A Vertical Graph Clustering Technique: Using CS0 and DONOT ISOLATE with 3CLIQUE deletion 158 7 6 3 4 4 4 5 1 3 1 2 4 1 1 2 2 1 2 1 2 1 4 2 2 2 2 2 4 3 4 5 14 1 2 3 4 5 6 7 8 9 10111213141516171819202122232425262728293031323334 E 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 2 1 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 3 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 4 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 11 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 13 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 14 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 17 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 18 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 19 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 20 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 22 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 23 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 25 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 26 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 27 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 29 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 31 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 32 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 1 33 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 34 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 1 1 1 1 1 0 0 G7 Common Siblings and 3cliques Thm: An edge, (h,k) has No Common Sibs i.e., CSh,kShSk= iff Eh&Ek pure0 iff that edge is not in a 3clique. Proof: An edge (g,f) has a common sibling, k, iff (g,f,k) is a 3clique. Thus, removing all edges with NoCommonSibs leaves only 3cliques (if the DONOT ISOLATE rule is in place, it also leaves the isolates.). This initial graph shows the result of CS0NCS. Then we do the 2nd round. Keep a list of vertices with 1 or 2 remaining siblings (edges they participate in the DO NOT DELETE): 10 12 13 15 16 17 18 19 20 21 22 23 25 26 27 28 29 5 11 6 7 31 9 2.  S1P rd2 pairwise AND (of vertices of an edge) with count=2, if the two common siblings do not form and edge themselves (and thus, the 4 form a 4vertex 1plex = two 3cliques with a common edge, namely the original pair) delete the edge of that original pair. If count=1, deleted the edge of that original pair. m f g k CS(1,5)={7,11} not an edge, so delete 1,5 CS(1,11)={5,6} not an edge, so delete 1,11 CS(9,31)={33,34} not an edge, so delete 9,31 CS(1,6)={7,11} not an edge, so delete 1,6 CS(3,9)={1,33} not an edge, so delete 3,9 CS(24,30)={33,34} not an edge, so delete 24,30 CS(1,7)={5, 6} not an edge, so delete 1,7 CS(3,33)={9}, so delete 3,33 CS(30,34)={27}, so delete 30,34 CS(1,9)={3}, so delete 1,9 CS(6,7)={17}, so delete 6,7 CS(32,34)={29}, so delete 32,34 S1Prd2 pairwise ANDs 7 5 5 2 2 2 3 1 2 0 1 3 1 1 1 4 4 3 3 1 1 1 4 3 2 3 1 3 1 3 1 1 2 1 1 1 2 2 1 0 1 2 1 2 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 5 5 6 6 6 7 9 9 9 10 24242424 2525 26 2727 28 2929 3030 3131 32 2 3 4 5 6 7 8 9 11121314182022 3 4 8 14182022 4 8 9 1433 8 1314 7 11 7 1117 17 313334 34 28303334 2632 32 3034 34 3234 3334 3334 34 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 1 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 That ends Round-2. If we would do a Round-3 of CS=0 again, (29,34) deletes since there are no common siblings. The result is very very close to GN! 1 2 3 4 5 6 7 8 9

