Top-K Structural Diversity Search in Large Networks

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1 Top-K Srucural Diversiy Search in Large Neworks Xin Huang, Hong Cheng, Rong-Hua Li, Lu Qin, Jeffrey Xu Yu The Chinese Universiy of Hong Kong Guangdong Province Key Laboraory of Popular High Performance Compuers, Shenzhen Universiy ABSTRACT Social conagion depics a process of informaion (e.g., fads, opinions, news) diffusion in he online social neworks. A recen sudy repors ha in a social conagion process he probabiliy of conagion is ighly conrolled by he number of conneced componens in an individual s neighborhood. Such a number is ermed srucural diversiy of an individual and i is shown o be a key predicor in he social conagion process. Based on his, a fundamenal issue in a social nework is o find op-k users wih he highes srucural diversiies. In his paper, we, for he firs ime, sudy he op-k srucural diversiy search problem in a large nework. Specifically, we develop an effecive upper bound of srucural diversiy for pruning he search space. The upper bound can be incremenally refined in he search process. Based on such upper bound, we propose an efficien framework for op-k srucural diversiy search. To furher speed up he srucural diversiy evaluaion in he search process, several carefully devised heurisic search sraegies are proposed. Exensive experimenal sudies are conduced in 13 real-world large neworks, and he resuls demonsrae he efficiency and effeciveness of he proposed mehods. 1. INTRODUCTION Recenly, online social neworks such as Facebook, Twier and LinkedIn have araced growing aenion in boh indusry and research communiies. Online social neworks are becoming more and more imporan medias for users o communicae wih each oher and o spread informaion in he real world [14]. In an online social nework, he phenomenon of informaion diffusion, such as diffusion of fads, poliical opinions, and he adopion of new echniques, has been ermed social conagion [22], which is a similar process as epidemic diseases. Tradiionally, he models of social conagion are based on analogies wih biological conagion, where he probabiliy ha a user is influenced by he conagion grows monoonically wih he number of his or her friends who have been affeced already [9, 3, 24]. However, such models have recenly been challenged [19, 22], as he social conagion process is ypically more complex and he social decision can depend more subly on he nework srucure. Ugander e al. [22] sudy wo social conagion processes in Permission o make digial or hard copies of all or par of his work for personal or classroom use is graned wihou fee provided ha copies are no made or disribued for profi or commercial advanage and ha copies bear his noice and he full ciaion on he firs page. To copy oherwise, o republish, o pos on servers or o redisribue o liss, requires prior specific permission and/or a fee. Aricles from his volume were invied o presen heir resuls a The 39h Inernaional Conference on Very Large Daa Bases, Augus 26h - 3h 213, Riva del Garda, Treno, Ialy. Proceedings of he VLDB Endowmen, Vol. 6, No. 13 Copyrigh 213 VLDB Endowmen /13/13... $ 1.. a b c d e f g (a) G h i a e f g (b) G N(f) Figure 1: A running example Facebook: he process ha a user joins Facebook in response o an inviaion from an exising Facebook user, and he process ha a user becomes an engaged user afer joining. They find ha he probabiliy of conagion is ighly conrolled by he number of conneced componens in a user s neighborhood, raher han by he number of friends in he neighborhood. A conneced componen represens a disinc social conex of a user, and he mulipliciy of social conexs is ermed srucural diversiy. A user is much more likely o join Facebook if he or she has a larger srucural diversiy, i.e., a larger number of disinc social conexs. This finding reveals ha he srucural diversiy of a user is an imporan facor in he social conagion process. As suggesed in [22], he analysis of srucural diversiy in a social nework can be beneficial o a wide range of applicaion domains, for example, poliical campaign, he promoion of healh pracices, markeing, and so on. Among all of hese applicaions, a fundamenal problem is o find he individuals in a social nework wih high srucural diversiy. Given a social nework, we sudy a problem of finding op-k individuals wih he highes srucural diversiy in his paper. Following he definiion in [22], he srucural diversiy of a node u is he number of conneced componens in a subgraph induced by u s immediae neighbors. Take he nework in Figure 1 (a) as an example. The srucural diversiy of verex f is 2, as he induced subgraph by f s neighbors shown in Figure 1 (b) has wo conneced componens. To find he op-k verices wih he highes srucural diversiy, a naive mehod is o compue he srucural diversiy for all he verices and hen reurn he op-k verices. Clearly, such a naive mehod is oo expensive. To efficienly find he op-k verices, he idea of radiional op-k query processing echniques [12] can be used, which finds he op-k answers according o some heurisic search order, and prunes he search space based on some upper bound score. Following his framework, in our problem, we have o address wo key issues: (1) how o develop an effecive upper bound for he srucural diversiy of a verex, and (2) how o devise a heurisic search order in he compuaion. In his paper, we propose several efficien and effecive echniques o address hese issues. We find ha for wo verices conneced by an edge, some srucural informaion of hem can be shared. For example, in Figure 1 (b), verex e forms a compo- i 1618

2 nen of size 1 in f s neighborhood. From his fac, we can infer ha verex f also forms a componen of size 1 in e s neighborhood. Based on his imporan observaion, he srucural diversiy compuaion for differen verices can also be possibly shared. To achieve his, we design a Union-Find-Isolae daa srucure o keep rack of he known srucural informaion of a verex so as o avoid he compuaion of srucural diversiy for every verex. A novel upper bound of he srucural diversiy is developed for pruning unpromising verices effecively. Ineresingly, he upper bound can be incremenally refined in he search process o become increasingly igher. The main conribuions are summarized as follows. We develop a novel Top-k-search framework o ackle he op-k srucural diversiy search problem. We design a Union- Find-Isolae daa srucure o keep rack of he known srucural informaion during he compuaion, and an effecive upper bound for pruning. We devise a heurisic search order o raverse he componens in a verex s neighborhood. According o his search order, we propose a novel A -search-based algorihm o compue he srucural diversiy of a verex. We also design efficien echniques o handle frequen updaes in dynamic neworks and mainain he op-k resuls. We use he Union-Find-Isolae srucure and a spanning ree srucure o efficienly handle edge inserions and deleions respecively. We conduc exensive experimenal sudies on large real neworks o show he efficiency of our proposed mehods. The res of his paper is organized as follows. We formulae he op-k srucural diversiy search problem in Secion 2. We presen a simple degree-based algorihm in Secion 3. A novel Top-k-search framework is proposed in Secion 4. We design wo heurisic search sraegies in Secion 5 and discuss updae in dynamic neworks in Secion 6. Exensive experimenal resuls are repored in Secion 7. We discuss relaed work in Secion 8 and conclude his paper in Secion PROBLEM STATEMENT Consider an undireced and un-weighed graph G = (V, E) wih n = V verices and m = E edges. Denoe by N(v) he se of neighbors of a verex v, i.e., N(v) = {u V : (v, u) E}, and by d(v) = N(v) he degree of v. Le d max be he maximum degree of he verices in G. Given a subse of verices S V, he induced subgraph of G by S is defined as G S = (V S, E S ), where V S = S and E S = {(v, u) : v, u S, (v, u) E}. The neighborhood induced subgraph is defined as follows. DEFINITION 1 (NEIGHBORHOOD INDUCED SUBGRAPH). For a verex v V, he neighborhood induced subgraph of v, denoed by G N(v), is a subgraph of G induced by he verex se N(v). Consider a graph in Figure 1 (a). For verex f, he se of neighbors is N(f) = {a, e, g, i}. The neighborhood induced subgraph of f is G N(f) = ({a, e, g, i}, {(a, g), (g, i)}), as shown in Figure 1 (b). We define he srucural diversiy of a verex as follows. DEFINITION 2 (STRUCTURAL DIVERSITY [22]). Given an ineger where 1 n, he srucural diversiy of verex v V, denoed by score(v), is he number of conneced componens in G N(v) whose size measured by he number of