Discovering Frequent Graph Patterns Using Disjoint Paths

Transcription

1 Disovring Frqunt Grph Pttrns Using Disjoint Pths E. Gus, S. E. Shimony, N. Vntik Dpt. of Computr Sin, Bn-Gurion Univrsity of th Ngv, Br-Shv, Isrl Astrt Whrs t-mining in strutur t fouss on frqunt t vlus, in smi-strutur n grph t mining th issu is frqunt lls n ommon spifi topologis. Hr, th strutur of th t is just s importnt s its ontnt. W stuy th prolm of isovring typil pttrns of grph t, tsk m iffiult us of th omplxity of rquir su-tsks, spilly su-grph isomorphism. In this ppr, w propos nw priori-s lgorithm for mining grph t, whr th si uiling loks r rltivly lrg - isjoint pths. Th lgorithm is provn to soun n omplt. Empiril vin shows prtil vntgs of our pproh for rtin tgoris of grphs Kywors: H.2.8 Dts Applitions: H.2.8. Dt mining, H.2.8.i Mining mthos n lgorithms, H.2.8.m W mining, Grph mining. 1 Introution Du to inrsing mounts of strutur n unstrutur t ollt y vrious ompnis n institutions, th importn of t mining hs grown signifintly ovr th lst svrl yrs. Whrs in th pst, t mining ws minly ppli to strutur t n flt fils, thr is growing intrst in mining n isovring frqunt pttrns in smi-strutur t suh s w t ([24],[36],[2]), hmil ompouns t [8, 28] or iologil t [30]. Th fous of this ppr is on isovring suh frqnt pttrns, in th form of (possily ll) grphs, n nw lgorithm for this iffiult tsk. Smistrutur t pprs whn th sour os not impos rigi strutur on th t, suh s th w, or whn t is omin from svrl htrognous sours. Unlik unstrutur rw t (lik img n soun), smistrutur t os hv som strutur, ut unlik strutur t (suh s rltionl or ojt-orint tss), smistrutur t hs no solut shm or lss fix in vn. For xmpl, in th movi XML ts [29], som movis hv mor tors thn othrs; som fils (.g., Awr) r missing for som movis; som tors hv irthy ror n som o not; t. As rsult, th stru- Prtilly support y th KITE onsortium unr ontrt to th Isrli Ministry of Tr n Inustry, n y th Pul Ivnir Cntr for Rootis n Proution Mngmnt. tur of ojts is irrgulr n qury ovr th strutur is s importnt s qury ovr th t. This struturl irrgulrity, howvr, os not imply tht thr is no struturl similrity mong smistrutur ojts. On th ontrry, it is ommon for smistrutur ojts sriing th sm typ of informtion to hv similr struturs. For xmpl, vry movi ojt hs Titl n Dirtor lls; vry Ator ojt hs Nm ll; 50% of Ator ojts hv Ntionlity ll, t. This phnomnon is ommon in othr typs of smi-strutur t s wll [20]. Whil in th fil of strutur t-mining frqunt t vlus n thir ommon pprns r of intrst, in mining smi-strutur t, th fous is on frqunt lls n ommon pprns of su-sts of suh lls (in trms of XML, on woul look for frqunt ourrns of struturs of lmnts or ttriuts, isrgring th ttriut vlus). Thrfor, frquntly, ommon mol is grph, with lls on nos, or on gs, or on oth (th trnsformtion twn ths typs of mols is quit simpl). In this ppr w ssum th mol of irt or unirt grph (or st of grphs) with no lls, n our tsk is to fin frqunt pttrns in suh grph. For xmpl, s Figur 1, piting som frqunt grph pttrns foun y our lgorithm in n XML movi ts [29]. G2 st titl movi filminst G7 movi irtor irtor movi tor movi irtor movi movi G14 Figur 1: Pttrn xmpls 1.1 Grph mining pplitions Disovring n unrstning frqunt pttrns tht rprsnt suffiintly lrg prt of smistrutur ts n usful in svrl pplition rs: Improving ts storg n sign [10, 32]. Smistrutur t sts ( typil xmpl ing XML t), rry thir own shm informtion. Though rquir for t xhng n intgrtion, suh shm inorportion ntils onsirl sp ovrh, sin th shm informtion is stor with th t (.g. lmnt nms in XML). Bus of this 1

2 rltionl tss. Smistrutur t n lwys stor s trnry rltion, sin th t is n g-ll grph, ut this is no ttr thn storing th shm with th t. A goo mpping (in trms of isk sp or frgmnttion) from smistrutur t instn into rltionl shm is sirl. Frqunt pttrns isovr in th smistrutur t n us for tht purpos sin thy n hlp gnrt th si rltions, whil th nonfrqunt pttrns woul stor s ovrflow rltions (s [10]). Effiint inxing n qurying. Qurying smistrutur ts is n importnt n ommon tivity. Numrous qury lngugs wr propos for this purpos (s [9, 3]). To sp up qury prossing svrl inxing thniqus wr propos for smistrutur n XML t [13, 26, 14]. Rntly, it ws rliz tht full inxing on ll possil lls n pths in smistrutur t is not prtil. Th APEX inxing shm [5] suggsts inxing minly on frqunt pths, whr th frqunt pths r foun y t mining thniqus. This i of using mining for inxing n nturlly gnrliz to gnrl grphs, s propos in [31], whr pths r us for fining ll ourrns of qury grph in th ts. Usr prfrn-s n usr moling pplitions[11, 39]. An importnt gol for w-pg sign is to provi viwr-orint prsonliztion of w-pg ontnt. Dsignrs oftn striv to onition w-pg ontnt n pprn on th urrnt prfrns of th viwr, n proly on som unrlying strutur of th w-pg ontnt. In orr to optimiz suh ontnt, on oftn rfrs to t mining. Whn th smistrutur ts is olltion of usr trvrsl pttrns, on n riv xpt usr hvior from knowlg out frqunt trvrsl pttrns of th sm usr ollt ovr rtin prio of tim. This rsults in usful pplitions,.g. pling vrtismnts in propr pls n ttr ustomr/usr lssifition n hvior nlysis. In pst work, w nvigtion pttrns wr usully rprsnt s pths or trs, n for this typ of prolms tr n pth mining r most rlvnt [4]. Howvr, if on looks t sts of rlt w nvigtion pttrns, or t hviour ovr tim, on gts mor omplx pttrns whih n rprsnt y grphs, motivting th us of grph mining. Anothr pplition rlt to usr hviour is th r of soil ntworks, nlysing whih is n importnt fil in ommunition n in surity pplitions [12]. An xmpl soil ntwork ts s on -mils is us in our stuy. In th pst, most work on in this fil lt with ithr singl pth pttrns [2] or tr-lik pttrns [24, 36, 4]. Howvr, muh of th t on th w is grph-lik, ithr yli or yli, motivting th mining of gnrl grph t. Th fil of grph mining riv muh ttntion in rnt yrs n svrl wll known lgorithms wr vlop, suh s: AGM [18], FSG [21], gspn [40], ClosGrph [41], n AiMin [35]. In this ppr w prsnt nw lgorithm for mining frqunt pttrns in smi-strutur t, whr th t is mol s ll grph. Our lgorithm hnls gnrl unrstrit grphs, irt or unirt. Thr r two istint prolm formultions for frqunt pttrn mining in grph tsts. In th formr, known s th grph-trnstion stting, th input to th pttrn mining lgorithm is st of (usully) rltivly smll grphs, n pttrn is onsir frqunt if it pprs in lrg numr or frtion of th grphs. Not tht pttrn ourn is ount only on pr trnstion, inpnnt of posil multipl ourns in th sm trnstion. A typil pplition for this formultion is fining frqunt su-grphs in molulr trnstions [21]. In th ltrnt stting, th frquny of pttrn is s on th numr of its ourrns (i.., mings) of pttrn in ll th t, ounting multipl ourns pr trnstion. For this stting, on n ssum without loss of gnrlity tht th input is singl grph, us on n lwys trt multipl grphs s singl grph with isonnt omponnts. For historil rsons, w rfr to this formultion s th singl-grph stting [22], lthough nithr th prolm formultion nor th lgorithms r limit in this mnnr. Du to th inhrnt iffrns in hrtristis of th prolm formultion, lgorithms vlop for th grph-trnstion stting nnot hnl th singlgrph stting, whrs th lttr lgorithms n us to solv th formr prolm. In rnt yrs, numr of ffiint n sll lgorithms hv n vlop to fin pttrns in th grph-trnstion stting [20, 21, 40, 19, 16, 17, 6]. Ths lgorithms r omplt in th sns tht thy r gurnt to isovr ll frqunt sugrphs n wr shown to sl to vry lrg grph tsts. Howvr, vloping lgorithms tht r pl of fining pttrns in ss whr h trnstion is lrg grph, n spilly th singl-grph stting, hs riv muh lss ttntion, spit th ft tht this prolm stting is mor gnri n pplil to wir rng of tsts n pplition omins thn th formr prolm. Othr thn our own pprs [33, 34], th most rnt ppr ling with th singl grph stting is [22], isuss in Stion 5. 2

