Efficient Pruning of Large Knowledge Graphs

Transcription

1 Efficint Puning of Lag Knowldg Gaphs Stfano Faalli 1, In Finocchi, Simon Paolo Ponztto and Paola Vladi 1 Univsity of Rom Unitlma Sapinza Univsity of Rom Sapinza Univsity of Mannhim stfano.faalli@unitlma.it, {finocchi,vladi}@di.unioma1.it, simon@infomatik.uni-mannhim.d Abstact In this pap w psnt an fficint and highly accuat algoithm to pun noisy o ov-ambiguous knowldg gaphs givn as input an xtnsional dfinition of a domain of intst, namly as a st of instancs o concpts. Ou mthod climbs th gaph in a bottom-up fashion, itativly laying th gaph and puning nods and dgs in ach lay whil not compomising th connctivity of th st of input nods. Itativ laying and potction of p-dfind nods allow to xtact smantically cohnt DAG stuctus fom noisy o ov-ambiguous cyclic gaphs, without loss of infomation and without incuing in computational bottlncks, which a th main poblm of statof-th-at mthods fo claning lag, i.., Wbscal, knowldg gaphs. W apply ou algoithm to th tasks of puning automatically acquid taxonomis using bnchmaking data fom a SmEval valuation xcis, as wll as th xtaction of a domain-adaptd taxonomy fom th Wikipdia catgoy hiachy. Th sults show th supioity of ou appoach ov stat-of-at algoithms in tms of both output quality and computational fficincy. 1 Intoduction In th ag of infomation, th Wb povids a goldmin of data fom which knowldg can b havstd on an unpcdntd scal. As a matt of fact, ffots in infomation xtaction and knowldg acquisition fom th past dcad hav bn abl to poduc knowldg soucs on a scal that was aguably had to imagin a fw yas ago [Calson t al., 010; Wu t al., 01; Fad t al., 011; Gupta t al., 014; Dong t al., 014, int alia]. Lag covag, howv, coms with nw challngs associatd with th nois in th input data, as wll as os in th output knowldg bas. In this pap, w addss th poblm of puning lag knowldg gaphs, moving noisy dgs and lations. W spcifically focus on a high-pfoming, yt fficint algoithm sinc in this task w a typically facd with complxity Contibutions mad whil h was still at th Univsity of Mannhim. issus that ais fom th lag siz of th gaph to b pund. Exampls includ opn-domain Wb mining [Sitn t al., 016] o puning lag cowdsoucd knowldg bass.g., thos considd in [Faalli t al., 015a] and [Kapanipathi t al., 014]. W psnt a nw algoithm, namd CRUMB- TRAIL, to fficintly and ffctivly min a taxonomic stuctu hiddn within a lag gaph, stating fom a numb of constaints (o cumbs, as in th faiy tal of Hansl and Gtl) that a us can slct to chaactiz a domain of intst, which hlp idntifying pomising paths. Poblm Statmnt W stat by fomally dfining th task of puning a knowldg gaph, along with an intuition and al-cas xampls to justify th utility of such a task. W dfin a knowldg gaph as a typd dictd gaph KG = (V, E, T ): nods V idntify a st of concpts o ntitis and E is a st of lations acoss nods such that (a, b, t) E, wh a, b V and t T is th lation typ. Rlations typs may vay du to th modl spcification and th scops of th psntation. Dfinition 1 (Stict wak od lation). A lation of typ t is a stict wak od (SWO) if it is iflxiv, antisymmtic, and tansitiv [Robts and Tsman, 009]. Th dgs of a knowldg gaph KG induc a stict wak od if th dgs in th tansitiv closu of KG satisfy th th poptis of Dfinition 1. In this pap, w stict to th cas in which all dgs a of a givn typ t, wh t is a SWO lation. Common typs of SWO lations in a knowldg gaph a hypnymy, monymy and topical [Ponztto and Stub, 011] lations. Givn a knowldg gaph KG, w idntify two undsid phnomna, which w dnot as nois : i) infingmnts of SWO lations: if, fo som lation, at last on of th th conditions in Dfinition 1 dos not hold; ii) unssntial nods and dgs: assuming that KG is th gaph psntation of a spcific knowldg domain (lik, fo xampl, halth, touism, o achitctu), unssntial nods and dgs a thos which a ith dundant o vn hamful to dscib th domain smantics, fo instanc du to thi ambiguity and th sulting potntial fo smantic shifts. Accodingly, w dfin a noisy knowldg gaph NKG as a gaph wh on o both undsid phnomna occu. 4055