verices is larger han or equal o. is called he componen size hreshold. G N(f) in Figure 1 (b) conains a size-1 conneced componen {e} and a size-3 conneced componen {a, g, i}. If = 1, hen score(f) = 2. Alernaively, if = 2, score(f) = 1 as here is only one componen {a, g, i} whose size is no less han 2. We define he op-k srucural diversiy search problem as follows. Problem definiion: Given a graph G and wo inegers k and where 1 k, n, he goal of op-k srucural diversiy search is o find a se of k verices in G wih he highes srucural diversiy w.r.. he componen size hreshold. Le us re-consider he example in Figure 1. Suppose ha k = 1 and = 1. Then, {e} is he answer, as e is he verex wih he highes srucural diversiy (score(e) = 3). I is imporan o noe ha alhough we focus on he op-k srucural diversiy search, he proposed echniques can also be easily exended o process he iceberg query, which finds all verices whose srucural diversiy is greaer han or equal o a pre-specified hreshold. Unless oherwise specified, in he res of his paper, we assume ha a graph is sored in he adjacency lis represenaion. Each verex is assigned a unique ID. In addiion, for convenience, we assume ha m Ω(n), which does no affec he complexiy analysis of he proposed algorihms. Similar assumpion has been made in [15]. 3. A SIMPLE DEGREE-BASED APPROACH In his secion, we presen a simple degree-based algorihm for op-k srucural diversiy search. We use a procedure bfs-search o compue he srucural diversiy score(v) for a given verex v. I performs a breadh-firs search in G N(v) o find conneced componens and reurns he number of componens whose sizes are no less han. For breviy, he pseudocode of bfs-search is omied. Nex we inroduce a useful lemma which leads o a pruning sraegy in he degree-based algorihm. LEMMA 1. For any verex v in G, score(v) d(v) holds. PROOF. We prove his lemma by conradicion. Suppose o he conrary ha score(v) > d(v). By he definiion of srucural diversiy, G N(v) has d(v) + 1 or more componens whose size is greaer han or equal o. Then, he oal number of verices in hese componens is ( d(v) + 1) > d(v) = d(v), which conradics o he fac ha he number of verices in G N(v) is d(v). Hence, he lemma is esablished. We denoe d(v) by bound(v). Equipped wih Lemma 1 and he bfs-search procedure, we presen he degree-based approach in Algorihm 1, which compues he srucural diversiy of he verices in descending order of heir degree. Afer iniializaion (lines 1-2), Algorihm 1 sors he verices in descending order of heir degree (line 3). Then i ieraively finds he unvisied verex v wih he maximum degree, and calculaes bound(v ) (lines 5-6). If he answer se S has k verices and bound(v ) min v S score(v), he algorihm erminaes (lines 7-8). The raionale is as follows. By Lemma 1, we have score(v ) bound(v ) min v S score(v). For any verex w V wih a smaller degree, we have score(w) bound(w) bound(v ) min v S score(v). Therefore, we can safely prune he remaining verices and erminae. On he oher hand, if such condiions are no saisfied, hen he algorihm compues score(v ) by invoking bfs-search, and checks wheher v should be added ino he answer se S (lines 1-13). Finally, he algorihm oupus S. The following example illusraes how Algorihm 1 works. EXAMPLE 1. Consider he graph in Figure 1 (a). Suppose ha k = 1 and = 1. The running process on his graph is illusraed in Figure 2. The sored verex lis is c, a, b, f, h, i, d, e, g in descending order of heir degree. The algorihm compues he srucural diversiy of hese verices in urn, and erminaes before compuing score(g). This is because min v S score(v) = score(e) = 3 and bound(g) = 3 min v S score(v). Therefore, Algorihm 1 can save one srucural diversiy compuaion. 1619

3 Algorihm 1 degree (G, k, ) Inpu: G = (V, E), he op-k value k, he componen size hreshold. Oupu: Top-k search resul S. 1: S ; 2: for v V do score(v) 1; 3: sor all verices in he descending order of heir degree; 4: while v V s.. score(v) = 1 5: v arg max v V, score(v)= 1 d(v); 6: bound(v ) d(v ) ; 7: if S = k and bound(v ) min v S score(v) hen 8: break; 9: score(v ) bfs-search (G,, v ); 1: if S < k hen S S {v }; 11: else if score(v ) > min v S score(v) hen 12: u arg min v S score(v); 13: S (S {u}) {v }; 14: reurn S; v c a b f h i d e g bound score S {c} {a} {e} Figure 2: Illusraion of he degree algorihm THEOREM 1. For 1 k n and 1 n, Algorihm 1 performs op-k srucural diversiy search in O( v V (d(v))2 ) ime and O(m) space. PROOF. The algorihm firs sors all verices in O(n) ime using he bin-sor algorihm [8]. I has o calculae he srucural diversiy for every verex o answer a op-k query in he wors case. Consider a verex v. When he algorihm compues score(u) for each neighbor u N(v), i has o scan he adjacency lis of v in O(d(v)) ime. Since here are N(v) = d(v) neighbors, he oal cos for scanning v s adjacency lis is O((d(v)) 2 ). Thus, i akes O( v V (d(v))2 ) ime o calculae he srucural diversiies for all verices. In addiion, one can mainain he op-k resuls in O(n) ime and O(n) space using a varian of bin-sor lis. Pu i all ogeher, he ime complexiy of Algorihm 1 is O( v V (d(v))2 ). For he space complexiy, he graph sorage akes O(n + m) space, and S akes O(n) space. Thus, he space complexiy of Algorihm 1 is O(n + m) O(m). REMARK 1. The wors-case ime complexiy of Algorihm 1 is bounded by O( v V d(v) dmax) = O(mdmax) O(mn). 4. A NOVEL TOP-K SEARCH FRAMEWORK The degree algorihm is no very efficien for op-k search because he degree-based upper bound in Lemma 1 is loose. To improve he efficiency, he key issue is o develop a igher upper bound. To his end, in his secion, we propose a novel framework wih a igher pruning bound and a new algorihm called boundsearch o compue he srucural diversiy score. Before inroducing he framework, we presen wo srucural properies in a graph, which are very useful for developing he new bound. 4.1 Two Srucural Properies PROPERTY 1. For any verex v V, if a verex u N(v) and u forms a size-1 componen in G N(v), hen v also forms a size-1 componen in G N(u). PROOF. We prove i by conradicion. Suppose ha in G N(u), v is conneced wih anoher verex w in a componen. Then we can infer ha w N(u) and w N(v). As u and w are conneced and boh are in N(v), u and w form a size-2 componen in G N(v), which conradics o he fac ha u forms a size-1 componen in G N(v). This complees he proof. As an example, in Figure 1 (b), verex e forms a size-1 componen in G N(f). Symmerically, verex f also forms a size-1 componen in G N(e). PROPERTY 2. If hree verices u, v, w form a riangle in G, hen we have he ses {u, v}, {v, w}, and {u, w} belong o he same componen in G N(w), G N(u), and G N(v) respecively. PROOF. This propery can be easily derived by definiion, hus we omi he proof for breviy. For insance, in Figure 1 (a), verices a, f, g form a riangle in G. We can observe ha {a, g} belong o a conneced componen in G N(f) in Figure 1 (b). Similarly, {a, f} ({f, g}) belong o a conneced componen in G N(g) (G N(a) ). Based on hese wo properies, we can save a lo of compuaional coss in compuing he srucural diversiy scores. For example, if we find ha verex u forms a size-1 componen in G N(v), hen we know ha v also forms a size-1 componen in G N(u) by Propery 1. Thus, when we compue score(u), we do no need o perform a breadh-firs search from v, because we already know v forms a size-1 componen in G N(u). If we can efficienly record such srucural informaion of v s neighbors when we compue score(v), we can save a lo of compuaional coss. More imporanly, such srucural informaion can help us o ge a igher upper bound of he srucural diversiy. In he following subsecion, we shall inroduce a modified disjoin-se fores daa srucure o mainain such srucural informaion efficienly. 4.2 Disjoin-Se Fores Daa Srucure We modify he classical disjoin-se fores daa srucure and he Union-Find algorihm [8] o mainain he srucural informaion for each verex efficienly. The modified srucure consiss of four operaions: Make-Fores, Find-Se, Union, and Isolae. Compared o he classical disjoin-se fores daa srucure, he new srucure includes an addiional operaion Isolae which is used o record he srucural informaion described in Propery 1, i.e., a verex forms a size-1 componen. Thus he modified srucure is called Union- Find-Isolae. Algorihm 2 describes he four operaions. Make-Fores : For each verex v V, we creae a disjoin-se fores srucure, denoed as g[v], for is neighbors N(v) using he Make-Fores (v) procedure in Algorihm 2. Specifically, For each u N(v), we build a single-node ree T [u] wih hree fields: paren, rank and coun. The paren is iniialized o be u iself, he rank is se o and he coun is se o 1, as here is only one verex u in he ree. In addiion, we also creae a virual node T [] which is used o collec all size-1 componens in G N(v). The paren of T [] is se o and he coun is se o because here is no size-1 componen idenified ye. For convenience, we refer o he operaion of creaing a single-node ree (line 4) or a virual node (line 5) as a Make-Se operaion. Find-Se : Following [8], he Find-Se (x) procedure is o find he roo of T [x] using he pah compression heurisic. Union : Following [8], he Union(x, y) procedure applies he union by rank heurisic o union wo rees T [fx] and T [fy] which x and y belong o respecively. fx and fy are he roos of hese wo rees. If fx and fy have unequal rank, he one wih a higher rank is se o be he paren of he oher wih a lower rank. Oherwise, we arbirarily choose one of hem as he paren and increase is rank by 1. For boh cases, we updae he coun of he roo of he new ree. 162

