1. Girvan-Newman Edge Betweenness Clustering
1,1
Ekey
1,2
1,3
1,4
1,5
2,1
2,2
2,3
2,4
2,5
3,1
3,2
3,3
3,4
3,5
4,1
4,2
4,3
4,4
4,5
5,1
5,2
5,3
5,4
5,5 E 1
SPPC 4
G1_2 1 2 3 4 5
G1_2 1 2 3 4 5
1,1
Ekey
1,2
1,3
1,4
1,5
2,1
2,2
2,3
2,4
2,5
3,1
3,2
3,3
3,4
3,5
4,1
4,2
4,3
4,4
4,5
5,1
5,2
5,3
5,4
5,5 E 1
SPPC 5 4
G1_3 1 2 3 4 5
Ekey
1,1
1,2
1,3
1,4
2,1
2,2
2,3
2,4
3,1
3,2
3,3
3,4
4,1
4,2
4,3
4,4 E 1
G1_1 2 3 4 S 1 P 2 3 4 G1 1 2 3 4 1 S P 2 3 4 S 1 P 2 4 3 5 S 1 P 2 3 4 5
Compute all edge betweenesses (SPPCs)
Remove edge with largest betweeness
Recalc betweenesses; Repeat.
null
nul
SPPC 3 2 4 1
null
nul
Ekey
1,1
1,2
1,3
1,4
2,1
2,2
2,3
2,4
3,1
3,2
3,3
3,4
4,1
4,2
4,3
4,4 E 1
SPPC 1 2 3
nul 2 S P 1 3 S 2 P 4 1 3 S 2 P 3 4 1 S 2 P 4 3 1 2 S P 1 3 4 5 2 S P 1 3 5 4
Check SPPC(34)=SPPC(43)
(verify SPs backwards from hk get counted.)
(34)E so ct=1
+ CountS2P(34)=1
+ CountS2P(43)=1 so ct=3
+ CtS3P(34g)=0
+ CtS3P(g34)=1, g=1 ct=4
GN says delete (3,4)!
GN says delete any edge! 2 S P 1
2Pkey
1,1,1
1,1,2
1,1,3
1,1,4
1,2,1
1,2,2
1,2,3
1,2,4
1,3,1
1,3,2
1,3,3
1,3,4
1,4,1
1,4,2
1,4,3
1,4,4
2,1,1
2,1,2
2,1,3
2,1,4
2,2,1
2,2,2
2,2,3
2,2,4
2,3,1
2,3,2
2,3,3
2,3,4
2,4,1
2,4,2
2,4,3
2,4,4
3,1,1
3,1,2
3,1,3
3,1,4
3,2,1
3,2,2
3,2,3
3,2,4
3,3,1
3,3,2
3,3,3
3,3,4
3,4,1
3,4,2
3,4,3
3,4,4
4,1,1
4,1,2
4,1,3
4,1,4
4,2,1
4,2,2
4,2,3
4,2,4
4,3,1
4,3,2
4,3,3
4,3,4
4,4,1
4,4,2
4,4,3
4,4,4 2 P 1 3 S P 1 4 S 3 P 2 4 1 S 3 P 1 2 5 4
GN says
delete
12 | 25
| 34 | 36
G1_4 1 2 3 4 5 6
To construct SPPC(hk) =SPPC(kh) (Shortest Path Participation Count)
if (hk)E count 1
+ OneCountS2P(hk)
+ OneCountS2P(kh)
+ OneCountS3P(hkg)
+ OneCountS3P(ghk), g
+ OneCountS4P(hkfm)
+ OneCountS4P(fhkm)
+ OneCountS4P(fmhk)
f,m. Etc.
GN: delete
12 | 23 | 25
not 34, 45 1 S P 2 3 4 5 6
Ekey
1,1
1,2
1,3
1,4
1,5
1,6
2,1
2,2
2,3
2,4
2,5
2,6
3,1
3,2
3,3
3,4
3,5
3,6
4,1
4,2
4,3
4,4
4,5
4,6
5,1
5,2
5,3
5,4
5,5
5,6
6,1
6,2
6,3
6,4
6,5
6,6 E 1
G1_4 2 3 4 5 6
G1_3 1 2 3 4 5
G1_4 1 2 3 4 5 6
not 23, 16, 45
SPPC 7 5 6 4
G1_3 1 2 3 4 5 2 S P 1 3 5 4 6
G1_3 1 2 3 4 5
SPPC recalculation and repeat steps? Anyone see a shortcut? Or do we just start the calculation over on the reduced graph?
Do the pointers help? Since in
S2P(hk) one has to search out S2P(kh) and in
S3P(hk) one has to find all S3P(hkg) snf D3P(ghk) g
In the appendix I begin work on uniquely representing shortest k paths using both a fore and aft pTree.
Consider that in G1_4 S3P(16)=2.
G1_3 1 2 3 4 5
Notes:
If any OneCount=0, no subsequence exist.
It might be useful to use ptrs to make this proc easier.
GN edge betweenness specifies pruning (2,4) S 3 P 1 2 5 4 6
G1_3 1 2 3 4 5