2 Puning a noisy knowldg gaph N KG is viwd thoughout th pap as th pocss of saching a subgaph of NKG such that: 1) its dgs induc a stict wak od and ) it dos not contain unssntial nods and dgs. Condition 1 is satisfid if th subgaph dos not contain cycls: in fact, fo ach pai of nods u and v in a cycl, both u v and v u hold by tansitivity (u v mans that dg (u, v) xists), sinc th two nods can ach ach oth along cycl dgs, violating th popty of bing antisymmtic. This can b fomally vifid using only topological poptis of th gaph. On th contay, tsting condition should in pincipl involv th us of knowldg-basd mthods, which a huistic in natu. Ou challng in this pap is to adopt a notion of (un)ssntiality which can b fomally tstd on th gaph topology without soting to mthods fo assssing loosly dfind constaints such as domain smantics. To this nd, ou appoach is to assum th availability of an initial st of nods P, dfind as follows: Dfinition (Pimitiv ssntial nods). Givn a noisy knowldg gaph NKG = (V, E), lt P V b a st of pimitiv ssntial nods containing: a) tminological nods: a subst of nods of N KG which blong to a tagt domain of intst (i.., domain-spcific instancs and tms); a) catgoical nods (th cumbs ): som (vn fw) domain-ptinnt concpts of N KG xpctd to b locatd in high positions in th stict wak od; c) oot: a concpt that is a common ancsto of all nods in P, i.., a nod fom which all nods of P can b achd (in shot, w call this popty -connctivity). Givn P, w chaactiz ssntial nods in tms of connctivity of P with spct to th oot : Dfinition (Essntial nods). Essntial nods in a NKG a thos nods that a stictly ndd to psv st P duing th puning pocss, i.., thos nods which, if movd, would compomis -connctivity. Th idntification of (un)ssntial nods is at th co of th CRUMBTRAIL algoithm. Th intuition is that, povidd that st P implicitly dfins on o vn mo domains of intst, noisy nods and dgs ith oiginatd by xtaction os o out-of-domain should mostly li outsid th lvant paths conncting tminological and cumb nods to th oot, i.., thy a not ssntial. Futh not that th output subgaph is diffnt fom th subgaph of NKG inducd by P : in paticula, it might contain nods that a not in P. Similaly, whil smbling th maximal acyclic subgaph [Bg and Sho, 1990] and th minimum fdback ac st [Dmtscu and Finocchi, 00] poblms, which a oftn usd fo untangling th stuctu of complx ntwoks [Sugiyama t al., 191; Dmtscu and Finocchi, 001], computing such subgaphs would not b appopiat fo ou puning task, sinc connctivity poptis would not b ncssaily psvd. On th oth sid dictd Stin t [Chaika t al., 1999] algoithms, bsids computational complxity issus (s Sctions and 5), would maintain connctivity but would pun N KG vy aggssivly, moving all multipl paths fom to any nod in P, at th isk of omitting nods that could b smantically lvant to dscib domain-consistnt and qually valid classifications of a concpt. Exampls of nois. In NKG, th psnc of nois might b th sult of automatic [Vladi t al., 0; Mitchll t al., 015] o collaboativ [Ponztto and Stub, 011] knowldg gaph constuction. In fact, it is acknowldgd that th pocss of cating lag and dynamically updatd knowldg soucs [Hoffat t al., 016] cannot b ntustd to a small tam of domain xpts, and is thfo subjct to os. Som common xampls of nois, which illustat th utility of NKG claning, includ: Infingmnt of SWO lations: cycls, which psnt an infingmnt of SWO lations, a vy common in th Wikipdia catgoy gaph 1 (.g., P sian books Ianian books P sian books) and in dictionais. Ciculaity of dfinitions is a wll-known issu in lxicogaphy and is considd to b a poblm sinc th ali histoy of computational linguistics [Richns, 00]. Extaction os: th poblm of automatd taxonomy laning is commonly addssd by xtacting SWO lations such as hypnymy lations fom glossais [Navigli and Vladi, 010] o dfinitional pattns [Hast, 199] lik: x a y ( cats a flins ). Although mo o lss sophisticatd, all algoithms a pon to xtaction os. As an xampl, th sntnc cats a xampls of highly find adaptation and volution may lad to xtacting th following hypnymy lations: cats xampl, cats f ind, cats adaptation, which a all wong accoding to commonsns knowldg. Out-of-domain nods: in Wikipdia, Fbas, DBpdia and oth lag knowldg bass, catgoical infomation is fly gnatd by contibutos with limitd ditoial vification, which lads to an xcssiv multipl inhitanc, a poblm that may cancl th advantag of adding smantics [Matykiwicz and Duch, 014]. A typical xampl fom Wikipdia s catgoy gaph is shown in Tabl 1: not that two vy diffnt ntitis, a dicto (David Lynch) and an ducation institution (Univsity of Tokyo), nd up almost in th sam st of upp Wikipdia catgois (oddly, thy both ach th catgois Education and P opl). Th poblm howv is not so much th quality of smantic lations, but ath th coxistnc of many diffnt pspctivs. Fo xampl, th pag Univsity of Tokyo is classifid (s column of Tabl 1) as National univsitis and Visito attactions in Tokyo. Ths a both asonabl classifications, howv th fist would b an appopiat catgoy fo an Education taxonomy, and th scond fo a Touism and placs taxonomy. Futhmo, ths diffnt smantic thads do not main spaatd whil climbing towads upp catgois, but thy intwav ov and ov again in th catgoy gaph, making th (usful) task of gnating domain viws paticulaly complx. 1 Dump_pots/Catgoy_cycls 4056

