Design Options for Weighted Directed Graphs

My question in brief: what are some reasonable design options for displaying data in weighted, directed, cyclical graphs?

The longer version: I have some data that details shipments of goods from one place to another over time, and am searching for an elegant way to visualize the data.

In the case of undirected graphs, one can use chord diagrams, or matrices with colored cell values, to represent this data nicely. My graph is directed though, so these are out.

One technique I've seen for directed weighted graphs is to represent each observation as a discrete primitive (line, arc, etc), as in the Charting Culture visualization:

Maps of shipping routes and airline traffic often use this strategy. That said, it can be difficult to see aggregate trends in this kind of visualization.

Another strategy is to try to encode the weight between vertices i, j visually, e.g. through color, width, or intensity. Visualizations of neural network weights sometimes use this strategy [link]:

That said, it can be hard to appreciate the strength of a weight by virtue of the color, width, or intensity of the geometry that connects two vertices.

I'm trying to come up with some novel, strange ways to visualize weighted, directed, cyclic graphs. Do others have notions on how one might achieve this goal? Any insights or suggestions would be very helpful!