2. OSPPE (One Sortest Path per Pair): Always taking 1st occurence
OSPPE (One Sortest Path per Pair): Always taking 1st occurence. Hoping from green vertex on 1st line, vertices you arrive at after: 1hop is shown in green, 2hops is black, 3hops is maroon, 4hops in blue-green, 5hops in brown. SPPC(,)= vertices below ? (shown in red on top of the edge). Then SPPC{,}=SPPC(,)+SPPC(,) 2 1 v 11 1 3
astx f g j l n u 1 4 1 5 2 1 6 h 1 7 1 8 3 1 9 y r 1 b 1 c 1 d 1 e 1 i 1 k 1 m 4 1 w pq o 12 2 1
5679bcdw
h p q 11 2 3
astx
o f g j l n 1 2 4 1 2 8 3 2 e y r 1 2 i 1 2 k 1 2 m 1 2 v 12 3 1
567bcdikmw h 2 3 v 2 3 4 d 1 3 8 3 9 y r 1 3 a 1 3 e 4 3 s op q 2 3 t 8 3 x
fgjlnu r 13 4 1
5679bcikmw
h q 2 4 v 7 4 3
astx o p 1 4 8 1 4 d 9 4 e y f g J l n r u 30 5 1
234689cdeikmw
va yr p
s qo t
xfgjlnu 2 5 7 h 1 5 b 30 6 1
234589cdeikmw
va yr p
s qo t
xfgjlnu 1 6 7 1 6 b 1 6 h 30 7 1
23489bcdeikmw
va yr p
s qo t
xfgjlnu 1 7 5 1 7 6 1 7 h 16 8 1
5679bcdeikmw
h y q 2 8 v 14 8 3
astx
o f
p g j l n ur 1 8 4 16 9 1
245678bcdeikmw h 5 9 3
ast p 1 9 v 9 x
fgjlnou q 2 9 y r 33 c 1
bdeikmw
va h yr p
s qo t
xfgjlnu 32 d 1
bceikmw
va hyr p
s qo t
xfgjlnu 1 d 4 20 a 3
12489estx
5 p 6 7h b c d i m 13 a y
fgjklnoruvw q 30 b 1
234789cdeikmw
va yr p
s qo t
xfgjlnu 1 b 5 2 b 6 h 15 e 1
5679bcdikmw
h y q 2 e v 14 e 3
astx
o f
p g j l n ur 1 e 4 25 f x
39gjlnouvw
: q :
1567bcdim
2 h 4 8 8 f y
aekrst p 25 g x
39fjlnouvw
: q :
1567bcdim
2 h 4 8 9 g y
aekrst pv 31 h 6 1b : 2v
3astx
4 f
8 g
9y j
cr l
d n
e u i k m
wpq o 2 h 7 5 31 i 1
bcdekmw
a h yr p
s qo t
xfgjlnu 2 i v 25 j
x q
39fglnouvw :
1567bcdim
2 h 4 8 8 j y
aekrst p 17 k 1
bcdeimw
x h 2 k v 14 k y
afgjlnorstu
q p G7 25 L
x q
39fgjnouvw :
1567bcdim
2 h 4 8 8 L y
aekrst p 17 m 1
bcdeiw
a h y 16 m 2
kstvx
o qf
p g j l n ur 25 n x
39fgjlouvw
: q :
1567bcdim
2 h 4 8 3 o q pw 15 o s 3 :
1567bcdim
2 h 4 8 2 o u r 8 o x
9fgjlnv 5 o y
aekt 8 n y
aekrst p 3 p q o u 5 p s
3-y
a r 25 p w
1t-x
2 f
4 g
5 j
6h l
7 n
8 v 9 b c d e i k m 13 q o
suxy
rfa g j l n v 1 q p 19 q w 1t 2 3 4 5 6h 7 8 9 b c d e i k m 4 r u ox q 27 r y
9aefgijklnstvw :
15678bcdim
3 h 26 s 3
1289aetx
4v f g
6h j l
b n c d i k m 4 s o qu r 1 s p 2 s y w 1 2 3 4 5 6 7 8 9 a b c e d g h f j k l m n o p q r s t u v w x y i 1 2 3 4 5 6 7 8 9
10 a
11 b
12 c
13 d
14 e
15 f
16 g
17 h
18 i
19 j
20 k
21 l
22 m
23 n
24 o
25 p
26 q
27 r
28 s
29 t
30 u
31 v
32 w
33 x
34 y 3 u o qs 1 u r 23 u x
9fgjlnvw
: 2p :
15678bcdim
3 h 6 u y
aekt 4 15 v 2
138ikm 4 5 6h 7 b c d 1 v 9 10 v x
fgjlnouw q 7 v y
aerst p 19 t 3
12489aesx 5 6h 7 b c d i m 3 t w q p 11 t y
fgjklnoruv 17 w 1 :
bcdeikm h 2 w p s 2 w q o 1 w t 8 w x
fgjlnuv 3 w y ar 19 x 3
1248aest 5 6h 7 b c d i k m 1 x 9 1 x f 1 x g 1 x j 1 x l 1 x n 2 x o q 3 x u ry 1 x v 2 x w p 14 y 9
13x 5 6h 7 8 b c d i m 1 y a 3 y e 24 1 y f 1 y g 1 y j 1 y k 1 y l 1 y n 2 y o q 1 y r 2 y s p 1 y t 1 y u 1 y v 1 y w

3. OSPPE (One Shortest Path Per Edge) always taking 1st one occursing
OSPPE (One Shortest Path Per Edge) always taking 1st one occursing. Hoping from green vertex on 1st line the vertices you arrive at after: 1hop is shown in green, 2hops is black
3hops is maroon, 4hops in blue-green, 5hops in brown. SPPC(,)= vertices below ? Then BETWEEN-NESS(,)=SPPC{,}=SPPC(,)+SPPC(,)
2 13 SPPCs a b c d e f g h i j k l m n o p q r s t u v w x y
a b c d e f g h i j k l m n o p q r s t u v w x
Max SPPC=34 is at edge {1,12}DNI
MSPPC=33 {1,13}
MSPPC=32 {6,17} G7
=9 {2,8} {3,9}+DNI{15,34}{16,34}{19,34}{2134}{23,34}
MSPPC=31 {1,5} {1,6} {1,7} {1,11}
MSPPC=29 {3,28} {27,34}
=6 {2,4} {25,28} {32,34 DNI{27,30}
MSPPC=27 {3,28} {27,34}
MSPPC=26 {3,33} {15,33} {16,33} {19,33} {23,33}
=5 {2,14} DNI{24,30}
MSPPC=25 {21,33} {30,33}
MSPPC=24 {1,3}
MSPPC=19 {24,28}
MSPPC=21 {1,32} {3,10} {3,29} {26,32}
=18 {1,20} {1,22}
=17 {1,8} {2,22}DNI
=16 {1,9} {1,14} {2,31} {9,34} {24,26}
=15 {20,34} {3,8} {3,14}
=14 {20,34} {3,8} {3,14}DNI
=13 {1,2}
=12 {1,4} {29,34}
=11 {9,33}
=10 {4,14} {32,33} {24,33}
=8 {31,34}
=7 {24,34} {30,34} {31,34} DNI{3,4}
I’m not impressed with this. Mainly because some of the critical between-nesses are so close, but also we don’t get the GN partition as claimed. Maybe if we recorded ALL shortest paths it would give us the GN partition? But that is a tremendous amount of additional work. b 1 3
astx f g j l n o u 2 6 h 7 4 9
vxy e y w
pqtxy v 5 8 c d i m k r
Start of ASP. Prohibitive amount of calculation? hh
5679bcdw xy fa gf jg lj nl
qon
uoq sp t
OSPPE: Measure between-ness using One Shortest Path per Edge. Delete edges with highest between-ness then next highest, etc.
ASP: Measure between-ness using All Shortest Paths. Delete edge(s) with highest between-ness”, then next highest, etc.
DOSPPE: (Divisive OSPPE) Peal off first OSPPE Connectivity Cluster that forms. Recalcedge between-ness of the edges in remaining graph. Peal off the next OSPPE connectivity cluster, … This, of course, can be done using ASP also ( call it DASP). Rationale: For a strung out series of clusters with one edge connecting consecutive pairs, if the cluster on one end peals off, we don’t want it’s edges to continue to contribute to between-ness counts because that really throws off the between-ness count of the rest of the graph.
OSPPEL: a method that uses OSPPE but only counts participations in SPs of a given threshold length or longer (e.g., length=3 or longer). Rationale: longer SPs are more likely to run thru “between” edges. SPs within a community (low k k-plex) tend to contain mostly short SP. This, of course, can be done using ASP also (ASPL). There is, of course, also DOSPPEL and DASPL G7
DOSPPE 1st Rnd. Del edges with MaxSPPC until a Connected Comp peals off. Recalc SP Cts,
Delete edges with highest, etc
2 13 a b d c e f g h j i k l n m o q p r s t u w v x
a b c d e f g h i j k l m n o p q r s t u v w x y
Already identified 1st Connectivity Component: Max SPPC=34 at edge {1,12}DNI
MSPPC=33 is at edge {1,13}
MSPPC=32 is at edge {6,17}
MSPPC=31 at {1,5}. {5,6,7,11,17}=CC. Del 5,6,7,11,17. Recalc SP cts.
Start over