4 Isolae : Procedure Isolae(x) unions a size-1 ree T [x] ino he virual ree T []. I ses T [x].paren o, and increases T [].coun by 1. Isolae(x) essenially labels x as a size-1 componen if we find x is no conneced wih any oher node in a neighborhood induced subgraph. We can apply he disjoin-se fores srucure o mainain he conneced componens in G N(v). For any verex v V, we creae a rooed ree for every neighbor u N(v) iniially. If we find ha u and w are conneced in G N(v), we process i by g[v].union(u, w). If we idenify ha u forms a size-1 componen in G N(v), we process i by g[v].isolae(u). Take G N(f) in Figure 1 (b) as an example again. Firs, we creae g[f] by Make-Fores (f) as shown in Figure 3 (a). Since verices a and g are conneced, we invoke g[f].union(a, g) and he resuled srucure is shown in Figure 3 (b). The combined ree is rooed by g and has 2 verices. Verex e forms a size-1 componen, hus we invoke g[f].isolae(e) and he resul is shown in Figure 3 (c). paren rank coun paren rank coun paren rank coun T[] = {,, } T[] = {,, } T[] = {,, 1 } T[a] = { a,, 1 } T[a] = { g,, 1 } T[a] = { g,, 1 } T[e] = { e,, 1 } T[e] = { e,, 1 } T[e] = {,, 1 } T[g] = { g,, 1 } T[g] = { g, 1, 2 } T[g] = { g, 1, 2 } T[i] = { i,, 1 } T[i] = { i,, 1 } T[i] = { i,, 1 } (a) Make-Fores(f) (b) g[f].union(a,g) (c) g[f].isolae(e) Figure 3: Disjoin-Se Fores Daa Srucure g[f] The ime complexiy of he Union-Find-Isolae algorihm is analyzed in he following lemma. LEMMA 2. A sequence of M Make-Se, Union, Find-Se and Isolae operaions, N of which are Make-Se operaions, can be performed on a disjoin-se fores wih union by rank and pah compression heurisics in wors-case ime O(Mα(N)). α(n) is he inverse Ackermann funcion, which is incredibly slowly growing and a mos 4 in any conceivable applicaion. Thus, he ime complexiy of he Union-Find-Isolae algorihm can be regarded as O(M). PROOF. The proof is similar o ha in [8], hus is omied. In he following, for simpliciy, we rea α(n) as a consan in he complexiy analysis. 4.3 A Tigher Upper nd Wih he disjoin-se fores daa srucure g[v], we can keep rack of he srucural informaion of he conneced componens in G N(v) and derive a igher upper bound of score(v) han he degree-based bound in Lemma 1. Before inroducing he upper bound, we give a definiion of he idenified size-1 se as follows. DEFINITION 3. In he disjoin-se fores srucure g[v], if u N(v) and T [u].paren =, we denoe S u = {u} as an idenified size-1 se, and S u = 1. If u N(v), T [u].paren = u, we denoe S u = {w N(v) : Find-Se(w) = u} as an unidenified se, and S u = T [u].coun. By Definiion 3, we know ha each idenified size-1 se is resuled from an Isolae operaion, and he oal number of he idenified size-1 ses is T [].coun. According o Propery 1, all hese ses do no union wih oher ses. On he oher hand, unidenified ses may furher union wih oher ses or become an idenified size-1 se. Consider he example in Figure 3 (c). S e = {e} is an idenified size-1 se and T [].coun = 1. Boh S g = {a, g} and S i = {i} are unidenified ses. Le S = {S u : u N(v), T [u].paren = u or T [u].paren = } denoe all disjoin ses in g[v], excluding he virual se T []. Afer raversing all he verices and edges in G N(v), S conains all acual ses corresponding o he conneced componens in G N(v), and Algorihm 2 Union-Find-Isolae 1: procedure Make-Fores (v) 2: g[v] = {T [u] : u N(v)} {T []}; 3: for u N(v) do 4: T [u].(paren, rank, coun) (u,, 1); 5: T [].(paren, rank, coun) (,, ); 6: procedure Find-Se (x) 7: if x T [x].paren hen 8: T [x].paren Find-Se (T [x].paren); 9: reurn T [x].paren; 1: procedure Union (x, y) 11: fx Find-Se (x); fy Find-Se (y); 12: if fx fy hen 13: if T [fx].rank > T [fy].rank hen 14: T [fy].paren fx; 15: T [fx].coun T [fx].coun + T [fy].coun; 16: else 17: T [fx].paren fy; 18: T [fy].coun T [fx].coun + T [fy].coun; 19: if T [fx].rank = T [fy].rank hen 2: T [fy].rank T [fy].rank + 1; 21: procedure Isolae (x) 22: T [x].paren ; 23: T [].coun T [].coun + 1; we have score(v) = {S u : S u S, S u }. However, before raversing he neighborhood induced subgraph G N(v), S may no conain all he acual ses corresponding o he conneced componens, bu includes some inermediae resuls. Even wih such inermediae resuls mainained in S, we can sill use hem o derive an upper bound. Specifically, we have he following lemma. LEMMA 3. Le S = {S 1,..., S l } be he disjoin ses of g[v], a be he number of idenified size-1 ses, b be he number of ses whose sizes are larger han or equal o, and c be he oal size of hese b ses. Then, we have an upper bound of score(v) as follows. If = 1, bound(v) = b; if > 1, bound(v) = b + d(v) c a. PROOF. Firs, i is imporan o noe ha he curren disjoin ses in S are no final, if we have no raversed all verices and edges of G N(v). Tha is, some of hem may be furher combined by he Union operaion and he number of ses may be reduced. Second, we consider he following wo cases. If = 1, we have bound(v) = b, as he curren number of ses whose sizes are greaer han or equal o 1 is b and his number can only be reduced wih he Union operaion. If > 1, he curren number of ses whose sizes are greaer han or equal o is b and his number can only be reduced wih he Union operaion. In addiion, besides a idenified size-1 ses and c verices from he above b ses, here are sill d(v) c a verices which may form ses whose sizes are greaer han or equal o. The maximum number of such poenial ses is d(v) c a. Thus we have bound(v) = b + d(v) c a. For any verex v V, a he iniializaion sage, each neighbor verex u N(v) forms a size-1 componen. Thus bound(v) = + d(v) = d(v), he same as he bound in Lemma 1. As he disjoin ses are gradually combined, bound(v) is refined owards score(v) and becomes igher. For example, in Figure 3 (c), suppose = 2, we obain S = {S e, S g, S i} and he hree parameers in Lemma 3 are a = 1, b = 1 and c = 2. I follows ha bound(f) = = 1, which is equal o score(f) = 1. This bound based on he disjoin-se fores is obviously igher han he degree-based bound 4 = 2 derived in Lemma

5 Algorihm 3 Top-k-search Inpu: G = (V, E), he op-k value k, he componen size hreshold, gradien raio θ 1. Oupu: Top-k search resul S. 1: H ; S ; 2: for v V do 3: score(v) 1; 4: Make-Fores (v); 5: H.push((v, d(v) )); 6: while H = 7: (v, opbound) H.pop(); 8: compue bound(v ) according o Lemma 3; 9: if θ bound(v ) < opbound hen 1: if S < k or bound(v ) > min v S score(v) hen 11: H.push((v, bound(v ))); 12: coninue; 13: if S = k and opbound min v S score(v) hen 14: break; 15: score(v ) bound-search (G,, v ); 16: if S < k hen S S {v }; 17: else if score(v ) > min v S score(v) hen 18: u arg min v S score(v); 19: S (S {u}) {v }; 2: reurn S; 4.4 Top-K Search Framework Based on he disjoin-se fores daa srucure and he igher upper bound, we propose an advanced search framework in Algorihm 3 for op-k srucural diversiy search. Advanced Top-k framework: For each verex v V, he algorihm iniializes he disjoin-se fores daa srucure g[v] by invoking Make-Fores (line 4). I also pushes each verex v wih he iniial bound d(v) ino H which is a varian of bin-sor lis. Then he algorihm ieraively finds he op-k resuls (lines 6-19). I firs pops he verex wih he larges upper bound value from H. Such a verex and is bound are denoed as v and opbound respecively (line 7). The algorihm re-evaluaes bound(v ) from g[v ] based on Lemma 3, as he componen informaion in g[v ] may have been updaed. And hen, i compares he refined bound bound(v ) wih he old bound opbound. In order o avoid frequenly calculaing he upper bounds and updaing H, we inroduce a new parameer θ 1, and compare θ bound(v ) wih opbound. If θ bound(v ) < opbound, i suggess ha bound(v ) is subsanially smaller han opbound. Tha is, he old bound opbound is oo loose. Under his condiion, if S < k or bound(v ) > min v S score(v), he algorihm pushes v back o H wih he refined bound bound(v ) (lines 1-11). Oherwise, he algorihm can safely prune v. In boh cases, he algorihm coninues o pop he nex verex from H (lines 9-12). If θ bound(v ) opbound, i means ha bound(v ) is no subsanially smaller han opbound. In oher words, he old bound is a relaively igh esimaion. Then he algorihm moves o lines o check he erminaion condiion. If S = k and opbound min v S score(v), he algorihm can safely prune all he remaining verices in H and erminae, because he upper bound of hose verices is smaller han opbound. If he early erminaion condiion is no saisfied, he algorihm invokes bound-search (Algorihm 4) o compue score(v ). Afer compuing score(v ), he algorihm uses he same process o updae he se S by v as he degree algorihm does (lines 16-19). nd-search: Algorihm 4 shows he bound-search procedure o compue score(v). Based on he disjoin-se fores g[v], we Algorihm 4 bound-search (G,, v) Inpu: G = (V, E), he componen size hreshold, verex v. Oupu: score(v). 