we show that the spread of infection can be significantly sped up even by a few virtually mobile agents acting randomly. More specifically, for general networks with bounded virulence (e.g., a single or finite number of random virtually mobile agents), we derive upper bounds on the order of the time taken (as a function of network size) for infection to spread. Conversely, for certain common classes of networks such as linear graphs, grids and random geometric graphs, we also derive lower bounds on the order of the spreading time over all (potentially network-state aware and adversarial) virtual mobility strategies. We show that up to a logarithmic factor, these lower bounds for adversarial virtual mobility match the upper bounds on spreading via an agent with random virtual mobility. This demonstrates that random, state-oblivious virtual mobility is in fact order-wise optimal for dissemination in such spatially constrained networks.

This paper presents the concept of incremental traceback for determining changes in the trace of a network as it evolves with time. A distributed algorithm, based on the methodology of algebraic traceback developed by Dean et al., is proposed that can determine a path of d nodes using O(d) marked packets, and subsequently determine the changes in it using O(log d) marked packets. The algorithm is established to be order-wise optimal, i.e. no other distributed algorithm can determine changes in the path topology using lesser order of bits (or marked packets). The algorithm is shown to have a computational complexity of O(d log d), which is significantly less than that of any existing non-incremental algorithm for algebraic traceback. The extension of the traceback mechanism to systems deploying network coding is also considered.

This paper considers the problem of network coding for multiple unicast connections in networks represented by directed acyclic graphs. The concept of interference alignment, traditionally used in interference networks, is extended to analyze the performance of linear network coding in this setup and to provide a systematic code design approach. It is shown that, for a broad class of three-source three-destination unicast networks, a rate corresponding to half the individual source-destination min-cut is achievable via alignment strategies.