Ranking of Closeness Centrality for Large-Scale Social Networks

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1 Raig of Closeess Cetrality for Large-Scale Social Networs Kazuya Oamoto 1, Wei Che, ad Xiag-Yag Li 3 1 Kyoto Uiversity, oia@uis.yoto-u.ac.jp Microsoft Research Asia, weic@microsoft.com 3 Illiois Istitute of Techology ad Microsoft Research Asia xli@cs.iit.edu Abstract. Closeess cetrality is a importat cocept i social etwor aalysis. I a graph represetig a social etwor, closeess cetrality measures how close a vertex is to all other vertices i the graph. I this paper, we combie existig methods o calculatig exact values ad approximate values of closeess cetrality ad preset ew algorithms to ra the top- vertices with the highest closeess cetrality. We show that uder certai coditios, our algorithm is more efficiet tha the algorithm that calculates the closeess-cetralities of all vertices. 1 Itroductio Social etwors have bee the subject of study for may decades i social sciece research. I recet years, with the rapid growth of Iteret ad World Wide Web, may large-scale olie-based social etwors such as Faceboo, Friedster appear, ad may large-scale social etwor data, such as coauthorship etwors, become easily available olie for aalysis [Ne04a, Ne04b, EL05, PP0]. A social etwor is typically represeted as a graph, with idividual persos represeted as vertices, the relatioships betwee pairs of idividuals as edges, ad the stregths of the relatioships represeted as the weights o edges (for the purpose of fidig the shortest weighted distace, we ca treat lower-weight edges as stroger relatioships). Cetrality is a importat cocept i studyig social etwors [Fr79, NP03]. Coceptually, cetality measures how cetral a idividual is positioed i a social etwor. Withi graph theory ad etwor aalysis, various measures (see [KL05] for details) of the cetrality of a vertex withi a graph have bee proposed to determie the relative importace of a vertex withi the graph. Four measures of cetrality that are widely used i etwor aalysis are degree cetrality, betweeess cetrality 4, closeess cetrality, ad eigevector cetrality 5. I this paper, we focus o shortest-path closeess 4 For a graph G = (V, E), the betweeess cetrality C B(v) for a vertex v is C B(v) = σ v(s,t) s,t:s t v where σ(s, t) is the umber of shortest paths from s to t, ad σ(s,t) σ v(s, t) is the umber of shortest paths from s to t that pass through v. 5 Give a graph G = (V, E) with adjacecy matrix A, let x i be the eigevector cetrality of the ith ode v i. The vector x = (x 1, x,, x ) T is the solutio of equatio

2 cetrality (or closeess cetrality for short) [Ba50, Be65]. The closeess cetrality of a vertex i a graph is the iverse of the average shortest-path distace from the vertex to ay other vertex i the graph. It ca be viewed as the efficiecy of each vertex (idividual) i spreadig iformatio to all other vertices. The larger the closeess cetrality of a vertex, the shorter the average distace from the vertex to ay other vertex, ad thus the better positioed the vertex is i spreadig iformatio to other vertices. The closeess cetrality of all vertices ca be calculated by solvig allpairs shortest-paths problem, which ca be solved by various algorithms taig O(m + log ) time [Jo77, FT87], where is the umber of vertices ad m is the umber of edges of the graph. However, these algorithms are ot efficiet eough for large-scale social etwors with millios or more vertices. I [EW04], Eppstei ad Wag developed a approximatio algorithm to calculate the closeess cetrality i time O( log ɛ ( log + m)) withi a additive error of ɛ for the iverse of the closeess cetrality (with probability at least 1 1 ), where ɛ > 0 ad is the diameter of the graph. However, applicatios may be more iterested i raig vertices with high closeess cetralities tha the actual values of closeess cetralities of all vertices. Suppose we wat to use the approximatio algorithm of [EW04] to ra the closeess cetralities of all vertices. Sice the average shortest-path distaces are bouded above by, the average differece i average distace (the iverse of closeess cetrality) betwee the ith-raed vertex ad the (i + 1)th-raed vertex (for ay i = 1,..., 1) is O( ). To obtai a reasoable raig result, we would lie to cotrol the additive error of each estimate of closeess cetrality to withi O( ), which meas we set ɛ to Θ( 1 ). The the algorithm taes O( log ( log + m)) time, which is worse tha the exact algorithm. Therefore, we caot use either purely the exact algorithm or purely the approximatio algorithm to ra closeess cetralities of vertices. I this paper, we show a method of raig top highest closeess cetrality vertices, combiig the approximatio algorithm ad the exact algorithm. We first provide a basic raig algorithm TOPRANK(), ad show that uder certai coditios, the algorithm ras all top highest closeess cetrality vertices (with high probability) i O(( + 3 log 1 3 )( log + m)) time, which is better tha Ax = λx, where λ is the greatest eigevalue of A to esure that all values x i are positive by the Perro-Frobeius theorem. Google s PageRa [BP98] is a variat of the eigevector cetrality measure. The PageRa vector R = (r 1, r,, r ) T, where r i is the PageRa of webpage i ad is the total umber of webpages, is the solutio of the equatio R = 1 d 1 + dlr. Here d is a dampig factor set aroud 0.85, L is a modified webpage-adjacecy matrix: l i,j = 0 if page j does ot li to i, ad ormalised such that, for each j, i=1 li,j = 1, i.e., li,j = a i,j d j where a i,j = 1 oly if page j has li to page i, ad d j = i=1 ai,j is the out-degree of page j.