1: R ; 2: for u N(v) and T [u].paren do R R {u}; 3: for u R do bound-bfs (u); 4: reurn coun-componens (g[v], ); 5: procedure bound-bfs (u) 6: Q ; UnionF lag false; 7: Q.EnQueue(u); R R {u}; 8: while Q 9: u Q.DeQueue(); 1: for w N(u) do 11: if w R hen 12: Q.EnQueue(w); R R {w}; 13: g[v].union (u, w); UnionF lag rue; 14: if score(u) = 1 hen g[u].union (v, w); 15: if score(w) = 1 hen g[w].union (v, u); 16: if UnionF lag = false hen 17: g[v].isolae (u); 18: if score(u) = 1 hen g[u].isolae (v); 19: procedure coun-componens (g[v], ) 2: score ; 21: for u N(v) do 22: if T [u].paren = u and T [u].coun hen 23: score score + 1; 24: if = 1 hen score score + T [].coun; 25: reurn score; know ha any verex u N(v) wih T [u].paren = corresponds o an idenified size-1 componen resuled from an Isolae operaion. So bound-search does no need o search hem again. I only adds he verices whose paren ino an unvisied verex hashable R (lines 1-2). This is an improvemen from bfs-search, as bound-search avoids scanning he idenified size-1 componens. For each verex u R, he algorihm invokes he procedure boundbfs (lines 5-18) o search u s neighborhood in a breadh-firs search manner. For u s neighbor verex w, if w R, i.e., w N(v), he algorihm unions u and w ino one se in g[v]. According o Propery 2, we also union v and w ino one se in g[u], and union v and u ino one se in g[w] (lines 11-15). If u does no union wih any oher verex, he algorihm invokes an Isolae operaion on u o mark ha u forms a size-1 componen in g[v] (lines 16-18). Symmerically, by Propery 1, he algorihm invokes an Isolae operaion on v o mark ha v forms a size-1 componen in g[u] oo. Afer he BFS search, he algorihm can compue score(v) using he procedure coun-componens (lines 19-25) o coun he number of ses in g[v] whose sizes are a leas. The following example illusraes how he Top-k-search framework (Algorihm 3) works. EXAMPLE 2. Consider he graph shown in Figure 1 (a). Suppose ha = 1 and k = 1. We apply he Top-k-search algorihm wih θ = 1 and he running seps are depiced in Figure 4. Firs, we push each verex v wih he upper bound d(v) ino H, as shown in Figure 4 (a). Second, we pop verex c from H wih opbound = 5. We calculae bound(c) = 5 according o Lemma 3. Then, we compue score(c) by bound-search. In G N(c), here is a single pah connecing all verices a, b, d, h, i in N(c), so score(c) = 1. When he algorihm raverses he edge (a, b), we perform wo operaions g[a].union (c, b) and g[b].union (c, a) in g[a] and g[b] respecively according o Propery 2. Third, we push verex c ino S, as shown in Figure 4 (b). In he nex ieraion, we pop verex a from H wih opbound = 4. Then, we updae bound(a) = 3 as we know ha verices b and c are in he same se in g[a]. Since 1622

6 H (c,5) (a,4) (d,3) (b,4) (e,3) (f,4) (g,3) (h,4) S=Ø (i,4) (a) Iniializaion H (c,5) score(c)=1 S={c} (f,4) (h,4) H (b,4) (d,3) (a,4) (f,4) (e,3) (h,4) (g,3) (i,4) (a,3) S={c} (c) Updae bound(a) ino H H H (a,4) (d,3) (h,4) (d,3) (b,2) (g,3) (b,2) (b,4) (e,3) (f,4) (i,4) (e,3) (e,3) (a,3) (g,3) (g,3) (i,4) (b) Compue score(c), and add c ino S score(f)=2 (a,3) S={f} (d) Compue score(f), and updae S by f H (e,3) (b,2) (d,3) (g,3) (a,3) S={f} (e) Pop ou h, i, d from H score(e)=3 S={e} (f) Compue score(e), and updae S by e Figure 4: Illusraion of Top-k-search wih bound-search running on he graph in Figure 1 (a). k = 1, = 1, and θ = 1. θ bound(a) < opbound and bound(a) > min v S score(v), we push (a, 3) ino H again, as shown in Figure 4 (c). When he algorihm goes o process verex f, we have θ bound(f) = opbound = 4 and opbound > min v S score(v). And hen we compue score(f) = 2 and replace verex c in S wih f, as shown in Figure 4 (d). Afer ha, we pop verices h, i, d from H in urn. One can easily check ha none of hem saisfies he condiion in line 1 of Algorihm 3. Thus, we do no push h, i, d back ino H again, as shown in Figure 4 (e). Nex we pop verex e, compue score(e) = 3 and updae S by e, as shown in Figure 4 (f). Since opbound in H is no greaer han score(e) = 3, we can safely erminae. In his process, we only invoke bound-search hree imes o calculae he srucural diversiy scores, while he previous degree algorihm performs eigh compuaions of srucural diversiy score which is clearly more expensive. 4.5 Complexiy Analysis LEMMA 4. The upper bound bound(v) defined in Lemma 3 for any verex v V can be compued in O(1) ime in Algorihm 3. PROOF. We need o mainain a, b and c in g[v] o compue bound(v). Obviously, a = T [].coun, and b, c can be easily mainained in he Union operaion of g[v] wihou increasing he ime complexiy. Thus bound(v) can be compued in O(1) ime. LEMMA 5. The oal ime o compue bound for all verices in Algorihm 3 is O( m ). PROOF. According o Lemma 4, bound(v) for a verex v can be compued in O(1) ime. The iniial upper bound of v is d(v), and bound(v) is updaed in non-increasing order. In line 9 of Algorihm 3, we compare θ bound(v) and opbound o check wheher v should be pushed ino H. Since opbound d(v), bound(v) can be updaed for a mos d(v) imes. Thus he oal ime cos is O( d(v) v V ) = O( m ). LEMMA 6. In Top-k-search, H can be mainained in O( m + n) ime using O(n) space. PROOF. H can be implemened by a varian of bin-sor lis which suppors a push operaion in consan ime and l pop operaions in O(l + n) ime (illusraed in Figure 4). Each ime, esimaing he upper bound bound in line 8 causes a mos one push operaion (line 11) in H. By Lemma 5, we know ha for each verex v V, here are a mos d(v) bound refinemens. Thus, here are a mos v V d(v) bound refinemens in oal for all verices. In addiion, here are n iniial push operaions. Therefore, he algorihm uses O( d(v) v V + n) = O( m + n) ime for all he push operaions. The number of pop operaions is no larger han he number of push operaions. Pu i all ogeher, he ime complexiy o mainain H is O( m + n). The space complexiy of H is O(n). Algorihm 5 fas-bound-search (G,, v) Inpu: G = (V, E), he componen size hreshold, verex v. Oupu: score(v). 1: R ; 2: for u N(v) and T [u].paren do R R {u}; 3: for u R do fas-bound-bfs (u); 4: reurn coun-componens (g[v], ); 5: procedure fas-bound-bfs (u) 6: Q ; UnionF lag false; 7: Q.EnQueue(u); R R {u}; 8: while Q 9: u Q.DeQueue(); 1: if d(u) > d(v) hen MinAdjL N(v); 11: else MinAdjL N(u); 12: for w MinAdjL do 13: if (w, u) E and w R hen 14: Q.EnQueue(w); R R {w}; 15: g[v].union (u, w); UnionF lag rue; 16: if score(u) = 1 hen g[u].union (v, w); 17: if score(w) = 1 hen g[w].union (v, u); 18: if UnionF lag = false hen 19: g[v].isolae (u); 2: if score(u) = 1 hen g[u].isolae (v); THEOREM 2. Algorihm 3 akes O ( v V (d(v))2 ) ime and O(m) space. PROOF. Since he ime o access he adjacency liss in boundsearch is O( v V (d(v))2 ), and all Union operaions are in he loop of accessing adjacency liss (lines of Algorihm 4), he number of Union operaions is O( v V (d(v))2 ). The algorihm invokes n Make-Fores operaions (line 4 of Algorihm 3), which includes v V (d(v) + 1) = 2m + n Make-Se operaions. Nex, all Isolae operaions are in he procedure bound-bfs (lines of Algorihm 4). The number is no greaer han v V 2d(v) = 4m. No Find-Se operaion is direcly invoked. Thus, Union-Find- Isolae includes O( v V (d(v))2 ) Make-Se, Union, Find-Se, Isolae operaions, 2m + n of which are Make-Se. By Lemma 2, he ime complexiy of Union-Find-Isolae is O( v V (d(v))2 ). By Lemma 6, H akes O( m +n) ime. S mainains he op-k resuls using O(n) ime. By Lemma 5, compuing he upper bounds for all verices akes O( m ) ime. Therefore, he ime complexiy of Algorihm 3 is O ( v V (d(v))2 ). Nex, we analyze he space complexiy. For v V, g[v] conains d(v) + 1 iniial disjoin singleon rees, in which each node akes consan space. Hence, he disjoin-se fores srucure akes O(m) space for all verices. S and H boh consume O(n) space. In summary, he space complexiy of Algorihm 3 is O(m). Hence, Theorem 2 is esablished. 5. FAST COMPUTATION OF STRUCTURAL DIVERSITY SCORE In his secion, on op of he Top-k-search framework, we propose wo mehods for fas compuing he srucural diversiy score for a verex. The firs mehod is fas-bound-search which improves bound-search and achieves a beer ime complexiy using he same space. The second is an A -based search mehod which uses a new search order and a new erminaion condiion. 5.1 Fas nd-search We presen fas-bound-search in Algorihm 5, which is buil on bound-search. The major difference is in procedure fas-boundbfs for raversing a conneced componen. When accessing he adjacency lis of verex u having d(u) > d(v), we will access he 1623