4. DOSPPE 2nd round on G7 2 1 v 11 1 3 astx f g j l n u 1 4 1 8 3 1 9 y rc 1 d 1 e 1 i 1 k 1 m 4 1 w pq o 5 2 1
cdw q 14 2 3
9astx
o f
p g j l n u 1 2 4 1 2 8 3 2 e y r 1 2 i 1 2 k 1 2 m 1 2 v 7 3 1
cdikmw 2 3 v 1 3 4 1 3 8 2 3 9 y 1 3 a 1 3 e 4 3 s op q 1 3 t 8 3 x
fgjlnu r 8 4 1
9cikmw q 2 4 v 14 4 3
astx
o f
p g j l n r u 1 4 8 1 4 d 3 4 e y r 11 8 1
9cdeikmw
y q 2 8 v 14 8 3
astx
o f
p g j l n ur 1 8 4 11 9 1
248cdeikmw 5 9 3
ast p 1 9 v 9 x
fgjlnou q 2 9 y r 15 a 3
12489estx
c p d i m 13 a y
fgjklnoruvw q 28 c 1
23489deikmw
va : p
s yr qo t
xfgjlnu 27 d 1
2389ceikmw
va : p
s yr qo t
xfgjlnu 1 d 4 9 e 1
89cdikmw 2 e v 6 e 3
astx p 1 e 4 10 e y
fgjlnoru q 20 f x
39gjlnouvw
: q
1cdim 2 4 8 8 f y
aekrst p 20 g x
39fjlnouvw
: q
1cdim 2 4 8 8 g y
aekrst p 26 i 1
3489cdekmw
a yr p
s qo t
xfgjlnu 2 i v 20 j x
39fglnouvw
: q
1cdim 2 4 8 8 j y
aekrst p 3 p q o u 5 p s
3-y
a r 20 p w
1t-x
2 f
4 g
8 j
9 l
c n
d v e i k m 12 k 1
3489cdeimw x 2 k v 14 k y
afgjlnorstu
q p 20 L x
39fgjnouvw
: q
1cdim 2 4 8 8 L y
aekrst p 1 q p 14 q w 1t 2 3 4 8 9 c d e i k m 26 m 1
3489cdeikw
a yr p
s qo t
xfgjlnu 10 o s 3 :
1cdim 2 4 8 4 r u ox q 24 r y
9aefgjklnstvw
: p
: 4 :
18cdim 3 2 m v 8 n y
aekrst p 3 o q pw 2 o u r 8 o x
9fgjlnv 5 o y
aekt 13 q o
suxy
rfa g j l n v 14 s 3
12489aetx c d i m 4 s o qu r 1 s p 20 n x
39fgjlouvw
: q
1cdim 2 4 8 9 s y f g j kl nv w 14 t 3
12489aesx c d i m 3 t w qp 11 t y
fgjklnoruv 3 u o qp 1 u r 19 u x
39fgjlnvw :
1cdim 2 4 8 s 5 u y
aekt 10 v 2
1348ikm c d 1 v 9 10 v x
fgjlnouw q 7 v y
aerst p 12 w 1 :
23489cdeikm 2 w p s 2 w q o 1 w t 8 w x
fgjlnuv 3 w y ar 14 x 3
1248aest c d i k m 1 x 9 1 x f 1 x g 1 x j 1 x l 1 x n 3 x o qy 2 x u r 1 x v 2 x w p 1 y a 3 y e 24 1 y f 1 y g 1 y j 1 y k 1 y l 1 y n 2 y o q 1 y r 2 y s p 1 y t 1 y u 1 y v 1 y w 2 1 3 4 8 9
10 a
12 c
13 d
14 e
15 f
16 g
18 i
19 j
20 k
21 l
22 m
23 n
24 o
25 p
26 q
27 r
28 s
29 t
30 u
31 v
32 w
33 x
34 y 5 6 7 a b c e d g h f j k l m n o p q r s t u v w x y i
MSPPC=29 {15,34} {1,12}DNI
=28 {1,13}
=20 {31,33}
=18 {31,33}
=16 {2,3} {1,32} {26,32} {32.33} {3,10}
=15 {3,4} {3,8} {3,29} {20,34}
=25 {27,34}
=22 {3,33} {25,32}
=14 {1,9} {24,28} {10,34}DNI
=13 {1,20} {14,34
=12 {1,12DNI {29,34
=11 {2,31} {24,33} {28,34} {28,33}
=10 {1,14} {9,33}
=9 {1,4} {16|19|21|23,34}DNI
=8 {1,4} {31.34} {16|19|21|23,34}DNI
=7 {3,9} {1,2} {24,34} {3,14}DNI
2 7 1 a c d e f i g j l k m o n p r q s t u v w x y
a c d e f g i j k l m n o p q r s t u v w x
=21 {15,33}DNI {16|19|21|23|30,33}
=27 {1,18} {1,22}