3 Th lgorithm prsnt in this ppr uss rthfirst numrtion, n is s on th Apriori lgorithm [1]. Ths lgorithms us n missiility proprty (fin low) of th support msur in orr to prun nit pttrns, without hking thir support irtly, whil nsuring ompltnss. Sin pttrn is onsir to frqunt in tst grph if its support msur is grtr thn usrprovi thrshol, thn on pttrn hs support smllr thn th thrshol, ll of its suprpttns n prun, or potntilly not vn gnrt in th first pl. Lt th support msur S funtion from grph pttrns n tst grphs to rl numrs (usully in [0, 1]). As usully th tst grph is unrstoo, this rgumnt to S is omitt. S oys th missiility onstrint (lso ll nti-monotoniity, or ownwr losur) if vry sugrph of frqunt pttrn is lso frqunt [1, 15]. Formlly: Dfinition (missil support msur) A support msur S is missil if for vry pttrn P, S(P ) 0 n for ll pttrns P 1, P 2 suh tht P 1 P 2 w hv S(P 1 ) S(P 2 ). Apriori-s lgorithms ompos nit pttrns from uiling loks tht vry twn lgorithms. In our lgorithm, th uiling lok is omplt pth (s th nxt stion for pris finitions) - s sn in th following (xtrmly simplifi) outlin of our lgorithm: 1. Fin ll pttrns ompos of singl pth, y irtly ounting th numr of ourrns of ths pttrns in th tst. Elimint th nonfrqunt pttrns. 2. Fin ll nit pttrns ompos of two frqunt pths, n limint th non-frqunt pttrns. 3. At h sussiv stp n: () Construt nit pttrns from smllr frqunt ons, whih hv ommon or. Spifilly, gnrt pttrns with n + 1 pths y mrging two pttrns with n pths, tht hv ommon or with n 1 pths. A simpl xmpl for n = 2, is shown in Figur 8: two grphs, h onsisting of two pths, with n intil or onsisting of on pth, r mrg to rt grph with thr pths. In gnrl, this onstrution, th hrt of th lgorithm, is quit omplx. () Prun nits tht r not frqunt. () Stop whn no mor frqunt pttrns n gnrt. priori s lgorithms, th ruil iffrn ing tht whil for itmst mining th uiling lok is n itm, n for most grph mining lgorithm (suh s FSG or gspn) th uiling loks r gs or nos, in our lgorithm th uiling lok is th (typilly muh lrgr) g-isjoint pths. Mking th uiling lok lrgr llows for smllr numr of itrtions, s wll s for smllr numr of nit pttrns tht n to tst for support - th min gol of our shm. Sin tsting support of pttrn is xpnsiv, spilly for grph t, it is importnt to improv pruning, vn if it ntils onsirl ovrh ovr th niv mthos of gnrting n tsting pttrns. Anothr omplition is tht hiving ompltnss oms non-trivil, n onsirl sp is vot in this ppr to how ompltnss n provly mintin. 1.4 Contriutions n outlin of th ppr Th i of using pths s uiling loks in grph mining lgorithm ws prsnt rifly in n rlir, onfrn vrsion of this ppr [33]. Th oprtors us to fin grph omposition, tht llow for ffiint implmnttion of th grph mrg in prti, r nw ontriution of this vrsion. A mjor issu in this rspt is proving tht our g-isjoint pth s lgorithm is omplt. This proof of orrtnss (Stion 3) is prviously unpulish, nontrivil min ontriution of this ppr. In ttmpting to fin frquntly ourring sugrph pttrns within grph, omputing th frquny of ourrn of th pttrn in th lrgr grph (th ts), n th support msur, is n intnsiv omputtionl stp. This is u to involvs multipl omputtions of th su-grph isomorphism prolm whih is hr prolm. In orr to rs th numr of xtrmly xpnsiv support omputtions, w must isr, s rly s possil, s mny nit pttrns s possil. This is gnrl proprty of our lgorithm. Minimizing th numr of xpnsiv support omputtions is th son mjor ontriution of this ppr. This vntg is mor prominnt whn th trnstion-grphs r lrg, n vn mor so in th singl-grph stting, whr th support omputtion tns to xtrmly hr. To prov th fsiility of our shm w implmnt th propos lgorithms, tst thm on som XML tss n synthti grphs, n ompr thm to othr pprohs for ounting grphs pttrns, minly th niv n th FSG lgorithms. Not tht th lgorithm prsnt hr is orthogonl to th support msur n thrfor n us for oth ss, n is ompr xprimntlly to FSG in oth ss. Th rsults show tht whil in th trnstion stting, th two lgorithms r omprl, in th singl-grph stting our lgorithm shows signifint rution in 3

4 th numr of support omputtions. In our xprimnts w lt with mium-siz grph tss (upto 20,000 nos) sin for lrgr sizs th singlgrph s support omputtion ws too hvy omputtionlly for oth th FSG lgorithm n ours. Th xprimntl vlution of our lgorithm (Stion 4) is th thir ontriution of this ppr. Th rst of this ppr is orgniz s follows. W gin with rvisiting som grph thorti nottion n rsults (Stion 2), follow y forml finition of our grph-mining prolm, n nw finitions us in spifying th lgorithm. Th grph mining lgorithm n its orrtnss proof r thn prsnt (Stion 3). Empiril vlution of our lgorithm on oth synthti n rl t is xmin in Stion 4. Stion 5 isusss rlt work, s wll s ppliility of our lgorithm to othr sttings. 2 Prliminris W gin with rvisiting stnr trms from th litrtur. A forml sttmnt of our grph-mining prolm is m, follow y finition of omposition oprtors ssntil for gnrting nit grphs in th lgorithm. Importnt si proprtis of th oprtors r stt n prov. 2.1 Pths n pth ovrs W gin y rvisiting som grph-thorti trms n proprtis. Nottion n ltr on is lso introu. A pth is n ltrnting squn of nos n (thir inint) gs, tht gins n ns with no, n tht os not ontin ny g mor thn on. For irt grphs, w rquir pth to rspt th irtion of th gs, rsulting in irt pth. A st P of g-isjoint pths ovring ll gs of of grph G xtly on is ll pth ovr of G. A pth ovr P is ll miniml if it hs th smllst rinlity of ll pth ovrs of G. Clrly, in gnrl th miniml ovr is not uniqu. Th pth numr p(g) is th rinlity of ny miniml pth-ovr of G. In this ppr w us pths s th uiling loks, in orr to rt lrgr grphs, ut r not onrn out how to trvrs th pths on thy hv n rt. Hnforth, w ignor th orring inhrnt to th pth finition, n rprsnt pth simply s th st of nos n gs in th pth, i.. s grph. Two iffrnt pths tht hv th sm st of nos n gs r thus inistinguishl in our mtho. Not tht w still rquir tht suh grph trvrsl s singl pth, vn though th trvrsl os not hv to uniqu. Rmoving pth P from grph G, not y G \ P, onsists of rmoving ll gs of P from G, follow y rmoving ll stn-lon nos. To omput th pth numr w rly on wll-known fts: lrin (n ovr y singl yli pth) iff for vry v V, (v) is vn. A onnt igrph G = (V, E) is Eulrin (n ovr y singl irt yli pth) iff for vry v V, + (v) = (v). (Throughout, w not th in-gr of v y + (v), n th out-gr y (v).) 2. For vry onnt unirt grph G = (V, E), p(g) = 1 if G is Eulrin, n p(g) = {v v V, (v) is o} /2 othrwis. For vry onnt irt grph G = (V, E), p(g) = 1 if G is Eulrin n p(g) = ( v V + (v) (v) )/2 othrwis. Osrv tht th pth numr of grph is nvr grtr thn th numr of gs, ing in ft muh smllr in most ss - spilly for unirt grphs. Thus, pths s uiling loks shoul rs th numr of itrtions in th lgorithm, s wll s improv th pruning. 2.2 Prolm sttmnt A ll grph is grph tht hs ll ssoit with h no v, not y ll(v). W ssum without loss of gnrlity tht th tst (s wll s th pttrn) grph is ll (othrwis, ssign to ll nos in th grph th sm ritrry ll). Givn two grphs G = (V, E ) n G = (V, E ), llprsrving isomorphism twn G n G is grph isomorphism φ : V V suh tht for vry v V ll(v) = ll(φ(v)). Whn suh n isomorphism xists, not y G G th ft tht th grphs r isomorphi. P is grph pttrn in grph G if it is isomorphi to onnt sugrph of G. Our prolm is formlly fin s follows. Givn tst ll grph G, n support msur S ovr pttrn grphs, n support thrshol σ, fin ll pttrn grphs P with support S(P ) σ in G. Rll tht th input n st of grphs s wll s singl grph w.l.o.g. throught. 2.3 Lxiogrphi orring To filitt ffiint inxing of pth ovrs in grph, w us nonil rprsnttion of pths n pth squns. Th lxiogrphil orring ovr pths uss no lls n grs of nos in pths, s follows. A pth P uniquly fins th grph (V (P ), E(P )): th nos trvrs y th pth, n th gs trvrs y th pth, rsptivly. For no v V (P ) th pth grs + P (v) n P (v) r th in-gr n out gr, rsptivly, of v in (V (P ), E(P )). For unirt pths, th pth gr P (v) of v is simply th gr of v in (V (P ), E(P )). Lt P irt pth, n lt v V (P ). A no in pth is rprsnt y rprsnting tupl (s 4