3 Wikipag: David Lynch 1st lvl catgois nd lvl catgois Top catgois Amican popl of Filmmaks fom Califonia, Gogaphy, Popl, Hu- Finnish dscnt, Amican Expimntal mans, Wold, Histoy, mal voic actos, filmmaks by nationality, Infomation, Educa- Sualist filmmaks, Mal voic actos by tion, Knowldg, Ats, Amican tlvision nationality, (mo... ) Industy, Languag, dictos, (mo... ) (mo... ) Wikipag: Univsity of Tokyo 1st lvl catgois nd lvl catgois Top catgois Educational institutions Spot in Japan by spot, Gogaphy, Humans, stablishd in 177, Visito attactions in Scinc, Histoy, 1964 Summ Olympic Tokyo, Collg athltics Knowldg, Popl, vnus, Univsitis confncs in Japan, Industy, Education, and collgs in Tokyo, National univsitis, Tchnology, Employmnt, (mo... ) (mo... ) (mo... ) Tabl 1: Exampl of multipl inhitanc in th Wikipdia catgoy gaph: not that top catgois a almost compltly ovlapping. Ou hunch is that both xtaction os and out-of-domain nods a xpctd to b unssntial fo psving th connctivity of th st P. Fo xampl, if w aim at building an animal taxonomy, with cumbs such as mammal and/o animal, and a txt mining algoithm xtacts multipl hypnymy lations (som of which a ith wong o out-of-domain) such as cats flin, cats xampl, cats musical, it is vy unlikly that th nods xampl and musical li on hypnymy chains conncting cat with mammal o animal, and vn mo unlikly that thy a ssntial to psv th connctivity btwn ths nods. Rlatd Wok Appoachs to knowldg gaphs claning in litatu diff both in th mthod to idntify lvant and ilvant nods, and in th high o low impact of th puning policy. 1) Domain-awa soft puning. Most taxonomy puning appoachs qui that th us slcts th lvant concpts by hand [Swatout, 1996] o dscibs in som way th domain of intst [Bst and Lbi, 010]. Ths appoachs a ath consvativ, and th numb of dltd nods is actually quit low [Kim t al., 007]. Futhmo, non of ths mthods is abl to dtct and mov cycls, and complxity may also b an issu, sinc it is ncssay to comput all th paths top-down fom oot to laf nods. ) Domain-awa aggssiv puning. A numb of paps ly on mo aggssiv puning policis basd on topological gaph puning mthods. [Kozava and Hovy, 010] popos an algoithm to induc a taxonomy fom a gaph stuctu. It uss a oot tm, a st of sd xampls of hypnymy lations (.g., cats f lin) and lxicosyntactic pattns to lan automatically fom th Wb hyponyms and hypnyms subodinatd to th oot. It thn uss cycl puning and longst path huistics to induc a DAG stuctu. Similaly, In Ontolan Rloadd [Vladi t al., 0] th algoithm stats fom a st of automatically xtactd tms and itativly xtacts hypnym lations thus building an hypnymy gaph. To induc a taxonomy fom th gaph, th authos us a vaiant of Chu-Liu Edmonds (CLE) optimal banching algoithm [Edmonds, 1967], in which th nod wighting statgy is basd on psving both longst paths and th highst covag of input tms. ) Domain-unawa soft puning appoachs. Studis lik [Kapanipathi t al., 014] and [Faalli t al., 015a] look at th poblm of moving cycls in Wikipdia in a us commndation task. Th fist pap uss simulatd annaling to idntify lvant upp catgois stating fom a st of Wikipags psnting uss intsts. Th latt uss an fficint vaiant of Tajan s topological soting [Tajan, 197] fo cycl puning. To avoid a andom slction of dgs to pun, [Sun t al., 017] combin diffnt huistics to appoximat an aggssiv nod anking function and xpimnt diffnt statgis to slct an dg to b movd and bak cycls. All pvious mthods, stting asid th pcision of th puning pocss, psnt at last on of th following two poblms, if not both: 1) Computational complxity: Som of th abov appoachs ly on tim o spac xpnsiv tchniqus. Fo xampl, th complxity of CLE, and oth Stin algoithms, is affctd by th nd to comput th wight of altnativ banchs, which in gnal implis a dpth fist sach (DFS). Simila poblms ais in soft puning mthods, sinc thy nd to comput all paths fom oot to laf nods using DFS. Paalllization tchniqus such as MapRduc would not hlp and would ath b hamful, du to th paalllization ovhad, sinc DFS is inhntly squntial [Rif, 195]; ) Infomation loss: Edg o path puning basd ith on topological (nods outdg, longst o shotst paths, tansitiv closu, tc.) o andom slction, may caus th loss of possibly lvant hiachical lations (spcially if th shotst path huistics is adoptd, lik in [Kapanipathi t al., 014]), and vn th disconnction of slctd sd nods [Faalli t al., 015a]. In makd contast, as shown in this pap, ou CRUMB- TRAIL algoithm dos not incu in spac and tim limits (vn whn puning xtmly lag and dns gaphs such as th full Wikipdia), and is both domain awa and aggssiv, whil psving all th availabl infomation on th domain (th st P ). 