The non-uniform demand network coding problem is posed as a single-source and multiple-sink network transmission problem where the sinks may have heterogeneous demands. In contrast with multicast problems, non-uniform demand problems are concerned with the amounts of data received by each sink, rather than the specifics of the received data. In this work, we enumerate non-uniform network demand scenarios under which network coding solutions can be found in polynomial time. This is accomplished by relating the demand problem with the graph coloring problem, and then applying results from the strong perfect graph theorem to identify coloring problems which can be solved in polynomial time. This characterization of efficiently-solvable non-uniform demand problems is an important step in understanding such problems, as it allows us to better understand situations under which the NP-complete problem might be tractable.
A technique which we use is that of transmitting data along paths, or through flows. This approach has been widely used in the network coding literature, and has enabled many significant results. In Jaggi @cite_8 , the polynomial-time algorithms for multicast problems rely on the concept of sending data down [perhaps overlapping] paths. @cite_16 , Fragouli and Soljanin give a decomposition of networks into flows, in order to model data transmission in a network more simply. Using this decomposition and a graph coloring formulation, alphabet size bounds for any network code are then proven. Although the flow-based and path-based approaches are similar in many ways, the two techniques differ in that the flow-based approach creates a new flow every time a piece of data is transformed by coding, whereas the path-based approach keeps track of each piece of data as it is sent individually down a path, even if any transformations get applied to the data. We shall use a path-based approach.
@cite_16 @cite_8
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Internet service provider faces a daunting challenge in provisioning network efficiently. We introduce a proactive multipath routing scheme that tries to route traffic according to its built-in properties. Based on mathematical analysis, our approach disperses incoming traffic flows onto multiple paths according to path qualities. Long-lived flows are detected and migrated to the shortest path if their QoS could be guaranteed there. Suggesting nondisjoint path set, four types of dispersion policies are analyzed, and flow classification policy which relates flow trigger with link state update period is investigated. Simulation experiments show that our approach outperforms traditional single path routing significantly. We introduce a new class of problems called network information flow which is inspired by computer network applications. Consider a point-to-point communication network on which a number of information sources are to be multicast to certain sets of destinations. We assume that the information sources are mutually independent. The problem is to characterize the admissible coding rate region. This model subsumes all previously studied models along the same line. We study the problem with one information source, and we have obtained a simple characterization of the admissible coding rate region. Our result can be regarded as the max-flow min-cut theorem for network information flow. Contrary to one's intuition, our work reveals that it is in general not optimal to regard the information to be multicast as a "fluid" which can simply be routed or replicated. Rather, by employing coding at the nodes, which we refer to as network coding, bandwidth can in general be saved. This finding may have significant impact on future design of switching systems. Network coding is often explained by using a small network model called Butterfly. In this network, there are two flow paths, s_1 to t_1 and s_2 to t_2, which share a single bottleneck channel of capacity one. So, if we consider conventional flow (of liquid, for instance), then the total amount of flow must be at most one in total, say 1 2 for each path. However, if we consider information flow, then we can send two bits (one for each path) at the same time by exploiting two side links, which are of no use for the liquid-type flow, and encoding decoding operations at each node. This is known as network coding and has been quite popular since its introduction by Ahlswede, Cai, Li and Yeung in 2000. In QIP 2006, showed that quantum network coding is possible for Butterfly, namely we can send two qubits simultaneously with keeping their fidelity strictly greater than 1 2. In this paper, we show that the result can be extended to a large class of general graphs by using a completely different approach. The underlying technique is a new cloning method called entanglement-free cloning which does not produce any entanglement at all. This seems interesting on its own and to show its possibility is an even more important purpose of this paper. Combining this new cloning with approximation of general quantum states by a small number of fixed ones, we can design a quantum network coding protocol which simulates'' its classical counterpart for the same graph. Many real-world networks are so large that we must simplify their structure before we can extract useful information about the systems they represent. As the tools for doing these simplifications proliferate within the network literature, researchers would benefit from some guidelines about which of the so-called community detection algorithms are most appropriate for the structures they are studying and the questions they are asking. Here we show that different methods highlight different aspects of a network's structure and that the the sort of information that we seek to extract about the system must guide us in our decision. For example, many community detection algorithms, including the popular modularity maximization approach, infer module assignments from an underlying model of the network formation process. However, we are not always as interested in how a system's network structure was formed, as we are in how a network's extant structure influences the system's behavior. To see how structure influences current behavior, we will recognize that links in a network induce movement across the network and result in system-wide interdependence. In doing so, we explicitly acknowledge that most networks carry flow. To highlight and simplify the network structure with respect to this flow, we use the map equation. We present an intuitive derivation of this flow-based and information-theoretic method and provide an interactive on-line application that anyone can use to explore the mechanics of the map equation. The differences between the map equation and the modularity maximization approach are not merely conceptual. Because the map equation attends to patterns of flow on the network and the modularity maximization approach does not, the two methods can yield dramatically different results for some network structures. To illustrate this and build our understanding of each method, we partition several sample networks. We also describe an algorithm and provide source code to efficiently decompose large weighted and directed networks based on the map equation.