The problem of multi-function computation over a directed acyclic network is investigated in this paper. In such a network, a sink node is required to compute with zero error multiple vector-linear functions, where each vector-linear function has distinct inputs generated by multiple source nodes. The computing rate tuple of an admissible code is defined as a tuple consisting of the average number of zero-error computations for each vector-linear function when the network is used once jointly. From the information theoretic point of view, we are interested in characterizing the rate region, which is defined as the closed set of all achievable computing rate tuples. In particular, when the sink node is required to compute a single vector-linear function, the network multi-function computation problem degenerates to the network function computation problem. We prove an outer bound on the rate region by developing the approach of the cut-set strong partition. We also illustrate that the obtained outer bound is tight for a typical model of computing two vector-linear functions over the diamond network. Furthermore, we establish the relationship between the network multi-function computation rate region and the network function computation rate region. Also, we show that the best known outer bound on the rate region for computing an arbitrary vector-linear function over an arbitrary network is a straightforward consequence of our outer bound.

The model of multi-source multicast network coding is investigated in this paper,where each sink node desires to decode correctly the source messages generated by multiple source nodes.For this model,the rate region is defined as the set of rate tuples of source nodes at which the source nodes can multicast source messages to all sink nodes for one use of the network on average.In this paper,we fully characterize the rate region and investigate the code construction for the model of multi-source multicast network coding.We first prove that for any subset of source nodes,the sum rate is upper bounded by the smallest minimum cut capacity separating a sink node from the subset of source nodes.We further develop a systematic construction of vector-linear network codes by using a transformation approach.This code construction is applicable to an arbitrary rate tuple that is achievable.Based on the upper bounds obtained and the developed code construction,we fully characterize the rate region for any model of multi-source multicast network coding problem.

Network function computation is investigated in the letter. In the model, a target function, of which the inputs are generated at multiple source nodes, is required to be computed with zero error at a sink node over a network. Toward this end, distributed coding by integrating communication and computation in networks is regarded as an efficient solution. We are interested in its fundamental computing capacity of importance in theory and applications. In the letter, we explicitly characterize the capacities of computing all the vector-linear functions over the diamond network. The diamond network has an important topology structure which not only is typical for many multi-terminal information-theoretic problems but also illustrates the combinatorial nature of the computing problem. By applying the computing capacities thus obtained, we solve the solvability problem of vector-linear functions over the diamond network. We determine all the solvable vector-linear functions and obtain an enhanced result that the remaining vector-linear functions are not only linearly non-solvable but also non-linearly non-solvable.

Random linear network coding (RLNC) is a promising network coding solution when the network topology information is not fully available to all the nodes. However, in practice, nodes have partial knowledge of the network topology information. Motivated by this, we investigate the performance of RLNC and obtain different upper bounds on the failure probability of RLNC for the network constrained by different partial network topology information. These upper bounds not only improve the existing ones in the literature, but also show that the partial network topology information can bring benefits to the performance analysis of RLNC. On the other hand, it is observed that if more network topology information can be utilized, tighter upper bounds can be obtained, as expected. The upper bounds on two classical networks are compared for demonstration. To obtain a deeper understanding about the performance of RLNC, the asymptotic behavior of RLNC as the field size goes to infinity is also investigated.

In practice, since many communication networks are huge in scale or complicated in structure even dynamic, the predesigned network codes based on the network topology is impossible even if the topological structure is known. Therefore, random linear network coding was proposed as an acceptable coding technique. In this paper, we further study the performance of random linear network coding by analyzing the failure probabilities at sink node for different knowledge of network topology and get some tight and asymptotically tight upper bounds of the failure probabilities. In particular, the worst cases are indicated for these bounds. Furthermore, if the more information about the network topology is utilized, the better upper bounds are obtained. These bounds improve on the known ones. Finally, we also discuss the lower bound of this failure probability and show that it is also asymptotically tight.