Coherent Closed Quasi-Clique Discovery from Large Dense Graph Databases

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1 Coherent Closed Qusi-Clique Discovery from Lrge Dense Grph Dtbses Zhiping Zeng, Jinyong Wng, Lizhu Zhou, George Krypis Tsinghu University, Beijing, 00084, P.R.Chin University of Minnesot, Minnepolis, MN 55455, USA ABSTRACT Frequent coherent subgrphscn provide vluble knowledge bout the underlying internl structure of grph dtbse, nd mining frequently occurring coherent subgrphs from lrge dense grph dtbseshs been witnessed severl pplictions nd received considerble ttention in the grph mining community recently. In this pper, we study how to efficiently mine the complete set of coherent closed qusi-cliques from lrge dense grph dtbses, which is n especilly chllenging tsk due to the downwrd-closure property no longer holds. By fully exploring some properties of qusicliques, we propose severl novel optimiztion techniques, which cn prune the unpromising nd redundnt sub-serch spces effectively. Menwhile, we devise n efficient closure checking scheme to fcilitte the discovery of only closed qusi-cliques. We lso develop coherent closed qusi-clique mining lgorithm, Cocin. Thorough performnce study shows tht Cocin is very efficient nd sclble for lrge dense grph dtbses. Ctegories nd Subject Descriptors: H..8 [Dtbse Mngement]: Dt Mining Generl Terms: Algorithms Keywords: Grph mining, Qusi-clique, Coherent subgrph.. INTRODUCTION Recently severl studies hve shown tht mining frequent coherent subgrphs is especilly useful, where coherent subgrph cn be informlly defined s subgrph tht stisfies minimum cut bound (nd the forml definition cn be found in Section.), s the set of frequent coherent subgrphs mined from grph dtbse usully reflects the density distribution of the reltionships mong the objects in the dtbse, nd cn provide vluble knowledge bout the internl structure of the grph dtbse. Coherent subgrph mining hs lso been witnessed severl pplictions such s This work ws supported by Ntionl Nturl Science Foundtion of Chin (NSFC) under Grnt No nd Bsic Reserch Foundtion of Tsinghu Ntionl Lbortory for Informtion Science nd Technology(TNList). Cocin stnds for Coherent closed qusi-clique mining. Permission to mke digitl or hrd copies of ll or prt of this work for personl or clssroom use is grnted without fee provided tht copies re not mde or distributed for profit or commercil dvntge nd tht copies ber this notice nd the full cittion on the first pge. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission nd/or fee. KDD 06, August 0 3, 006, Phildelphi, Pennsylvni, USA. Copyright 006 ACM /06/0008 $5.00. highly correlted stock discovery[5], gene function nnottion nd functionl module discovery[3]. Severl coherent subgrph discovery lgorithms hve been recently developed [3, 4, 6, 5]. However, ech of the previously proposed lgorithms hs its own limittions. For exmple, the lgorithm proposed in [4] cn only mine qusi-cliques with exct 00% support threshold; the lgorithms proposed in [6] cn only work on reltionl subgrphs, where vertex lbels re distinct in ech grph; similrly, the CLAN lgorithm [5] mines fully connected frequent subgrphs (i.e., frequent cliques). In this pper, we study more generl problem formultion: mining frequent qusi-cliques from lrge dense grph dtbses, tht is, we neither limit the minimum support to be 00% nor require the input grphs to be reltionl. While the problem becomes more generl, it gets more tough. As we will see lter, the downwrd-closure property[] no longer holds, devising some effective serch spce pruning techniques is especilly chllenging. By fully exploring some properties of qusicliques, we propose severl novel optimiztion techniques, menwhile, we devise n efficient closure checking scheme to fcilitte the discovery of only closed qusi-cliques. We lso develop n efficient coherent closed qusi-clique mining lgorithm, Cocin. The remining of this pper is orgnized s follows. Section introduces the problem formultion. In Section 3, we present lgorithm Cocin by focusing on severl optimiztion techniques. The empiricl results re exmined in Section 4. Finlly, we conclude our study in Section 5.. PROBLEM FORMULATION In this section, we introduce some preliminry concepts, nottions, nd terms in order to simplify our discussion. We lso