4 Th CumbTail Algoithm In this sction w summaiz ou puning algoithm, calld CRUMBTRAIL. In lin with th poblm dsciption povidd in Sction 1, givn a (dictd) noisy knowldg gaph NKG(V, E) (haft dnotd G fo bvity), a st P V of tminological nods and cumbs to b psvd (fd to as potctd nods), and a oot P, CRUMBTRAIL puns G to obtain an acyclic subgaph G P that contains all nods of P, as wll as possibly oth nods to guaant connctivity poptis as xplaind blow. Unssntial nods, namly dundant nods that hind xposing th domainfocusd stuctu du to th psnc of multipl altnativ paths (Sction 1), a liminatd by CRUMBTRAIL, sulting in a mo aggssiv puning. Th output gaph G P is layd, with top-down dgs conncting upp to low lays. Nods of P can appa on any lay, including intmdiat ons: this is in lin with th fact that P contains both tminological nods and cumbs. Moov, und th assumption 4057

4 Algoithm 1: CRUMBTRAIL Input: ppocssd gaph G(V, E), P V, oot P Output: acyclic and -connctd gaph G P 1 Initializ sts Gound, Intmdiat, and postpond tabl l = 0 V 0 = Gound 4 pat 5 lt P l = V l P 6 Gound = Gound P l 7 Intmdiat = Intmdiat \ P l l = l catnwlay(g, Gound, Intmdiat,, V l 1, postpond) 10 postponnods(g,v l, postpond) 11 pununssntial(g, V l, Gound, Intmdiat, ) 1 until Gound and postpond is mpty that G is -connctd with spct to P (i.., contains dictd paths fom th oot to vy oth nod of P ), CRUMB- TRAIL also psvs th -connctivity in G P. Ovviw. Th algoithm is calld aft th faiy tal of Hansl and Gtl: intuitivly, it finds its path towads hom th oot following a tail of badcumbs th st P of potctd nods. Diffntly fom pvious appoachs, CRUMBTRAIL climbs G bottom-up, ath than top-down, tavsing cumbs fom a bottom lay V 0 containing a subst of P up to a lay V h containing th oot. Th numb h of lays of G P is not known a pioi. Instad, du to th psnc of cycls and multipl paths btwn nods, lays V i of G P a unfoldd incmntally whil climbing G upwads, accoding to th following citia: a) no path in G P can connct any pai of nods in V i. If such a path is found duing th constuction of V i, its stat nod is dfd (postpond) to an upp lay; b) cycls involving dgs incidnt to nods of V i a bokn, taking ca of liminating cycl dgs whos moval dos not disconnct any of th nods P fom th oot ; c) fo ach nod in th nw lay V i, incoming dgs stat fom unpocssd o postpond nods and outgoing dgs nd in low lays (i.., lays V t such that t < i); d) nods in V i (and thi incidnt dgs) a pund whnv unssntial to psv th -connctivity. Tavsing and puning G bottom-up guaants that th numb of altnativ paths dcass pogssivly as th gaph is incmntally unfoldd, du to citia b and d. As dmonstatd in Sction 5, this sults in low spac consumption and fast xcution with spct to pvious appoachs, vn whn pocssing lag and highly connctd gaphs. Ppocssing. As a pliminay stp, w liminat fom G all slf loops, all dgs lading to th oot (if any), and all nods of V \ P with indg o outdg qual to 0. Fo ach nod in P, w also bak cycls passing though its outgoing dgs. Non of ths opations hams th connctivity btwn and nods in P. Data stuctus. CRUMBTRAIL maintains a lay count l 0 and a hash tabl of postpond nods consisting of pais v, i, wh i is an stimat of th lay at which nod v will b analyzd and is usd as ky in th dictionay. With a slight abus of notation, thoughout this sction w dnot by postpond(i) th subst of nods tmpoaily postpond to lay i. Thoughout th xcution of CRUMB- TRAIL, P is patitiond into two sts, calld Gound and Intmdiat = P \ Gound. At th bginning, Gound contains only P nods with outdg 0. Th maining P nods a addd to Intmdiat as wll as to th postpond tabl, using as a tntativ lay th lngth of a shotst path to a gound nod. In subsqunt itations, P nods not yt analyzd a tansfd fom Intmdiat to Gound as thy a pocssd and assignd to lays. Th fist lay, V 0, coincids with Gound nods (lin of Algoithm 1). Main itation. Aft data stuctu initialization (lins 1 of Algoithm 1), gaph puning is pfomd by an itativ pocdu (lins 4 1) that pats th following stps, until th oot has bn visitd and th postpond tabl is mpty (tmination condition at lin 1): 1) catnwlay: build a nw lay V l and mov cycls passing though dgs outgoing fom nods of V l (lin 9); ) postponnods: postpon nods of th cunt lay V l that ach ach oth (lin 10). Namly, if two nods of V l a connctd by a path of lngth k, th stating nod of th path is tmpoaily postpond to lay V l+k ; ) pununssntial: mov unssntial nods in V l (lin 11) whil psving th -connctivity with spct to nods in P. In mo dtails, th th suboutins wok as follows. Algoithm CatNwLay. Bsids slcting candidat nods to b addd to th cunt lay V l, this suboutin also movs cycls passing though incidnt dgs. Nods addd to V l can b ith stating nods of dgs whos tagt is in V l 1 o postpond nods. Fist, vy nod u with an outgoing dg to V l 1 is addd to V l, povidd that th is no dictd path fom u to any postpond nod p. This