How to implement hive plot for graphs in Mathematica?

In Graph you can use EdgeShapeFunction to make curved edges.

Graph[{1 -> 2, 2 -> 3, 3 -> 1}, 
 EdgeShapeFunction -> (
    BezierCurve[{#1[[1]], 
                (#1[[1]] + #1[[2]])/ 2 + {-1, 1}/5 (#1[[1]] - #1[[2]])[[{2, 1}]], 
                 #1[[2]]}] &)
]

In GraphPlot you would use EdgeRenderingFunction.