Supporting information How to concatenate the local attractors of subnetworks in the HPFP

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1 n Effcen lgorh for Idenfyng Prry Phenoype rcors of Lrge-Scle Boolen Newor Sng-Mo Choo nd Kwng-Hyun Cho Depren of Mhecs Unversy of Ulsn Ulsn 446 Republc of Kore Depren of Bo nd Brn Engneerng Kore dvnced Insue of Scence nd Technology (KIST) Deeon 44 Republc of Kore Supporng nforon How o cene he l rcors of subnewors n he HPFP Correspondng uhor E-l: ch@scr Phone: Fx: Web: hp://sbescr

2 In he Mehod secon we used sple exple for how o sequenlly cene l rcors In hs suppleenry we expln he cenon n generl Le x x n denoe he se of nodes for soe posve neger n nd x represen he se of he node x e sep for ech posve neger where he se es he vlue of eher (ncve) or (cve) The ses of ll nodes re upded he se e ccordng o he ls ses of he Boolen newor The updng rules re whch re wren n he vecor for s We defne cycle of lengh p s he sequence nd x f x xn n X f X p such h p re prwse dsnc p p p f f f p nsed of he rcor p nd cycles For venence we wll use he sybol p n soe suon Fxed pons p re clled rcors of he Boolen newor For splcy we cll he rcors rcors of Consder he subses n h ssfy s pron of where s denoed by Then he se Denoe he se of every node n of x x Slrly denoe he se of every node n The upde rules for where x x h hs les n ougong edge o soe of nodes n n n n x x by h hs les n ougong edge o soe of nodes n ou ou ou x x n becoe x f x xn n f f fn Snce ll nodes n f we cn equvlenly wre x f x xn s nd defnng he funcons by do no pper n he expresson

3 Hence he nodes x n n n n x n f x x x x or X f X X n cn be sdered o genere npu sgnl no X n X X f nd so we cll ech node n oupu node fro n n npu node o or n npu sgnl no The pr n Now we re redy o defne he herrchcl pron Slrly we cll ech node n becoes subnewor wh he upde rules X n X X f ou n SDefnon We cll pron neger f he pron ssfes he hree dons of herrchcl pron of for soe posve (C) here re no npu nodes o nd no nercon beween nd (C) he se of ll npu nodes o s nonepy subse of he se of ll oupu nodes fro There s lso no nercon beween nd (C) here re no oupu nodes fro The se s clled he subnewor wh he upde rules X f X nd n subnewor wh he upde rules X f X X he - h cegory The pr n becoes he for Noe h ll re srongly neced coponens nd here exss unque herrchcl pron of gven newor n SExple In he cse of SFg below here exss subnewor whch hs he epy se The subnewors nd upde rules n he frs sed hrd nd ls cegores re denoed by ou nd 4 x6 x7 x8 x9 X f X f x6 x7 x8 x9 n n x x x x X f X X f x7 x x4 x5 n x x x X f n X X f x x x n n x x x x x X f X X f x x x6 x x

4 SFg Herrchcl pron Now we descrbe he wy o sruc rcors off he gven newor bsed on he herrchcl pron SDefnon The defnon of rcors of he newor snce here re no npu nodes o rcors of re lso clledd l rcors of or sply rcors of The se of rcors of s denoed by : : Le nd he se off rcors of be wren s Due o he don C bove he se of rcors of becoes Heree for wo rcors = n nd n n nd b nd b denoes he rcor b b lc n n lc n n b wh he les coon ulpler lc n n of The operor B for he wo ses of rcors nd B s lso defned s 4 su : b b= b ubnewors : : n b n s he n se s h of he whole : n we defne

5 B b n n for n nd B b b n 4 For exple ng = nd b= wee hve b 4 SExple In he cse of SFg here exss only one subnewor n he s s cegory Usng he upde rules n ddonl fle 7() we hve n ddonl fle 7(b) nd c : :: : c : : : = SFg Process for sequenl cenon of l rcors o sruc he globl rcors Dfferenly fro he subnewors n he frs cegory he subnewors n hee sed cegory hve npu nodes nd hen we need o sder he npu sgnls for defnon of rcors of he su bnewors 5

6 SDefnon ssue h n n n ( ) ( ) lengh p n he frs cegory Then he sybol x n ( ) ( ) obned fro For exple ng n ses of he node x obned fro x x s nonepy nd sder cyclc rcor of x nd denoes he ses of he nodes o p of perod n SExple we hve whch gves Slrly we hve o o denoes he o SDefnon 4 Two rcors nd b of Boolen newor re equl n s se rnson grph f here exss such h b for ll For exple wo rcors = nd b= for nodes x x re equl snce b for ll be herrchcl pron of STheore Le For n rcor Le of perod p denoe p p o o denoe he se of rcors of wh he upde rules Then for b Proof X x f b We frs show h b Le p for soe whch ens h p re prwse dsnc nd Snce nd p p p p p f f f re prwse dsnc p 6

