How to style a graph according to the direction of the edges and the centrality of the vertices?

Question

I'm new to Mathematica and I'm using it to study graph theory and network data processing and analysis. I recently started to deal with graph data visualization and directed graphs. Which lead me to the wonder about ways of presenting directed graphs.

Lets suppose the following graph:

vlist = {1, 2, 3, 4, 5}
vrules = {1 -> 2, 1 -> 3, 1 -> 5, 2 -> 1, 2 -> 4, 2 -> 5, 3 -> 2, 4 -> 1, 4 -> 5, 5 -> 3, 5 -> 4}
g=Graph[vlist,vrules]

Which is rendered as follows:

Now I want to make bidirectional connections to be presented as a thick line with arrowheads in both ends, single direction connections to be presented with a thin line and the arrowhead indicating its direction.

I also want the vertices to reflect the centrality of the nodes, so nodes with higher centrality would be presented bigger in proportion to its centrality up to a maximum size.

Fo example, the betweeness and degree centralities of this graph are:

BetweennessCentrality[g]
{2., 3.5, 1., 1., 1.5}

DegreeCentrality[g]
{5, 5, 3, 4, 5}

I've looked into the documentation but didn't find much insight on how to do this kind of stuff, I know that I essentially have to deal with GraphStyle options, but most of them are either poorly documented or not documented at all.

Regards and thanks in advance,

Answer 1

A bit more styling:

color[x_, minmax_] := ColorData[{"GrayTones", "Reverse"}][Rescale[x, minmax, {.1, 1}]];

iEShapeFunction[UndirectedEdge[x_, y_], {vsrule_, vstyle_}, dcheck_:True] := 
  With[{sback = {vsrule[x], vsrule[y]}, acolor = {vstyle[x], vstyle[y]}}, 
   ({If[TrueQ[dcheck] && dcheck[x] != #[[1]], sback = Reverse[sback]; 
      acolor = Reverse[acolor]];Blend[acolor, Divide @@ Reverse[sback]], 
      Arrowheads[{-.05, .05}], Thickness[.01], 
      Arrow[Line[#, VertexColors -> acolor], sback]} &)];

iEShapeFunction[DirectedEdge[x_, y_], {vsrule_, vstyle_}, dcheck_:True] := 
  With[{sback = {vsrule[x], vsrule[y]}, acolor = {vstyle[x], vstyle[y]}}, 
   ({Blend[acolor, Divide @@ Reverse[sback]], 
      Arrowheads[{.03}], Thickness[.001], 
      Arrow[Line[#, VertexColors -> acolor], sback]} &)];

For example:

g = RandomGraph[{15, 35}, DirectedEdges -> True];

dc = DegreeCentrality[g];
{vmin, vmax} = {.02, .15};
vsrule = Association[Thread[VertexList[g] -> Rescale[dc, MinMax[dc], {vmin, vmax}]]];
vstyle = color[#, {vmin, vmax}] & /@ vsrule;
labelstyle = 
  Normal[Directive[White, Bold, Rescale[#, {vmin, vmax}, {8, 20}]] & /@ vsrule];

vlist = VertexList[g];
vcoords = 
  Association[Thread[vlist -> GraphEmbedding[Graph[vlist, edges]]]];

edges = GatherBy[EdgeRules[g], 
    Union] /. {{Rule[x_, y_], Rule[y_, x_]} :> 
     UndirectedEdge[x, y], {Rule[x_, y_]} :> DirectedEdge[x, y]};

Graph[edges, VertexSize -> Normal[{#, #} & /@ vsrule], 
 EdgeStyle -> Blue, VertexStyle -> Normal[vstyle], 
 EdgeShapeFunction -> ((# -> iEShapeFunction[#, {vsrule, vstyle}, vcoords]) & /@
     edges), VertexLabels -> Placed["Name", Center],
 VertexLabelStyle -> labelstyle
 ]

Answer 2

edges=Flatten[GatherBy[vrules, Union] /. {Rule[x_,y_],Rule[y_,x_]}:> 
  Property[UndirectedEdge[x,y], 
    EdgeShapeFunction->({Arrowheads[{-.05, .05}], Thickness[.01], Arrow@#}&)]];

vertices=Property[#,VertexSize->#2]& @@@ Transpose[{vlist,{"Scaled",#/2} & /@ 
   Normalize[DegreeCentrality[Graph[vlist, vrules]], Total]}];

Graph[vertices, edges, VertexLabels -> Placed["Name", Center],
   GraphLayout -> { "RenderingOrder" -> "EdgeFirst"}, 
   EdgeStyle -> Blue, VertexStyle->White]