5. DOSPPE 3rd round on G7 1 9 v 10 9 x fgjlnouw q 6 9 y arst p 17 a y
9fgjlnorstuvw
x q p 11 f x
9gjlnouvw q 6 f y
arst p 11 g x
9fjlnouvw q 6 g y
arst p 11 j x
9fglnouvw q 6 j y
arst p 11 L x
9fgjnouvw q 6 L y
arst p 11 n x
9fgjlouvw q 6 n y
arst p 3 o q pw 1 o s 2 o u r 8 o x
9fgjlnv 3 o y at 3 p q o u 4 p s y a r 10 p w tx 9 f g j l n v 14 q o
suxy
r9a f g j l n v 1 q p 2 q w t 4 r u ox q 13 r y
9afgjlnstvw p 4 s o
qux 2 s p w 11 s y
9afgjlnrtv 4 t w
pqx 13 t y
9afgjlnorsuv 4 u o qs p 1 u r 9 u x
9fgjlnvw 3 u y at 1 v 9 10 v x
fgjlnouw q 6 v y
arst p 2 w p s 2 w q o 1 w t 9 w x
9fgjlnuv 3 w y ar 3 x 9 y a 1 x f 1 x g 1 x j 1 x l 1 x n 3 x o qs 2 x u r 1 x v 3 x w pt 2 y 9 x 1 y a 1 y f 1 y g 1 y j 1 y l 1 y n 2 y o q 1 y r 2 y s p 1 y t 1 y u 1 y v 1 y w
MSPPC=17 at edge {24,26}
MSPPC=14 at edge {27, 34} {29,34}
MSPPC=13 at edge {9, 33} {28,34}
MSPPC=12 at edge {25,32} {28,34} {15|26|29|21|23|32,33}
MSPPC=11 at edge {24,33} {30,33} {31,33}DNI
MSPPC=8 at edge {9,34}
MSPPC=7 at edge {31,34} {15|16|19|21|23,34}DNI 2 1 3 4 8 9
10 a
12 c
13 d
14 e
15 f
16 g
18 i
19 j
20 k
21 l
22 m
23 n
24 o
25 p
26 q
27 r
28 s
29 t
30 u
31 v
32 w
33 x
34 y 5 6 7 a b c e d g h f j k l m n o p q r s t u v w x y i
15 f 0 0
10 a 0
19 j
16 g
21 l
24 o
23 n
25 p
27 r
26 q
28 s
30 u
29 t
31 v
33 x
32 w
34 y
9 a f g j l n o p q r s t u v w x

6. DOSPPE Round 4
a f g j l n o p q r s t u w
15 f 0
16 g 0 0
19 j
21 l
23 n
24 o
25 p
26 q
27 r
28 s
29 t
30 u
32 w
34 y
MSPPC=15 at edge {10|15|16|19|21|23,34}DNI
=14 {24,26}
=12 {27|29,34}
=11 {25,28} {28,34}
=10 {24|30|32,34}
=6 {24,30}DNI
=4 {24,32}DNI {25,26} {26,32} {29,32}DNI 14 a y
fgjlnorstuw
q p 3 p q o u 9 s f g j l n r 2 w t pw 8
afgjlnt 1 11
suy ra
afgjlnstw qu
afgjlnrt pq
afgjlnorsu 4 qs
afgjlnru
agjlnorstuw
afjlnorstuw
afglnorstuw L
afgjnorstuw
afgjlorstuw
DOSPPE gives a chopped up version of close to GN’s version (although the white and the light blue are chopped up and {27,30,24,28} is half white and half blue).
The next step (in the interest of duplicating GN’s partition) would be to recalculate between-nesses after every edge delete. This is far to much work for me (given that I’m not that happy with the GN partition anyway ;-)
In conclusion, the definition of edge between-ness as a count of the number of Shortest Paths participated in, seems less than optimal, since we noticed that many edges which did not stand “between” partitions in any sense have a high between-ness level (e.g., edges within large cliques such as and e and within the largest 1-plex 12348e).
Before leaving the topic though, we might examine thresholding Shortest Path Length from below. That is, define between-ness to be the count of the number of Shortest Paths of length at least Threshold.
We try that on the next slide.
Note that I have abandoned trying to achieve GN’s partition, since the edges that should have the highest between-ness very often do not and many that do have high between-ness are not at all between clusters but dead center within one. 2 1 3 4 8 9
10 a
12 c
13 d
14 e
15 f
16 g
18 i
19 j
20 k
21 l
22 m
23 n
24 o
25 p
26 q
27 r
28 s
29 t
30 u
31 v
32 w
33 x
34 y 5 6 7 a b c e d g h f j k l m n o p q r s t u v w x y i

7. OSPPE (One Sortest Path per Pair, 1st occurrence)
OSPPE (One Sortest Path per Pair, 1st occurrence). 1hop is 2nd green, 2hops=black, 3hops=maroon, 4hops=blue-green, 5hops=brown. Overall Ct in red 13 1 2 9 3 22 1 3 14 7 13 1 4 10 2 30 1 5 13 8 31 1 6 14 8 30 1 7 13 8 16 1 8 12 3 18 1 9 15 2 30 1 b 13 8 33 1 c 15 9 8 32 1 d 14 9 8 15 1 e 11 3 31 1 i 14 8 17 1 k 14 2 17 1 m 13 3 20 1 w 17 2 12 2 3 5 6 2 4 1 2 8 1 4 2 e 1 2 i 1 2 k 1 16 2 m 5 9 1 15 2 v 6 7 1 8 3 4 5 2 14 3 8 4 1 7 3 9 4 2 20 3 a 9 1 14 3 e 4 8 1 29 3 s 10 17 1 19 3 t 9 8 1 26 3 x 14 10 1 1 4 8 1 4 d 9 4 e 1 7 2 5 7 1 1 5 b 1 6 7 2 6 b 1 31 6 h 2 12 8 1 7 h 1 9 v 10 9 x 7 1 14 9 y 4 1 13 a y 11 1 3 e y 2 25 f x 10 5 8 1 8 f y 6 1
10 a
11 b
12 c
13 d
14 e
15 f
16 g
17 h
18 i
19 j
20 k
21 l
22 m
23 n
24 o
25 p
26 q
27 r
28 s
29 t
30 u
31 v
32 w
33 x
34 y 25 g x 10 6 8 9 g y 6 2 2 h 7 1 25 j x 10 5 8 1 8 j y 6 1 14 k y 11 2 25 L x 10 5 8 1 8 L y 6 1 25 n x 10 5 8 1 8 n y 6 1 15 o q 6 8 18 o s 3 5 8 1 4 o u 3 9 o x 8 6 o y 5 3 p q 1 5 p s 2 26 p w 4 20 1 20 q w 3 15 1 4 r u 2 1 27 r y 14 2 9 1 3 s y 2 3 t w 1 11 t y 10 25 u x 10 4 9 1 6 u y 4 1 10 v x 8 1 7 v y 5 1 9 w x 8 3 w y 2 1 y u
Delete by Decr by length and Decreasing by Count (break ties by next Count).
3 28
1 12
25 32
3 33
1 13
1 18
1 6
1 3
1 7
1 11
1 5
1 32
3 10
3 29
1 9
1 22
1 20
26 32
27 34
16 33
23 33
15 33
1 8
19 33
21 33
24 26
1 14
6 17
30 33
9 34
2 31
20 34
3 14
10 34
3 8
1 2
1 4
2 3
29 34
31 33
4 14
9 33
24 33
24 28
32 33
23 34
3 4
21 34
31 34
3 9
24 34
30 34
25 28
27 30
24 30
2 14
32 34
16 34
29 32
28 34
14 34 G7
All Browns.
All BlueGreens.
{3,9} should have high between-ness but doesn’t (will be deleted late). Why?
We only count one SP per vertex pair and even though {3,9} participates in many SPs we selected those of the same length that use {1,9} instead.
Also {14,34} {20,34} {1,9} {1,32} are the edges that should be deleted (should have high between-ness) but their maroon numbers are low so they’ll be deleted late too.
Next del high middle lengths(e.g., in decr order of Maroon + BlueGreen).
It starts out well, but degenerates later. Green is all chopped up and blue is starting to disintegrate … G7