5 RT P (v) := (ll(v), + P (v), P (v)) For unirt pths, th rprsnting tupl is likwis fin s: RT P (v) := (ll(v), P (v)) Assuming nturl omplt orring twn lls, s wll th nturl omplt orring twn intgrs, lxiogrphil orring twn th no rprsnttion tupls of u n v, not v L u, is unrstoo. Likwis, qulity oprtor v = L u nots qulity of th rsptiv rprsnting tupls. v 1 v 2 pth P pth Q v 3 v 5 Figur 2: Rprsnting tupls v 4 Pths r inx y pth sriptor, fin s follows. Givn pth P, sort V (P ) using th orr L. Th rsulting sort squn, not y (P ), is th pth sriptor of P. Th orr p twn pths is lxiogrphi orring twn pth sriptors, using L for lmnt-wis omprison. Whn th pth sriptors of P n Q r lxiogrphilly qul (whih ours just whn th squns r qul), w writ P = p Q. If (P ) n (Q) r squns of unqul lngth, suh tht th shortr squn (lt it (P ) without loss of gnrlity) is prfix of th longr squn, w will us th onvntion tht in this s P p Q. Not tht P = Q ntils P = p Q, ut not vi-vrs. Figur 2 shows pth ovr of siz 2 of grph, whr th pths r P = v 1, v 2, v 5, v 4, v 3 n Q = v 1, v 3, v 2. Th pth sriptors r (P ) = (, 0, 1), (, 1, 1), (, 1, 0), (, 1, 1), (, 1, 1), n (Q) = (, 0, 1), (, 1, 0), (, 1, 1). Sin (Q) lx (P ) (us (, 0, 1) = lx (, 0, 1) n (, 1, 0) lx (, 1, 1)), w hv Q p P. Osrvtion 1: p is trnsitiv n omplt (tht is, vry pir of pths is omprl). Finlly, multi-sts of pths (whih r us to rprsnt omposition of grph into pths) r inx y omposition sriptors, fin s follows. Lt P multi-st of pths. Th omposition sriptor of P, not (P), is th sort squn of th oring to th orring p. Th orring lx ovr multi-sts of pths is fin s lxiogrphi orring of thir omposition sriptors. A miniml pth ovr P of grph G is ll P miniml if thr is no miniml pth ovr Q for whih Q lx P. Osrv tht thr my mor thn on P miniml pth ovr for givn grph G, ut th omposition sriptors of ll th miniml pth ovrs of G r qul. 2.4 Proprtis of pth ovrs In our t mining lgorithm, w intn to kp in th nth frqunt nit st only grphs with pth numr n. Th pth numr of grph n omput in linr tim, s it n trmin uniquly from th multi-st of no grs. In orr to orrtly prou nits with pth numr (n + 1), y omining grph pirs with pth numr n, svrl si proprtis of th pth ovrs must shown, to gurnt ompltnss of our lgorithm: 1. Rmoving pth (in miniml pth ovr) from grph rus th pth numr y For vry onnt grph G n miniml pth ovr of siz n > 2, thr r t lst two pths in th ovr, h of whih n sutrt from G, lving th rsulting grph onnt. 3. If pth numr is grtr thn 1, ll pths in miniml ovr r non-yli. Ths proprtis r stt n prov low. Thorm 1 Lt G grph (irt or unirt) with pth numr n > 1, n P miniml pth ovr of G. Thn for vry pth P P, th grph G = G \ P hs pth numr n 1. Proof. Clrly, P \ P is pth ovr of G \ P, n thus p(g \ P ) n 1. Now, lt P pth ovr of G \ P of siz n < n 1. Thn P {P } is pth ovr of G of siz n + 1 < n, ontrition. Thorm 2 Lt G = (V, E) onnt grph with p(g) = n 2 n (P 1,..., P n ) miniml pth omposition (ssuming ny ritrry orring on th pths). Thn thr xist 1 i < j n suh tht grphs G \ P i n G \ P j r onnt. Proof. Dfin th unirt omposition grph G = (V, E ) of th omposition, s follows: V = {v i 1 i n}, n {v i, v j } E just whn P i, P j hv t lst on no in ommon. Clrly G is onnt if n only if G is onnt. This proprty lso hols for ny G \ P i n its orrsponing omposition grph, whr th lttr omposition grph is qul to G with no v i n its inint gs rmov. Sin G is onnt, so is th omposition grph G. It is wll known tht vry onnt grph with mor thn two nos hs t lst two nos, h of whih n rmov (togthr with thir inint gs), lving th grph onnt. Lt v i, v j two suh nos in G (with i j). By onstrution, this 5

6 grphs. Finlly, miniml pth ovr of onnt grph with pth numr grtr thn 1 onsists only of nonyli pths, i. pths whos strt n n vrtis r iffrnt. Tht is us ny yl P n t ny point v whr it intrsts nothr pth Q, n mrg into pth Q - thry ruing th siz of th ovr (ontriting th minimlity of th pth ovr). Thus, w n onstrut ll grphs with pth numr n > 1 just from non-yli frqunt pths. For unirt grphs, w lso n show this: Lmm 1 Lt G = (V, E) n unirt grph with miniml pth ovr P, with p(g) 2. Thn vry pth P P strts t no v of o gr, n ns t no u of o gr, n v u. Proof. From th ov rsult, ll pths in th pth ovrs r non-yli. Lt P P, n no v th strt of P (ltrntly, P ns t v, ut not oth), implying tht P hs n o numr of gs inint on v. Thn, for ll Q P, pth Q P ontins n vn numr of gs inint to v (us othrwis Q ithr strts or ns t v, n n mrg with P into singl pth, gin ontriting minimlity of P). Th gr of v is th sum of th numr of gs inint on v ovr ll pths in th ovr, whih (ing th sum of vn numrs plus xtly on o numr) is o. 2.5 Compositions n grph mrging - nottion n finitions In this stion w fin th si oprtions us to omin grphs with ommon or, pr y som rquir nottion Nottion For squns n tupls, w us th following stnr nottion. Lt t squn of lngth n (or n-tupl). Thn for 1 i j n w not th ith lmnt of t y t[i], n t[i : j] nots th susqun (sutupl) of t strting t i n ning t j, inlusiv. Th ov susripting n susqun oprtors r lso ppli to sts of tupls. Thus, if T is st of tupls, thn T [i : j] = {t[i : j] t T }. A st sutrtion oprtor insi th squr rkts inits rmovl of th sutrt lmnts from th tupl (rsp. st of tupls). Thus, t[1 : n \ j] inits n (n-1)-tupl onsisting of ll th lmnts of t xpt t[j], in th sm orr s in t. W us th ot oprtor s squn (rsp. tupl) ontntion oprtor. Appli to simpl lmnt, w mn ontntion with th rsptiv 1-tupl. For xmpl, whn is simpl lmnt, t. nots n (n+1)-tupl, with (t.)[1 : n] = t, n (t.)[n + 1] =. Whn rfrring to grph lmnts, w us to not null lmnt. By t = w mn tht in tupl t ll lmnts r qul to. W fin omposis follows. Lt non-null grph lmnt. Th oprtor is fin s follows: + = + = + = + = Th omposition of two iffrnt, non-null lmnts is unfin (s us in this ppr, suh omposition is ll inonsistnt). Th omposition oprtor is lso ppli s vtor oprtor, to pirs of n-tupls, noting lmnt-wis omposition, thus: (,,, )+ (,,, ) = (,,, ). A vtor omposition whr ny of th lmnt-wis ompositions is unfin is lso unfin (inonsistnt) Grph ompositions As on of th stps of our lgorithm, two grphs (h ompos of st of pths) r mrg to rt lrgr grph. In orr to filitt oprtions on suh omposit grphs, w fin th notion of omposition tupl-st ( omposition for short). S Tl 1 for n xmpl. Dfinition 1 Lt G st of grphs. A omposition tupl-st τ of with n ovr G is pir (G(τ), tupls(τ)), whr G(τ) is n n-tupl with h lmnt signting grph in G, n T = tupls(τ) is st of n-tupls, whr, for vry tupl t T, n vry 1 i n, th lmnt t[i] signts ithr no in th grph G(T )[i] or. A tupl t is llonsistnt if for ll 1 i, j n for whih t[i] n t[j] r non-null, th nos signt y t[i] n t[j] hv th sm ll. Th numr of lmnts in h tupl in tupls(τ) will not y with(τ). Osrv tht G(τ) my hv mor thn on lmnt rfrring to th sm grph. In our lgorithm, th st G will lwys ontin singl pths, i.. grphs tht hv pth numr of 1, ut th nottion n lso us for omposition of othr typs of grph. W will hnforth ssum tht G is th st of pths in our tst grph, n thus omit rfrn to G. (In prti, w tully tk this st to th st of just th frqunt pths, for rsons of ffiiny.) Th smntis of omposition is (omposit) grph, ll th inu grph, whih hs on no for vry tupl in T. In orr to fin th inu grph, w first wish to mk sur tht th omposition tupl-st fins n g-isjoint omposition of sugrphs, tht os not istort th sugrphs of whih it onsists. Dfinition 2 A omposition τ (ovr G) is onsistnt if ll th following onitions hol: 1. For vry 1 i with(t ) n vry no v V (G(T )[i]), thr xists uniqu t T suh tht t[i] = v. (Th no onsistny onition: thr is uniqu rprsnting tupl for vry no.) 6