formulte the problem we study. Tble summrizes the nottions we use in this pper nd their menings. Nottions V E L F G G G(S) N G (v) deg G (v) dis G (u, v) Description V = {v,v,, v k }, the set of vertices E V V, the set of edges the set of vertex lbels F :V L, the mpping function from lbels to vertices G =(V,E,L,F), n undirected vertex-lbeled grph G = V, the crdinlity of G the induced subgrph on S from G, S V (G) N G (v) ={u (v, u) E(G)} deg G (v) = N G (v) the number of edges in the shortest pth between u nd v Tble : Nottions used in this pper. Preliminries In this pper, we consider simple grph only, which does not contin self-loops, multi-edges, nd edge lbels. An undirected

2 vertex-lbeled grph trnsction, G, cn be represented by 4- tuple, G=(V, E,L,F). An induced subgrph of grph G is subset of the vertices of V (G) together with ny edges whose endpoints re ll in this subset. In the following discussions, the term grph mens the undirected vertex-lbeled grph, unless otherwise stted. DEFINITION.. (γ-qusi-clique) A k-grph(k ) G is γ- qusi-clique (0 γ ) if v V (G), deg G (v) γ (k ). From the definition we cn see tht qusi-cliques re subgrphs tht stisfy user-specifiedminimum vertex degree bound, γ (k ). Apprently, γ-qusi-clique must be fully connected grph when γ =, nd singleton grphs re considered s γ-qusi-cliques. If v V (G) such tht deg G (v)= γ (k ) nd γ (k ) γ k, v is clled criticl vertex of G w.r.t γ. Most of the existing frequent subgrph mining lgorithms re bsed on the downwrd-closure property. Unfortuntely, this property does not hold for qusi-clique ptterns. An induced subgrph of γ-qusi-clique my not be γ-qusi-clique. For instnce, in Figure, G ({v,v 3,v 4,v 5,u 6}) is 0.5-qusi-clique, but one of its induced subgrphs, G ({v,v 3,v 5,u 6}) is not 0.5-qusi-clique. DEFINITION.. (Edge Cut nd Edge Connectivity) Given connected grph G =(V, E), n edge cut is set of edges E c such tht G =(V,E E c) is disconnected. A minimum cut is the smllest set mong ll edge cuts. The edge connectivity of G, denoted by κ(g), is the size of the minimum cut. As shown in [6], the minimum vertex degree cn reflect the level of connectivity to some extent in grph, but they cnnot gurntee the grph is connected in blnced wy. However, the following lemm gives lower bound on the minimum cut when γ 0.5, which gurntees the coherency of γ-qusi-cliques. LEMMA.. (Minimum Edge Connectivity) Let n-grph Q = (V, E) be γ-qusi-clique (0.5 γ,n ). The edge connectivity of Q must be no smller thn n, i.e., κ(q) n. PROOF. Let us divide V into two sets, V nd V, nd suppose V V nd V =k, k n must hold. Since Q is γ-qusi-clique, v V, deg Q (v) γ (n ). However, v is t most djcent to other k vertices in V, nd k n < γ(n ). Therefore, v must be djcent to vertices belonging to V, nd the number of edges which connect v nd vertices in V must be no smller thn γ (n ) (k ). There re k vertices in V, so there exist t lest k ( γ (n ) (k )) edges between V nd V. Let f(k)=k ( γ (n ) (k )), then f(k) = k + k ( γ (n ) +) () d f(k) The second-order derivtive of f(k) (i.e., = ) isnegtive, f(k) chieves the mximum t its sole sttionry point k = dk γ (n ) +. As there exists only one sttionry point for qudrtic polynomil function f(k) nd k n, f(k) must get its minimum either t k= or t k= n. When k=, f()= γ (n ) n = n. While when k= n, ech vertex in V must be djcent to t lest one vertices in V, thus f( n ) n. From the bove result, we cn get tht k [, n ], f(k) n. According to the definition of edge connectivity, κ(q) n. Becuse we re more interested in mining tightly connected subgrphs, we do not expect tht the edge connectivity of subgrph is too smll in comprison with the minimum vertex degree. From Lemm., if γ 0.5 the vertices in the γ-qusi-clique re connected tightly nd reltively evenly. In this pper, γ-qusi-clique is sid to be coherent if γ 0.5, nd if not explicitly stted γ by defult hs vlue no smller thn 0.5. DEFINITION.3. (γ-isomorphism) A grph G ={V,E,L,F } is γ-isomorphic to nother grph G = {V,E,L,F } iff both of them re γ-qusi-cliques, G = G, nd there exists bijection f:v V such tht v V,F (v)= F (f(v)). According to the bove definition, we know tht the γ-isomorphism is quite different from the grph isomorphism in grph theory, which is defined s bijection f:v (G ) V (G ) from grph G to nother grph G, such tht (u, v) E(G ) iff (f(u),f(v)) E(G ). While the γ-isomorphism between two γ-qusi-cliques does not imply n exct bijection edge mpping. A multiset is defined s bg of vertex lbels in which the order is ignored, but multiplicity is explicitly significnt. Let M(G) indicte the multiset of G. From the γ-isomorphism definition, we cn derive the following lemm directly: two γ-qusi-cliques, G nd G, re γ-isomorphic to ech other iff M(G )=M(G ). For two γ-qusi-clique Q nd Q,ifM(Q) M(Q ), Q is clled subqusi-clique of Q, while Q is clled superqusi-clique of Q. We use Q Q or Q Q (i.e. Q Q but Q Q ) to denote the subqusi-clique or proper subqusi-clique reltionship.. Problem Definition A grph trnsction dtbse, D, consists of set of input grphs, nd the crdinlity of D is denoted by D. Figure shows n exmple of grph trnsction dtbse D, which consists of two input grphs, G nd G. u 5 d u u c Grph G u 4 b e u 6 u 3 b v 3 d v v 4 c v b v 6 Grph G Figure : An exmple of grph dtbse D For two grphs G nd G, let g be n induced subgrph of G, if M(g)=M(G ), we cll g n instnce of G in G. If there exists t lest one instnce of G in G, we sy tht grph G roughly supports G. Menwhile, if g is γ-isomorphic to nother γ-qusiclique Q, we cll g n embedding of Q in G. If there exists t lest one embedding of Q in G, G is sid to strictly support Q. The number of input grphs in grph dtbse D tht strictly (or roughly) support γ-qusi-clique Q (or grph G) is clled the bsolute strict-support (or bsolute rough-support) of Q (or G)in D, denoted by sup D s (Q) (or sup D r (G)), while the reltive strictsupport rsup D s (Q) (or reltive rough-support rsup D r (G)) is defined s rsup D s (Q)=sup D s (Q)/ D (nd rsup D r (G)=sup D r (G)/ D ). Given n bsolute support threshold min sup nd grph trnsction dtbse D, qusi-clique Q (or subgrph g) is clled frequent qusi-clique (or vice-frequent grph) if sup D s (Q) min sup (or sup D r (g) min sup). If there does not exist ny nother qusi-clique Q such tht Q Q nd sup D s (Q) sup D s (Q ), Q is clled closed qusi-clique in D. Here the definition of closed pttern is little different from the trditionl one, due to the bsence of the downwrd-closure property for qusi-clique ptterns, nd it is possible tht qusiclique s strict-support is greter thn tht of its subqusi-cliques. Also, in the cse tht Q Q nd sup D s (Q) sup D s (Q ), we sy Q cn subsume Q. b c v 5

3 In this pper, given D nd min sup, we study the problem of mining the complete set of γ-qusi-cliques in D tht re frequent, closed, nd lso coherent (i.e., γ 0.5). 3. Cocin: EFFICIENTLY MINING CLOSED COHERENT QUASI-CLIQUES In this section, we describe our comprehensive solution to frequent closed coherent qusi-clique mining, including n efficient cnonicl representtion of coherent qusi-clique, subgrph enumertion frmework, severl serch spce pruning techniques, nd qusi-clique closure checking scheme. We lso present the integrted lgorithm, Cocin. 3. Cnonicl Representtion of Subgrphs One of the key issues in grph mining is how to choose n efficient cnonicl form tht cn uniquely represent grph nd hs low computtionl complexity in order to fcilitte the grph isomorphism testing. Some previously proposed solutions to this problem re designed for generl grph nd my not be the most efficient representtion for qusi-clique. According to the Section., we cn see tht multiset preserves much informtion for qusi-clique. Given k-grph g, we cll ny sequence of ll elements in M(g) grph string. Assume there is totl order on the vertex lbels, we define the following totl order of ny two strings p nd q with size p nd q respectively. Let p i denote the i-th vertex lbel in string p, we define p<q if either of the following two conditions holds: () t (0<t min{ p, q }) such tht i [,t ], p i=q i nd p t<q t, () ( p < q ) nd i [, p ], p i=q i; otherwise p q. A string S = n is clled substring of nother string S b=b b b m, denoted by S S b (or S S b if m>n), if there exist n integers i <i < i n m such tht =b i, =b i,, n=b in. DEFINITION 3.. The cnonicl form of grph G is defined s the minimum string mong ll its strings nd denoted by CF(G). As we ignore the exct topology of qusi-clique, it is evident tht the bove definition is unique representtion of qusiclique. However, we note tht it does not hold for generl grph. After giving the definition of the cnonicl form of grph nd the substring reltionship, we cn derive the following two lemms to fcilitte γ-isomorphism checking nd subqusi-clique reltionship checking. LEMMA 3.. Two γ-qusi-cliques Q nd Q re γ-isomorphic to ech other iff CF(Q )=CF(Q ). LEMMA 3.. Given two γ-qusi-cliques Q nd Q, Q Q (or Q Q )iffcf(q ) CF(Q ) (or CF(Q ) CF(Q )). The bove two lemms cn be derived esily from the definition of γ-isomorphism nd subqusi-clique reltionship respectively, here we omit the proof. 