Achievable rate regions for fractional routing

The characterization of information rate region for a network is seemed as the holy grail of network information theory. Existing studies mostly focused on theoretical analysis on the bound of rate region. However, a set of implementable and feasible methods or algorithms to precisely calculate the rate region for a given network is still missing. Research community is still in dire need for an operative and efficient method to get a clear and precise depiction of the rate region. To this end, this article tackles the task of calculating the achievable rate region for fractional routing networks. Three classes of network patterns are discussed, including multiple unicast, multicast, and mixed patterns. We treat the first two classes completely by two implementable methods, within which a variety of techniques are exploited and utilized, including reduced graph, union reduced graph, virtual node, combinatorial design, and subtree decomposition, etc. The rate region for a mixed pattern is described partially. We propose a method to delineate the region for a mixed pattern composed of two traffic flows. Moreover, a technique based on the limit of information rate sequence is proposed to verify the validity of the calculated rate region. A great number of examples and figures are given to illustrate the implementation of these methods. Our work is helpful to both theorists and practitioners in the fields of graph theory and network information theory.