A Framework for Efficient Representative Summarization of RDF Graphs

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1 A Frmework for Efficient Representtive Summriztion of RDF Grphs Šejl Čebirić, Frnçois Gosdoué, Ion Mnolescu To cite this version: Šejl Čebirić, Frnçois Gosdoué, Ion Mnolescu. A Frmework for Efficient Representtive Summriztion of RDF Grphs. Interntionl Semntic Web Conference (ISWC), Oct 2017, Vienn, Austri. Interntionl Semntic Web Conference (ISWC) < <hl > HAL Id: hl Submitted on 28 Aug 2017 HAL is multi-disciplinry open ccess rchive for the deposit nd dissemintion of scientific reserch documents, whether they re published or not. The documents my come from teching nd reserch institutions in Frnce or brod, or from public or privte reserch centers. L rchive ouverte pluridisciplinire HAL, est destinée u dépôt et à l diffusion de documents scientifiques de niveu recherche, publiés ou non, émnnt des étblissements d enseignement et de recherche frnçis ou étrngers, des lbortoires publics ou privés.

2 A Frmework for Efficient Representtive Summriztion of RDF Grphs Šejl Čebirić 1 Frnçois Gosdoué 2,1 Ion Mnolescu 1 1 INRIA nd Ecole Polytechnique, Frnce 2 Univ. Rennes 1, Frnce August 28, 2017 Abstrct RDF is the dt model of choice for Semntic Web pplictions. RDF grphs re often lrge nd hve heterogeneous, complex structure. Grph summries re compct structures computed from the input grph; they re typiclly used to simplify users experience nd to speed up grph processing. We introduce forml RDF summriztion frmework, bsed on grph quotients nd RDF node equivlence; our frmework cn be instntited with mny such equivlence reltions. We show tht our summries represent the structure nd semntics of the input grph, nd estblish sufficient condition on the RDF equivlence reltion which ensures tht grph cn be summrized more efficiently, without mterilizing its implicit triples. 1 Introduction To fcilitte working with very lrge, complex-structure, heterogeneous grphs, mny grph summries hve been proposed, including some specificlly tilored for Resource Description Frmework (RDF) grphs [1, 5, 6]. A summry of n RDF grph G is smller grph (typiclly lso RDF), bsed on which questions bout G my be nswered more efficiently thn by using G directly. In this work, we define forml generic summriztion frmework for RDF grphs, bsed on the clssicl notion of grph quotients, nd on our notion of RDF node equivlence. While quotient-style summries hve been studied in the pst [1, 3], our first contribution is forml frmework for summrizing RDF grphs including possible RDF Schem constrints, which leds us to study the interply between summriztion nd sturtion with such constrints. Specificlly, our second contribution is sufficient condition on the RDF node equivlence reltion, which gurntees tht the summry of the sturtion of G cn be built through shortcut procedure, without ctully sturting G; this cn significntly speed up the summry construction. Our summries, representtive of the complete (sturted) grphs but often much smller, cn be used in GUIs to help users explore nd query RDF grphs, or to 1

3 RDF sttement Triple Shorthnd Clss ssertion (s, rdf:type, o) (s,, o) Property ssertion (s, p, o) with p rdf:type (s, p, o) RDFS sttement Triple Shorthnd Subclss (s, rdfs:subclssof, o) (s, sc, o) Subproperty (s, rdfs:subpropertyof, o) (s, sp, o) Domin typing (s, rdfs:domin, o) (s, d, o) Rnge typing (s, rdfs:rnge, o) (s, r, o) Nme Entilment rule rdfs2 (p, d, o), (b s1, p, o 1 ) (b s1,, o) rdfs3 (p, r, o), (b s1, p, o 1 ) (o 1,, o) rdfs5 (p 1, sp, p 2 ), (p 2, sp, p 3 ) (p 1, sp, p 3 ) rdfs7 (p 1, sp, p 2 ), (b s, p 1, o) (b s, p 2, o) rdfs9 (b s, sc, o), (b s1,, b s ) (b s1,, o) rdfs11 (b s, sc, o), (o, sc, o 1 ) (b s, sc, o 1 ) ext1 (p, d, o), (o, sc, o 1 ) (p, d, o 1 ) ext2 (p, r, o), (o, sc, o 1 ) (p, r, o 1 ) ext3 (p, sp, p 1 ), (p 1, d, o) (p, d, o) ext4 (p, sp, p 1 ), (p 1, r, o) (p, r, o) Tble 1: RDF & RDFS sttements (left) nd smple RDF entilment rules (right). optimize structured nd/or keyword queries etc. s hs been done in previous works [3, 4, 5, 6]. Due to spce constrints, proofs re delegted to our technicl report vilble t [2]. 2 Preliminries An RDF grph is set of triples (s, p, o) where s is termed the subject, p the property, nd o the object; such triple sttes tht s is described with the property p tht hs vlue o. Well-formed triples, s per the RDF specifiction, belong to (U B) U (U L B), where U is set of Universl Resource Identifiers (URIs in short), L is set of literls (constnts), nd B is set of blnk nodes, representing unknown URI or literl vlues. A triple (s, p, o) sttes tht its subject s hs the property p whose vlue is the object o. RDF llows mking clss ssertions, if p is the specil builtin RDF property rdf:type ( in short), nd property ssertions otherwise (Tble 1). RDF Schem sttements (t the bottom left of Tble 1, together with the shorthnd nottions of their properties) llow specifying ontologicl constrints relting clsses nd/or properties. The semntics of n RDF grph G is its sturtion (or closure) G, defined s the G triples together with ll the implicit triples tht cn be derived from them nd the entilment rules from the RDF stndrd. Tble 1 (right) shows rules tht use RDFS P ub t d P ub sc sc CP ub r 1 t JP ub tr 2 y r 3 t 1 t 1 t 2 y 1 t Figure 1: Smple RDF grph G. constrints to derive implicit fcts or implicit constrints. Figure 1 depicts smple publictions grph, where P ub stnds for publiction (CP ub in conferences nd JP ub 2