7 3. For vry pir of tupls t 1, t 2 T, w hv {i (t 1 [i], t 2 [i]) E(G(T )[i])} 1. (Th g isjointnss onition: h pir of (inu) vrtis hs n g in t most on of th grphs prtiipting in T.) Two omposition tupl sts r quivlnt if thy r qul, or on is qul to th othr unr prmuttion of th inis. (By unr prmuttion w mn ny ritrry prmuttion, ut with th sm prmuttion ppli to ll th tupls in tupls(τ) n to G(τ).) Th grph inu y omposition tuplst τ = (G(τ), T ) is not y Ω(τ), n fin s follows: Dfinition 3 Ω(τ) = (V, E), with V = {ν(t) t T } (whr ν is n ritrry funtion tht ssigns uniqu no to vry tupl t), n E = {(ν(t 1 ), ν(t 2 )) t 1, t 2 T i (t 1 [i], t 2 [i]) E(G(τ)[i])}. Tht is, th inu grph hs no for vry tupl in T, n n g twn pir of nos just whn on of th sugrphs omposing τ hs n g twn ths nos. Osrv tht th g isjointnss onition nsurs tht this sugrph is uniqu. Whn us to ompos nw grphs, th funtion ν vluts to nw uniqu no, i.. on tht os not ppr lswhr in th systm. Figur 3 shows grph onsisting of 3 pths: P 1, P 2, P 3 n Tl 1 prsnts orrsponing omposition tupl-st, i.. th grph is n inu grph of th tupl-st P P 2 1 P 3 3 f 2 1 v v 4 1 v 3 v 2 v 5 v 7 Figur 3: Grph G ompos from 3 g-isjoint pths: P 1, P 2, P 3. No P 1 P 2 P 3 v 1 1 v v 3 3 v 4 1 v v 6 1 v 7 2 Tl 1: Composition tupl-st τ on P 1, P 2, P 3. Osrvtion 2: Lt grph G ovr y n mutully g-isjoint sugrphs G 1, G 2,..., G n. Thn G is intil to th grph inu y th omposition tupl-st τ = (G(T ), T ) onstrut s follows: G(T ) = (G 1, G 2,..., G n ), n tupls(t ) onsists of V (G) tupls, on uniqu tupl t for h no in f v 6 n lt t[j] = µ(t) if µ(t) V (G(T )[j]), n othrwis t[j] =. A omposition tupl-st fin s ov is ll nturl omposition tupl-st w.r.t. G n its ovr. Proposition 1 Th omposition tupl-st τ is onsistnt, n Ω(T ) is isomorphi to G, unr th nturl isomorphism whr ν(t) µ(t) for ll t T. Proof: Osrv tht T oys th no-onsistny onition y onstrution. Sin th grph ovr of G is g-isjoint, n g in G implis n g in xtly on of th sugrphs, n thus T osrvs th g-isjointnss onition. Clrly ν(t) µ(t) s fin ov is n isomorphism twn Ω(T ) n G, y onstrution. Until this point, w i not onstrin th typ of sugrphs G(τ) in omposition. Hnforth, w will ssum tht ll ths sugrphs hv pth-ovr of siz 1, i.. h suh sugrph is singl pth. Finlly, w introu th notion of P-miniml ompositions, s n xtnsion of this notion in pth ovrs omposition tupl-st τ is P-miniml if thr is no τ suh tht Ω(τ ) = Ω(τ) n G(τ ) lx G(τ) Oprtors on ompositions W pro to fin oprtors on omposition tuplsts, n th rsptiv oprtions on th inu grph. Th first sir oprtion is projtion oprtor - kping only rtin prts of ll tupls (orrsponing to kping only som prts of th inu grph). This oprtion uss our prviously fin inx rng nottion. Thus, y: τ = τ[i : j] (= (G(τ)[i : j], tupls(τ)[i : j] \ )) w init tht τ is projtion of th omposition τ onto olumns i to j inlusiv. Osrv tht rmoving som lmnts of non-null tupl my rsult in null tupl, n tht suh tupls r ropp y th projtion oprtion. Likwis, to init rmovl of sugrph i from omposition T of with n: T = T [1 : n \ i] (= (G(T )[(1 : n \ i], tupls(t )[1 : n \ i]) \ ) Th rsulting τ is omposition of with n 1. In gnrl, projtion oprtions n us projt tupls to om qul, thus ruing th numr of tupls in th rsulting omposition. Howvr, for onsistnt ompositions, this n our only for tupls whih thn om null n r ropp in th projtion. This is u to th following proprty, whih follows immitly from th no onsistny onition: Proposition 2 Lt R [1, n] n ritrry squn of inis, τ onsistnt omposition, n t 1, t 2 tupls(τ) with t 1 t 2. Thn t 1 [R] = t 2 [R] implis t 1 [R] = t 2 [R] =. 7

8 tors us in our lgorithm r fin low. Crting lrgr grph from two smllr grphs is on using th ijtiv sum oprtor, fin s follows. Dfinition 4 (Bijtiv Sum) Lt τ 1 = τ 2 ompositions, h of with n 1, suh tht τ 1 [1 : (n 2)] = τ 2 [1 : (n 2)]. Lt T 1 = tupls(τ 1 ), n T 2 = tupls(τ 2 ). Th ijtiv sum of τ 1 n τ 2, not BS(τ 1, τ 2 ), is omposition τ of with n with G(τ) = G(τ 1 ).G(τ 2 )[n 1] n with tupls(τ) ing (th union of) th following sts of tupls: 1. {t 1.t 2 [n 1] t 1 T 1, t 2 T 2, t 1 [1 : (n 2)] = t 2 [1 : (n 2)] } 2. { n 2.t[n 1]. t T 1, t[1 : (n 2)] = } (whr i mns n ll- i-tupl). 3. { n 2..t[n 1] t T 2, t[1 : (n 2)] = } Th intuition for this finition is s follows, y onsiring th inu grphs of th omposition tupl-sts (s Figur 4). Now, mp (n onsir s th sm no) th nos in th inu grphs stning for th tupls tht inlu T 1 [1 : (n 2)] to thos inu y T 2 [1 : (n 2)], sing th mpping on tupl qulity. Tupls in (1) orrspon to nos ppring in th inu grphs of oth τ 1 n τ 2. Tupls in (2) orrspon to nos tht ppr in th grph inu y τ 1, ut o not ppr in τ 2. Likwis, tupls in (3) orrspon to nos tht ppr in th grph inu y τ 2, ut o not ppr in τ 1. Hnforth, th onstrution (1) ov will ll typ 1 onstrution n th rsptiv gnrt tupls r ll typ 1 tupls. Likwis for itms (2) n (3) ov. Osrv tht in som ss th rsult of ijtiv sum my inonsistnt u to violtion of th g isjointnss onition. Our lgorithm will isr th rsults of suh inonsistnt ijtiv sums. Th finition of ijtiv sum n sily gnrliz to llow for th quivlnt prt of τ 1 n τ 2 to ny sust of inis of siz n 2, not nssrily [1 : (n 2)], n not nssrily in sort orr. Howvr, this woul mk th nottion xingly umrsom. Equivlntly, on n viw this gnrliz finition s prmuting th lmnt positions of τ 1 n τ 2 in orr to gt τ 1 [1 : (n 2)] = τ 2 [1 : (n 2)], prforming th ijtiv sum, n ritrrily prmuting th positions of τ. In th sription of th lgorithm, w us this prmuttion shm, in orr to simplify th nottion. Tl 2 monstrts ijtiv sum T 3 = BS(T 1, T 2 ) of two omposition tls T 1 n T 2, n Figur 4 th rsptiv inu grphs G 1 = Ω(T 1 ), G 2 = Ω(T 2 ) n G 3 = Ω(T 3 ). Null vlus r shown s lnks. Osrv tht lifting th rstrition tht th with of th omposition sts qul rsults in mningful (s fr s th inu grph is onrn) n potntilly usful oprtor. But sin our lgorithm os G 1 2 h 2 2 f g 3 3 P 1 P 2 P 3 P 4 v v 4 1 v 3 v 3 v 2 v v 4 1 v 5 v 7 v 6 G 3 v v 2 8 v 5 v 7 f f v 6 h v v 4 1 G 2 v v 2 h 6 g v 3 v 5 Figur 4: Inu grph of ijtiv sum. not us suh gnrliztion, w shll not isuss this issu furthr. Our lgorithm lso rquirs n oprtor tht llows nos inu y tupls of typs (2) to mrg with nos inu y tupls of typ (3) ftr ijtiv sum. Th mrg nos r trmin y omposition of with 2. For this purpos, w fin th spli oprtion, s follows (rfr to Figur 5 s n xmpl). Dfinition 5 (Spli) Lt τ omposition of with n 3, n S omposition of with 2, with G(S) = G(τ)[(n 1) : n]. Th rsult of spliing τ y S, not Spli(τ, S), is omposition τ with G(τ ) = G(τ), n T = tupls(τ ) fin s follows. Dnot T = tupls(τ), s = tupls(s), n lt M st of mrg tupls: M = { t 1 + t 2 t 1, t 2 T, s s (1) ( t 1 [n 1] = s[1] t 2 [n] = s[2] v 9 t 1 [n] + s[2] = s[2] t 2 [n 1] + s[1] = s[1])} Lt M th st of ll tupls t 1, t 2 from T ing mrg ov (i.. tht prtiipt in th sum t 1 +t 2 in th ov finition of M). Th tupls in th rsulting omposition r T = M T \ M. Osrv tht t 1 = t 2 is llow in Eqution 1. Also, not tht it is possil to hv S n T suh tht som of th t 1 + t 2 r unfin. In this s th spli oprtion is unfin (inonsistnt). For xmpl, Tl 3 sris omposition tupl-sts T 1, T 2 n thir spli T 3 = Spli(T 1, T 2 ). Figur 5 shows th orrsponing inu grphs G 1 = Ω(T 1 ), G 2 = Ω(T 2 ) n G 3 = Ω(T 3 ). In this figur, th pths P 2 n P 3 in G 1 r spli using informtion on nos ommon to ths pths in G 2. 3 Th grph mining lgorithm This stion prsnts our lgorithm psuoo for mining frqunt grph pttrns, whih works for oth irt n unirt grphs. A proof of orrtnss n prtil omplxity nlysis r thn vlop. g v 7 8