3 O(( log + m)) (whe = o()), the time eeded by a brute-force algorithm that simply computes all average shortest distaces ad the ras them. We the use a heuristic to further improve the algorithm. Our wor ca be viewed as the first step toward desigig ad evaluatig efficiet algorithms i fidig top raig vertices with highest closeess cetralities. We discuss i the ed several ope problems ad future directios of this wor. Prelimiary We cosider a coected weighted udirected graph G = (V, E) with vertices ad m edges ( V =, E = m). We use d(v, u) to deote the legth of a shortest-path betwee v ad u, ad to deote the diameter of graph G, i.e., = max v,u V d(v, u). The closeess cetrality c v of vertex v [Be65] is defied as 1 c v = Σ u V d(v, u). (.1) I other words, the closeess cetrality of v is the iverse of the average (shortestpath) distace from v to ay other vertex i the graph. The higher the c v, the shorter the average distace from v to other vertices, ad v is more importat by this measure. Other defiitios of closeess cetralities exist. For example, 1 some defie the closeess cetrality of a vertex v as Σ u V d(v,u) [Sa66], ad some defie the closeess cetrality as the mea geodesic distace (i.e the shortest path) betwee a vertex v ad all other vertices reachable from it, i.e., Σ u V d(v,u) 1, where is the size of the etwor s coected compoet V reachable from v. I this paper, we will focus o closeess cetrality defied i equatio (.1). The problem to solve i this paper is to fid the top vertices with the highest closeess cetralities ad ra them, where is a parameter of the algorithm. To solve this problem, we combie the exact algorithm [Jo77, FT87] ad the approximatio algorithm [EW04] for computig average shortest-path distaces to ra vertices o the closeess cetrality. The exact algorithm iterates Dijstra s sigle-source shortest-paths (SSSP for short) algorithm times for all vertices to compute the average shortest-path distaces. The origial Dijstra s SSSP algorithm [Di59] computes all shortest-path distaces from oe vertex, ad it ca be efficietly implemeted i O( log + m) time [Jo77, FT87]. The approximatio algorithm RAND give i [EW04] also uses Dijstra s SSSP algorithm. RAND samples l vertices uiformly at radom ad computes SSSP from each sample vertex. RAND estimates the closeess cetrality of a vertex usig the average of l shortest-path distaces from the vertex to the l sample vertices istead of to all vertices. The followig boud o the accuracy of the approximatio is give i [EW04], which utilizes the Hoeffdig s theorem [Ho63]: 6 Pr{ 1 ĉ v 1 c v ɛ } ɛ l log ( 1 6 I this paper, log meas atural logarithm with base e. ), (.)

