Optimal connections: Strength and distance in valued graphs

We identify some issues in measuring the strongest path connecting pairs of actors that arise in attempts to generalize binary graph concepts to valued graphs. Neither Peay's path value nor Flament's path length indicators take into account the costs of interacting via long chains of intermediaries. We proposed two alternative measures of optimal connections between dyads, respectively, dividing each measure by the distance between a pair (number of lines in a path). We illustrate these average path value (APV) and average path length (APL) measures with a hypothetical five-actor valued graph, observing instances where an indirect path may yield a stronger connection than a direct path. Computer programs to calculate these measures out to three steps for small graphs are available on request.