9 No P 1 P 2 P 3 No P 1 P 2 P 4 No P 1 P 2 P 3 P 4 v 1 1 u 1 1 w 1 1 v u w v 3 3 u 3 3 w 3 3 v 4 1 u w v u 5 3 w v 6 1 u 6 2 w 6 1 v 7 2 u 7 3 w 7 2 w 8 2 w 9 3 Tl 2: Bijtiv sum. T 1 T 2 T 3 No P 1 P 2 P 3 No P 2 P 3 No P 1 P 2 P 3 v 1 1 u w 1 1 v u 4 2 w v 3 3 u w 3 3 v 4 1 u 6 2 w v w v 6 1 w 6 2 v 7 2 Tl 3: Spli. G P 1 v 2 v P 2 v 5 v 7 f 1 P 3 G 3 3 f v v 4 1 v 3 v v v 4 1 v 3 v 2 G 1 f v 6 v 5 Figur 5: Inu grph of spli. 3.1 Dsription of th lgorithm v 5 v 7 Th lgorithm onsists of thr phss. In phs I, w fin ll frqunt pths (inluing pths with yls), strting with frqunt nos n frqunt gs. In phs II, w fin ll grphs ompos of two pths, in othr wors, w fin ll possil intrstions twn pirs of pths from phs I. In phs III w mrg pirs of frqunt grphs, h onsisting of n 1 pths, suh tht th grphs hv ommon or of n 2 pths, in n ttmpt to prou grphs with n pths. Throughout, w ssum tht som missil support msur is us. In phss I n III w onstrut frqunt grph pttrns rursivly, using th Apriori pproh [1]. Phs I (Algorithm 1) onstruts th frqunt pths onsiring ll frqunt pths foun in th prvious itrtion, n potntilly ing frqunt g. Aing th gs is on using th ExpnPth funtion. First, onsir th s for irt grphs in ExpnPth, whih onsirs ing n outgoing g from som nos in th pth. If th pth is yli f v 6 + = () ing g to non-yli pth ln no + = () ing g to yli pth unln no Figur 6: Phs I xmpl. (not nssrily simpl yl) w n th outgoing g nywhr, provi th no lls mth (s Figur 6 for xmpls). Othrwis, w n only n outgoing g t no tht hs n in-gr grtr thn th out-gr - thr n only on suh no if P is pth (Figur 6). W us th no st X to not th nos whr n g n. Thr r now two ss: ing n itionl no to th pth (stp 1, n s Figur 6 ( n 2) for n xmpl), n ing n g to no lry on th pth (stp 2, s Figur 61 for n xmpl) In th grph, on oul n g t ny no tht hs unqul in-gr n out-gr (n unln no), ut it is suffiint to just th outgoing g, s shown in th proof of orrtnss ltr on. Trtmnt of unirt grphs is prtilly th sm, iffring only in tht ExpnPth for unirt 9

10 i pttrns tht r pths with i gs; ψ(.) is funtion tht rts nw no with th sm ll s its rgumnt. C i is nit st for 1-pth pttrns with i gs. Output: L 1, sort st ontining th frqunt pths. 1. Fin ll frqunt nos n thm to F Fin n to F 1 ll frqunt gs y snning th t st n st k := St C k :=, F k :=. 4. For vry pth P = (V, E) F k 1 n for vry F 1 o: C k := C k ExpnPth(P, ). 5. For vry G C k, G to F k if G is frqunt, n F k ontins no grph isomorphi to G. 6. If F k st k := k + 1 n goto stp Output L 1 = k 1 i=1 F i sort oring to p. Funtion ExpnPth(P, ) for irt grphs Lt Rsult =. Dnot y (v, u). If thr is no x V s.t. (x) < + (x), lt X = {x}. Othrwis (i.. P is yli), lt X = V. 1. For vry x X s.t. ll(x) = ll(v) G = (V {ψ(u)}, E {(x, ψ(u))}) to Rsult. 2. For vry x X s.t. ll(x) = ll(v), n vry y V \ x s.t. ll(y) = ll(u) n (x, y) / E, grph G = (V, E {(x, y)}) to Rsult. Funtion ExpnPth(P, ) for unirt grphs Lt Rsult =. Dnot y {v, u}. Lt X th st of nos of o gr in P. If X is mpty, (i.. P is yli), lt X = V. 1. For vry x X s.t. ll(x) = ll(v), G = (V {ψ(u)}, E {{x, ψ(u)}}) to Rsult. 2. For vry x X, s.t. ll(x) = ll(u) G = (V {ψ(v)}, E {{x, ψ(v)}}) to Rsult. 3. For vry x, y V, x y s.t. {x, y} / E, ll(x) = ll(v), ll(y) = ll(u) s.t. t lst on of x, y is in X, G = (V, E {{x, y}}) to Rsult. Algorithm 1: Frqunt pths - Phs I grphs onsirs ing n unirt g. Hr, n g n nywhr if th pth is yli, or t on of th two o-gr nos if th pth is nonyli. Whn ing n itionl no (stps 1, 2 in Algorithm 1, ExpnPth for unirt grphs) th nw no n t ithr n of th g. Osrv tht only in phs I thr xists signifint iffrn twn irt n unirt grphs, xpt for o hin in omputing th numr of pths th support msur (whih is xtrnl n lrgly inpnnt of our lgorithm). Nottion: L 2 is st tht ontins omposition tupl-sts of frqunt grph pttrns with pth numr 2. C 2 is nit st for th ov omposition tupl-sts. 1. Lt C 2 =, L 2 =. 2. For vry pir of pths P 1, P 2 L 1 n vry onsistnt omposition tupl-st τ with G(τ) = (P 1, P 2 ), s.t. Ω(τ) is onnt n p(ω(τ)) = 2, tupl-st τ to C Rmov from C 2 ll tupl-sts tht r not P-miniml. 4. For vry tupl-st τ C 2, if Ω(τ) is frqunt, τ to L Output grphs {Ω(τ) τ L 2 }. Algorithm 2: Frqunt pth pirs - Phs II Phs II (s Algorithm 2) onstruts th frqunt grphs with pth numr 2, y omining on-pth grphs. Th non-trivil stps r stps 2, 3 n 4, whr in stp 2 ll possil ompositions of th two pths r onsir, n in stp 4 oth th pth numr n th support msur r lult; in stp 3 ll non-p-miniml isomorphi grphs r rmov. Figur 7 shows (in trms of th lxiogrphi orr w fin rlir) how svrl iffrnt 2-pth grphs r onstrut from two pths. + orinry nos join nos = Figur 7: Phs II xmpl Phs III (s Algorithm 3) onstruts th frqunt grphs with pth numr n from grphs with pth numr n 1. Th non-trivil stp is stp 2. In s 2 th grph is onstrut y fining th ommon n 1 sugrph strutur n ing th rmining two pths P 1, P 2 (on from h grph), using th ijtiv-sum oprtion. Not tht th spifition of n ritrry prmuttion is just nottionl onvnin, n is not tully implmnt this wy (it woul rquir n xponntil numr of tsts). Inst, th omposition tupl-sts τ i r rprsnt in sort orr of th pths in G(τ i ), whr h pth is 10

11 whthr two ompositions n unrgo ijtiv sum, simply ompr th strings of sort inis of pths in G(τ1), G(τ2), llowing for up to on sustitution, whih n on vry ffiintly. Th numr of ss mting this rquirmnt is typilly mny orrs of mgnitu smllr thn th numr of possil prmuttions, whih r not xpliitly gnrt 1. Only ftr th ov tst psss o w n to ompr th tupls in th projt tupl-sts. In s 2, ny omintion S of th two pths P 1, P 2 whih is frqunt n isomorphi to th rmining pths is foun from L 2. Although not stt in th psuoo, this stp is fst us L 2 n inx for fst rtrivl of ompositions ontining spifi pths. Th pths P 1, P 2 in th grph r omin (using Spli) with th gnrt nit. This lttr stp is n us mrging two pttrns irtly (using ijtiv sum) my ovrlook ss whr som nos in th rmining pths r shr. Stp 3 rmovs runnt isomorphi grphs, whil stp 4 hks th support of th nits, s in phss I n II. An optionl finl stp in th lgorithm (not shown hr) is rmoving ll frqunt su-grphs whih r not mximl, i.. ontin in lrgr frqunt grphs. Figur 8 monstrts mrging two 2-pth grphs tht hv on pth in ommon, into on 3-pth grph. Nottion: L n : st of omposition tupl-sts of with n. C n is nit st for ths ompositions. 1. St n = 3, C n =, L n =. 2. For vry pir τ 1, τ 2 of (ritrrily prmut) omposition tupl-sts from L n 1 s.t. τ 1 [1 : n 2] = τ 2 [1 : n 2], o: () Construt τ = BS(τ 1, τ 2 ). If Ω(τ) is onnt n hs pth numr n, τ to C n. () For vry omposition tupl-st S L 2, if Ω(Spli(τ, S)) is onnt n hs pth numr n, Spli(τ, S) to C n. 3. Rmov from C n ll omposition tupl-sts tht r not P-miniml. 4. For vry τ C n, τ to L n if Ω(τ) is frqunt. 5. If L n =, hlt. 6. Output {Ω(τ) τ L n }, thn st n := n + 1 n go to stp 3. Algorithm 3: Frqunt grphs - Phs III 1 Th ft tht non-isomorphi pths n hv th sm sriptor is omplition, ut not srious prolm, spilly in ll n irt grphs, whr suh ss r lss likly to our. + = pths tht r iffrnt ommon pth Figur 8: Phs III xmpl 3.2 Proof of orrtnss It is ovious y onstrution, tht our lgorithm is soun, sin (in ll phss) only frqunt pttrns r kpt t th n of th omputtion. Thrfor, showing ompltnss of th lgorithm, i.. tht ll frqunt pttrns r in foun y th lgorithm, is suffiint to prov orrtnss. Sin ll phss of th lgorithm r sprt (n run squntilly), ompltnss of h will formlly stt n prov sprtly. Thorm 3 Whn phs I (Algorithm 1) omplts, L 1 ontins ll frqunt singl-pth grph pttrns. Proof outlin: Not tht from vry pth with k gs n g n rmov so tht th rmining grph is pth with k 1 gs. Using th missiility of th support msur, n th ssumption tht ll frqunt pths with k 1 gs wr foun in th prvious itrtion, th thorm follows y inution. Thorm 4 Phs II (Algorithm 2) outputs ll onnt frqunt grphs pttrns with pth numr 2. Aitionlly, t th n of phs II, th st L 2 ontins ll P-miniml omposition tupl-sts, for vry onnt frqunt grph pttrn with pth numr 2. Proof. Lt G frqunt grph pttrn with p(g) = 2. Thn G n ompos into two gisjoint pths, n hs P-miniml omposition P 1, P 2. Sin w r using n missil support msur, P 1 n P 2 r frqunt, n y Thorm 3 n isomorphi opy of h of thm is in L 1 t th n of phs I. Dnot th isomorphisms of P 1, P 2 y P 1, P 2, rsptivly. During phs II, ll possil onsistnt omposition tupl-sts τ with G(τ) = (P 1, P 2 ) r onstrut, inluing th omposition τ for whih Ω(τ) is isomorphi to G unr th nturl isomorphism. Sin th pth sriptors r invrint unr isomorphism, n th omposition of G into P 1, P 2 is P-miniml, thn τ is lso P-miniml, n thus not prun from C 2 t stp 3. Sin G is frqunt, τ is stor in L 2 in stp 4, n G is output t stp 5. Thorm 5 Phs III outputs ll frqunt onnt grph pttrns G with pth numr p(g) 3. Proof outlin: W show th invrint tht t th n of h itrtion n, if G is frqunt grph with 11