7 whch ples so h we hve Slrly we cn hve f f b f f p b b b p p p p p f p f b whch coplees he proof p b b p p S Exple () For n SFg wee hve nn x ndd nd hen n ddonl fle 7(c) x x whch gves Slrly leng x x b b we hve n ddonl fle 7(c) nd hen () For fxed rcors nd b d c n SFg we hve b b nd hen 7

8 b b SRer STheore ples h he srng se vlue of npu sgnl fro rcors o he subnewor wh preservng he order of he npu sgnl does no chnge he se of l rcors of cn defne rcors of Fnlly we For n rcor SDefnon 5 ( pr o( ) The sybol wh lengh defne p p o o for soe posve neger denoes n rcor of p whch ens h re prwse dsnc o p p p p p p o o o f f f Here he rcor s lso clled prl rcor of In hs cse we cn obn he perodc sequence dvsor of q of wh perod q whch s one p The perodc sequence s clled l rcor of The ses of prl rcors nd l rcors of respecvely re denoed by pr nd SExple 4 For nd pr x x4 pr x5 x6 x7 pr x x4 pr x5 x6 x7 nd n SFg we hve n ddonl fle 7(d) pr pr x x 4 x x x

9 SDefnon 6 The sybol p p s clled cened rcor of for The se of cened rcors of wh ( ) : : Three nds of rcors of eleens of he ses pr nd respecvely Le nd s denoed by hve been defned: cened prl nd l rcors whch re be he se of cened rcors of Then ( ) : : : : where he operor s defned s pq pq pq bc xy xy x y for b= p p x x nd c= q q y y lc lc lc SExple 5 For Slrly we hve n SFg we hve : : 9

10 Here he sybol : : : : = denoes he se rnson SExple 6 For nd nd nd n SFg : : : : : : : : : : : : : = : : : : : : : = for s denoed SDefnon 7 The se of cened rcors of

11 by SExple 7 Fro SFg we hve = SExple 8 For n SFg we hve n ddonl fles 7(e) nd 7(f) : ; : : : : = SExple 9 Slrly for we hve : : : : : : =

12 SExple For we hve : : : : : : : : : : : : : = SExple For we hve : : : : : : = SExple For we hve : : :

13 : : : : : : : = SExple For we hve : : : : : : : : : : = SExple 4 Usng SExple 8-- we hve = he orgnl newor hs globl rcors

14 SRer We surze he sybols for l rcors used for cenon o sruc globl rcors s n STble In prculr he ls l rcors becoe he globl rcors Ne rcor : : : : : : : : : : : : : : : : : : : : : : : : STble Sybols for l rcors SRer We surze he process of how we sruc globl rcors by cenon of l rcors n SFg 4

15 s : : : : : : : : : : : () : : : () : : : nd nd : : : : : : : ( ) : : : : : : : : : nd h : : SFg Sury of he process o cenon l rcors In he followng we wre heore h he ls l rcors becoe he globl rcors be herrchcl pron of STheore Le cened rcors of for he -h cegory nd () s he se of rcors of () s he se of rcors of () s he se of rcors of Proof () The se of npu nodes o s epy he se of 5

16 s for soe s we hve he upde rules for Leng y y : for soe n s Snce he se f f s nd s s s y f y y y f y y s he unon of he upde rule funcons for nd s wh he se of cened rcors of s Slrly we cn prove () nd () QED Fnlly we cn descrbe he lgorh for srucon of globl rcors s follows lgorh for fndng rcors bsed on he herrchcl pron of he frs ype be herrchcl pron of Le Sep Consrucon of rcors n he s -cegory Sep- Fnd he se of l rcors of sr subnewors nd denoe by : : Sep- Fnd he se of l rcors of he subnewor s -cegory nd denoe by he unon of ll subnewors n he where he operor s defned n SDefnon 6 Sep Consrucon of rcors n he -h cegory for Sep- Fnd he hree ses of cened prl nd l rcors of sr subnewors pr denoe he by nd respecvely where ( ) : : Sep- Fnd he se of l rcors of he subnewor whch s he unon of ll subnewors n he -h cegory Denoe he se by nd 6

17 where he operor s defned n SDefnon 6 Sep- Fnd he se of l rcors of he subnewor nd denoe by S Rer 4 In order o fnd he rcor we frs choose he hrd rcor n he s -cegory nd use s he npu sgnl o he nd -cegory for fndng he cened rcors n he nd - cegory Nex we choose he sed cened rcor whch s used s he npu sgnl o he rd - cegory for fndng he cened rcors n he rd -cegory wo rcors nd cn be founded herrchclly nd ndependenly fro he frs cegory so h s possble o use prllel copuon for fndng globl rcors n our frewor In he cse of nd hey cn be founded herrchclly nd ndependenly fro he nd -cegory 7

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