8. OSPPE G7 G7 Next del in decr order 2*Maroon + BlueGreen + Black
1 12
3 28
25 32
1 13
1 18
1 6
1 11
1 7
1 5
3 33
16 33
6 17
3 10
19 33
1 3
15 33
23 33
21 33
27 34
30 33
1 22
3 29
9 34
24 28
24 26
3 8
1 32
3 14
2 31
1 8
1 9
1 20
2 3
1 14
26 32
1 2
4 14
20 34
10 34
1 4
31 33
9 33
29 34
3 4
24 33
21 34 G7
1 12
3 28
3 33
25 32
1 6
1 13
1 18
1 7
1 11
1 5
1 3
1 32
3 10
1 9
3 29
27 34
26 32
16 33
1 20
21 33
1 22
23 33
19 33
15 33
1 8
6 17
30 33
24 26
1 14
9 34
2 31
20 34
3 14
3 8
10 34
1 2
1 4
2 3
29 34
31 33
24 28
4 14
9 33
32 33
3 4
24 33
23 34
31 34
21 34
3 9
30 34
24 34
25 28
2 14
27 30
24 30
16 34
14 34
25 26
Next del in decr order Maroon + Black
We suspend this algorithm because we note that {3,9} and {14,34} are quite far down in the delete ordering and therefore it will get very chopped up before we get there. We’ve already chopped the blue in half.
Looking back over the attempts, we see that {3,9} and {14,34} are always a problem (do not delete even though they are between white and green).
Possibly some Genetic Algorithm approach to optimize the coefficients of the counts?
There are other more serious concerns however.
We really should be using all SPs, as noted before.
The example at right points out that we aren’t even counting participations correctly as it is!!!!!
The SPs (taking only one per vertex pair as we have been doing): 1 2 3 4 5 4 1 2 3 5 1 2 3 2 4 5 2 3 1 2 3 4 5 3 4 2 1 1 4 5 4 5 3 2 1
So the Total Participation Counts are:
1hop=green, 2hops=black, 3hops=maroon, 4hops=blue-green. Overall Ct in red 4 1 2 3 5 G7 1 2 3 4
That fails to count the fact that, e.g., {2,3} participates in
So two things probably should be done. We should find ALL Shortest Paths between a pair of vertices (not just One Per Pair) and we should count total edge SP participations (edge between-ness) correctly.

9. ASP (All Shortest Paths 1hop is shown in green, 2hops is black, 3hops is maroon, 4hops in blue, 5hops in brown. 1 2 v 8 4 1 3
astx
o f g j l n o u 1 4 1 5 1 6 h 1 7 h 1 8 8 3 1 9
vxy rf q j l n o u 1 b 1 c 1 d 1 e y 1 i 1 k y 1 m 16 5 1 w
pqtxy
o ff gg jj ll nn oo ur u 4 8 2 1
5679bcdw
hh p q 9 5 2 3
9astx
o f
p g j l n o u 1 2 4 d 8 1 2 e y f g j l n o r u 8 1 2 k y f g j l n o r u 15 3 2 v
9xy ff gg jj ll nn oo ur u 3 10 1
567bcdikmw
hh q 1 4 3 2
ikmv r 1 3 4 d 3 8 1 2 3 9 vy r 1 3 a y r 1 3 e y r 5 3 s
opy
qqr rr 2 3 t wy qr 3 9 x
fgjlnouvw
q q r 2 8 2 i 2 m 55 15 13 5 1
234689cdeikmw
va v y y p
so x qo
t y t
: x
: yfgjlnoru :
xfgjlnou 55 16 13 6 1
234789cdeikmw
va v y y p
so x qo
t y t
: x
: yfgjlnoru :
xfgjlnou 4 10 1
5679bcikmw
hh p q 4 2
ikmv 9 4 3
9astx
o f
p g j l n o u 4 8 4 d 8 1 4 e y f g j l n o r u 2 5 7 6h 1 5 b 6 1 6 7 5 1 6 b 6 h i c m 8 2 1 3 4 e k d 5 6 7 b h p r w q s t o g u f n l j x a v 9 y G7 55 16 13 7 1
234689cdeikmw
va v y y p
so x qo
t y t
: x
: yfgjlnoru :
xfgjlnou 3 6 12 8 1
5679bcdeikmw
hhyr yryrq 1 5 8 2
eikmv yr 6 14 8 3
9aestx
yyyoyf
p g
y j l n
r o ur 1 8 4 de yr 4 14 9 1
245678bcdeikmw
hh p q 1 7 9 3
248aest p 9 v 1 7 9 x
fgjlnou q 4 15 9 y
aefgjklnorstuvw
q p p q 1 7 5 b 7 h 2 12 9 a 3
12489estx
5id p
h6m h7 b c d i m 4 15 a y
9efgjklnorstuvw
q p p q 55 15 13 b 1
234789cdeikmw
va v y y p
so x qo
t y t
: x
: yfgjlnoru :
xfgjlnou 33 16 15 c 1
bdeikmw
ka hh y y y p
vso qo
t t
xfgjlnou x
yfgjlnoru 2 b 5 7 1 b 6 7h 1 2 3 4 5 6 7 8 9 a b c e d g h f j k l m n o p q r s t u v w x y i 1 2 3 4 5 6 7 8 9 a b c d e f g h i j k l m n o p q r s t u v w x y 55 17 14 d 1
bceikmw
va hh v y y p
so x qo
t y t
: x
xfgjlnou yfgjlnoru 8 4
238e y 4 11 e 1
5679bcdikmw
hh p q 5 e 2
8ikmv 1 6 e 3
89astx p 2 e 4 8d 4 16 e y
9afgjklnorstuvwx
q p p q 6 29 10 f x
39gjlnouvw
:1 q 21
: p
: q
1567bcdim
im2 hh d4 8 15 y
9aegjklnorstuvw
q p 21 p
4d q 6 29 10 g x
39fjlnouvw
:1 q 21
: p
: q
1567bcdim
im2 hh d4 8 15 y
9aefjklnorstuvw
q p 21 p
4d q