3. Vice-Frequent Subgrph Enumertion According to the definition of vice-frequent grph, it is evident tht ny induced subgrph of vice-frequent grph must be lso vice-frequent. Thus, the downwrd-closure property cn be exploited for vice-frequent subgrph enumertion. Bsed on the definition of embedding nd instnce of subgrph, we cn see tht the embedding is specil type of the instnce, n embedding of qusi-clique q must be n instnce of q. Given γ-qusi-clique Q, ssup D s (Q) sup D r (Q) holds, if Q is frequent, Q must be vice-frequent. Consequently, we cn discover the complete set of frequent γ-qusi-cliques from the set of vicefrequent subgrphs. By conceptully orgnizing the vice-frequent subgrphs into lttice-like structure in the wy we used in [5], the problem of mining frequent qusi-cliques becomes how to trverse the lttice-like structure to enumerte vice-frequent subgrphs nd discover frequent γ-qusi-cliques. In our running exmple in Figure, ssume the totl order mong vertex lbels is b c d e, ll the vice-frequent subgrphs re orgnized into structure s shown in Figure. Note tht here we use cnonicl form:rough-support:strict-support to represent subgrph. In ddition, ll nodes with yellow color re vicefrequent subgrphs but not frequent qusi-cliques, nodes with blue color re non-closed frequent qusi-cliques, nd nodes with ornge color re closed qusi-cliques. Figure shows tht mong ll the vice-frequent subgrphs, only bd:: nd bcd:: re closed qusicliques. We lso dopt the DFS serch strtegy s we used in [5] to trverse the lttice-like structure. In this wy, we cn get rudimentry lgorithm to discover frequent closed γ-qusi-cliques. However, due to gret del of redundncy during the enumertion, this rudimentry lgorithm is too expensive nd costs too much spce nd runtime. :: b:: c:: d:: b:: c:: d:: bb::0 bc:: bd:: cd:: bb:: γ = 0.5 min_sup= bc:: bd:: cd:: bbc::0 bbd::0 bcd:: bbc::0 bbd::0 φ bbcd::0 bcd::0 bbcd::0 Figure : A lttice-like structure built from the vice-frequent subgrphs of our running exmple Structurl Redundncy Pruning. As shown in [5], much redundncy exists if we just simply use the DFS serch strtegy. In order to eliminte the structurl redundncy while mintining the completeness of the result set, we propose n efficient vicefrequent subgrph enumertion method. Given n m-grph G nd CF(G)= 0 m, we require G cn only be generted by growing the subgrph g with cnonicl form CF(g)= 0 m. In this wy, except the node φ, ech node in the lttice-like structure hs only one prent, nd this lttice-like structure would turn to tree structure. Let LAS(g) be the lst element in CF(g) (i.e., LAS(g)= m), obviously, in the enumertion tree structure ll descendnts of the node g would be in the form of CF(g) b 0b b k, where b 0 LAS(g) nd i [,k], b i b i. 3.3 Serch Spce Pruning In this subsection, we propose severl novel optimiztion techniques to prune futile serch subspces bsed on some nice properties of qusi-cliques Preliminries Before we elborte on the pruning techniques, let us first introduce the following two importnt lemms which form the foundtion of severl pruning techniques. Level 3 4 5

4 LEMMA 3.3. If m+u< γ (k + u) (where m, u, k 0, nd 0.5 γ ), then m< γ k nd i [0,u], m + i< γ (k + i). PROOF. First, we ssume m γ k, then m + u γ k + u γ k + γ u γ (k + u), which contrdicts with the fct m+u< γ (k + u). Thus, m< γ k holds. Second, let t=u i, then m+i=m+u t < γ (k + u) t γ (k + u) γ t γ ((k + u) t) = γ (k + i). LEMMA 3.4. (Mximl Dimeter) If grph Q is γ-qusiclique, then u, v V (Q), dis Q (u, v). PROOF. Let Q =n (n ). u, v V (Q), ifu nd v re djcent to ech other, then dis Q (u, v)=. While if u nd v re not djcent, then N Q (u) N Q (v) (n ). Furthermore, s Q is γ-qusi-clique, N Q (u) γ (n ) nd N Q (v) γ (n ) hold. Therefore, N Q (u) N Q (v) φ, otherwise N Q (u) N Q (v) = N Q (u) + N Q (v) γ (n ). Also, since γ 0.5, wehve γ (n ) (n ). Thus, N Q (u) N Q (v) (n ) holds, which contrdicts with the fct N Q (u) N Q (v) (n ). As result, there must exist t lest one vertex which is djcent to both u nd v, nd dis Q (u, v)= must hold. Therefore, from the bove nlysis, we cn get u, v V (Q), dis Q (u, v) Pruning Methods From Lemm 3.4, we cn derive the following lemm to help us prune some futile brnches. LEMMA 3.5. (Dimeter Pruning) Let G be grph nd S V (G),ifG(S) is γ-qusi-clique, then u,v S, dis G (u, v). PROOF. AsG(S) is subgrph of G, dis G(S) (u, v) dis G (u, v) holds. In ddition, ccording to Lemm 3.4, we hve dis G(S) (u, v). Therefore, dis G (u, v) must hold. Given grph G, let S V (G) nd v V (G) S, from Lemm 3.5 we know tht if G(S) nd v cn form bigger qusi-clique, u S, dis G (v, u) must hold. Accordingly, strightforwrd nd resonble ppliction of Lemm 3.5 is to discover the extensible vertices of subgrph which cn be used to form qusicliques lter. First, we clculte the extensible vertex set E(u) for ech vertex u in G(S), then conjoin ll E(u) s to obtin the globl extensible vertex set. Obviously, E(u) cn be divided into two subsets. One is denoted by D(u), which consists of the vertices tht re djcent to u, i.e., D(u) ={v dis G (u, v) =}; nother subset is denoted by I(u), where I(u) ={v u D(u), dis G (u,v)= nd v/ D(u)}. The set D(u) cn be obtined esily, nd I(u) cn be computed from D(u) in the wy of discovering vertices which re djcent to t lest one vertex in D(u). In this wy, we cn discover extensible vertex set for n instnce of subgrph G(S) nd cn prune some unpromising vertices. Combintion Pruning. We cn combine the structurl redundncy pruning with Dimeter Pruning to further shrink the the extensible vertex set. As stted in structurl redundncy pruning, only those vertices whose lbels re no smller thn LAS(g) cn be used to grow g, where g is the current prefix subgrph. Therefore, when clculting E(u) we cn remove the vertices whose lbels re smller thn LAS(g), nd the removl of some vertices in E(u) my mke some vertices left in E(u) violte the condition of n extensible vertex introduced in Lemm 3.4. For exmple, suppose v 0 I(u) nd there exists only one vertex v such tht v D(u) nd dis G (v 0,v )=, ifv / V (g) nd the lbel of v is smller thn LAS(g), then v will never pper in ll descendnts of g, thus the descendnt g of g which contins vertex v 0 cnnot be qusi-clique, s dis g (u, v 0)> must hold. Therefore, we cn remove the vertex v 0 from E(u) sfely. In order to eliminte these vertices efficiently, fter getting D(u), we remove the vertices in D(u) whose lbels re smller thn LAS(g) nd in the mentime do not belong to V (g). After we compute the finl set of extensible vertices, E(u), for ech vertex u in subgrph g, we cn then conjoin ll the E(u) s to get the globl extensible vertex set w.r.t. g. We cll n element in the globl extensible vertex set n extensible cndidte w.r.t. g, nd use Vcd(g) G to denote the set of extensible cndidtes w.r.t. g in G. In the following, we propose other three optimiztion techniques bsed on Lemm 3.3 which cn be used to prune the unpromising serch spce effectively. For v Vcd(g), G we define the internl set V g in (v)=n G (v) V (g) nd externl set Vex(v)=N g G (v) Vcd(g). G Let indeg g (v)= V g in (v) be the inner degree of v, nd exdeg g (v)= Vex g be the extern degree of v. Tke Figure for n exmple, ssume g=g ({v 5}), V G cd (g)={v3, v4 } (note v nd v6 hve been pruned by Combintion Pruning), indeg g (v 3)=0, exdeg g (v 3)=. LEMMA 3.6. (Vertex Connectivity Pruning) Suppose g is k-subgrph of G, if v Vcd(g) G such tht indeg(v) < γ k nd indeg g (v)+exdeg g (v)< γ (k + exdeg g (v)), there does not exist qusi-clique Q in G such tht V (Q) V (g) {v}. PROOF. Assume there exists qusi-clique Q such tht V (Q) V (g) {v}, nd let Q =l. Since Q is qusi-clique, V (Q) V (g) Vcd(g). G We define R=V (Q) Vex(v) g nd denote R by m, then l k m nd m Vex(v) =exdeg g g (v). Becuse indeg g (v) < γ k nd indeg g (v) +exdeg g (v)< γ (k + exdeg g (v)), ccording to the Lemm 3.3, we cn get tht i [0,exdeg g (v)], indeg g (v)+i< γ (k + i). Thus, deg Q (v)= indeg g (v)+m< γ (k + m) γ (l ), i.e. deg Q (v)< γ ( Q ). This contrdicts with the ssumption tht Q is qusi-clique. In the cse of indeg g (v)+exdeg g (v)< γ (k + exdeg g (v)), v is clled n invlid extensible cndidte, otherwise it is clled vlid extensible cndidte. Obviously, the invlid