12 τ L n suh tht Ω(τ) is isomorphi to G. Proof of th invrint is s on th invrint holing for grphs with pth numr n 1 t th ginning of th itrtion, whih hols for n = 2 u to Thorm 4. Using th missiility of th support msur, w show tht if G is frqunt thn thr xist P-miniml ompositions in L n 1 with ommon or of with n 2. Ths ompositions inu frqunt sugrphs G 1, G 2 with pth numr n 1, tht r ompos in th itrtion y using ijtiv sum n spli to form G. 3.3 Complxity isussion Th omplxity of our lgorithm is ompos of two omponnts. Th first omponnt hs to o with th prolm finition, n not with th spifi lgorithm. This omplxity is xponntil in th siz of th pttrn, n inhrnt to priori-lik lgorithms. Th omplxity of Apriori is u to th ft tht th numr of frqunt pttrns n xponntil, n th omplxity of ny grph mining lgorithm is onstrin y th n to fin ll sugrphs of ts isomorphi to givn pttrn in orr to vlut its support. Th min gol of mining lgorithm shoul thus to rs th numr of nit pttrns, n y oing so rs th numr of support omputtions. Our pproh is fsil, us th numr of pttrns rmining from on phs to th nxt is ru onsirly, oring to our xprimnts. Th gnrtion of nit st C n 1 in th worst s, tks tim: O( L n 2 2 n 2 L 2 ). In rl lif ss, frqunt pttrns from th st L n usully hv iffrnt pth strutur n lling, n th numr of nit pttrns rt is muh smllr. Evn though th omplxity is oun y n xponntil in n, in rlity for lrg tss, th sn of th ts whos omplxity is n N my wors, n in ths ss, th pproh of priori-tid my nfiil. Support omputtion. Th son omponnt of omplxity is u to th n to fin ll sugrphs isomorphi to th givn pttrn, whih is xponntil in th siz of th pttrn s wll. Whil th lrg numr of support omputtions is inhrnt to th si Apriori lgorithm ([1]), omplxity of singl support omputtion is signifintly highr for smistrutur tss. Suh omputtion rquirs () fining ll sugrphs of ts isomorphi to givn grph pttrn; () vluting support using n missil support msur. Fining ll sugrphs of ts grph isomorphi to givn pttrn pns strongly on th topology of ts grph. For ns grph, th numr of suh sugrphs n xponntil in th siz of pttrn. For omsugrph of omplt grph n frqunt! Howvr, for sprs grph (or th s for th rl-lif smi-strutur tss) th numr of instns of pttrn is muh smllr. In ition, ts grph with lrg numr of iffrnt lls is likly to prou smllr numr of pttrn instns thn similr grph with smll numr of iffrnt lls. A forml omplxity nlysis of th ntir lgorithm is vry iffiult, n thus not pursu hr. Although th omplxity is xponntil in th worst s, th xprimnts in th nxt stion suggst tht for nonns grphs th lgorithm is still rsonl for lrg grphs. 4 Empiril vlution In th mpiril vlution, two sts of xprimnts wr prform. Th first st of xprimnts omprs our lgorithm to n g-s lgorithm. Two typs of tss wr us: synthti, whr w n ontrol oth th topology n lling of grphs, n rl-lif XML movis ts [29]. Only th singl grph stting ws tst in this st of xprimnts. Th son st of xprimnts ompr our lgorithm to FSG for oth trnstions n singl grph sttings. This st of xprimnts us lso two tss, on synthti, n rl-lif soil ntwork ompos of ltroni mils. Th ts rors mils ovr prio of wk mong usrs of th BGU mil systm. Sour, stintion n siz of th mssg wr ror. Th mssg siz is us s n pproximt ll on th g. 4.1 Exprimntl stting Th xprimntl nvironmnt is Sun Ultr-30 worksttion running t 247 MHz n 128 MB of min mmory. Th rl XML fil w us is portion of th movis ts. XML lmnts r trt s nos, n inhritn rltionships n rfrns - s gs Th support msur Th stnr msur of support for trnstion tss in th litrtur is s follows. Th support S for n itm st I =< i 1,..., i k > in tst of trnstions D is: S(I) = {t t D, < i 1,..., i k > t} D (2) Howvr, if th pplition mks it nssry to ount th totl numr of ourns of pttrn, th ov shm is inpproprit. An ltrnt finition of support, tking th multipl ourns into ount, must fin, non-trivil issu u to possil ovrlps twn instns. For xmpl, on trivil support msur is th numr of instns of frqunt pttrn. This msur, howvr, is not missil. Figur 9 shows ts 12

13 stn of pttrn B, whil B A. Anothr pproh is to tk into ount ll utomorphisms of pttrn in qustion. Agin, th ts in figur 9 is ountrxmpl, sin Aut(B) = 6 n Aut(A) = 4, mking th totl ount of A s instns 12 whih is still grtr thn 6. B - pprs on A - pprs 3 tims, ut not inpnntly Figur 9: Grph pttrn support Dts grph Th only non-trivil provly missil msur w oul fin for th singl-grph -stting is fin s follows [33]. Lt D ts grph, n G grph pttrn for whih w wish to omput support. Lt A 1, A 2,..A n ll instns of G in D. W rt nw grph ll th instn grph, in whih h of th A i is no, n thr is n g twn A i n A j if th two su-grphs A i n A j hv t lst on ommon g. Th mximum inpnnt st (MIS) msur is fin s th siz of th mximum inpnnt st ovr th instn grph, n ws shown in [33] to missil. Using th MIS msur, w must omput th mximum inpnnt st of th instn grph I G. Thortilly, this n tk tim xponntil in th siz of I G, sin th inpnnt st prolm is NP-hr. Howvr, for rl-lif ss of sprs ts grph with rsonl numr of lls, this tsk is usully muh sir. In our xprimnts, tim for omputing th mximum inpnnt st ws tully ngligil ompr to th tim to fin th instns. Thrfor, th prformn of th lgorithms is not strongly pnnt on th spifi (MIS) support msur. In ition, pproximtion thniqus n us in this s (s [15] for tils) s usr usully os not r out pris support vlu. S furthr isussion on omputing th MIS msur in Stion Th implmnt lgorithms W implmnt th mining lgorithm for fully ll grphs sri in Stion 3, s wll s th two typs of g-s lgorithms isuss low. Th lttr wr us in orr to ompr th numr of gnrt nit pttrns with our lgorithm. 2 Th sm missil MIS support msur ws us for ll 2 Th rson our omprison is on opposit simpl gs lgorithms, rthr thn to FSG or GSPAN, is tht th lttr lgorithms us th trnstion-grph stting, mking irt omprison inpplil. Aitionlly, littl rsrh xists on lgorithms tht us th mximum inpnnt st (MIS) support msur, th only non-trivil missil support msur w know for th singl-grph stting (s Stion 5). tim vrgs wr tkn to limint ftors of systm lo. Th first lgorithm is s on fining ll frqunt grphs G with k gs, n thn xtning h grph G into grphs with k + 1 gs y ithr ing nw no n n g to G frqunt grph or y ing n g twn two xisting nos of G. Th pross is rpt until no grphs xtn in this mnnr r frqunt. For th omprison with FSG, w hv implmnt vrsion of FSG for th singl grph stting, s on [21]. 4.3 Exprimntl rsults First st W invstigt th hvior of th lgorithms using th following prformn prmtrs: (1) numr of nit pttrns prou y n lgorithm uring t mining; (2) numr of isomorphism omputtions uring t mining, n ovrll numr of support omputtions; (3) totl tim spnt on t mining (not CPU tim) n on support omputtions. Tl 5 prsnts rsults for tsting on synthti trs n synthti sprs grphs. Th nottion us in ll thr tls is xplin in Tl 4. For our lgorithm th numr of nit pttrns n somtims lss thn th numr of frqunt pttrns sin frqunt nos n gs r omput irtly without gnrting nit pttrns. Our implmnt- N, E, L # of nos, gs n lls in th ts S support thrshol in % C, FP # of nit n frqunt pttrns I, SC # of isomorphism n support omputtions TT totl tim (sons) spnt on t mining ST tim in s. of support omputtions EA g ition lgorithm PM Pth Mining, nots our lgorithm # sril numr of ts grph CR nit rtio Tl 4: Nottion us in rsults tls tion ns to gnrt ll pproprit sugrphs of ts grph, fin mong thm ll sugrphs tht r isomorphi to th pttrn in qustion, n uil n instn grph n fin its mximum inpnnt st siz. Thus, tsting our lgorithm on ns grphs sms to xtrmly tim onsuming. An itionl onsirtion ws th ft tht most rl-lif tss rprsnt sprs grphs rthr thn ns ons. Thrfor, w i to limit our tsts to trs n sprs grphs n to hoos support thrshol tht, on th on hn, will not limit th output to trivil grphs (nos n gs) n on th othr hn, will not mk vry onnt sugrph of th ts frqunt. From Tl 5 w onlu tht our lgorithm runs fstr vn though it onuts mor isomorphism 13