10. ASP (All Shortest Paths 1hop is shown in green, 2hops is black, 3hops is maroon, 4hops in blue, 5hops in brown. 51 15 12 2 h 6 1b : 2v
3-astx
4 o f
8 g
9vxy j
c l
d n
ey o
I u ky m
wpqtxy
o f g j l n o r u 51 15 12 2 h 7 : 2v
3-astx
4 o f
8 g
9vxy j
c l
d n
ey o
I u ky m
wpqtxy
o f g j l n o r u 6 29 10 j x
39fglnouvw
:1 q 21
: p
: q
1567bcdim
im2 hh d4 8 15 y
9aefgklnorstuvw
q p 21 p
4d q 52 14 i 1
bcdekmw
a h x yy pr
s y qo
t t
xfgjlnu x
yfgjlnoru 46 13 8 i 2
348ekmvw
a yy xpr
s yq
t t
x xfgjlnu
yfgjlnru 7 14 k 1
bcdeimw
x hh x p q x 2 7 k
348eimv
x x 14 15 k y
9aefgjlnorstuvw
x xxxxxp p xxp
x q x i c m 8 2 1 3 4 e k d 5 6 7 b h p r w q s t o g u f n l j x a v 9 y G7 29 10 l x
39fgjnouvw
:1 q 21
: p
: q
1567bcdim
im2 hh d4 8 15 y
9aefgjknorstuvw
q p 21 p
4d q 52 14 m 1
bcdekiw
a h x yy pr
s y qo
t t
xfgjlnu x
yfgjlnoru 46 13 8 2
348ekivw
a yy xpr
s yq
t t
x xfgjlnu
yfgjlnru 6 29 10 n x
39fgjlouvw
:1 q 21
: p
: q
1567bcdim
im2 hh d4 8 15 y
9aefgjklorstuvw
q p 21 p
4d q 1 2 o q pw u r 18 3 9 x
39fgjlnvw
: :1
: 2im
1567bcdim 20 4 13 y
9aefgjklnrtvw
1 :1
2im :
2im 11 s 3p :
: hh 4d 8 6 52 17 14 r y
9aefgjklnostvw
: q3328im
: 28im28imp : :
: 48d p
: q :
15678bcdim
: hh
38im 1 2 3 4 5 6 7 8 9 a b c e d g h f j k l m n o p q r s t u v w x y i 2 34 4 p w
1-t-x-y
4 f a
5 g e
6h j f
7h l g
8 n j
9 u k
b v l
c n
d r
e u
I v k m 2 14 4 q w
1t-xy
23 39
3 9a
4 fe
5 gf
6h jg
7h lj
8 nk
9 vl
b n
c r
d v e i k m 1 2 r u
o-x
q-38 1 p q o u 6 3 p s
3oy
2ur 4 8 9 e 19 4 q o
suxy
3r39 9a fe gf jg lj nk vl n r v 1 q p s 3 1 2 3 4 5 6 7 8 9 a b c d e f g h i j k l m n o p q r s t u v w x y 2 11 9 s 3
12489aetx
:id 5m 6h 7h b c d i m 2 11 9 t 3
12489aesx
:id :m 5 6h 7h b c d i m 8 4 t w
1pqx 5 6 7 b c d i m 3 s o
qux 2 s p qw 15 s y
9aefgjklnrtuvwx 15 t y
9aefgjklnorsuvx

11. ASP (All Shortest Paths 1hop is shown in green, 2hops is black, 3hops is maroon, 4hops in blue, 5hops in brown. 2 u o qs pp 8 37 7 9 x
39fgjlnvw
:: :11
:1 :
: 2im :
1567bcdim
2 hh 4d 32 5 13 y
9aefgjklnstvw
p 11 r 2 7 v
134eikm
5 d 6h 7h b c d 2 6 v 9 13 5 6h 7h b c d 3 10 v x
39fgjlnouw
q p q 4 15 v y
9aefgjklnorstuw
q p p q 2 15 w 1
bcdeikm hh 1 w p s 1 w q o 1 w t 3 10 w x
39fgjlnouv 13 w y
9aefgjklnorsuv i c m 8 2 1 3 4 e k d 5 6 7 b h p r w q s t o g u f n l j x a v 9 y G7 2 13 8 x 3
1248aest
:id :k :m : 5 6h 7h b c d i k m 2 10 x 9 1y :k : 5 6h 7h b c d i k m 2 10 5 x w
1pqty
: k 6h 7h b c d i k m 2 9 y 1x : 5 6h 7h 8 b c d i m 2 15 4 y e
1234
:888
:i d :m 5 6h 7h 8 b c d i m 2 12 y k :8 :i :m 5 6h 7h 8 b c d i m 2 9 4 y w
1pqx : 5 6h 7h 8 b c d i m 1 x f y k 1 x g y k 1 x j y k 1 x l y k 1 x n y k 1 3 x o
qsy k 1 2 x u ry k 4 2 x v 2y ik k m 3 2 y v 2x 8 i m 1 y a 3 1 y f x 1 y g x 1 y j x 1 y l x 1 y n x 2 y o qx 1 y r x 1 y s 3p 8 1 y t 3 1 y u x
MSPPC=90 r y
=67 2i =66 1m,1i DNI
MSPPC=86 1 c 1 d
=64 xu =57 x,f|g||jn
MSPPC=
=55 y,f|g||j|l|nDNI
MSPPC=83 1 b
=51 xl =46 uy =45 ky
MSPPC=80 6h 1m 7hDNI
=41 ey ,pw =39 oy =38 1w
MSPPC= I
=34 2v,ox =33 3x =32 ylDNI 1 2 3 4 5 6 7 8 9 a b c e d g h f j k l m n o p q r s t u v w x y i
23 a
83 b
86 c
86 d
16 e
0 f
0 g
80 h
67 i
0 j
22 k
0 l
80 m
0 n
0 o
0 p
26 q
0 r
30 s
24 t
5 u
34 v
41 w
64 x
90 y
a b c d e f g h i j k l m n o p q r s t u v w x
=30 3s =29 19 xw =26 yw qo 38 1 2 3 4 5 6 7 8 9 a b c d e f g h i j k l m n o p q r s t u v w x y
= =24 3t yv =23 2a =22 1k 9x
=21 ayDNI =19 23 xvDNI =18 2kDNI
= 17 wq =16 ys 1e =14 2e 14 34
= 13 4dDNI 12 tw