chck is don to satisfy th nod laying citia: all dgs should flow top-down, but adding u to V l would cat at last a bottom-up dg if u is connctd to a postpond nod. It may b th cas that aft this phas th cunt lay V l mains mpty. If this is th cas, and if th a no nods postpond to lay l, th algoithm skips to th fist non-mpty lay. Nxt, catnwlay movs cycls passing though dgs outgoing fom nods in V l. Th cycl baking pocdu itats ov ths dgs and dtcts a cycl acoss an dg (x, y) whnv it finds a path π = y x stating at nod y and nding in x. To implmnt this chck, th algoithm fist computs a BFS t T ootd at. Notic that, if an dg f dos not blong to T, it can b safly movd fom th cycl involving (x, y) without compomising -connctivity. Moov, sinc T is acyclic, such an dg must ncssaily xist in cycl x, y x idntifid by th cycl baking pocdu. Hnc, vy cycl can b safly bokn whil maintaining -connctivity. W mak that, aft th xcution of this suboutin, it may b possibl that th a still dgs btwn nods in V l : if such an dg (x, y) xists, w a guaantd that it is not pat of a cycl, and x will b postpond to an upp lay immdiatly lat by suboutin postponnods. It may b also th cas that th a dgs outgoing fom V l and 405

5 aching nods that a not yt assignd to a lay: if such an dg (x, y) xists, with x V l and y unlayd, it can happn nith that y is postpond no that th is a path fom y to a postpond nod. In that cas, x would hav not bn addd to V l. Sinc postpond nods includ intmdiat nods (s th data stuctu dsciption), if upwad dgs xist stating fom V l, thy must lad to nods that a unssntial to psv -connctivity and can b lat movd by suboutin pununssntial. Algoithm PostponNods. This suboutin is invokd by CRUMBTRAIL immdiatly aft building a nw lay and idntifis nods of V l to b postpond to subsqunt lays, updating th postpond tabl. As pviously obsvd, CRUMBTRAIL aims at cating lays so that dgs flow topdown. Hnc, nods of th cunt lay V l who a th stating points of paths nding in V l itslf must b shiftd to high lays. In mo dtails, givn two nods u and v in V l connctd by a path π = u v, u is postpond π lvls high than v, wh π dnots th path lngth. Sinc th could b multipl paths conncting u and v, w do not attmpt at stablishing a pioi th xact lay distanc among th two nods. Instad, π is chosn abitaily and th lay at which u is postpond might b lat incasd du to th discovy of additional (long) paths oiginating fom u. Th postpond tabl is updatd accodingly and all th schduld nods a movd fom V l. Algoithm PunUnssntial. Nods that a not ssntial to psv th connctivity btwn intmdiat and gound nods can b movd by th pocdu pununssntial. This algoithm finds nods of th cunt lay ith achabl fom intmdiat nods o aching gound nods. Nods of V l that cannot b movd unlss compomising th connctivity btwn Intmdiat and Gound a calld ssntial. Fo any v V l, lt G(v) b th st of gound nods achabl fom v and lt I(v) b th st of intmdiat nods fom which v is achabl. A nod v V l is ssntial if th is at last on pai of nods, x I(v) and y G(v), that can b connctd only though v: i.., x and y would b disconnctd by dlting v. It follows that, if G(v) = o I(v) =, thn v dos not connct any intmdiat-gound pai and can b safly movd. Unssntial nods a thn ankd basd on G(v) and I(v) and th highst ankd nod is movd (tis a bokn abitaily). Sinc th moval of any nod changs th ank of th oths, th pocdu is itatd until no mo nods can b dltd. Hnc, only ssntial nods suviv in V l. As a last stp, th algoithm dlts nods with indg o outdg qual to 0 that might hav bn catd duing th pvious stps. Exampl. Figu 1 psnts a stp-by-stp xcution of an itation of CRUMBTRAIL on a 1-nod noisy gaph. As shown in Figu 1.i, P = {, a, b, c, d, }. Duing ppocssing (Figu 1.ii), slf-loops on nods and 9, as wll as nods 10, 11, and 1, whos in/out-dg is 0, a movd. Edg (6, ) points to th oot and is thus dltd. Edgs (, ) and (d, 1) a liminatd in od to bak cycls, 4,, and d, 1, d, spctivly. Th potctd nods and a makd as intmdiat and postpond to upp lays, basd on th lngth of a shotst path to gound nods {a, b, c, d}. In Figu 1.iii, catnwlay builds a nw lay V 1 containing all nods with an outgoing dg to V 0. Th algoithm thn baks th cycls 7, 6,, 7, 6,, 6, and 14, 4, 14 by moving dgs (7, 6), (6, ), and (14, 4), spctivly. Notic that aft dg dltion th connctivity btwn intmdiat and gound nods is still maintaind. In Figu 1.iv, postponnods movs nod 4 out of V 1, dfing its pocssing to lay V du to th xistnc of paths 4, 5, b and 4,, c. Nods 7, 6, 5,, and 14 a unssntial to psv th connctivity of and with gound nods, and a thus liminatd by pununssntial in Figu 1.v. Aft th fist itation, only nods 1 and 15 main in V 1, as shown in Figu 1.vi. 5 Evaluation In od to bnchmak CRUMBTRAIL (haft, CRU), w consid th