A Vertical Graph Clustering technique Minimum Common Cousins, McC: Delete edge(s) with Min # Common 1st Cousins (CFCh,kS2Ph & S2Pk). G7 Del McC0: (1,5) (1,6) (1,11) (1,12) incorrect. CC0: Delete edge(s) with zero Common 1st Cousins (CCh,kS2Ph & S2Pk). h k a b d c e f g i j McS0-McC0: Unless it results in an isolated singleton or doubleton (keep 1,12) Delete all common Siblings=0 (McS0) and all common Cousins=0 (McC0). Del 1,32 2,31 3,10 3,28 3,29 14,34 20,34 15,33 16,33 19,33 21,33 23,33 24,26 25,28 This is McS0-McC0 So do the 1time SiblingANDs (S1Ph&S1Pk) and CousinANDs (S2Ph&S2Pk). Then in one pass reading counts McS0-McC0 deletes 12 edges (whereas Girvan-Neuman makes 1 pass per edge deletion and recalculates each new pass). Next we could delete more edges with our current counts or recaculate counts and redo McS0-McC0 S2Ph= blue and orange Use DelThresh=1 on Siblings (recalculating nothing): Delete additionally: S2Pk= red and green 1,9 1,13 1,18 1,20 1,22 3,33 6,11 6,17 9,34 24,28 24,33 25,26 27,30 29,32 30,33 31,33 31,34 (but not 2,18 2,20 2,22 4,13 5,7 5,11 7,17 25,32 26,32 27,34 28,34 29,34; DONOT ISOLATE rule). This is McS1-McC0. Use DelThresh=1 on Cousins: del 1,4 (but not 7,17 15,34 16,34 19,34 23,34 27,34 due to the DONOT ISOLATE rule.) . This is McS1-McC1 Likely, next round (after recalculating CS and CC), 1,7 and 3,9 will delete. Note: {10 15 16 19 21 23 24 27 28 29 30 34} has already separated as a component. Then the other clusters would be: {9 25 26 31 32 33} TheGreens TheYellows S2P pairwise ANDs 3 6 1 0 0 5 3 4 0 0 5 4 4 4 4 6 8 1212128 118 4 11111710189 102 151015121312121 1 188 2 4 1 5 1 5 1 5 1 2 5 6 5 1 1 13135 2 4 3 2 2 131 5 176 5 2 8 3 7 3 counts 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 5 5 6 6 6 7 9 9 9 101415151616191920212123232424242424252525262727282929303031313232 S2P-AND-OP-1 2 3 4 5 6 7 8 9 11121314182022323 4 8 14182022314 8 9 10142829338 13147 117 111717313334343433343334333434333433342628303334262832323034343234333433343334 S2P-AND-OP-2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 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1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 1 0 1 0 1 0 0 1 0 1 0 1 1 1 1 0 1 1 0 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S1P pairwise ANDs 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 5 5 6 6 6 7 9 9 9 10 14 1515 1616 1919 20 2121 2323 2424242424 252525 26 2727 28 2929 3030 3131 32 2 3 4 5 6 7 8 9 1112131418202232 3 4 8 1418202231 4 8 9 1014282933 8 1314 7 11 7 1117 17 313334 34 34 3334 3334 3334 34 3334 3334 2628303334 262832 32 3034 34 3234 3334 3334 34 7 5 5 2 2 2 3 1 2 0 1 3 1 1 1 0 4 4 3 3 1 1 1 0 4 3 2 0 3 0 0 1 3 1 3 1 1 2 1 1 1 2 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 1 2 1 0 1 1 1 1 1 1 1 1 2 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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A Vertical Graph Clustering technique Min Common Siblings 2; Min Common Cousins 0 (McS2-McC0) G7 Unless singleton/doubleton is isolated, del CommonCousins0 and CommonSiblings2 Del CC0 (1,5) (1,6) (1,11) (1,12) saved by the DNI rule. Del CS2 1:5,6,7,9,11,12,13,18,20,22,32 2:18,20,22,31 3:9,10,28,29,33 4:13 5:7,11 6:11,17 7:17 9:34 10:34 14:34 15:33,34 16:33,34 19:33,34 20:34 21:33,34 23:33,34 24:26,28,33 25:26,28,32 26:32 27:30,34 28:34 29:32.34 30:33 31:33.34 32:34 We get Yellow Green(-20) {20, 24, 28, 29 ,10,15,16,19,21,23,27,30,34)} {9, 31, 33,25,26,32} So again Black and Blue are a confused, but Yellow and Green are almost perfect. At this point we have looked at serveral threshold combinations for siblings and cousins. I think McS0-McC0 followed by a recalculation and then a reapplication of McS0-McC0 might be best. S2P pairwise