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I loaded my dependency data in as directed graph data from a .gv file with the following format of pairwise relations:
digraph {
01245 -> 01251
01246 -> 01251
01249 -> 01251
01245 -> 01252
01246 -> 01252
01249 -> 01252
01293 -> 02029 [comment=?]
01296 -> 02029 [comment=?]
01778 -> 01838
01782 -> 02102
01799 -> 01782
01800 -> 01782
01784 -> 02007
01793 -> 01833
01806 -> 01099
01292 -> 01825
}

Yet when I get an Overview visualization, some of the edges displayed automatically by GEPHI are really thick (i.e. wide) and some are really thin, even though the weights are all set to 1.0 in the Data Laboratory (which seems to have no effect on the display). I cannot discern any particular pattern for how these edge thicknesses are being set by GEPHI - it is not a function of the number of connections or node density, for example. Any suggestions how I can get the same edges that are all the same width (or thickness) when the weights are all set to 1.0?

I have resolved the problem previously described.

It turned out to be a GEPHI bug, at least I think so. I found by accident that when I exited GEPHI, due to another anomaly, re-executed and restored the saved session, all edges were displayed with the same weights. Realizing that I had kept the GEPHI session (v 0.9.1) up for over a week on my desktop, my conclusion is that GEPHI sessions do not age well and accumulate hiccups like the one I described. My lesson is to save my workspace and exit GEPHI at the end of each workday.