Comparing multiple network files (edge lists)

I want to compare 4 edge-lists to basically see if an edge is present in all 4 networks. The issue is that an edge A-B in one file can be present as B-A in another file.

Example:

Input 1:  net1.txt

A B 0.1
C D 0.65
D E 0.9
E A 0.7

Input 2:  net2.txt
A Z 0.1
C D 0.65
E D 0.9
E A 0.7

Input 3:  net3.txt
Y Z 0.1
C D 0.65
D E 0.9
W R 0.7


Input 4:  net4.txt
F Z 0.1
D C 0.65
D E 0.9
W Q 0.7

Intersection of net1.txt, net2.txt, net3.txt and net4.txt:

C D 0.65
D E 0.9

Which sequence do you want printed: C D or D C ? How do you determine the one preferred, by the count of occurrences? What if each has a count of 2?

---------- Post updated at 22:45 ---------- Previous update was at 22:28 ----------

Assuming it's the count, try

awk '
FNR==1  {FCNT++}

        {T[$1,$2,$3]++
         T[$2,$1,$3]++
         C[$1,$2]++
        }

END {for (t in T)       {split (t, X, SUBSEP)
                         if (T[t]==FCNT && C[X[1],X[2]] >= FCNT/2) print t}}

' SUBSEP=" " file[1-4]
C D 0.65
D E 0.9

This has a small drawback which to find out I leave as a challenge to you.