Finding the dangling free part of a cluster connecting two nodes

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SeedRandom[123]
n = 15;
m = 20;
(*conductances=1/RandomReal[{0,1},m];*)

conductances = ConstantArray[1., m];
G = RandomGraph[{n, m}, VertexLabels -> "Name"];


grad = With[{edges = UpperTriangularize[AdjacencyMatrix[G]]["NonzeroPositions"]},
   With[{m = Length[edges]},
    SparseArray @@ {Automatic, {m, n}, 0, {1, {
        Range[0, 2 m, 2],
        Partition[Flatten[edges], 1]
        },
       Flatten[Transpose[{ConstantArray[1., m], ConstantArray[-1., m]}]]}}
    ]
   ];
L = grad\[Transpose].DiagonalMatrix[SparseArray[conductances]].grad;

Now with source s and target t:

s = 1;
t = 2;
(* currents inserted at the nodes *)
Inodes = SparseArray[{{s}, {t}} -> {1., -1.}, {VertexCount[G]}, 0.];
a = SparseArray[ConstantArray[1., {1, n}]];
A = ArrayFlatten[{{L, a\[Transpose]}, {a, 0.}}];
S = LinearSolve[A];

(* potentials at the nodes *)
Unodes = S[Join[Inodes, {0.}]][[;; -2]];

(* currents through edges *)
Iedges = conductances grad.Unodes;

ϵ = 1. 10^-8;
stylefun = x \[Function] Directive[Thickness[0.0001 + x 0.02], Opacity[1.], ColorData["DarkRainbow"][x]];

Graph[G, EdgeStyle -> (
    Thread[EdgeList[G] ->stylefun /@ Normalize[Threshold[Abs[Iedges], ϵ], Max]]
    )
 ]

This paper was extremely helpful for me in order to set this up:

https://arxiv.org/pdf/1712.10263.pdf