following compting appoachs: 1) th algoithm fom [Faalli t al., 015a] (TAR), which is basd on Tajan s topological soting [Tajan, 197]; ) th algoithm fom [Vladi t al., 0] (CLE), which is basd on Chu-Liu/Edmond s optimal banching [Edmonds, 1967] and an ad-hoc nod wighting policy; ) th Wikipdia hiachical dg puning algoithm poposd in [Kapanipathi t al., 014] (HPA). To th bst of ou knowldg, ths th algoithms a th only ons that tackl th poblm of fully automatd puning of lag Wb-siz knowldg gaphs. As summaizd in Sction : TAR and HPA a soft mthods only aimd at liminating cycls, whil CLE and CRU blong to th catgoy of aggssiv puning algoithms. Futhmo, both TAR and CLE a lossy, i.., thy a not guaantd to psv th connctivity of nods in P. HPA is not lossy with spct to laf nods, but givn th simpl puning statgy, it may nd up liminating lvant dgs. Expimntal stting. W valuat th pfomanc of th afomntiond mthods fo two diffnt tasks, namly ontology puning and ontology domain adaptation. W a givn a noisy knowldg gaph NKG(V, E) (G fo bvity) a gold-standad gaph GS(V, E ) and a fnc gaph R(V, E ) = G(V, E) GS(V, E ). In ontology laning G(V, E) is an automatically land stuctu that may hav noisy and missing lmnts with spct to th gold standad. Instad, in ontology adaptation G(V, E) is a possibly vy dns knowldg gaph, and th aim is to xtact a domain taxonomy which is fully mbddd in G. Pfomanc mtics. W apply ou fou puning algoithms to G(V, E) and obtain a pund gaph G p (V p, E p ). Impotantly, fo th sak of compaison with CRU (s lin of Algoithm 1), w mov fom th pund gaphs obtaind fom ach of th compad algoithms th pnding laf and oot nods. W call that pnding laf nods in a noisy gaph a thos laf nods not in P and pnding oots a thos diffnt fom. As a consqunc, vn though HPA is not lossy sinc its simpl dg puning statgy guaants that all nods v V a psvd in ou implmntation a numb of piphal nods a vntually pund. 4059

6 oot Potctd P stict wak odings movd intmdiat nods gound nods V 0 initial lay to lay i Initial noisy gaph G=(V,E) ii Ppocssing {} iii V V nods {} Building V 1 and moval of cycls passing though outgoing dgs. V V V 14 1 {} {} a b 1 c d a b 1 c d V 0 a b c d iv Postponing nods of V 1 v Rmoval of unssntial nods vi End of fist itation V {,4} V {,4} {} {} V V V V {,4} {} V V V V 0 a b c d V 0 a b c d V 0 a b c d Figu 1: A complt walkthough xampl of th application of CRUMBTRAIL to a noisy gaph. Idally, w would lik to hav th pund gaph to pfctly match th fnc gaph, namly G p (V p, E p ) = R(V, E ) o to b abl to quantify diffnt dgs of similaity btwn th two gaphs whn no pfct match is attaind. To this nd, w us th Jaccad distanc ov th two nod sts as valuation masu: J V = V V p V V p V (1) V p Sinc in both tasks of ontology puning and domain adaptation, th algoithms a povidd with a numb of initial nods P, a scond objctiv is that th ontology puning task is not lossy, i.. that all nods in P a psvd in G p. Accodingly, w comput th covag of P nods as: P Vp C P = P. Nxt, in od to povid a masu of th difficulty of th puning task, w dfin th following indxs to comput th amount of nois of G(V, E) with spct to th fnc gaph R(V, E ): Nois V = (1 V V ), Nois E = (1 E E ). Finally, w comput th familia mtics of pcision, call and balancd F-masu of th nod P V, R V and F 1 V and dg P E, R E and F 1 E puning tasks. Expimnt 1: Claning automatically land taxonomis. In ou fist xpimnt, w us th gold-standad taxonomis and th taxonomy laning systms fom Task 17 of th SmEval 015 challng. In this shad task, compting Taxonomy typ Avg. # nods Avg. # dgs Gold (GS) Rfnc (R) 55 5 SmEval submittd uns (NKGs) avag Nois V 0.6% avag Nois E 0.4% Puning algoithm Avg. # nods Avg. # dgs HPA 9 79 TAR 764 CLE CRU 99 Tabl : Stuctual analysis of th datast usd fo ontology puning. systms w quid to lan a taxonomic stuctu givn an input tminology T and a oot nod (P = T ). Fou domains w considd, namly Chmical, Food, Equipmnt and Scinc. Fo ach domain, th paticipants w askd to automatically induc thi own taxonomis, using th tminology P as th laf nods of th taxonomy. Th paticipating systms thus output automatically built hypnymy gaphs, with som amount of noisy and missing nods and dgs with spct to th fou gold-standad taxonomis. W apply th fou pviously listd puning algoithms (HPA, TAR, CLE and CRU) to th output of ach of th ontology laning systms paticipating in th SmEval 015 challng, giving a total of 1 uns (not that not all systms submittd an NKG fo ach of th 4 domains). Tabl povids an ovviw of th chaactistics of th data usd in ou 4060

