Using a graph similar to the weighted graph example in the documentation FindGraphCommuinities >> Scope:
vl = {1, 2, 3, 4, 5, 6, 7, 8};
el = {1 <-> 2, 1 <-> 3, 2 <-> 4, 3 <-> 4, 3 <-> 5, 4 <-> 6,
5 <-> 6, 5 <-> 7, 6 <-> 8, 7 <-> 8};
vc = {{3.24, 0.86}, {3.24, 0.02}, {2.2, 0.88}, {2.2, 0.}, {1.04, 0.88},
{1.04, 0.}, {0., 0.86}, {0., 0.02}};
ew1 = {0.3, 0.9, 0.1, 0.4, 0.2, 0.5, 1., 0.6, 0.8, 0.7};
g1 = Graph[vl, el, VertexLabels -> Placed["Name", Center], VertexShapeFunction -> "Name",
EdgeWeight -> ew1, EdgeLabels -> Thread[el -> ew1], ImagePadding -> 30, ImageSize -> 500];
Row[{g1, Style[FindGraphCommunities[g1], 16, "Panel"]}, Spacer[5]]

Now, randomly re-shuffle the edge weights:
ew2 = RandomSample[ew1];
g2 = Graph[vl, el, VertexLabels -> Placed["Name", Center], VertexShapeFunction -> "Name",
EdgeWeight -> ew2, EdgeLabels ->Thread[el -> ew2], ImagePadding -> 30, ImageSize -> 500];
Row[{g2, Style[FindGraphCommunities[g2], 16, "Panel"]}, Spacer[5]]

Update: Addressing the question in the comments:
Vertices connected with larger EdgeWeights
are considered closer or farther (that is, larger EdgeWeights pull vertices to the same community or to distinct communities)?
Holding everything else fixed, observe how graph communities change as the EdgeWeight
of the edge 3 <-> 5
changes from 0
to 5
:
Manipulate[
ew2 = {0.3`, 0.9`, 0.1`, 0.4`, ew35, 0.5`, 1.`, 0.6`, 0.8`, 0.7`};
g2 = Graph[vl, el, VertexLabels -> Thread[vl ->
(Placed[Style[#, "Panel", 16, Black, Background -> None], Center] & /@ vl)],
VertexShapeFunction -> "Circle", VertexSize -> .2,
EdgeWeight -> ew2, EdgeStyle -> Thick,
EdgeLabels -> Flatten[{Thread[el -> (Style[#, "Panel", 16, Red] & /@ ew1)],
el[[5]] -> Style[ew2[[5]], Bold, "Panel", 20, Purple]}],
ImagePadding -> 30, ImageSize -> 500];
gc = FindGraphCommunities[g2];
Column[{Style[gc, 16, "Panel"], HighlightGraph[g2, gc]}, Alignment -> Center],
{ew35, 0, 5}]