12. ASP (with weighting options for between-ness definition) n r f a
Between-ness = 4H SP partic count.
edge 2H edge 3H edge 4H edge 5H
1 w p w b h 51
e y k y h 51
w y v d u x 8
v y g x j x 6
3 x f x r y 6
9 y o q i n x 6
k y w x m g x 6
e y r y f x 6
s y w i l x 6
j y r y m o s 4
a y d u x 37
g y x c 33
n y u y 32
t y c n y 29
l y s f x 29
f y f y 29
w x b g x 29
1 k l y 29
o y q w l x 29
r y i g y 29
1 c m j x 29
u y n x 29
t j y 29
1 d m o y 20
1 i a o x 18
y h 15
1 m i h 15
o s 11
h s 9
1 b h d 8
n x
1 e
o x l x a 2
v x j x e y 2
3 s x x 2
j x v 2
g x l y x 2
l x n y y 2
3 t k w y 2
n x g y p w 2
u x f y w x 2
f x j y k y 2
w y q w 2
3 a t 2
2 v e v 2
e r u 1
t w
9 x u x
v x 7
2 i k 7
2 m v y 7
2 k
3 e v 6
2 e p s 6
o q u y 5
p w
a y 4
q w e 4
o y 4
t w o x 4
o s o s 4
o u 2
p s
4 d r u 2
p q 2
o u e 2
5 b
4 e d 1
6 b
23H edge
43 k y
39 e y
39 p w w
34 w x x
32 r y v
31 g x
31 f x c d y b i m s
26 o q
26 n y
26 j y
26 l y
26 f y
26 g y
26 w y
24 v y t a k
22 l x
22 j x
22 n x i m x
20 a y
19 o y
19 v x
19 u y
19 u x
19 q w k
16 t y e
16 s y
16 o x e h h
13 t w
11 p s e
9 o s
9 3 e
8 9 v
5 o u
5 4 d
4 r u
4 p q
3 5 b
2 6 b r f a
Follow by = 2H SP partic count? g v k d u y e 4 l 9 5 x 6
Try wts (5hop SPs only) isolates edges at the outside (not between). Put 5H on DONOT DEL list immediately? j 3 h 1 o 2 7 t b s 8 q w i m p c n
End of
Edge 2H
7 h 2
6 h 2
p q 2
r u 2
9 v 2 f
Between-ness = 3H SP partic count. r a
Follow by = 2H SP partic count? g v k d u y e 4 l 9 5 x 6 j 3 h o 2 1 7 t b s
Conclusion: 2H alone is as good as any (including weighted combos like 2+3 (wts 00110) and is by far the easiest to calculate!
However, we are not counting all participations (see next slide) 8 q w i m p c n
Between-ness = 2H SP partic count. r f n a r f a g g v k d v k d u y e u y e 4 4 l 9 5 l 9 5 6 x 6 x j h j h 3 3 2 1 2 1 o 7 o 7 t t b b s s 8 8 q w i m q w i m p c p c

13. ASP-2 (with weighting options for between-ness definition)c m 8 2 1 3 4 e k d 5 6 7 b h p r w q s t o g u f n l j a x v 9 y
Edge 2H
7 h 2
6 h 2
p q 2
r u 2
9 v 2

14. Maximal Base CliqueTrees for H3
aoa then oaa on the 6 cTrees (removing duplicates - no covers since aoa then oaa gives Maximal Cliques only). We get 34 MCs below.
Theorem: These 34 MCs are the only Maxmal Cliques.
General thm: {a..ao(a..oa(…oa..a(B)|B=BaseClique} is the MaxCliqueSet.
Thus, for a bipartite graph, ao(B) is MCS. 1 2 5 3 4 B A C D E      B A C D E 1 4 3 2 5      2 1 3 4 5 1 2 5 4 3 B A C D E      1 3 4 2 5 B A C D E      1 5 2 4 3 B A C D E      1 2 3 4 5 B A C D E      1 4 3 2 5 B A C D E      1 5 2 4 3 B A C D E      1 3 2 4 5 B A C D E      1 2 5 3 4 B A C D E      1 2 5 3 4 B A C D E     