7 nods dgs C P J V P V R V F 1 V P E R E F 1 E HPA TAR CLE CRU Tabl : Pfomanc in th ontology puning task (bst sults fo ach valuation mtic a boldd). fist xpimnt, coving stuctual poptis of th goldstandad, fnc noisy (i.., SmEval submittd uns) and pund taxonomis. All th data shown in th tabl hav bn avagd ov ach typ of taxonomy, considing fo ach submittd N KG only th main connctd componnt (th on which contains th oot nod). Not that, as mntiond in th tabl, in som cass TAR and CLE could not guaant th connctivity of any of th nods in P. Also not that, as shown by th avag siz of th fnc gaph, in th avag, N KGs submittd to th SmEval challng includ th majoity of nods in th gold taxonomy (V V ) but thy a unabl to captu many hypnymy lations (E E ). Tabl shows th pfomanc of th diffnt puning algoithms. In th SmEval challng, th submittd noisy hypnymy gaphs a not vy lag (cf. Tabl ): consquntly non of th algoithms incus in complxity poblms. Howv, both TAR and CLE a lossy i.., som of th nods in P a disconnctd fom th sulting taxonomy (cf. th covag of P nods C P in column ). In paticula, TAR and CLE a lossy in 11 and 19 out of 1 uns, spctivly. Concning th quality of th sulting pund taxonomis, th bst pfomanc figus a obtaind by CRU and HPA. HPA shows a high call, pimaily du to th fact that it pfoms only cycl puning. CRU instad achivs th bst ovall sults whn using pcision, F 1 masu and th Jaccad distanc as valuation mtic in th cas of Jaccad, th small th valu, th btt th taxonomy, sinc w masu h th distanc of th pund taxonomy fom th goldstandad on. W not that all appoachs pfom wost on dg puning than on nod puning. This is bcaus fnc taxonomis R do not cov many of th dgs of th gold-standad taxonomis in th fist plac, as shown in Tabl. That is, sinc puning algoithms cannot, and a not mant to b abl to tiv missing nods and dgs in GS\R, limitd covag of th dgs of th gold-standad taxonomy GS in th fnc taxonomy R havily impacts ovall pfomanc on dg stuctuing. Missing dgs a also th main caus fo puning psvd nods in lossy algoithms (i.., TAR and CLE). In Tabl, fo th sak of fai compaison, w do not tst th uniqu fatu of CRUMBTRAIL, which is abl to idntify pomising paths in th noisy gaph, whn givn a numb of intmdiat nods as additional hints (th cumbs ). Howv, w found that, whn adding to th st P an incasing but small numb of cumbs andomly slctd fom th intmdiat nods of th fnc taxonomis R, pfomanc indicatos that w alady high achiv only vy small impovmnts, whas pfomanc on dg puning, which was low, incass of about 0% bfo satuating. Expimnt : Domain adaptation of th Wikipdia catgoy hiachy. In ou scond xpimnt, w apply th fou puning algoithms to th nti Wikipdia catgoy gaph, thus tsting th full pow of th CRUMBTRAIL algoithm. Fo this, w us diffnt subgaphs of th Wikipdia catgoy gaph as silv-standad datasts, namly th catgoy hiachis ootd in th catgois Sings, Enttains and Companis. Th silv-standad catgoy hiachis a obtaind as follows: 1) stating fom th full Wikipdia catgoy gaph, w mov all incoming dgs in th slctd oot nod (.g., fo Sings, w mov dgs stating in Singing, Vocal Ensmbls and Musicians); ) w comput th tansitiv closu of th oot nod,.g., Closu(Sings); ) w add to th gold-standad all th nods in Closu() and all dgs (v i, v j ) such that v i, v j Closu(). Not that this appoach is not guaantd to poduc an o-f, gold-standad catgoy hiachy. Fo xampl, back to Tabl 1, slcting th oot nod Gogaphy, w would oddly ach both David Lynch and Univsity of Tokyo. Howv, fo vy focusd intmdiat concpts such as thos slctd in this xpimnt, w manually vifid on lag sampls of th datasts that nods a indd mostly goldn (.g., thy can b considd as spcializations of th th oots accoding to commonsns). Th fou compad algoithms a povidd with: i) th st P = T, wh T a th Wikipags at th laf nods of Closu() (.g., Diamanda Galás und catgoy Sings), and ii) th full Wikipdia catgoy gaph ootd in Main topic classifications. Th task fo ach algoithm is thn to induc fom th noisy Wikipdia catgoy gaph a domain-focusd hiachy, i.., a dictd acyclic gaph mbddd in it, with oot and laf nods T. Th sult of ach algoithm is thn compad with th silv standad, which, in contast to th pvious xpimnt, is compltly includd in th Wikipdia catgoy gaph. Tabl 5 compas th fou algoithms and shows a stiking supioity of CRU ov th oth mthods, both in tms of Jaccad distanc and F-masu, fo all domains. Though th Jaccad distanc is always makably low, th lativ ankings (not shown fo sak of spac) flct th od of gnality of th domain, and thus, th impact of multipl inhitanc: fom th most focusd domain Sings, to th most gnic Enttains. Tabl 4 shows that th amount of nois in this scond xpimnt is much high than in th pvious xpimnt using SmEval, and that th dimnsions of both noisy and silv taxonomis a od of magnitud high. Th Tabl also shows that CRU and CLE (th latt, whnv it dos not un out of mmoy) a much mo aggssiv in puning nods and dgs, as xpctd, givn that HPA and TAR a soft algoithms. In tms of fficincy CRU is always fast, followd by th naiv HPA puning statgy. Instad, fo CLE it was ncssay to limit th maximum dpth h of dpth-fist sach to 5 (dnotd as CLE(5) in Tabl 4), and vn with this limit, a solution was actually poducd only fo th fist domain, namly Sings. In th oth two cass, CLE uns out of mmoy on a multi-co machin. CRU is batn in tms of call by som of th oth mthods, sinc, as w patdly makd, TAR and HPA only mov cycls. Not that, in tms of Jaccad distanc, CRU almost pfctly tivs th silv catgoy hiachis fom th noisy 4061