7 adjacency lis of v insead (lines 1-13), i.e., we always selec he verex wih a smaller degree o access. Checking wheher (w, u) E in line 13 can be done efficienly by keeping all edges in a hashable. Moreover, R can also be implemened by a hashable. Thus line 13 can be done in expeced consan ime by hashing. p d(p)=1 r q d(q)=1 Figure 5: G N(r) has wo verices p and q wih degree 1 and 1 To show he effeciveness of his improvemen, we consider an example G N(r) in Figure 5. Suppose ha r has wo neighbors p and q wih degree 1 and 1 respecively. To compue score(r), bound-search needs o access he adjacency liss of p and q, and check N(p) + N(q) = 11 verices. In conras, fas-boundsearch accesses N(r) insead of N(q) because d(q) > d(r), hus he number of visied verices is reduced o N(p) + N(r) = 3. Complexiy Analysis: Using fas-bound-search o compue srucural diversiy scores, we achieve a beer ime complexiy of he Top-k-search framework shown in he following heorem. THEOREM 3. The Top-k-search framework using fas-boundsearch akes O( (u,v) E min{d(u), d(v)}) ime and O(m) space. PROOF. For a verex v, he ime cos of accessing he adjacency liss is u N(v) min{d(u), d(v)} for compuing score(v). To compue scores for all verices, accessing he adjacency liss consumes O( v V u N(v) min{d(u), d(v)}) = O( (u,v) E min{d(u), d(v)}). Since he number of Union operaions is bounded by he number of accessing adjacency liss, he number of Union operaions is O( (u,v) E min{d(u), d(v)}). Moreover, here are 2m + n Make-Se operaions, O(m) Isolae operaions and no direc Find-Se operaion invoked by he algorihm. By Lemma 2, Union- Find-Isolae akes O( (u,v) E min{d(u), d(v)}) ime in oal. The oher seps in he loop of accessing adjacency lis ake consan ime. Therefore, i akes O( (u,v) E min{d(u), d(v)}) ime o calculae all verices srucural diversiy scores using he fas-bound-search algorihm. By Lemma 5, he oal ime of esimaing upper bound is O( m ) O(m), and by Lemma 6, he oal ime o mainain H is O( m + n) O(m). Compared wih bound-search, fas-bound-bfs needs exra O(m) space for soring he edge hashable. Thus, he space consumpion is sill O(m). Hence, Theorem 3 is esablished. REMARK 2. According o [6], O( (u,v) E min{d(u), d(v)}) O(ρm) where ρ is he arboriciy of a graph G and ρ min { m, d max} for any graph G. Thus he wors-case ime complexiy of he Top-k-search framework using fas-bound-search is bounded by O( (u,v) E min{d(u), d(v)}) O(ρm) O(m1.5 ). 5.2 ased nd-search In his subsecion we design a new search order and a new erminaion condiion o compue he srucural diversiy score for a verex. Take Figure 6 as an example which shows he neighborhood induced subgraph of r. Suppose ha before examining G N(r), he algorihm has compued he srucural diversiy scores for r s neighbors,...,. Then, by Propery 2, he verices,..., are combined ino one componen P in G N(r). There is anoher componen Q in G N(r) wih only one verex q. To compue score(r), he algorihm needs o furher check wheher he componens P p 2 r q Figure 6: G N(r) conaining wo componens P and Q and Q are conneced or no. If he algorihm firs checks verex q in he componen Q, hen i will go hrough q s adjacency lis N(q) o verify wheher q connecs wih any verices in,...,. If q is no conneced wih any one of hem, we can conclude ha Q forms a size-1 componen and P forms a size-4 componen in G N(r). Thus, he algorihm does no need o raverse he adjacency liss of,...,, and i can erminae early. In conras, if he algorihm firs checks he componen P, hen i needs o go hrough he adjacency liss of verices,..., o verify wheher hey connec wih q or no. This is clearly more expensive han saring from he componen Q. Moivaed by his observaion, we propose an A -based heurisic search mehod o efficienly compue he srucural diversiy in he neighborhood induced subgraph of a verex. Below, we firs give he definiion of componen cos which is used as a heurisic funcion o deermine he componen visiing order in he A search process. DEFINITION 4. Given a componen S in a neighborhood induced subgraph, he componen cos of S is he sum of degree of he unvisied verices in S, denoed as cos(s) = unvisied v S d(v). Suppose ha in Figure 6 all verices in N(r) are unvisied. The componen coss are cos(p ) = 16 and cos(q) = 1. The componen cos measures he cos of accessing he adjacency liss of a componen. If we check he low-cos componens firs and he high-cos componens laer, we can poenially save more compuaion. Thus in A search, we always pick a componen T [x] in G N(v) which has he leas cos o raverse. To record he cos, we add he componen cos as a field in he Union-Find-Isolae daa srucure. Specifically, for a verex u, when we creae a single-node componen T [u], we iniialize T [u].cos = d(u). When we union wo componens T [u] and T [v], we add up heir coss, i.e., T [u].cos + T [v].cos. The Algorihm: A -bound-search uses he componen cos for deermining a heurisic search order o raverse he componens in G N(v) unil here is only one unvisied componen lef. In raversing a componen, he algorihm accesses he adjacency liss of he unvisied verices in increasing order of heir degrees unil he componen is conneced wih oher componens or raversed. Algorihm 6 shows A -bound-search. For a verex v, he algorihm uses a minimum heap T C o mainain all he unidenified componens in G N(v) ordered by heir componen coss. For a componen rooed by a verex u, he algorihm makes use of a minimum heap C[u] o mainain all verices in his componen ordered by heir degree. Iniially, for each verex u whose paren is no, he algorihm pushes u wih cos d(u) ino he minimum heap C[T [u].paren], and adds u ino he hashable R which sores all he unvisied verices (line 5). Moreover, if u is he roo of T [u], he algorihm pushes he componen of u and is componen cos T [u].cos ino he heap T C (lines 6-7). Le us consider an example. Figure 7 (a) shows he neighborhood induced subgraph G N(r), and Figure 7 (b) shows he degree and he paren in T [.] for each verex in N(r). We know ha, p 2, are in a componen rooed by, and q 1, q 2 are in a componen rooed by q 1, and s is in a componen rooed by s iself. Afer iniializaion, he minimum heaps T C, C[s], C[ ], C[q 1 ] and he hashable R are illusraed in Figure 7 (c). 1624

8 Algorihm 6 A -bound-search (G,, v) Inpu: G = (V, E), he componen size hreshold, verex v. Oupu: score(v). 