15. DOSPPE: Measure between-ness using One Shortest Path per Edge
DOSPPE: Measure between-ness using One Shortest Path per Edge. Delete edges with highest betweenness until a Connected Component peals off, recalculate SP Counts, delete edges with highest, etc G7
SPPCs a b d c e f g h j i k l n m o q p r s t u w v x
a b c d e f g h i j k l m n o p q r s t u v w x y
Max SPPC=34 is at edge {1,12}DNI
Next Max SPPC=33 is at edge {1,13}
Next Max SPPC=32 is at edge {6,17}
Next Max SPPC=31 is at edges {1,5}
{5,6,7,11,17}=CC. Delete. Recalc.
Start over
SPPC after deleting
a c d e f g i j k l m n o p q r s t u v w x 1
2 13 c a d e f g j i k l n m o q p r s t u w v x y 1 2 3 4 5 6 7 8 9 a b c e d g h f j k l m n o p q r s t u v w x y i 1 2 3 4 5 6 7 8 9
10 a
11 b
12 c
13 d
14 e
15 f
16 g
17 h
18 i
19 j
20 k
21 l
22 m
23 n
24 o
25 p
26 q
27 r
28 s
29 t
30 u
31 v
32 w
33 x
34 y
Max SPPC=34 is at edge {1,12}DNI
Next Max SPPC=13 is at edges {1,2} {1,4} {29,34}
Next Max SPPC=31 is edges {1,18} {1,13}DNI
Next Max SPPC=11 is at edges {9,33}
Next Max SPPC=29 is edges {3,29} {27,34}
Next Max SPPC=10 is at edges {4,14} {24,33} {31,33}
Next Max SPPC=27 is at edge {25,32}
Next Max SPPC=9 edges {2,8} {3,9} DNI{15,34} {16,34} {19,34} {23,34}
Next Max SPPC=26 is at edges {3,33} {15,33} {16,33} {19,33} {23,33}
Next Max SPPC=8 edges {31,34}DNI {32,33}
Next Max SPPC=25 is at edges {21,33} {30,33}
Next Max SPPC=7 edges {3,4}{30,34}DNI {24,34}
Next Max SPPC=24 is at edges {1,3} {2,3}
Next Max SPPC=6 edges {25,28}DNI {2,14}
Next Max SPPC=21 is at edges {3,10} {1,32} {26,32} {24,28} {3,28}
{1,2,3,4,8,12,13,18,20,22}=CC.
Delete. Recalc. Start over
Next Max SPPC=18 is at edges {1,20} {1,22}
Next Max SPPC=17 is at edges {1,8} {2,22}
Next Max SPPC=16 is at edges {1,9} {1,14} {24,26} {2,31} {9,34}
Next Max SPPC=15 is at edges {3,8} {3,14} {10,34}DNI {20,34}

16. DOSPPE: Measure between-ness using One Shortest Path per Edge
DOSPPE: Measure between-ness using One Shortest Path per Edge. Delete edges with highest betweenness until a Connected Component peals off, recalc, del next highest, etc G7
SPPC after deleting
a c d e f g i j k l m n o p q r s t u v w x 1
2 13 c a d e f g j i k l n m o q p r s t u w v x y
Max SPPC=29 is at edge {27,34}
Max SPPC=27 is at edge {15,33} {16,33} {19,33} {23,33}
Max SPPC=25 is at edge {21,33} {30,33}
Max SPPC=21 is at edge {21,33} {30,33}
Max SPPC=19 is at edge {24,28}
Max SPPC=16 is at edge {24,26} {9,34}
Max SPPC=14 is at edge {10,34}DNI (but del would remove only green)
Max SPPC=12 is at edge {29,34}
Max SPPC=11 is at edge {9,33}
Max SPPC=10 is at edge {24,33}
Max SPPC=9 is at edge {15,33} {16,33} {19,33} {21,33} {23,33} all DNI 1 2 3 4 5 6 7 8 9 a b c e d g h f j k l m n o p q r s t u v w x y i
Max SPPC=8 is at edge {32,33} {31,34}DNI 1 2 3 4 5 6 7 8 9
10 a
11 b
12 c
13 d
14 e
15 f
16 g
17 h
18 i
19 j
20 k
21 l
22 m
23 n
24 o
25 p
26 q
27 r
28 s
29 t
30 u
31 v
32 w
33 x
34 y
Max SPPC=7 is at edge {24,34} {30,34}DNI
Max SPPC=6 is at edge {25,26}DNI
Max SPPC=5 is at edge {24,30}DNI
Max SPPC=4 is at edge {28,34} {29,32}DNI

17. DOSPPE: Measure between-ness using One Shortest Path per Edge
DOSPPE: Measure between-ness using One Shortest Path per Edge. Delete edges with highest betweenness until a Connected Component peals off, recalc, del next highest, etc G7 1 2 3 4 5 6 7 8 9
10 a
11 b
12 c
13 d
14 e
15 f
16 g
17 h
18 i
19 j
20 k
21 l
22 m
23 n
24 o
25 p
26 q
27 r
28 s
29 t
30 u
31 v
32 w
33 x
34 y
SPPCs a b d c e f g h j i k l n m o q p r s t u w v x
a b c d e f g h i j k l m n o p q r s t u v w x y
Max SPPC=34 is at edge {1,12}DNI
Next Max SPPC=33 is at edge {1,13}
Next Max SPPC=32 is at edge {6,17}
Next Max SPPC=31 is at edges {1,5}
{5,6,7,11,17}=CC. Delete. Recalc.
Start over
SPPC after deleting
a c d e f g i j k l m n o p q r s t u v w x 1
2 13 c a d e f g j i k l n m o q p r s t u w v x y 1 2 3 4 5 6 7 8 9 a b c e d g h f j k l m n o p q r s t u v w x y i 1 2 3 4 5 6 7 8 9
10 a
11 b
12 c
13 d
14 e
15 f
16 g
17 h
18 i
19 j
20 k
21 l
22 m
23 n
24 o
25 p
26 q
27 r
28 s
29 t
30 u
31 v
32 w
33 x
34 y
Max SPPC=34 is at edge {1,12}DNI
Next Max SPPC=13 is at edges {1,2} {1,4} {29,34}
Next Max SPPC=31 is edges {1,18} {1,13}DNI
Next Max SPPC=11 is at edges {9,33}
Next Max SPPC=29 is edges {3,29} {27,34}
Next Max SPPC=10 is at edges {4,14} {24,33} {31,33}
Next Max SPPC=27 is at edge {25,32}
Next Max SPPC=9 edges {2,8} {3,9} DNI{15,34} {16,34} {19,34} {23,34}
Next Max SPPC=26 is at edges {3,33} {15,33} {16,33} {19,33} {23,33}
Next Max SPPC=8 edges {31,34}DNI {32,33}
Next Max SPPC=25 is at edges {21,33} {30,33}
Next Max SPPC=7 edges {3,4}{30,34}DNI {24,34}
Next Max SPPC=24 is at edges {1,3} {2,3}
Next Max SPPC=6 edges {25,28}DNI {2,14}
Next Max SPPC=21 is at edges {3,10} {1,32} {26,32} {24,28} {3,28}
{1,2,3,4,8,12,13,18,20,22}=CC.
Delete. Recalc. Start over
Next Max SPPC=18 is at edges {1,20} {1,22}
Next Max SPPC=17 is at edges {1,8} {2,22}
Next Max SPPC=16 is at edges {1,9} {1,14} {24,26} {2,31} {9,34}
Next Max SPPC=15 is at edges {3,8} {3,14} {10,34}DNI {20,34}