4 in journls), for uthor, t for title nd y for yer; clss nodes nd RDFS triples pper in blue, for instnce, the domin of t (title) is P ub. Solid rrows correspond to explicit G triples, nd dotted rrows to implicit triples; ll together, they depict G. 3 Summriztion frmework We strt by reclling the clssicl notion of grph quotient. Let G = (V, E) be lbeled directed grph whose edges E hve lbels from set A. Let be n equivlence reltion over the grph node set V. The quotient of G through, denoted G /, is lbeled directed grph hving (i) node n S for ech set S of equivlent V nodes, nd (ii) n edge n S1 ns2 for some lbel A iff there exist two V nodes n 1 S 1 nd n 2 S 2 such tht the edge n 1 n2 E. When summrizing n RDF grph, clss nd schem informtion (e.g., the blue prt of Figure 1) should be preserved, s they encode its semntics. Thus, we define: Definition Let be binry reltion between the nodes of n RDF grph. We sy is n RDF equivlence reltion iff (i) is reflexive, symmetric nd trnsitive, (ii) ny clss node is only to itself, nd (iii) ny property node is only to itself. We define n RDF summry s grph quotient w.r.t. given RDF node equivlence: Definition Given n RDF grph G nd n RDF node equivlence reltion, the summry of G by, which is n RDF grph denoted G /, is the quotient of G by. G / dt nodes use fresh URIs, one for ech set of equivlent G dt nodes. Different RDF equivlence reltions led to different summries. Figures 2 illustrtes two of them on the sturted G from Figure 1; circles denote new-uri summry nodes, ech of which represents set of G nodes. For instnce, t left, single node represents r 1, r 2, r 3 ; t right, they re seprted by their sets of types. Below, we do not discuss ny prticulr summry further; insted, we focus on our summriztion frmework, nd its interply with sturtion. CP ub P ub sc sc t JP ub y t d P ub sc CP ub P ub sc JP ub t t y t t d P ub Figure 2: Smple summries. For summry to reflect (represent) grph G, queries hving nswers on G should lso hve nswers on the summry. Given n RDF query lnguge Q, we define: Definition Let G be ny RDF grph. G / is Q-representtive of G if nd only if for ny query q Q such tht q(g ), we hve q((g / ) ). We trget summries representtive of ny query over the grph structure of G, including imprecise queries using vribles in some property positions. Thus, we instntite Q into Extended Reltionl Bsic Grph Pttern Queries (RBGP*, in short), core frgment of SPARQL, defined s follows. A query triple pttern belongs to V (U V) V or V {} U, where V is set of vribles. An RBGP* query q is of the 3

