Methods, systems and apparatus for characterizing networks are presented. For example, a method of characterizing a network represented by a plurality of nodes and a plurality of edges is provided. The method may be implemented on a processor device and includes calculating, for example, by the processor device, a passthrough count of at least a portion of the network. The passthrough count includes a count of a number of passthroughs in the at least a portion of the network. A passthrough includes one of the plurality of nodes, a directed edge of the plurality of edges coupled to the one of the plurality of nodes, and another edge of the plurality of edges coupled to the one of the plurality of nodes. At most one of the directed edge and the other edge is directed towards the one of the plurality of nodes. At most one of the directed edge and the other edge is directed away from the one of the plurality of nodes.