14 # N, L, S, FP Alg C, I, SC ST TT N, E, L, S, FP Alg C, I, SC ST TT % 15 EA % 14 EA PM PM % 16 EA % 17 EA PM PM % 37 EA % 28 EA PM PM % 27 EA % 16 EA PM PM % 15 EA % 27 EA PM PM % 44 EA % 27 EA PM PM % 14 EA % 32 EA PM PM Tl 5: Exprimntl rsults for trs n sprs grphs hks thn th g ition lgorithm. Th lttr ours us our lgorithm prous fwr nit pttrns, n thus lss tim is wst on support omputtion. Tl 6 ontins th numr of frqunt pttrns foun in six iffrnt susts of th movi ts with iffrnt support vlus. Th strutur of th ts ( tr s in st #6 or sprs grph) n sn to hv mor impt on th numr of frqunt pttrns thn th support vlu. As sn from Tl 6, for th sm vlus of support, th numr of frqunt pttrns is smllr, n thus th xution tim is muh smllr in th movi ts thn in th synthti tst. This inits th fsiility of our lgorithm in rl-lif ss. As th grph oms lrgr, th numr of frqunt pttrns for th sm support vlu rss sin lrgr numr of g-isjoint instns is rquir for h pttrn in orr to pss th support thrshol. Not tht ths pttrns o not ontin titls of movis or nms of irtors, sin ths r prsnt only s ttriuts n not s tgs in th XML ts. Rlt rsrh [25] ttmpts to trt ttriuts n vlus of n XML ts s wll. W u th following fts from our xprimnts: 1. Our lgorithm prous fwr nit pttrns n thrfor prforms fwr support omputtions thn th g ition lgorithm. 2. Support omputtion is sir if th ts is tr u to fwr nit pttrns. 3. Synthti grphs r not vry rgulr. As th numr of istint lls in synthti ts inrss, th hn of fining non-trivil frqunt pttrns in tht ts rss rstilly. 4. Lrg rl-lif grph tss r highly rgulr n ontin omplx pttrns Son st - omprison with FSG In this st of xprimnts, w ompr FSG with our lgorithm for oth trnstion stting n singl grph stting. For th trnstion stting, th rsults wr omprl n r not shown hr. For th singl grph stting, w msur oth th tim n th numr of support omputtions. Sin th running tim ws omint y th numr of support omputtions, w i not to rport it t ll, n inst rport th numr of support omputtions, whih is qul to th numr of nits gnrt. Thrfor in ll th tls n grphs low, th msur of ffiiny is th numr of nits gnrt. Tl 7 shows numr of nit n frqunt pttrns gnrt y oth lgorithms for vrious support vlus on two susts (5000 n 2000 nos) of Bn- Gurion univrsity -mil trffi ts. Th ntir ts is lrg (ovr nos) n quit ns, whih mks it iffiult to min. In ll tls, PM stns for Pth Mining n nots rsults hiv y our lgorithm. Tl 8 shows numrs of nits n frqunt pttrns gnrt y oth lgorithms for vrious support vlus on rnom grphs with 3000 nos, 4000 gs n iffrnt numrs of lls: 30, 40 n 50. Ths rsults show tht our lgorithm prous fwr nit pttrns tht FSG n thrfor prforms fwr support omputtions. Figur 10 shows numrs of nits gnrt y oth lgorithms for vrious support vlus on rnom grphs with 1000 nos, 2000 gs n iffrnt numrs of lls: 10 n 20 rsptivly. W lrn from our xprimnts tht support omputtion is th ftor hving th most impt on th omputtion tim us of th n for multipl sugrph isomorphism omputtions, in oth singl n multipl grph sttings. Ruing support omputtion is signifintly mor importnt thn omputing DFS o of pt- 14

15 # FP # FP # FP # FP # FP # FP Tl 6: Movi DB: support vs. frqunt pttrns 5000 nos 2000 nos S C FSG C PM FP CR S C FSG C PM FP CR 1% % % % % % % % % % % % % % % % % % % % Tl 7: BGU -mil ts rsults 50 lls 40 lls 30 lls S C FSG C PM FP S C FSG C PM FP S C FSG C PM FP Tl 8: Rnom grph with 3000 nos n 4000 gs Figur 10: Rnom grph with 1000 nos n 2000 gs trn or liminting isomorphi nits, sin frqunt pttrns r not vry lrg ompr to th ts siz. 5 Disussion n rlt work This stion rifly prsnts rlt work, n isusss our ontriution in th ontxt of prior rsrh in th fil. As mntion in th introution, most of th work on on grph mining is omprtivly rnt. Th si work rlt to this sujt is frqunt itm-st mining in strutur tss n th Apriori lgorithm n its vritions [1]. For onisnss, rfrn to th signifint oy of xisting work on trnstion ts mining is omitt. Pprs tht l with mining topologilly simpl pttrns, suh s pths n trs, r irtly rlt to our work n thus rviw low. [2] prsnts two lgorithms for mining frqunt irt simpl pth pttrns in w nvironmnt. Both lgorithms r s on n lgorithm ll MF, tht fins ll mximl forwr rfrns in st of trvrsl squns ontin in th ts. Th gol of th two mining lgorithms is to fin frqunt squns in ths pths. Th min iffrns twn th lgorithm of [2] n ours is tht th formr hnls only linr pths, mking its support msur omputtionlly simpl. Th simpl pths mining prolm is gnrliz in [36], whih sris n lgorithm for fining mximl frqunt tr-lik pttrns in smi-strutur oumnts, rprsnt in th stnr OEM mol. Although this lgorithm srhs only for tr-lik pttrns, it n lso hnl pttrns ontining yls y trnsforming thm into trs. On importnt rstrition in this ppr is tht only root trs r onsir, i.. trs whos root is th sm s th root of th ntir w ts. Chi, Nijssn, Muntz n Kok hnl th prolm of tr mining in wir sns in [28]. 15

16 90 s. A rnt survy of grph mining, y Wshio n Moto [38], prsnts som of th rlir works on th sujt lik SUBDUE [7] n GBI [42]. It thn lssifis th mining lgorithms into two mjor tgoris. Gry srh lgorithms whih srh xhustivly for ll th frqunt grph pttrns, n Inutiv (ILP) pprohs whih prgnrt mny grph pttrns oring to som logi onstrints n kgroun knowlg, n thn us qury lngug to rtriv th intrsting pttrns [27]. Sin our ppr uss th gry pproh, w o not furthr isuss ILP hr. Rgring th gry pproh, two tgoris of lgorithms wr mntion in th introution: trnstion grphs n singl grph sttings. To-t, most work ws for th trnstion grph stting, with lgorithms r ivi roughly into two lsss: rthfirst srh (or Apriori-s) n pth-first srh. Most BFS lgorithms us th si i mploy in th Apriori lgorithm. Th min iffrn twn th vrious lgorithms of this tgory is in th typ of th uiling lok us to gnrt th itm of lvl K. [18] uss vrtis. An lgorithm y Kurmohi n Krpis [20] uss gs s th min uiling lok, n ws xtn n improv in [21] y ing svrl lvr huristis whih mk mining n support omputtion mor ffiint. This lttr vrsion, ll FSG, is urrntly on of th st known n oftn ompr to vrsion of BFS grph mining lgorithms for th grph-trnstion stting s. FSG introus th finition of nonil lling of grphs s on th jny mtrix, us to limint isomorphi nits. To inrs th ffiiny of riving th nonil lls, th pproh uss som grph vrtx invrints, suh s th gr of h vrtx in th grph. FSG lso inrss th ffiiny of th nit frqunt sugrph gnrtion y introuing th trnstion ID (TID) mtho. Furthrmor, FSG limits th lss of th frqunt sugrphs to onnt grphs. Unr this limittion, FSG introus n ffiint srh lgorithm using or, whih is shr prt of siz k 1 in th two frqunt sugrphs of th siz k. FSG inrss th joining ffiiny y limiting th ommon prt of th two frqunt grphs to th or. On th nits r otin, thir frquny ounting is onut y hking th rinlity of th intrstion of oth TID lists. FSG is fst u to th introution of numrous thniqus, ut its mmory onsumption is hvy (storg for TID lists of mssiv grph t). Som is similr to thos in FSG,.g. thos rlt to joining of two su-grphs, r prsnt in this ppr s wll. Howvr, th mtho in this ppr ws riv inpnntly, n our us of g-isjoint pths s uiling lok is nw. Othr works whih us th BFS pproh r [19, 16, 17, 6]. Th son pproh, ll gspn [40], grows first srh strtgy. Th lgorithm mps h pttrn to uniqu nonil ll, n ssigns h grph uniqu minimum DFS o. By using ths lls, omplt orr rltion is impos ovr ll possil pttrns. This lxiogrphi orr is lso us to impos tr-hirrhy orr ovr ll pttrns, rsulting in hirrhil srh tr. This srh-tr is trvrs in DFS mnnr, pruning on th wy ll sugrphs with non-miniml DFS o. This lgorithm lso uss th TID pproh. Sin th lgorithm xplors th srh sp in DFS mnnr, it nls us of svrl mining thniqus whih r spilly pplil to DFS lgorithms, suh s mintining n ming st for h frqunt sugrph, lik [54]. [40] lso prsnts n xprimntl vlution whr thy ompr gspn with FSG n show th ttr prformn of gspn on svrl molulr tss. Svrl of th is of [40] wr us ltr on in n pproh whih is intrmit twn BFS n DFS, in [18]. In summry, th is prsnt in th ov pprs, hv influn our work onsirly. Howvr, using pths s uiling loks, n n ffiint mtho for mrging grphs rprsnt s ompositions of pths, r originl ontriutions of this ppr. From our xprimns w i not s n inhrnt prolm of sling up th lgorithm to vry lrg grphs, howvr, w nountr mmory prolms with lrg grphs. Ths my hnl similrly to [35]. 6 Conlusion An priori-lik lgorithm for rtriving frqunt grph pttrns from givn st of grphs is th ntrl issu in this ppr. In ontrst with most xisting work, th pttrn n ithr irt or n unirt grph, n my ontin yls. Th funtionlity n support t mining on th inrsing frtion of on-lin oumnts, whih onsist of loks onnt y rfrns. Knowlg out typil strutur of oumnts is hlpful in nlyzing omplx rpositoris of smistrutur t (.g. XML tss, th w), n is potntilly usful for qurying t, inxing it n storing it ffiintly. In srhing for frqunt pttrns, nits r onstrut using frqunt pths. Th shm is vlut mpirilly n is promising, s it shows i vntg ovr othr lgorithms. Th shm propos hr n xtn in svrl wys, suh s using prtilly ll pttrns, using mor omplx uiling loks (trs), pting th lgorithm to th ynmi ts mol n using priori-tid thniqu. Aknowlgmnts Prtilly support y th KITE onsortium unr ontrt to th Isrli Ministry of Tr n Inus- 16

