The interaction of intelligent agents, i.e. the important part of cognitive systems, implies the existence of an environment to support such interaction. The usual representations of this environment are graphs with certain properties. Metric tasks often arise when simplifying complex and important problems on graphs. Reliability is one of the most important characteristics of such graphs. Traditional metrics, such as the usual shortest paths and minimal cuts, form the basis of the traditional reliability indices. The Euclidean metric is used to obtain advanced results, such as the flow description of the edge weights for the reliability of an arbitrary graph in the form of the best quadratic approximation to a similar representation for a complete graph. It can also be used for the synthesis of highly reliable graphs under resource constraints using a single abstract quadratic measure of the closeness in throughput and reliability of an arbitrary graph to a complete graph. The analytical capabilities of the Euclidean metric are used here to further explore flow representations in an arbitrary graph. The theoretical results are illustrated with numerical examples.