1: R ; T C ; 2: for u N(v) do C[u] ; 3: for u N(v) do 4: if Find-Se (u) hen 5: C[T [u].paren].push((u, d(u))); R R {u}; 6: if T [u].paren = u hen 7: T C.push((u, T [u].cos)); 8: while T C 9: (x, cos x ) T C.pop(); UnionF lag false; 1: if x Find-Se (x) hen coninue; 11: if cos x T [x].cos hen 12: T C.push((x, T [x].cos)); coninue; 13: if R = C[x] hen goo Se3; 14: while C[x] and UnionF lag = false 15: (u, cos u ) C[x].pop(); R R {u}; 16: T [x].cos T [x].cos cos u; 17: w.l.o.g, we assume d(u) < d(v); 18: for w N(u) do 19: if (w, v) E and w R hen 2: fu Find-Se (u); fw Find-Se (w); 21: if fu fw hen 22: Q Heap Merge(C[fu], C[fw]); 23: g[v].union (u, w); 24: C[Find-Se (x)] Q; UnionF lag rue; 25: if score(u) = 1 hen g[u].union (v, w); 26: if score(w) = 1 hen g[w].union (v, u); 27: if UnionF lag = rue and Find-Se (x) = x hen 28: T C.push((x, T [x].cos)); 29: if UnionF lag = false and C[x] = hen 3: if T [x].coun = 1 hen 31: g[v].isolae (x); 32: if score(x) = 1 hen g[x].isolae (v); 33: reurn coun-componens (g[v], ); The algorihm ieraively pops a componen wih he minimum cos from T C, denoed as x wih cos cos x (line 9). If he componen is rooed by verex x and cos x = T [x].cos, he algorihm will examine he verices in he componen of x. Oherwise, if he componen is no longer rooed by x or cos x T [x].cos, i means ha he componen of x has been combined wih anoher componen in he previous ieraion. Then, he algorihm pops he nex componen from T C. If R = C(x) holds, hen all he unvisied nodes in R are from he same componen rooed by x, and his componen is he las o be raversed in G N(v) (line 13). By he early erminaion condiion, he algorihm does no need o raverse his componen and can direcly go o coun he number of componens in G N(v) (line 33). For a popped componen rooed by x, he algorihm ieraively examines he verices in he componen in increasing order of heir degree (lines 14-26). For such a verex u, we will access is adjacency lis N(u) o find ou hose verices denoed as w ha are also in N(v). Then we will union he componens which conain u and w respecively ino one. This process is very similar o he previous algorihms. So we omi he deails for breviy. Coninuing wih our example in Figure 7. Afer iniializaion, we pop he firs componen (s, 8) from T C, as shown in se (Figure 7 (d)). Then, we examine verex s in his componen and find ha i is no conneced wih oher componens in G N(r). Nex, we move o sep 2 (Figure 7 (e)) o pop he componen (, 12). In his componen, we firs examine he adjacency lis of, i.e., N(). We find ha is conneced wih q 1, so we union he componens rooed by and q 1. Assume ha he new componen is rooed by. Then we se T [q 1 ].paren = and merge C(q 1 ) p 2 q 1 op (s,8) (p1,12) (q1,15) r s q 2 id deg paren p1 3 p1 p2 4 p1 p3 5 p1 q1 7 q1 q2 8 s 8 TC C[s] C[p1] C[q1] (s,8) op (p1,12) (q1,15) (a) GN(r) (s,8) (c) Iniializaion (p1,3) (p2,4) (p3,5) (p1,3) (p2,4) (p3,5) (q1,7) (q2,8) TC C[s] =Ø C[p1] C[q1] (s,8) (d) Se R={p1,p2,p3,q1,q2,s} R={p1,p2,p3,q1,q2} (q1,7) (q2,8) q1 s (b) Verex degree and paren in T[.] TC (p1,12) op (q1,15) TC (p1,12) op (q1,15) (p1,24) TC (q1,15) op (p1,24) T[q1].paren q1 (e) Sep 2 C[p1] (p1,3) (p2,4) (p3,5) C[p1] (p2,4) (p3,5) (q1,7) (q2,8) C[p1] (p2,4) (p3,5) (q1,7) (q2,8) R={p2,p3,q1,q2} (f) Se (Terminaion) C[q1] (q1,7) (q2,8) Union T[p1] and T[q1], and se he new roo as p1. R={p2,p3,q1,q2} C[q1] =Ø Figure 7: A -bound-search example for compuing score(r) ino C( ). We push he new componen (, 24) ino T C again. In se (Figure 7 (f)), we pop he componen (q 1, 15) and find ha T [q 1 ].paren q 1, as he componen of q 1 has been combined wih ha of in sep 2. In his sep, here is only one componen in T C, which mees he early erminaion condiion. Complexiy Analysis: In he componen union process (line 22 of Algorihm 6), we need o merge wo heaps C[fw] and C[fu] ino one. We can implemen C[.] by he mergeable heap such as lefis heap or binomial heap [8], which can suppor he merge of wo heaps in O(log n) ime and a push/pop operaion in O(log n) ime for a heap wih n elemens. LEMMA 7. In Algorihm 6, he operaions for T C and all C[.] ake O(d(v) log d(v)) ime and O(d(v)) space in oal. PROOF. Since he number of componens in G N(v) is no greaer han d(v), we perform a mos d(v) 1 Union operaions before erminaion. Hence, here are a mos d(v) 1 new componens o be pushed ino T C (lines 12 and 28). In addiion, for iniializaion T C d(v) holds, which indicaes ha T C 2d(v) always holds. As here are a mos 2d(v) push and pop operaions respecively, and each operaion akes O(log d(v)) ime, overall T C akes O(d(v) log d(v)) ime using O(d(v)) space. For iniializaion, all C[.] heaps ake d(v) push operaions in oal (line 5), and he ime cos of each operaion is O(log d(v)) as he size of he larges heap is smaller han d(v). Hence, he iniializaion ime is O(d(v) log d(v)). As analyzed above, here are a mos d(v) 1 heap merging operaions and each operaion coss O(log d(v)), he oal ime cos in line 22 is O(d(v) log d(v)). Moreover, here are a mos d(v) pop operaions in line 15, he ime cos of which is O(d(v) log d(v)). All C[.] heaps conain a mos d(v) verices oally cosing O(d(v)) space. As a resul, all C[.] heaps ake O(d(v) log d(v)) ime and O(d(v)) space. THEOREM 4. The Top-k-search framework using A -boundsearch akes O( (u,v) E(min{d(u), d(v)}+log d(u))) ime and O(m) space. PROOF. The proof is similar o he proof of Theorem 3. A difference is ha we use he Find-Se operaions in A -bound-search. Since he Find-Se operaions in lines 2 and 24 are in he loop of accessing adjacency lis, he oal number of such operaions is O( (u,v) E min{d(u), d(v)}) for he whole process. Consider he process of compuing score(v) for a verex v, we ake d(v) 1625

9 Find-Se operaions in line 4. Since lines 1 and 27 are boh in he ouer while loop (line 8), and T C has a mos 2d(v) pop operaions according o Lemma 7, he algorihm akes a mos 2d(v) Find-Se operaions in lines 1 and 27 respecively. Hence, i akes O( (u,v) E min{d(u), d(v)} + v V 5d(v)) = O( (u,v) E min{d(u), d(v)}) Find-Se operaions. By Lemma 2, Union-Find- Isolae akes O( (u,v) E min{d(u), d(v)}) ime in oal. Anoher difference is ha we mainain wo ypes of heaps T C and C[.]. By Lemma 7, he oal ime of T C and C[.] are O ( u V d(u) log d(u)) = O( u V v N(u) log d(u)) = O( (u,v) E log d(u)). The addiional space overhead is O(m). Hence, Theorem 4 is esablished. REMARK 3. The wors-case ime complexiy of he Top-k-search framework using A -bound-search is bounded by O( (u,v) E ( min{d(u), d(v)} + log d(u))) O((ρ + log d max)m) O(m 1.5 ), where ρ is he arboriciy of he graph as menioned in Remark 2. Complexiy Comparison: We compare he ime complexiy of algorihms degree and Top-k-search. According o Theorem 1, degree akes O( v V (d(v))2 ) ime, which can be equivalenly rewrien as O( v V u N(v) d(u)) = O ( (u,v) E (d(u)+d(v))) = O( (u,v) E (max{d(u), d(v)} + min{d(u), d(v)})) = O( (u,v) E max{d(u), d(v)}). For Top-k-search using fas-bound-search, according o Theorem 3, i akes O( (u,v) E min{d(u), d(v)}) ime, which is obviously beer han O( (u,v) E max{d(u), d(v)}), he ime complexiy of degree, and (u,v) E min{d(u), d(v)} = (u,v) E max{d(u), d(v)} only if all verices in he graph have he same degree. In a power-law graph such as a social nework, he degrees of verices have a large variance, hus Top-k-search using fasbound-search is much beer han degree in a social nework. For example, on a sar graph wih n nodes, Top-k-search using fasbound-search akes O(n) ime while degree akes O(n 2 ) ime. For Top-k-search using A -bound-search, according o Theorem 4, is ime O( (u,v) E(min{d(u), d(v)}+log d(u))) is also beer han O( (u,v) E max{d(u), d(v)}) of degree. The firs par O( (u,v) E min{d(u), d(v)}) is he same as Theorem 3, and he second par O( (u,v) E log d(u)) is obviously beer han O( (u,v) E max{d(u), d(v)}) as log d(u) max{d(u), d(v)}. 6. UPDATE IN DYNAMIC NETWORKS Many real-world neworks undergo frequen updaes. When he nework is updaed, he op-k srucural diversiy resuls also need o be updaed. The challenge, however, is ha insering or deleing a single edge (u, v) can rigger updaes in a series of neighborhood induced subgraphs including G N(u), G N(v) and G N(w) where w N(u) N(v). This can be a cosly operaion because he corresponding srucural diversiy scores need o be recompued, and he op-k resuls need o be updaed oo. In he following, we will show ha our Top-k-search framework can be easily exended o handle updaes in dynamic graphs. We consider wo ypes of updaes: edge inserion and edge deleion. Verex inserion/deleion can be regarded as a sequence of edge inserions/deleions preceded/followed by he inserion/deleion of an isolaed verex, while i is rivial o handle he inserion/deleion of an isolaed verex. 