17 Proution Mngmnt. W wish to thnk Mrin Litvk for implmnting signifint frtion of th o for th xprimnts, n th nonymous rviwrs for usful ommnts tht ontiut to th finl vrsion of th mnusript. Rfrns [1] R. Agrwl n R. Sriknt, Fst Algorithms for Mining Assoition Ruls, Pro. of th 20th Int l Conf. on VLDB, Sntigo, Chil, Sptmr [2] M. S. Chn, J. S. Prk, P. S. Yu, Effiint Dt Mining for Pth Trvrsl Pttrns, IEEE Trnstions on Knowlg n Dt Enginring, 10(2), 1998: [3] D. Chmrlin, XQury: A Qury Lngug for XML, Proings of SIGMOD Confrn 2003 [4] Y. Chi, S. Nijssn, R. R. Muntz, n J. N. Kok. Frqunt sutr mining: n ovrviw, Funmnt Informti, Spil Issu on Grph n Tr Mining, 2005 [5] C. Chung, J. Ki Min, K. Shim: APEX: n ptiv pth inx for XML t, Proings of SIGMOD Confrn 2002: , 2002 [6] M. Cohn, E. Gus, Digonlly Sugrphs Pttrn Mining, Proings of th 9th ACM SIGMOD Workshop on Rsrh Issus in Dt Mining n Knowlg Disovry, Pris, Frn, 2004 [7] J. Cook, L. Holr, Sustrutur isovry using minimum sription lngth n kgroun knowlg, J. of Artifiil Intll. Rsrh, pgs , 1994 [8] L. Dhsp, H. Toivonn, n R. D. King, Fining frqunt sustruturs in hmil ompouns, Proings of th 4th Intrntionl Confrn on Knowlg Disovry n Dt Mining (KDD-98), pgs Nw York, Nw York: 1998 [9] A. Dutsh, M. Frnnz, D. Florsu, A. Lvy, D. Mir, D. Suiu, Qurying XML t, IEEE Dt Enginring Bulltin 22(3), 1999: [10] A. Dutsh, M. F. Frnnz, D. Suiu, Storing Smistrutur Dt with STORED, Proings of SIGMOD Confrn 1999: [11] C. Domshlk, R. Brfmn n S. E. Shimony, Prfrns Configurtion of W Pg Contnt, Proings of IJCAI, August [12] Grton, L., Hythornthwit, C. n Wllmn, B. StuyingOnlin Soil Ntworks. Journl of Computr-Mit Communition, 3(1), 2004 [13] R. Golmn n J. Wiom, DtGuis: Enling Qury Formultion n Optimiztion in Smistrutur Dtss, Pro. of 23r VLDB Conf., Athns, Gr, [14] R. Golmn, J. Wiom, DtGuis: Enling Qury Formultion n Optimiztion in Smistrutur Dtss. Proings of VLDB 1997: , 1997 smistrutur t. To ppr in DAMI journl, 2006 [16] M. Hong, H. Zhou, W. Wng, B. Shi, An Effiint Algorithm of Frqunt Connt Sugrph Extrtion Proings of PAKDD, 2003 [17] J. Hun, W. Wng, J. Prins, Effiint Mining of Frqunt Sugrphs in th Prsn of Isomorphism. Proings of ICDM 2003: , 2003 [18] A. Inokuhi, T. Wshio, H. Moto, An priori s lgorithm for mining frqunt sustruturs from grph t, Proings of PKDD00, 2000 [19] A. Inokuhi, T. Wshio, H. Moto: Complt Mining of Frqunt Pttrns from Grphs, Mining Grph Dt. Mhin Lrning 50(3): [20] M. Kurmohi n G. Krypis, Frqunt Sugrph Disovry, Proings of IEEE ICDM, [21] M. Kurmohi n G. Krypis, An ffiint lgorithm for isovring frqunt sugrphs, IEEE Trns on Knowlg n Dt Eng., Vol 16, No. 9, Spt [22] M. Kurmohi n G. Krypis, Fining Frqunt Pttrns in Lrg Sprs Grph Proings 2004 SIAM Dt Mining Confrn, Orlno, Flori, 2004 [23] V. Lipts n E. Gus, An Effiint Algorithm for Sugrph Isomorphism, 4th Hif workshop on grph thory n lgorithms, Hif, Isrl, [24] X. Lin, Ch. Liu, Y. Zhng n X. Zhou, Effiintly Computing Frqunt Tr-Lik Topology Pttrns in W Environmnt, Proings of 31st Int. Conf. on Th. of Ojt-Orint Lngug n Systms, [25] A. Misls, M. Orlov, T. Mor, Disovring ssoitions in XML t, BGU Thnil rport, [26] T. Milo n D. Suiu: Inx Struturs for Pth Exprssions, Proings of ICDT 1999: D, Jruslm, Isrl, 1999 [27] S. Mugglton, L. DRt, Inutiv logi progrmming: Thory n mthos, J. of Logi progrmming, 19(2), pgs , 1994 [28] Nijssn, S., J. N. Kok: Frqunt grph mining n its pplition to molulr tss, proings of IEEE Int. Conf. on Systms, Mn n Cyrntis, pp , 2004 [29] Movi ts, [30] X. Pnn, N. Ayh, A gomtri lgorithm to fin smll ut highly similr 3D sustruturs in protins, Bioinformtis 14(6): ,1998 [31] D. Shsh, J. T. L. Wng n R. Guigno, Algorithmis n Applitions of Tr n Grph Srhing, Proings of th 21st ACM SIGMOD-SIGACT-SIGART Symposium on Prinipls of Dts Systms, 2002, pp

18 n Prformn Evlution of Altrntiv XML Storg Strtgis, Thnil rport, CS Dpt., Univrsity of Wisonsin, [33] N. Vntik, E. Gus, S. E. Shimony, Computing Frqunt Grph Pttrns from Smistrutur Dt, Proings ICDM 2002 : [34] N. Vntik n E. Gus, Mining Frqunt Ll n Prtilly Ll Grph Pttrns, Proings of ICDE 2004: , Boston, MA, 2004 [35] C. Wng, W. Wng, J. Pi, Y. Zhu, B. Shi, Sll mining of lrg isk s grph ts, Proings of KDD2004, Stl, 2004 [36] K. Wng, H. Liu, Disovring Typil Struturs of Doumnts: A Ro Mp Approh, Proings of SIGIR 1998 : [37] X. Wng, J. T. Li Wng, D. Shsh, B. Shpiro, I. Rigoutsos, K. Zhng, Fining Pttrns in Thr-Dimnsionl Grphs: Algorithms n Applitions to Sintifi Dt Mining, IEEE Trns. on Knowlg n Dt Eng. 14(4): , 2002 [38] T. Wshio, H. Moto, Stt of th rt of grph-s t mining,, SIGKDD xplortions, July, 2003 [39] S. Wssrmn, K. Fust, D. Ioui, Soil Ntwork Anlysis: Mthos n Applitions (Struturl Anlysis in th Soil Sins S.) Cmrig univrsity prss, 1994 Ms from th Thnion, n his PhD in Computr n Informtion Sin from th Ohio Stt Univrsity in Following his Ph, h work oth in mi (Pnn Stt Univrsity, Bn-Gurion Univrsity,), whr h i rsrh in th rs of ts systms n t surity, n in inustry (Wng Lortoris, Ntionl Smionutors, Elron, IBM Rsrh), whr h vlop qury lngugs, CAD softwr, n xprt systms for plnning n shuling. H is urrntly n ssoit profssor in omputr sin t BGU, n his rsrh intrsts r: knowlg n tss, t surity n t mining, spilly grph mining. Ntli Vntik riv th BS gr in mthmtis n omputr sin from Bn-Gurion Univrsity in 1996, th MS gr in mthmtis n omputr sin from Bn-Gurion Univrsity in Sh is urrntly PhD stunt t th Dprtmnt of Computr Sin, Bn- Gurion Univrsity. Hr rsrh intrsts inlu omintoril optimiztion, grph thory n grph mining. [40] X. Yn n J. Hn, gspn: Grph-Bs Sustrutur Pttrn Mining, Proings of ICDM 2002 : [41] X. Yn n J. Hn, ClosGrph: mining los frqunt grph pttrns, Proings of KDD03, Wshington, 2003 [42] K. Yoshi, H. Moto n N. Inurkhy, Grph-s inution s unifi lrning frmwork, J. of Appli Intllign, pgs , 1994 Solomon Eyl Shimony riv th BS gr in Eltril Enginring from th Thnion in 1982, PhD in Computr Sin from Brown Univrsity in 1991, ftr whih h join th prtmnt of Computr Sin t Bn-Gurion Univrsity. At prsnt, h is puty h of CS prtmnt t BGU, hir of th Pul Ivnir Cntr for Rootis n Proution Mngmnt, n n ssoit itor of IEEE Systms, Mn, n Cyrntis, stion B. His rsrh intrsts r rtifiil intllign, proilisti rsoning, knowlg isovry in tss, rootis, flxil omputtion, n sptil t mols. 18