ANDs 3 6 1 0 0 5 3 4 0 0 5 4 4 4 4 6 8 1212128 118 4 11111710189 102 151015121312121 1 188 2 4 1 5 1 5 1 5 1 2 5 6 5 1 1 13135 2 4 3 2 2 131 5 176 5 2 8 3 7 3 counts 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 5 5 6 6 6 7 9 9 9 101415151616191920212123232424242424252525262727282929303031313232 S2P-AND-OP-1 2 3 4 5 6 7 8 9 11121314182022323 4 8 14182022314 8 9 10142829338 13147 117 111717313334343433343334333434333433342628303334262832323034343234333433343334 S2P-AND-OP-2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 1 0 1 0 1 0 0 1 0 1 0 1 1 1 1 0 1 1 0 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S1P pairwise ANDs 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 5 5 6 6 6 7 9 9 9 10 14 1515 1616 1919 20 2121 2323 2424242424 252525 26 2727 28 2929 3030 3131 32 2 3 4 5 6 7 8 9 1112131418202232 3 4 8 1418202231 4 8 9 1014282933 8 1314 7 11 7 1117 17 313334 34 34 3334 3334 3334 34 3334 3334 2628303334 262832 32 3034 34 3234 3334 3334 34 7 5 5 2 2 2 3 1 2 0 1 3 1 1 1 0 4 4 3 3 1 1 1 0 4 3 2 0 3 0 0 1 3 1 3 1 1 2 1 1 1 2 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 1 2 1 0 1 1 1 1 1 1 1 1 2 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Agglomerative Graph Clustering with similarity=DegDif DegDif(v)=0-deg(v) since 0=intdeg(v) DegDif(12)= -1 (max). Agglomerate with siblings{1,12} DegDif=1 - 15 = -14 DegDif(10)= -2 (max). Agglomerate with siblings{3 10 34} DegDif=2 - 25 = -23. DegDif(13)= -2 (max). Agglomerate with siblings{1 4 12 13} DegDif=4 - 16 = -12. DegDif(15,16)= -2 (max). Agglomerate with siblings{3 10 15 16 33 34} DegDif=7 - 28 = -25. DegDif(17)= -2 (max). Agglomerate with siblings{6 7 17} DegDif=3 - 1 = 2. DegDif(6 7 17)= 2 (max). Agglomerate with siblings{5 6 7 11 17} DegDif=6 - 4 = 2. DegDif(18)= -2 (max). Agglomerate with siblings{1 2 4 12 13 18} DegDif=8 - 17 = -9. DegDif(19,21)= -2 (max). Agglomerate with siblings{3 10 15 16 19 21 33 34} DegDif=11 - 24 = -13. DegDif(22)= -2 (max). Agglomerate with siblings{1 2 4 12 13 18 22} DegDif=10 - 15 = -5. DegDif(23,27,30)= -2 (max). Agglomerate with siblings{3 10 15 16 19 21 23 27 30 33 34} DegDif=6 - 19 = -13. 1 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 20 1 2 3 4 5 6 7 8 9 30 1 2 3 4 SP1 16 9 10 6 3 4 4 4 5 2 3 1 2 5 2 2 2 2 2 3 2 2 2 5 3 3 2 4 3 4 4 6 11 16=1deg DegDif(25)=-3 Aggl w sibs{25 26 28 32} DegDif=4-7 = -3. -12 -14 -5 -9 -23 2 -25 -13 -3 Even though there is no cluster overlap here, our method does not follow the usual agglomeration methodology, in which there is a similarity measure between pairs (starting out with all subclusters being points, so the initial similarity is between points and then involves similarity between a point and a subset and also between two subsets. There needs to be a consistent definition of similarity across all these types of pairs, which we do not have here. Therefore, let’s start over trying to define a correct similarity. Given a similarity there are two standard clustering approaches, k means and agglomerative. Agglomerative requires the above complete similarity (between pairs of subsets, one or both of which can be singletons), while k means simply requires a similarity between pairs of points. One similarity we might consider is some weighted sum of common cousins. E.g., let c0 be the # of common 0th cousins (siblings), c1=# of 1st cousins, etc. If we sum the common cousin counts with weights, w0, w1,… (presumably decreasing), then we have a similarity measure which is complete. We try this similarity on the next slide, first for agglomeration, then k means.

