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    $\begingroup$ Don't know if there is a GraphLayout which does this but at the end you could play with EdgeShapeFunction see e.g.: 66784. Anyway, could you provide something to working with? That is a small multiedge graph? $\endgroup$ Commented Aug 3, 2017 at 10:53
  • $\begingroup$ Thanks! I like this, except that I can't keep the middle edge "flat" since you specify the curvature for all edges between a particular two vertices simultaneously: Graph[{1, 2}, {1 <-> 2, 1 <-> 2, 1 <-> 2}, EdgeShapeFunction -> {1 <-> 2 -> GraphElementData[{"CurvedArc", "Curvature" -> 1}]}] $\endgroup$ Commented Aug 3, 2017 at 11:03