extensible cndidtes do not mke ny contribution to the genertion of bigger qusi-cliques. Therefore, fter getting the extensible cndidte set Vcd(g), G we could remove ll invlid extensible cndidtes from Vcd(g). G Due to the removl of these vertices, some originlly vlid extensible cndidtes my turn to be invlid, so we cn do this pruning itertively until no vertex cn be removed from Vcd(g). G We denote the remining set by Vvd(g). G Hence, if G(S) is qusiclique in G nd S V (g), S V (g) Vvd(g). G Accordingly,we only need to use the vertices in V vd(g) G to grow g, which cn further improve the lgorithm s efficiency. Assume g is subgrph of grph G, for vertex u V (g), let Vext(u)=N g G (u) Vvd(g), G nd the extensible degree be the crdinlity of Vext(u), g i.e., extdeg g (u)= Vext(u). g If there exists criticl vertex v in g such tht extdeg g (v) = 0, we cll v filed-vertex of g. LEMMA 3.7. (Criticl Connectivity Pruning) If there exists filed-vertex v in k-subgrph g of G, there will be no such n induced subgrph Q of G tht V (Q) V (g) nd Q is qusiclique. PROOF. We prove this by contrdiction. Let Q be such n induced subgrph of G nd Q = m. Obviously m > knd V (Q) V (g) Vvd(g). G Since v is criticl vertex in g, deg g (v) = γ (k ) nd γ (k ) < γ k. Furthermore, becuse deg Q (v) deg g (v) +extdeg g (v) nd extdeg g (v) =0, deg Q (v) < γ k γ (m ), which contrdicts with the ssumption. Thus, there must exist no such n induced subgrph.

5 Given k-subgrph g of grph G, if u V (g) such tht deg g (u) < γ (k ) nd deg g (u)+extdeg g (u) < γ (k +extdeg g (u)), we cll u n unpromising vertex in g. LEMMA 3.8. (Subgrph Connectivity Pruning) If k-subgrph g of G contins n unpromising vertex u, there will be no induced subgrph Q of G such tht V (Q) V (g) nd Q is qusi-clique. PROOF. Let Q be such n induced subgrph nd Q = l. Since Q is γ-qusi-clique, V (Q) V (g) Vvd(g) G holds. Let V = V (Q) Vext(u) g nd V = m, then l m+ V (g) =m+k. Becuse deg g (u) < γ (k ) nd deg g (u) +extdeg g (u) < γ (k +exdeg g (u)), from Lemm 3.3 we cn get tht i [0,exdeg g (u)],deg g (u) +i< γ (k +i). Thus, deg Q (u) =deg g (u)+ V = deg g (u)+m < γ (k +m) γ (l ), tht is, deg Q (u) < γ ( Q ). This contrdicts with the ssumption tht Q is γ-qusi-clique. According to Lemm 3.7 nd Lemm 3.8, if subgrph contins filed-vertex or n unpromising vertex, it does not mke ny contribution to the genertion of qusi-cliques. Thereby, fter getting the vlid extensible cndidte set V vd(g), G once we inspect the existence of filed-vertices or unpromising vertices in g, then there is no hope to grow instnce g to generte qusi-cliques, nd thus we cn stop growing g. 3.4 Closure Checking Scheme By integrting the pruning techniques proposed in this pper with the vice-frequent subgrph enumertion frmework, we cn discover the complete set of frequent qusi-cliques. How do we discover the closed qusi-cliques? A strightforwrd pproch is to store ll the frequent qusi-cliques tht we hve found, nd when inserting qusi-clique q to the result, we do super-clique-detecting which checks whether there lredy exists n element q such tht cn subsume the current q nd sub-clique-detecting which checks if there exists ny lredy mined qusi-clique q tht cn be subsumed by q. Obviously, this nïve pproch is very costly, we will introduce more efficient closure checking scheme. Given two subgrphs, G nd G, with cnonicl forms CF(G ) = n nd CF(G )=b b b m (where n<m) respectively, if CF(G ) CF(G ), there must exist n integers i <i << i n m such tht =b i, =b i,, n=b in.ifi n=n, the reltionship of CF(G ) nd CF(G ) in the enumertion tree cn be illustrted in Figure 3(), tht is, the node corresponding to CF(G ) is n ncestor of the node corresponding to CF(G ). Otherwise, let k = min{j i j j}, then CF(G ) nd CF(G ) hve the sme prefix k (if k=, the prefix is empty) nd k>b k (s k=b ik,i k>k, nd b k<b ik hold), thus CF(G )>CF (G ) holds, nd this reltionship in the enumertion tree is shown in Figure 3(b). n n b n + () Descendnt CF(G ) b nbn + bn + m b b k k k _ b k m k _ (b) Non-descendnt k _ k CF(G ) CF(G ) CF(G ) Figure 3: Two Cses of CF(G ) CF(G ) in Vice-Frequent Subgrph Enumertion Tree One strtegy to speed up the pttern closure checking is tht we postpone the closure checking for G until ll its descendnts hve n been processed. In this wy, G must be discovered before G for the first cse