Agglomerative Graph Clustering with sim=DegDif cont Agglomerative Graph Clustering with sim=DegDif cont. DegDif(v)=0-deg(v) since 0=intdeg(v) G7 DegDif(12)= -1 (max). Agglom with siblings{1,12} DegDif=1 - 15 = -14 DegDif(10)= -2 (max). Agglom with siblings{3 10 34} DegDif=2 - 24 = -22 DegDif(12)= -1 (max). Agglom w siblings{1,4,12,13} DegDif=4 - 13 = -9 DegDif(15)= -2 (max) Agglom w sibs{3 10 15 33 34} DegDif=4 - 22 = -18 DegDif(16)= -1 (max) Aggl w sibs{3 10 15 16 33 34} DegDif=6 - 20 = -14 DegDif(19)= -1 mx Aggl w sibs{3 10 15 16 19 33 34} DegDif=8 - 18 = -10 DegDif(21)= -1 mx Ag w sbs{3 10 15 16 19 21 33 34} DegDif=10-16= -6 DegDif(23)= -1 Ag w sbs{3 10 15 16 19 21 23 33 34} DegDif=12 - 14 = -2 DegDif(27,30)= -2 {3 10 15 16 19 21 23 27 30 33 34} DegDif=16 - 10 = 6 DegDif(17)= -2 {6 7 17} DegDif=3 - 2 = 1 DegDif(22)= -2 (max). Agg w siblings{1,4,12,13 22} DegDif=5 - 12 = -7 DegDif(18)= -2 (max). Agg w sibls{1,4,12,13 18 22} DegDif=6 - 11 = -5 1 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 20 1 2 3 4 5 6 7 8 9 30 1 2 3 4 SP1 16 9 10 6 3 4 4 4 5 2 3 1 2 5 2 2 2 2 2 3 2 2 2 5 3 3 2 4 3 4 4 6 11 16=1deg DegDif(29)= -2 (max). Agg w sibls{29 32} DegDif= 1 - 5 = -4 DegDif(31)= -3 {3 9 10 15 16 19 21 23 27 30 31 33 34} DgDf=22 - 6 = 16 DegDif(25 26 28)= -2. Agg w sibls{25 26 28 29 32} DegDif= 5 - 4 = 1 Even though there is no cluster overlap here, our method does not follow the usual agglomeration methodology, in which there is a similarity measure between pairs (starting out with all subclusters being points, so the initial similarity is between points and then involves similarity between a point and a subset and also between two subsets. There needs to be a consistent definition of similarity across all these types of pairs, which we do not have here. Therefore, let’s start over trying to define a correct similarity. Given a similarity there are two standard clustering approaches, k means and agglomerative. Agglomerative requires the above complete similarity (between pairs of subsets, one or both of which can be singletons), while k means simply requires a similarity between pairs of points. One similarity we might consider is some weighted sum of common cousins. E.g., let c0 be the # of common 0th cousins (siblings), c1=# of 1st cousins, etc. If we sum the common cousin counts with weights, w0, w1,… (presumably decreasing), then we have a similarity measure which is complete. We try this similarity