6.1 Handling Edge Inserion Consider he inserion of an edge (u, v). Le L = N(u) N(v) denoe he se of common neighbors of u and v. The inserion of (u, v) causes he inserions of verex v and a se of L edges p 5 p 5 p 2 r (a) G p 2 q s (d) Spanning Tree of Componen P in GN(r) as TP p 5 p 5 p 2 r (b) GN(r) r s s (e) Updae GN(r) wih edge deleion of (r,p2) p 5 p 2 r q s (c) Updae GN(r) wih edge inserion of (r,q) p 5 (f) Updae TP wih edge deleion of (r,p2) Figure 8: Illusraion of updaes in a dynamic graph {(v, w) w L} ino u s neighborhood induced subgraph G N(u). For each w L, we perform a Union operaion g[u].union(v, w) o updae he componens and score(u). For verex v, G N(v) is updaed in a similar way. The inserion of (u, v) also affecs G N(w) for each w L. We check he disjoin-se fores srucure g[w]. If u, v belong o he same conneced componen before he edge inserion, hen all componens remain unchanged and so does score(w). If u, v are in differen componens before he edge inserion, we merge he wo componens ino one wih a Union operaion g[w].union(u, v) and updae score(w) accordingly. Consider he graph G in Figure 8 (a) as an example. Suppose ha = 2 and he insered edge is (r, q). L = N(r) N(q) = {s, }. Figure 8 (c) shows he updaed G N(r) wih he edge inserion. G N(r) has wo new edges (, q) and (s, q), bu score(r) = 1 remains unchanged. For verex s L, verices r, q are now conneced in he same componen in G N(s) wih he inserion of (r, q), so we updae score(s) from o Handling Edge Deleion Consider he deleion of an edge (u, v). To handle he edge deleion, we mainain a spanning ree for each conneced componen in he affeced subgraphs G N(u), G N(v) and G N(w) where w L. For example, consider he componen P = {,..., p 5 } of G N(r) in Figure 8 (b) and he corresponding spanning ree T P in Figure 8 (d). The edges in he spanning ree are called ree edges, and oher edges in he componen are called non-ree edges, e.g., (, p 2) is a ree edge and (, p 5) is a non-ree edge. For each w L, we consider updaing G N(w) wih he deleion of (u, v). We check wheher (u, v) is a ree edge in he spanning ree of he componen. If (u, v) is a non-ree edge, score(w) remains unchanged because verices u, v are sill in he same componen conneced by he corresponding spanning ree. Coninuing wih he example above, he deleion of he non-ree edge (, p 5 ) will no spli he componen P in G N(r), and, p 5 are sill in he same componen. If (u, v) is a ree edge, hen he deleion of (u, v) splis he spanning ree ino wo rees denoed as T u and T v. We will search for a replacemen edge so as o reconnec T u and T v. If a replacemen edge (u, v ) exiss, we inser (u, v ) o connec T u, T v ino a new spanning ree. Then he original componen is sill conneced, and score(w) remains unchanged. If he replacemen edge does no exis, he deleion of (u, v) splis he original conneced componen ino wo componens, and he corresponding spanning rees are T u and T v. So we updae score(w) accordingly. Mainaining he spanning ree can be implemened easily wih he Union operaion by keeping rack of he bridge edge beween wo differen componens. In he example above, if a ree edge (, p 2) is deleed, we can find a replacemen edge (, ) o reconnec he spanning ree in Figure 8 (d). 1626

10 The deleion of (u, v) also affecs G N(u) and G N(v). Consider u as an example. For all w L, we remove hose non-ree edges (v, w) from G N(u), and remove hose ree edges (v, w) from he spanning ree which is hen spli ino muliple rees. Then we search for replacemen ree edges o reconnec he spanning ree. Finally, we remove v from G N(u) and updae score(u). Figures 8 (e) and (f) show he updaes of G N(r) and T P wih he deleion of (r, p 2 ). The above echniques apply o updaing boh he acual score and he upper bound in our Top-k-search framework given edge inserions/deleions. In updaing an upper bound bound(v) for verex v, given an edge deleion as a ree edge, we only spli he original spanning ree ino wo, bu do no have o search for he replacemen edge. This will only relax bound(v) wihou affecing he resul correcness. This sraegy can avoid he cos of finding he replacemen edge and achieve higher efficiency. Summary: Handling edge inserion is rivial using our disjoin-se fores srucure, while handling deleion is more cosly as i mainains he spanning ree. In he real-world neworks, edge inserions are usually more frequen han deleions. Our updae echniques do no increase he space complexiy of Top-k-search. 7. EXPERIMENTS We conduc exensive performance sudies o evaluae he algorihms proposed in his paper. All algorihms are implemened in C++ and all he experimens are conduced on he Linux operaing sysem wih 2.67GHz six-core CPU and 5GB main memory. Comparison mehods: To he bes of our knowledge, we are he firs o sudy op-k srucural diversiy search. In he lieraure, no algorihms have been proposed o address his problem ye. Thus, we compare our algorihms wih he degree-based approach (Algorihm 1) which serves as a baseline. We evaluae four algorihms. : The degree-based approach in Algorihm 1. : Top-k-search equipped wih bound-search (Algorihm 4) and θ = 1. : Top-k-search equipped wih fas-bound-search (Algorihm 5) and θ = ( n ) 1 m. A -B : Top-k-search equipped wih A -bound-search (Algorihm 6) and θ = ( n ) 1 m. In our experimens, we find ha θ = ( n ) 1 m which is close o 1 always yields a good performance in he Top-k-search framework. For and A -B, heir performances are no very sensiive o he value of θ as long as θ (1.1, 1.5) on all daases. Due o he lack of space, we do no show he curves by varying θ, and simply se θ = ( n ) 1 m for boh and A -B. Evaluaion merics: We use he running ime and he number of verices whose srucural diversiy scores are compued in he search process as wo merics. The laer evaluaes he number of verices ha are pruned by he algorihm. Daases: We use 13 publicly available real-world neworks covering social, communicaion, collaboraion neworks, and webgraphs. The nework saisics are shown in Table 1. Excep for Epinions, Digg and KDDTrack1 1 which are from heir respecive websies, he oher 1 neworks are downloaded from he Sanford Nework Analysis Projec (snap.sanford.edu). We rea all he neworks as undireced. 7.1 Efficiency Comparison In his experimen, we compare he efficiency of differen mehods over all neworks. We se k = 1 and = 2. Similar resuls can be observed for oher k and values. Table 2 repors he resuls. 1 hps://www.kddcup212.org Table 1: Nework saisics (K = 1 3 and M = 1 6 ) Name V G E G d max Descripion WikiVoe 5K 14K 165 Epinions 76K 59K 344 Slashdo 82K 948K 2552 Social Gowalla 196K 1.9M 1473 neworks Digg M KDDTrack1 1.9M 1.2M Enron 37K 368K EuAll 265K 42K 7636 Communicaion WikiTalk 2.4M 5.M 129 neworks HepPh 12K 237K 491 Collaboraion AsroPh 19K 396K 54 neworks NoreDame 326K 1.5M 1721 Web graph Flickr 8K 11.8M 576 Flickr We can see ha A -B is he mos efficien, followed by,, and. Noice ha he performance of A -B,, and which adop he Top-k-search framework is subsanially beer han ha of he degree-based algorihm. The speedup raio beween and A -B defined as R s = / A B is beween 2.1 and 69.1 (column 6 in Table 2). The resul conforms wih he complexiy analysis in Secion 5. In addiion, we define he pruning raio beween and A -B as R p = S /S A B, where S and S A B denoe he number of verices whose srucural diversiy scores are compued by he respecive mehods. The pruning raio is beween 2.1 and 11.1 over all neworks (column 11 in Table 2). This resul suggess ha he upper bound derived in Lemma 3 is indeed igher han he degree-based upper bound in Lemma 1. When we compare and, he reducion of running ime and search space by demonsraes he effeciveness of he igher upper bound in Lemma 3 and he Union-Find-Isolae daa srucure. When we compare and, he reducion of running ime by shows he effeciveness of he fas-bound-search mehod. Finally we observe ha A -B is more efficien han, which proves he effeciveness of he A search order. 7.2 Performance Evaluaion by Varying k In his experimen, we evaluae he performance of all he mehods by varying he parameer k. We se = 2 and focus on six neworks Digg, WikiTalk, AsroPh, Gowalla, NoreDame and Flickr. Similar resuls can be observed for oher values and on oher neworks. Figures 9 (a)-(f) depic he running ime of differen algorihms. Again, we can see ha A -B is he mos efficien and is he leas efficien in mos neworks. The running ime of A -B is very sable as k increases. Figures 1 (a)-(f) show he number of verices whose srucural diversiy scores are compued by differen mehods in he six neworks. A -B is he clear winner by pruning he larges number of verices, and performs wors. In addiion, we find ha and achieve very similar performance in erms of he number of verices ha are pruned. This is because θ = ( n ) 1 m in is very close o 1 (as lised in he las column of Table 2), and θ in is se o 1 in our experimen. Thus, he pruning condiion in and is very similar. Bu on he oher hand, runs much faser han as shown in Figure 9, which conforms wih he ime complexiy analysis in Theorems 2 and Performance Evaluaion by Varying We evaluae he performance of all mehods by varying he parameer. In his experimen, we se k = 1 and similar resuls can be observed for oher k values. Figures 11 (a)-(f) show he running ime of differen algorihms. Once again, A -B is he mos efficien algorihm, and is he leas efficien one. We also observe ha in many cases, he running ime of all mehods increases wih increasing a firs, bu i may drop slighly when furher in- 1627