shown in Figure 3(), s the node CF(G ) is descendnt of CF(G ). In the second cse s shown in Figure 3(b), it is evident tht G is lso discovered before G ccording to the DFS trverse strtegy. In summry, if the current qusi-clique G cn be subsumed by nother frequent closed γ-qusi-clique G (i.e., CF(G ) CF(G ) nd sup D s (G ) sup D s (G )), the insertion of G into the result set will occur before the closure checking of G. Accordingly, when we check if frequent qusi-clique q is closed or not, there is no need to perform sub-clique-detecting, s there does not exist ny qusi-clique q in the result set such tht CF(q ) CF(q) nd sup D s (q ) sup D s (q). Although there is no need for sub-clique-detecting, we still hve to perform super-clique-detecting. As shown in Figure 3, there re two cses for super-clique-detecting. In the first cse, we need to check if there exists descendnt qusi-clique G of the current qusi-clique G tht cn subsume G in the enumertion tree. According to our strtegy described bove, we know G must be mined fter ll its descendnts, the super-clique-detecting in this cse becomes reltively simple. After the processing of ll descendnts of G, we let the recursive mining procedure return the mximum strict-support (denoted by r) of ll frequent qusi-clique nodes under the subtree rooted with CF(G ) (if there does not exist ny frequent qusi-clique, then it returns vlue zero). If sup D s (G ) r, we know G is non-closed nd will not insert G into the result set. However, if sup D s (G )>r, we still need to check if there exists ny non-descendnt super-clique of G tht cn subsume G (i.e., the second cse shown in Figure 3(b)). In order to ccelerte the non-descendnt super-clique-detecting process, we divide the elements in the result set into different groups ccording to their bsolute strict-support. In ech group, we first order them by the size of the qusi-cliques in descending order, nd mong the qusi-cliques with the sme size in the sme group, we then order them by their cnonicl form in descending order. This processing cn ccelerte the comprison steps. 3.5 The Algorithm In the following we describe the Cocin lgorithm by integrting vrious techniques discussed erlier. Let us first introduce the SUB- ALGORITHM :Vlid, which is clled by Cocin in order to compute the vlid extensible cndidtes for n instnce subgrph. For ech vertex u in current instnce g, we scn the grph G in which g resides to find the set D(u) (line 06), nd refine D(u) bsed on the combintion pruning technique (line 07). Then we generte the set I(u) from D(u), obtin the extensible cndidte set T of u (lines 08-0), nd conjoin ech discovered extensible cndidte set to get the globl extensible cndidtes rs (line ). Finlly, we pply the vertex connectivity pruning (line ), criticl connectivity pruning, nd subgrph connectivity pruning (lines 3-4) to rs to generte the finl set of vlid extensible cndidtes w.r.t. g. Before running Cocin s shown in ALGORITHM, we first compute the set of vice-frequent vertex lbels nd their corresponding instnces by scnning the originl dtbse, nd remove from the grph dtbse the vertices with non-vice-frequent vertex lbels. This procedure cn reduce the size of input grphs significntly, especilly when min sup is high. After this preprocessing, we use Cocin to mine the complete set of frequent closed coherent qusi-cliques. For the current prefix vice-frequent subgrph g, we first use procedure Vlid to get the set of vlid extensible cndidtes V vd for ech instnce of g (lines 7-8), from which we cn further clculte the vice frequent extensible lbels (line 9). For ech vice-frequent extensible lbel, we invoke Cocin to discover descendnts of g (lines -3). After ll recursive invo-