on the next slide, first for agglomeration, then k means. -14 -9 -10 -6 -3 -4 -4 -4 -5 -2 -3 -2 -5 -2 -2 -2 -2 -2 -3 -2 -2 -2 -5 -3 -3 -2 -4 -3 -4 -4 -6 -11 -16 =dgdf -14 -9 -10 -6 -3 -4 -4 -4 -4 -22 -3 -2 -5 -2 -2 -2 -2 -2 -3 -2 -2 -2 -5 -3 -3 -2 -3 -2 -4 -4 -6 -11 -16 =dgdf -9 -8 -10 -6 -3 -4 -4 -3 -3 -2 -3 -2 -3 -2 -1 -2 -2 -1 -3 -1 -2 -1 -4 -3 -3 -2 -3 -2 -3 -4 -6 -10 -16 =dgdf -5 -6 6 -6 -3 -4 -4 -3 -3 -2 -3 -2 -3 -2 -1 1 -2 -1 -3 -1 -2 -1 -3 -3 -3 -2 -3 -4 -3 -3 -4 -10 -16 =dgdf -4 -5 16 - -3 -4 -4 -3 -3 -2 -3 -2 -3 -2 -1 1 -2 -1 -3 -1 -2 -1 -3 -3 -3 -2 -3 -4 -3 -3 -4 -10 -16 =dgdf -4 -5 17 - -3 -4 -4 -3 -3 -2 -3 -2 -3 -2 -1 1 -2 -1 -3 -1 -2 -1 -2 -3 -3 -2 -3 1 -3 -3 -4 -10 -16 =dgdf -9 -8 -2 -6 -3 -4 -4 -3 -3 -2 -3 -2 -3 -2 -1 -2 -2 -1 -3 -1 -2 -1 -3 -3 -3 -2 -3 -2 -3 -3 -4 -10 -16 =dgdf -9 -8 -14-6 -3 -4 -4 -3 -3 -2 -3 -2 -3 -2 -1 -2 -2 -1 -3 -1 -2 -1 -4 -3 -3 -2 -3 -2 -3 -4 -6 -10 -16 =dgdf -9 -8 -18-6 -3 -4 -4 -3 -3 -2 -3 -2 -3 -2 -1 -2 -2 -1 -3 -1 -2 -1 -4 -3 -3 -2 -3 -2 -3 -4 -6 -10 -16 =dgdf -5 -6 6 -6 -3 -4 -4 -3 -3 -2 -3 -2 -3 -2 -1 1 -2 -1 -3 -1 -2 -1 -3 -3 -3 -2 -3 -2 -3 -3 -4 -10 -16 =dgdf -9 -8 -10-6 -3 -4 -4 -3 -3 -2 -3 -2 -3 -2 -1 -2 -2 -1 -3 -1 -2 -1 -4 -3 -3 -2 -3 -2 -3 -4 -6 -10 -16 =dgdf -8 -8 6 -6 -3 -4 -4 -3 -3 -2 -3 -2 -3 -2 -1 1 -2 -1 -3 -1 -2 -1 -3 -3 -3 -2 -3 -2 -3 -3 -4 -10 -16 =dgdf -9 -8 6 -6 -3 -4 -4 -3 -3 -2 -3 -2 -3 -2 -1 -2 -2 -1 -3 -1 -2 -1 -3 -3 -3 -2 -3 -2 -3 -3 -4 -10 -16 =dgdf -4 -5 17 - -3 -4 -4 -3 -3 -2 -3 -2 -3 -2 -1 1 -2 -1 -3 -1 -2 -1 -2 -3 -3 -2 -3 1 -3 -3 -4 -10 -16 =dgdf

1 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 20 1 2 3 4 5 6 7 8 9 30 1 2 3 4 E Agglomerative Clustering Method on G7 based on weighted sum of SPk counts to identify 1 and 34 as centers. Then among their individual nbrs, 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 20 1 2 3 4 5 6 7 8 9 30 1 2 3 4 E 16 9 10 6 3 4 4 4 5 2 3 1 2 5 2 2 2 2 2 3 2 2 2 5 3 3 2 4 3 4 4 6 11 16 SP2 9 13 19 16 14 13 13 17 25 19 14 15 14 25 15 15 3 15 16 26 15 16 16 15 6 6 14 20 21 15 20 26 11 6 SP3 8 11 4 11 8 8 8 11 3 11 8 9 9 3 6 6 12 8 6 4 6 8 6 4 23 23 6 8 8 5 8 1 10 10 SP4 0 0 0 0 8 8 8 1 0 1 8 8 8 0 9 9 8 8 8 0 8 8 8 8 1 1 10 1 1 8 1 0 1 1 SP5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 8 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 wt V#> 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 2 SP1 16 9 10 6 3 4 4 4 5 2 3 1 2 5 2 2 2 2 2 3 -1 SP2 9 13 19 16 14 13 13 17 