5 form: q( x) t 1,..., t n where ech t i is query triple pttern, {t 1,..., t n } is noted body(q), nd x, clled the nswer vribles, is subset of the vribles in body(q). A smple RBGP* query is: q (x 1, x 3 ) :- (x 1,, Book), (x 1, uthor, x 2 ), x 2 y x 3. We show (the proof is in [2]) tht for ny RDF equivlence reltion : Proposition 3.1 An RDF summry G / is RBGP*-representtive. RBGP* representtiveness ensures tht ny query specifying certin grph pttern in G nd/or querying the structure itself (by mens of vribles in property positions, such s y in the smple query bove) which hs nswers on G, lso hs nswers on G /. In the presence of n RDF Schem, the semntics of G is its sturtion G. Thus, representtive summry must reflect both the explicit nd the implicit triples of G. For instnce, the summries in Figure 2 show tht some G resources (e.g., r 1, r 2, r 3 ) re of type P ub, but the sme summries computed from G lone do not, s the corresponding triples re implicit in G. A simple wy to obtin (G ) / is to compute G nd then summrize it. We define novel lterntive shortcut method, which voids sturting G, yet it constructs n RDF grph strongly isomorphic to (G ) /, s follows: Definition A strong isomorphism between two RDF grphs G 1, G 2, noted G 1 G 2, is n isomorphism which is the identity for the clss nd property nodes. Definition Summriztion through the RDF node equivlence reltion dmits shortcut iff for ny RDF grph G, (G ) / ((G / ) ) / holds. The shortcut summrizes G, sturtes the result, then summrize it gin (the three green edges in Figure 3). Its result is essentilly (G ) /, s the two hve identicl grph structures (gurnteed by the strong isomorphism), on which RBGP* representtiveness is defined. They only differ in the new URIs of their nodes (circles in Figure 2). Wht is the interest of the shortcut? If G / is much smller thn G, it is much fster to sturte G / (on the shortcut) thn to sturte G; (G / ) is lso likely to be smll, thus fst to summrize. Further, summrizing G is fster thn summrizing G, given tht G is t lest s lrge s G. Summing up these inequlities, the time spent on the shortcut my be (much) shorter thn the time spent to build (G ) / directly. By the summry definition, to every node in G corresponds exctly one node in the summry G. We cll representtion function nd denote f the function ssociting summry node to ech G node; we sy f (n) represents n in the summry. An importnt structurl property reltes G, G nd the function f (see Figure 3): Lemm 3.2 (Summriztion Homomorphism) Let G be n RDF grph, G / its summry nd f the corresponding representtion function from G nodes to G / nodes. f defines homomorphism from G to (G / ). Bsed on the Lemm, we estblish the sufficient condition [2] (see Figure 3): Theorem 3.3 (Sufficient condition for shortcuts) Given n RDF node equivlence reltion, nd n RDF grph G, let G / be its summry nd f the corresponding representtion function from G nodes to G / nodes. If stisfies: for ny RDF grph G nd ny pir (n 1, n 2 ) of G nodes, n 1 n 2 in G iff f (n 1 ) f (n 2 ) in (G / ), then (G ) / ((G / ) ) / holds. 4

6 Lemm 3.2 G f representtive function of summrizing G G/ Theorem 3.3 G f homomorphism by Lemm 3.2 (G/ ) (G )/ by Theorem 3.3 ((G/ ) )/ Figure 3: Illustrtion for Lemm 3.2 nd Theorem 3.3. The summry illustrted t left Figure 2 turns out to dmit the shortcut; in our experiments, the shortcut ws up to 20x fster thn sturting G nd then summrizing G. The summry illustrted t right in Figure 2 does not dmit the shortcut. Conclusion nd perspectives Finding necessry (nd sufficient) condition for the shortcut is currently open. We hve instntited our frmework into mny summries, nd implemented summriztion tool vilble online (together with mny smple summries) t We re currently working on summry-bsed query pruning, where we decide bsed on (G ) / whether query my hve nswers on G or not. References [1] Stéphne Cmpins, Renud Delbru, nd Giovnni Tummrello. Efficiency nd precision trde-offs in grph summry lgorithms. In IDEAS, [2] Šejl Čebirić, Frnçois Gosdoué, nd Ion Mnolescu. A frmework for efficient representtive summriztion of RDF grphs. Inri Reserch Report no. 9090, vilble t [3] Qun Chen, Andrew Lim, nd Kin Win Ong. D(K)-index: An dptive structurl summry for grph-structured dt. In SIGMOD, [4] Sirm Gurjd, Stephn Seufert, Iris Milirki, nd Mrtin Theobld. Using grph summriztion for join-hed pruning in distributed RDF engine. In SWIM, [5] Thnh Trn, Günter Ldwig, nd Sebstin Rudolph. Mnging structured nd semistructured RDF dt using structure indexes. IEEE TKDE, 25(9), [6] Georgi Troullinou, Hridimos Kondylkis, Evngeli Dsklki, nd Dimitris Plexouskis. RDF digest: Efficient summriztion of RDF/S KBs. In ESWC,