8 Wikipdia catgoy gaph (G) # nods # dgs oot:main topic classifications 5,9,7 1,006,4 Silv catgoy hiachy (GS = R) # nods # dgs oot: Sings 49,076 9,4 Nois V 99.09% Nois E 99.4% algoithms # nods # dgs un tim HPA 167,6 5, h TAR 167,79 59,564 6 h CLE(5) 4,17 4,945 5 h CRU 4,6 91, h Silv catgoy hiachy (GS = R) # nods # dgs oot: Enttains,5 44,70 Nois V 9.7% Nois E 95.0% algoithms # nods # dgs un tim HPA 59,9 1,59,16 1 h TAR 5,6,099, days CLE(5) n.a n.a days (fail) CRU 4, ,50 0. h Silv catgoy hiachy (GS = R) # nods # dgs oot: Companis by stock xchang 7,95 9,46 Nois V 99.5% Nois E 99.94% algoithms # nods # dgs un tim HPA 5,54 177,79 1 h TAR 54,1 159,4 1.5 days CLE(5) n.a n.a days (fail) CRU 7,94 9, h Tabl 4: Stuctual analysis of th Wikipdia catgoy gaphs. nods dgs C P J V P V R V F 1 V P E R E F 1 E HPA TAR CLE CRU Tabl 5: Pfomanc in th task of domain adaptation of th Wikipdia catgoy gaph. Masus a avagd on th domains. gaphs. Th only comptitiv systm in tms of quality of th pund gaph is CLE: howv, computational complxity pvnts fom obtaining a solution as th dimnsion of th taxonomy incass, a poblm that cannot b mitigatd with paalllization algoithms. Th high pfomanc obtaind by CRU in th Wikipdia xpimnt is also du to th fact that, in contast to th SmEval xpimnt, th silv catgoy gaph is fully mbddd in th NKG: thfo, all th ncssay infomation is availabl to th puning algoithm. Futh not that in this scond xpimnt w did not tst th additional fatu of CRUMBTRAIL of xpanding th st P with intmdiat catgois: howv, th pfomancs a alady xtmly good and th is quit a limitd spac fo impovmnts. To summaiz, ou sults indicat that, as th dimnsion and connctivity of th NKG and th amount of nois incas, so dos th supioity of CRUMBTRAIL ov th oth gaph puning mthods, both in tms of quality of th sults and spd of xcution. 6 Conclusion and Futu Wok To th bst of ou knowldg, CRUMBTRAIL is th fist algoithm that has bn shown to pfom wll in th task of moving multipl inhitanc in th Wikipdia gaph. This poblm has pvntd so fa fom fully xploiting Wikipdia hiachical lations in many lvant applications including us commndation [Kapanipathi t al., 014; Faalli t al., 015b; Elgohay t al., 011], documnt catgoization [Gabilovich and Makovitch, 006] o quy undstanding [Schuhmach t al., 015], to nam a fw. W also acknowldg som limits, which w aim to addss in th futu. Fist, th puning statgy is limitd to knowldg gaphs with a singl lation typ that satisfis th SWO constaint. Although SWO lations hav a pdominant ol, this dos not fit to mo gnal cass that might occu in th Wb of Data, in which mo lation typs a availabl. Moov, CRUMBTRAIL lis on a numb tminological and catgoical nods (th cumbs ) that a us can slct to chaactiz a domain of intst. Sinc th availability of catgoical nods is a tight constaint, a btt solution could b to automatically inf sd catgois fom tminological nods. Acknowldgmnts This wok has bn patially suppotd by gant PO 1900/1-1 of th Dutsch Foschungsgminschaft (DFG) und th JOIN-T pojct, by th IBM Faculty Awad # , and by th MIUR und gant Dipatimnti di ccllnza 01-0 of th Dpatmnt of Comput Scinc of Sapinza Univsity. Rfncs [Bg and Sho, 1990] Bonni Bg and Pt W. Sho. Appoximation algoithms fo th maximum acyclic subgaph poblm. In Poc. of SODA, pags 6 4, [Bst and Lbi, 010] Badly J. Bst and Chistian Lbi. Extacting th ontological stuctu of opncyc fo us and potability of cognitiv modls. In Poc. of BRiMS, pags 6, 010. [Calson t al., 010] Andw Calson, Justin Bttidg, Byan Kisil, Bu Sttls, Estvam R. Huschka, and Tom M. Mitchll. Towad an achitctu fo nv-nding languag laning. In Poc. of AAAI, pags 06, 010. [Chaika t al., 1999] Moss Chaika, Chanda Chkui, To yat Chung, Zuo Dai, Ashish Gol, Sudipto Guha, and Ming Liy. Appoximation algoithms fo dictd Stin poblms. Jounal of Algoithms, (1):7 91, [Dmtscu and Finocchi, 001] Camil Dmtscu and In Finocchi. Baking cycls fo minimizing cossings. ACM Jounal on Expimntal Algoithmics, 6:1 9, 001. [Dmtscu and Finocchi, 00] Camil Dmtscu and In Finocchi. Combinatoial algoithms fo fdback poblms in dictd gaphs. Infomation Pocssing Ltts, 6():19 6,

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