25 19 14 15 14 25 15 15 3 15 16 26 -1 SP3 8 11 4 11 8 8 8 11 3 11 8 9 9 3 6 6 12 8 6 4 -1 SP4 0 0 0 0 8 8 8 1 0 1 8 8 8 0 9 9 8 8 8 0 -1 SP5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 8 0 1 0 WeightSum 15 -6 -3 -15 -24 -21 -21 -21 -18 -27 -24 -30 -27 -18 -27 -27 -27 -27 -27 -24 Nbrs1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 Nbrs34 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 -20 1 1 1 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 -20 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 If ( WtSum>=-20 & Nbr(1) ) then 1 else 0. wt V#> 21 22 23 24 25 26 27 28 29 30 31 32 33 34 2 SP1 2 2 2 5 3 3 2 4 3 4 4 6 11 16 -1 SP2 16 16 16 15 6 6 14 20 21 15 20 26 11 6 -1 SP3 6 8 6 4 23 23 6 8 8 5 8 1 10 10 -1 SP4 8 7 8 8 1 1 10 1 1 8 1 0 1 1 -1 SP5 1 0 1 1 0 0 1 0 0 1 0 0 0 0 WeightSum -27 -27 -27 -18 -24 -24 -27 -21 -24 -21 -21 -15 0 15 Nbrs1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 Nbrs34 1 0 1 1 0 0 1 1 1 1 1 1 0 1 -20 0 0 0 0 0 0 0 0 0 0 0 1 0 0 -20 1 0 1 1 0 0 1 1 1 1 1 1 0 1 1 2 4 3 5 6 7 8 9 Using weights of 0,1,2,4,6 for SP1,2,3,4,5 resp. wt V#> 5 6 7 8 11 12 13 17 18 22 25 26 33 0 SP1 3 4 4 4 3 1 2 2 2 2 3 3 11 1 SP2 14 13 13 17 14 15 14 3 15 16 6 6 11 2 SP3 8 8 8 11 8 9 9 12 8 8 23 23 10 4 SP4 8 8 8 1 8 8 8 8 8 7 1 1 1 6 SP5 0 0 0 0 0 0 0 8 0 0 0 0 0 WeightSum 62 61 61 43 62 65 64 107 63 60 56 56 35 SP1|2(17) 1 1 1 0 1 0 0 1 0 0 0 0 0 60 1 1 1 0 1 0 0 1 0 0 0 0 0 Select their communities with thresh on weighted sum (=-20) giving light green “1community” and black “34community (overlapping). Next, excise those and iterate. When all are in a community probably do a k means reshuffle to improve? Using weights of5,5,1,1,0 for SP1,2,3,4,5 resp. wt V#> 8 12 13 18 22 25 26 33 5 SP1 4 1 2 2 2 3 3 11 5 SP2 17 15 14 15 16 6 6 11 1 SP3 11 9 9 8 8 23 23 10 1 SP4 1 8 8 8 7 1 1 1 0 SP5 0 0 0 0 0 0 0 0 WeightSum 117 97 97 101 105 69 69 121 SP1|2(8) 1 1 1 1 1 0 0 0 SP1|2(33) 0 0 0 0 0 1 1 1 97 1 1 1 1 1 0 0 0 69 0 0 0 0 0 1 1 1 Iterate again on the remaining Method uses site betweeness, not edge betweenenss (SPPC not computed) but gives a good overlapping clustering (close to the author’s). One could attempt a few kMeans rounds to try to improve it. 10,25,26,28,29, 31 33,34 not shown (only 17 on, 8 only 27 turned on 1 5 6 7 11 2 3 5 6 7 8 9 21 2 3 4 7 30 SP4 8 8 8 8 8 8 9 10 8 8 8 8 8 8 8 10 8=4dg 15,16,19,21,23,24,27,30 only 17 on, 5deg=1 17 SP5 8=5dg G7 1 2 3 4 6 5 7 9 8