Josh,
What you need for each junction is simply the adjacent junctions and cost to them. You do not need to use any solver (shortest path, OD, etc) to do this. You can simply add the network dataset to ArcMap and then use the VBA below. This VBA lists out the adjacent junctions to each junction in the network and the cost (based on an attribute called "minutes") to get to that junction. You can customize it as needed. For example, change the EID to OID and/or change the cost attribute you need to list out. Currently it is set to show the results in the immediate window of VBA but you can customize it to write to a file.
Regards.
Jay Sandhu
Public Sub List_Adjacent_Junctions()
On Error GoTo eh
Dim pMxDoc As IMxDocument
Set pMxDoc = ThisDocument
Dim pNLayer As INetworkLayer
Set pNLayer = pMxDoc.FocusMap.Layer(0) 'Assume the Network dataset is first layer in TOC
Dim pND As INetworkDataset
Set pND = pNLayer.NetworkDataset
Dim pNQ As INetworkQuery
Set pNQ = pND
Dim pEnumNE As IEnumNetworkElement
Set pEnumNE = pNQ.Elements(esriNETJunction)
Dim pNEdge As INetworkEdge
Set pNEdge = pNQ.CreateNetworkElement(esriNETEdge)
Dim pNEFromJunc As INetworkJunction
Set pNEFromJunc = pNQ.CreateNetworkElement(esriNETJunction)
Dim pNEToJunc As INetworkJunction
Set pNEToJunc = pNQ.CreateNetworkElement(esriNETJunction)
Dim pNE As INetworkElement
Set pNE = pEnumNE.Next
Dim pNEJunc As INetworkJunction
Set pNEJunc = pNE
Dim i As Integer
Do Until pNE Is Nothing
Debug.Print "Junction: " & pNEJunc.EID & " is adjacent to: " & pNEJunc.EdgeCount & " junctions."
For i = 0 To pNEJunc.EdgeCount - 1 'For each connected edge...
pNEJunc.QueryEdge i, True, pNEdge 'Get that connected edge
pNEdge.QueryJunctions pNEFromJunc, pNEToJunc 'Get To junction of current edge
Debug.Print " Adjacent Junction: " & pNEToJunc.EID & " Cost = " & pNEdge.AttributeValueByName("minutes") 'List the adjacency and cost to it
Next
Set pNE = pEnumNE.Next
Loop
Exit Sub
eh:
MsgBox "Error: " & Err.Description
End Sub