$Version
(* "13.1.0 for Mac OS X x86 (64-bit) (June 16, 2022)" *)
Clear["Global`*"]
Import the data
data = Import[
"/Users/roberthanlon/Downloads/nodeData.txt", "Data"]
(* {{"node0,", "node1", "0.04,", "node8", "11.11,", "node14", 72.21},
{"node1,", "node46", "1247.25,", "node6", "20.59,", "node13", 64.94}} *)
Clean up the data
data2 = data /.
str_String :> ToExpression@StringReplace[str, "," -> ""]
(* {{node0, node1, 0.04, node8, 11.11, node14, 72.21},
{node1, node46, 1247.25, node6, 20.59, node13, 64.94}} *)
Restructure the data
helper[{node_, nodes_List}] := Flatten[{node, #}] & /@ nodes
data3 = Flatten[
helper /@ ({#[[1]], Partition[Rest[#], 2]} & /@ data2),
1]
(* {{node0, node1, 0.04}, {node0, node8, 11.11}, {node0, node14, 72.21},
{node1, node46, 1247.25}, {node1, node6, 20.59}, {node1, node13, 64.94}} *)
edges = UndirectedEdge @@ Most[#] & /@ data3;
weights = UndirectedEdge @@ Most[#] -> Last[#] & /@ data3;
gr = Graph[edges, EdgeWeight -> weights,
VertexLabels -> "Name",
EdgeLabels -> "EdgeWeight"]

For your actual graph, then use FindShortestPath