Function Computation In Networks Research Articles

Secure Network Function Computation for Linear Functions—Part I: Source Security

Secure Network Function Computation for Linear Functions—Part I: Source Security

Computing Vector-Linear Functions on Diamond Network

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.

Improved Upper Bound on the Network Function Computing Capacity

The problem of network function computation over a directed acyclic network is investigated in this paper. In such a network, a sink node desires to compute with zero error a target function , of which the inputs are generated at multiple source nodes. The edges in the network are assumed to be error-free and have limited capacity. The nodes in the network are assumed to have unbounded computing capability and be able to perform network coding. The computing rate of a network code that can compute the target function over the network is the average number of times that the target function is computed with zero error for one use of the network. In this paper, we obtain an improved upper bound on the computing capacity, which is applicable to arbitrary target functions and arbitrary network topologies. This improved upper bound not only is an enhancement of the previous upper bounds but also is the first tight upper bound on the computing capacity for computing an arithmetic sum over a certain non-tree network, which has been widely studied in the literature. We also introduce a multi-dimensional array approach that facilitates evaluation of the improved upper bound. Furthermore, we apply this bound to the problem of computing a vector-linear function over a network. With this bound, we are not only able to enhance a previous result on computing a vector-linear function over a network but also simplify the proof significantly. Finally, we prove that for computing the binary maximum function over the reverse butterfly network, our improved upper bound is not achievable. This result establishes that in general our improved upper bound is non-achievable, but whether it is asymptotically achievable or not remains open.

Comments on Cut-Set Bounds on Network Function Computation

A function computation problem in directed acyclic networks has been considered in the literature, where a sink node wants to compute a target function with the inputs generated at multiple source nodes. The network links are error-free but capacity-limited, and the intermediate network nodes perform network coding. The target function is required to be computed with zero error. The computing rate of a network code is measured by the average number of times that the target function can be computed for one use of the network, i.e., each link in the network is used at most once. In the papers [1], [2], two cut-set bounds were proposed on the computing rate. However, we show in this paper that these bounds are not valid for general network function computation problems. We analyze the arguments that lead to the invalidity of these bounds and fix the issue with a new cut-set bound, where a new equivalence relation associated with the inputs of the target function is used. Our bound is qualified for general target functions and network topologies. We also show that our bound is tight for some special cases where the computing capacity is known. Moreover, some results in [11], [12] were proved using the invalid upper bound in [1] and hence their correctness needs further justification. We also justify their validity in the paper.

Reduced Complexity Sum-Product Algorithm for Decoding Nonlinear Network Codes and In-Network Function Computation

While the capacity, feasibility, and methods to obtain codes for network coding problems are well studied, the decoding procedure and complexity have not garnered much attention. In this paper, we pose the decoding problem at a sink node in a network as a marginalize product function (MPF) problem over the Boolean semiring and use the sum product (SP) algorithm on a suitably constructed factor graph to perform iterative decoding. The number of operations required to perform SP decoding is reduced using traceback . The number of operations required to perform SP decoding with and without traceback is obtained. For nonlinear network codes, we define fast decodability of a network code at sinks demanding all the messages and identify a sufficient condition for the same. Next, we consider the network function computation problem wherein the sink nodes demand a function of the messages. We present an MPF formulation for function computation at the sink nodes and use the SP algorithm to obtain the value of the demanded function. Though the proposed method can be used for decoding both linear and nonlinear network codes, it is advantageous only for the case of nonlinear network codes.

CONDENSE: A Reconfigurable Knowledge Acquisition Architecture for Future 5G IoT

In forthcoming years, the Internet of Things (IoT) will connect billions of smart devices generating and uploading a deluge of data to the cloud. If successfully extracted, the knowledge buried in the data can significantly improve the quality of life and foster economic growth. However, a critical bottleneck for realising the efficient IoT is the pressure it puts on the existing communication infrastructures, requiring transfer of enormous data volumes. Aiming at addressing this problem, we propose a novel architecture dubbed Condense, which integrates the IoT-communication infrastructure into data analysis. This is achieved via the generic concept of network function computation: Instead of merely transferring data from the IoT sources to the cloud, the communication infrastructure should actively participate in the data analysis by carefully designed en-route processing. We define the Condense architecture, its basic layers, and the interactions among its constituent modules. Further, from the implementation side, we describe how Condense can be integrated into the 3rd Generation Partnership Project (3GPP) Machine Type Communications (MTC) architecture, as well as the prospects of making it a practically viable technology in a short time frame, relying on Network Function Virtualization (NFV) and Software Defined Networking (SDN). Finally, from the theoretical side, we survey the relevant literature on computing "atomic" functions in both analog and digital domains, as well as on function decomposition over networks, highlighting challenges, insights, and future directions for exploiting these techniques within practical 3GPP MTC architecture.

Linear Function Computation in Networks: Duality and Constant Gap Results

In linear function computation, multiple source nodes communicate across a relay network to a single destination whose goal is to recover linear functions of the original source data. When the relay network is a linear deterministic network, a duality relation is established between function computation and broadcast with common messages. Using this relation, a compact sufficient condition is found describing those cases where the cut-set bound is tight. These insights are used to develop results for the case where the relay network contains Gaussian multiple-access channels. The proposed scheme decouples the physical and network layers. Using lattice codes for both source quantization and computation in the physical layer, the original Gaussian sources are converted into discrete sources and the Gaussian network into a linear deterministic network. Network codes for computing functions of discrete sources across the deterministic network are then found by applying the duality relation. The distortion for computing the sum of an arbitrary number of independent Gaussian sources over the Gaussian network is proven to be within a constant factor of the optimal performance. Furthermore, the constant factor results are extended to include asymmetric functions for the case of two sources.