6 ctions hve returned, we cn use the closure checking scheme to determine whether or not to insert g to the finl result set ccording to the strict-support of g nd the returned vlues of the recursive invoctions (lines 4-5). SUBALGORITHM : Vlid(g) INPUT: () g: n instnce subgrph. OUTPUT: () rs: the set of vlid extensible cndidtes w.r.t. g. BEGIN 0. set rs = V (G)(G is the grph in which g resides); 0. For ech vertex u in V (g) 03. If rs is empty 04. brek; 05. set T = D = I = φ; 06. D = {v dis G (u, v) =}; 07. Refine D using combintionpruning; 08. I = {v t D, dis G (t, v) =nd v/ D}; 09. T = D I; 0. Remove ech element v T which stisfies L(v ) <LAS(g);. rs = rs T ;. Remove invlid extensible cndidtes from rs; 3. If there exists filed or n unpromising vertex in g 4. rs = φ; 5. return rs; END ALGORITHM : Cocin(D, CF(g),INS(g),min sup,γ) INPUT: () D: the input grph dtbse, () CF(g): the cnonicl form of g, (3) INS(g): the set of instnces of g in the dtbse D, (4) min sup: the minimum support threshold, (5) γ: the edge density coefficient. OUTPUT: () rs: the set of frequent closed γ-qusicliques, () mx: the mximum strict-support of ll descendnt qusi-cliques of g. BEGIN 6. glbsup=0, rv=0; 7. For ech instnce ins INS(g) 8. V vd(ins) =Vlid(ins); 9. Clculte vice-frequent vlid cndidte lbel set VEX(g) ccording to ech V vd(ins); 0. Sort the lbels in VEX(g) in certin order;. For ech lbel l VEX(g). rv = Cocin(D, CF(g) l, INS(g l),min sup, γ); 3. glbsup = mx{glbsup, rv}; 4. If (sup s(g) min sup) nd (sup s(g) >glbsup) 5. Insert CF(g) into RS if g psses the non-descendntsuper-clique-detecting; 6. return mx{sup s(g),glbsup}; END 4. EMPIRICAL RESULTS We conducted n extensive performnce study to evlute vrious spects of the lgorithm. We implemented the lgorithm in C++, nd ll experiments were performed on PC running FC 4 Linux nd with.8ghz AMD Sempron CPU nd GB memory. In the experiments, we used the US stock mrket series dtbse [], which ws converted to set of grphs bsed on the sme method of [5]. Due to limited spce, here we only report the results w.r.t. correltion coefficient of Efficiency Test. We implemented one bseline lgorithm, Rw, which excludes three pruning techniques, combintion pruning, criticl connectivity pruning, nd subgrph connectivity pruning. By compring the runtime efficiency between Cocin nd Rw, we cn get n ide bout the effectiveness of the pruning techniques proposed in this pper. Figure 4 shows the runtime comprison between Cocin nd Rw by fixing γ t.0 nd vrying min sup, nd fixing min sup t 40% nd vrying γ respectively. We see tht Cocin is lwys fster thn Rw. The high performnce of Cocin in comprison with Rw lso demonstrtes tht the pruning techniques proposed for Cocin re extremely effective Runtime(sec) Cocin Rw Reltive Strict-Support Threshold (%) ) γ = Runtime(sec) Cocin Rw Edge Density Coefficient b) min sup=40% Figure 4: Efficiency Comprison (stock mrket dt) Sclbility Test. Menwhile, comprehensive sclbility study ws conducted in terms of the bse size. The results show tht Cocin hs liner sclbility in runtime ginst the number of input grphs in dtbse. 5. CONCLUSION In this pper we proposed novel lgorithm, Cocin, to mine frequent closed coherent qusi-cliques from lrge dense grph dtbses. By focusing on vertex lbels, we first introduced simple cnonicl form to uniquely represent qusi-clique pttern. By exploring some nice properties of qusi-clique ptterns, we proposed severl effective serch spce pruning techniques, dimeter pruning, combintion pruning, vertex connectivity pruning, criticl connectivity pruning, nd subgrph connectivity pruning, which could be used to ccelerte the mining process. We lso introduced n efficient pttern closure checking scheme to speed up the discovery of closed qusi-cliques. An extensive performnce study with both rel nd synthetic dtbses hs demonstrted tht Cocin is very efficient nd sclble. 6. ACKNOWLEDGEMENT The uthors re grteful to Vldimir L. Boginski, Pnos M. Prdlos, nd Sergiy Butenko for providing us the US stock mrket dtbse. We thnk Jsmine Zhou nd Hiyn Hu for sending us the yest microrry dtbse nd Michihiro Kurmochi for sending us the KEGG dtbse. Thnks lso go to Beng Chin Ooi, Anthony K. H. Tung nd Xifeng Yn for their kind help. 7. REFERENCES [] R. Agrwl, R. Sriknt. Fst Algorithms for Mining Assocition Rules in Lrge Dtbses. VLDB 94. [] V. Boginski, S. Butenko, P.M. Prdlos. On structurl properties of the mrket grph. In A. Ngurney (editor), Innovtions in Finncil nd Economic Networks, Edwrd Elgr Publishers, Apr [3] H. Hu, X. Yn, Y. Hng, J. Hn, X. Zhou. Mining coherent dense subgrphs cross mssive biologicl network for functionl discovery. Bioinformtics, Vol., Suppl., 005. [4] J. Pei, D. Jing, A. Zhng. On mining cross-grph qusi-cliques. SIGKDD 05. [5] J. Wng, Z. Zeng, L. Zhou. CLAN:An Algorithm for Mining Closed Cliques From Lrge Dense Grph Dtbses. ICDE 06. [6] X. Yn, X. Zhou, J. Hn. Mining closed reltionl grphs with connectivity constrints. SIGKDD 05.