11 Table 2: Comparison of running ime (wall-clock ime in seconds) and search space (he number of verices whose srucural diversiy score are compued in search process) of differen algorihms. Here k = 1 and = 2. Nework Running Time Number of Compued Verices θ = ( n 1 ) m A -B R s A -B R p WikiVoe Epinions Slashdo Gowalla Digg KDDTrack Enron EuAll WikiTalk HepPh AsroPh NoreDame Flickr K 2K 12K 8K 4K (a) Digg (b) WikiTalk (c) AsroPh (d) Gowalla (e) NoreDame Figure 9: Running ime (in second) of differen algorihms versus parameer k 8K 6K 4K 2K K 8K 4K K 4K 3K 2K (f) Flickr (a) Digg (b) WikiTalk (c) AsroPh (d) Gowalla (e) NoreDame (f) Flickr Figure 1: Number of verices whose srucural diversiy scores are compued versus parameer k 6K 5K 4K 3K 2K M K 6K 4K 2K K 2K K 4K 3K 2K (a) Digg (b) WikiTalk (c) AsroPh (d) Gowalla (e) NoreDame Figure 11: Running ime (in second) of differen algorihms versus parameer (f) Flickr K 8K 6K 4K 2K (a) Digg (b) WikiTalk (c) AsroPh (d) Gowalla (e) NoreDame (f) Flickr Figure 12: Number of verices whose srucural diversiy scores are compued versus parameer 1 8K 6K 4K 2K creases. A possible reason is ha when is large, he number of he qualified componens (i.e., he componens whose sizes are no less han ) reduces. Thus, by he esimaed upper bound, he search space can be quickly pruned. Figures 12 (a)-(f) show he number of verices whose srucural diversiy scores are compued in differen neworks by varying. We observe ha A -B prunes he mos number of verices, and prunes he leas number of verices. 7.4 Handling Updae in Dynamic Neworks In his experimen, we evaluae he ime for incremenally mainaining he op-k resuls when he inpu nework is updaed. For each nework, we randomly inser/delee edges, and updae he op-k resuls afer every edge inserion/deleion. The average updae ime per edge inserion/deleion is repored in Table 3. In addiion, we repor he bach updae ime for he edge inserions/deleions. We repea his experimen for 5 imes and repor he average performance. For comparison, we also repor he ime for compuing he op-k resuls from scrach when he nework is updaed wih an edge inserion/deleion. The resul in Table 3 shows ha handling edge inserions is highly efficien. The updae ime per edge inserion is.1 or.2 millisecond on mos neworks, and he bach updae ime for edge inserions is wihin 1 milliseconds on mos neworks. Handling edge deleions is more cosly, because an edge deleion may rigger o check wheher he wo endpoins of he deleed edge are sill in he same componen or no in a number of neighborhood in- 1628

12 Table 3: Updae Time (wall-clock ime in milliseconds). Here k = 1 and = 2. Nework Inserion Inserion Deleion Deleion Compuing Per Edge Edges Per Edge Edges from scrach WikiVoe Epinions Slashdo Gowalla Digg KDDTrack Enron EuAll WikiTalk HepPh AsroPh NoreDame Flickr duced subgraphs. The updae ime per edge deleion is wihin 1 millisecond on mos neworks, and he bach updae ime for edge deleions is less han 1 second on mos neworks. Finally we can see he incremenal updae (per edge as well as bach updae of edges) is several orders of magniude faser han recompuing he op-k resuls from scrach. 8. RELATED WORK To he bes of our knowledge, op-k srucural diversiy search has no been sudied before. In he following, we briefly review he exising work ha are relaed o ours. Firs, our work is closely relaed o he work on op-k query processing. The goal of op-k query processing is o find k objecs wih he highes rank based on some pre-defined ranking funcion. A commonly used framework for his problem is o examine he candidaes in a heurisic order and prune he search space using an upper bound. Afer he seminal work by Fagin e al. [1, 11], a large number of sudies on op-k query processing have been done for differen applicaion scenarios, such as processing disribued preference queries [4], keyword queries [17], se similariy join queries [25]. Recenly, many sudies ake he diversiy ino consideraion in op-k query processing, in order o reurn diversified ranking resuls [26, 18, 1, 16, 2, 27]. A comprehensive survey of op-k query processing can be found in [12]. Second, our proposed echniques are relaed o he algorihms for he riangle lising problem, which is o find all riangles in a graph. Iai and Rodeh in [13] firs proposed an O(m 1.5 ) algorihm for he riangle lising problem. In [15], Laapy proved ha he ime complexiy O(m 1.5 ) is opimal. Subsequenly, Schank and Wagner [21, 2] proposed a simpler and paricularly fas soluion wih he opimal complexiy based on he verex ordering and efficien lookup of he adjacency liss for neighborhood esing. Recenly, Chu and Cheng [7] proposed an I/O-efficien algorihm for riangle lising in a massive graph, which canno fi ino he main memory. 9. CONCLUSIONS In his paper, we sudy he op-k srucural diversiy search problem moivaed by a number of nework analysis applicaions. We develop a novel Top-k-search framework o ackle his issue. Specifically, we design a Union-Find-Isolae daa srucure o keep rack of he known srucural informaion of each verex, and an effecive upper bound for pruning. We evaluae he proposed algorihms on 13 large neworks, and he resuls demonsrae he effeciveness and efficiency of he proposed algorihms. Our sudy in his paper serves as he firs sep o he exciing opic of op-k srucural diversiy search. [22] gives wo more definiions of srucural diversiy based on k-core [5] and k-russ [23]. I would be ineresing o exend he proposed echniques o hese wo definiions as a fuure work. ACKNOWLEDGMENTS This work is suppored by he Hong Kong Research Grans Council (RGC) General Research Fund (GRF) Projec No. CUHK , 41131, , and he Chinese Universiy of Hong Kong Direc Gran No REFERENCES [1] R. Agrawal, S. Gollapudi, A. Halverson, and S. Ieong. Diversifying search resuls. In WSDM, pages 5 14, 29. [2] A. Angel and N. Koudas. Efficien diversiy-aware search. In SIGMOD, pages , 211. [3] L. Backsrom, D. P. Huenlocher, J. M. Kleinberg, and X. Lan. Group formaion in large social neworks: membership, growh, and evoluion. In KDD, pages 44 54, 26. [4] K. Chang and S. Hwang. Minimal probing: supporing expensive predicaes for op-k queries. In SIGMOD, pages , 22. [5] J. Cheng, Y. Ke, S. Chu, and M. T. Özsu. Efficien core decomposiion in massive neworks. In ICDE, pages 51 62, 211. [6] N. Chiba and T. Nishizeki. Arboriciy and subgraph lising algorihms. SIAM J. Compu., 14(1):21 223, [7] S. Chu and J. Cheng. Triangle lising in massive neworks and is applicaions. In KDD, pages , 211. [8] T. H. Cormen, C. E. Leiserson, R. L. Rives, and C. Sein. Inroducion o Algorihms. MIT Press, 29. [9] P. S. Dodds and D. J. Was. Universal behavior in a generalized model of conagion. Physical Review Leers, 92:21871, 24. [1] R. Fagin. Combining fuzzy informaion from muliple sysems. Journal of Compuer and Sysem Sciences, 58(1):83 99, [11] R. Fagin, A. Loem, and M. Naor. Opimal aggregaion algorihms for middleware. In PODS, pages , 21. [12] I. Ilyas, G. Beskales, and M. Soliman. A survey of op-k query processing echniques in relaional daabase sysems. ACM Compuing Surveys (CSUR), 4(4):11, 28. [13] A. Iai and M. Rodeh. Finding a minimum circui in a graph. SIAM Journal on Compuing, 7(4): , [14] H. Kwak, C. Lee, H. Park, and S. B. Moon. Wha is wier, a social nework or a news media? In WWW, pages 591 6, 21. [15] M. Laapy. Main-memory riangle compuaions for very large (sparse (power-law)) graphs. Theor. Compu. Sci., 47(1-3): , 28. [16] R.-H. Li and J. X. Yu. Scalable diversified ranking on large graphs. In ICDM, pages , 211. [17] Y. Luo, X. Lin, W. Wang, and X. Zhou. Spark: op-k keyword query in relaional daabases. In SIGMOD, pages , 27. [18] L. Qin, J. X. Yu, and L. Chang. Diversifying op-k resuls. PVLDB, 5(11): , 212. [19] D. M. Romero, B. Meeder, and J. M. Kleinberg. Differences in he mechanics of informaion diffusion across opics: idioms, poliical hashags, and complex conagion on wier. In WWW, pages , 211. [2] T. Schank. Algorihmic aspecs of riangle-based nework analysis. Ph.D. Disseraion, Universiy Karlsruhe, 27. [21] T. Schank and D. Wagner. Finding, couning and lising all riangles in large graphs, an experimenal sudy. In WEA, pages 66 69, 25. [22] J. Ugander, L. Backsrom, C. Marlow, and J. Kleinberg. Srucural diversiy in social conagion. Proceedings of he Naional Academy of Sciences, 19(16): , 212. [23] J. Wang and J. Cheng. Truss decomposiion in massive neworks. PVLDB, 5(9): , 212. [24] D. J. Was and P. S. Dodds. Influenials, neworks, and public opinion formaion. J. Consum Res, 34: , 27. [25] C. Xiao, W. Wang, X. Lin, and H. Shang. Top-k se similariy joins. In ICDE, pages , 29. [26] Y. Zhang, J. Callan, and T. Minka. Novely and redundancy deecion in adapive filering. In SIGIR, pages 81 88, 22. [27] X. Zhu, J. Guo, X. Cheng, P. Du, and H. Shen. A unified framework for recommending diverse and relevan queries. In WWW, pages 37 46,

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