4 for ay small positive value ɛ, where ĉ v is the estimated closeess cetrality of vertex v. Let a v be the average shortest-path distace of vertex v, i.e., a v = Σ u V d(v, u) 1 = 1 c v. Usig the average distace, iequality (.) ca be rewritte as Pr{ â v a v ɛ } ɛ l log ( 1 ), (.3) where â v is the estimated average distace of vertex v to all other vertices. If the algorithm uses l = α log ɛ samples (α > 1 is a costat umber) which will cause the probability of ɛ error at each vertex to be bouded above by 1, the probability of ɛ error aywhere i the graph is the bouded from above by 1 ( 1 (1 1 ) ). It meas that the approximatio algorithm calculates the average legths of shortest-paths of all vertices i O( log ɛ ( log + m)) time withi a additive error of ɛ with probability at least 1 1, i.e., with high probability (w.h.p.). 3 Raig algorithms Our top- raig algorithm is based o the approximatio algorithm as well as the exact algorithm. The idea is to first use the approximatio algorithm with l samples to obtai estimated average distaces of all vertices ad fid a cadidate set E of top- vertices with estimated shortest distaces. We eed to guaratee that all fial top- vertices with the exact average shortest distaces are icluded i set E with high probability. Thus, we eed to carefully choose umber > usig the boud give i formula (.3). Oce we fid set E, we ca use the exact algorithm to compute the exact average distaces for all vertices i E ad ra them accordigly to fid the fial top- vertices with the highest closeess cetralities. The ey of the algorithm is to fid the right balace betwee sample size l ad the cadidate set size : If we use a too small sample size l, the cadidate set size could be too large, but if we try to mae small, the sample size l may be too large. Ideally, we wat a optimal l that miimizes l +, so that the total time of both the approximatio algorithm ad the computatio of exact closeess cetralities of vertices i the cadidate set is miimized. I this sectio we will show the basic algorithm first, ad the provide a further improvemet of the algorithm with a heuristic. 3.1 Basic raig algorithm We ame the vertices i V as v 1, v,..., v such that a v1 a v a v. Let â v be the estimated average distace of vertex v usig approximatio algorithm based o samplig. Figure 1 shows our basic raig algorithm TOPRANK(), where is the iput parameter specifyig the umber of top raig vertices

5 the algorithm should extract. The algorithm also has a cofiguratio parameter l, which is the umber of samples used by the RAND algorithm i the first step. We will specify the value of l i Lemma. Fuctio f(l) i step 4 is defied as follows: f(l) = α log l (where α > 1 is a costat umber), such that the probability of the estimatio error for ay vertex beig at least f(l) is bouded above by 1 3, based o iequality (.3) (whe settig ɛ = f(l)). For example, we ca set α to be ay value at least 4.5/3, which wors for ay 4. Algorithm TOPRANK() 1 Use the approximatio algorithm RAND with a set S of l sampled vertices to obtai the estimated average distace â v for each vertex v. // Reame all vertices to ˆv 1, ˆv,..., ˆv such that âˆv1 âˆv âˆv. Fid ˆv. 3 Let ˆ = mi u S max v V d(u, v). // d(u, v) for all u S, v V have bee calculated at step 1 ad ˆ is determied i O(l) time. 4 Compute cadidate set E as the set of vertices whose estimated average distaces are less tha or equal to âˆv + f(l) ˆ. 5 Calculate exact average shortest-path distaces of all vertices i E. 6 Sort the exact average distaces ad fid the top- vertices as the output. Fig. 1. Algorithm for raig top- vertices with the highest closeess cetralities. Lemma 1. Algorithm TOPRANK() give i Figure 1 ras the top- vertices with the highest closeess cetralities correctly w.h.p., with ay cofiguratio parameter l. Proof. We show that the set E computed at step 4 i algorithm TOPRANK() cotais all top- vertices with the exact shortest distaces w.h.p. Let T = {v 1,..., v } ad ˆT = {ˆv 1,..., ˆv }. Sice for ay vertex v, the probability of the estimate â v exceedig the error rage of f(l) is bouded above by 1, i.e., Pr ( {a 3 v f(l) â v a v + f(l) }) 1, we have 3 ( Pr { ) â v a v + f(l) a v + f(l) } v T 3. Next, we wat to achieve a similar iequality as above for vertices ˆv ˆT. However, ˆT ad its odes are radom variables, ad thus to apply Hoeffdig iequality as give i (.3) we eed aother level of uio boud, as give below. For

6 every ˆv i ˆT, we have Pr ( {aˆvi âˆv + f(l) }) Pr ( {aˆvi âˆvi + f(l) }) ( ) = Pr {aˆvi âˆvi + f(l) } {ˆv i = v} v V v V Pr ( {a v â v + f(l) } {ˆv i = v}) v V Pr ( {a v â v + f(l) }) v V 1 3 = 1. The we tae aother uio boud for all ˆv i ˆT, ad obtai Pr { ˆv ˆT aˆv âˆv + f(l) }. The above iequality meas that, with error probability of at most, there are at least vertices whose real average distaces are less tha or equal to âˆv + f(l), which meas a v âˆv + f(l) with error probability bouded above by. The â v a v + f(l) âˆv + f(l) for all v T with error probability bouded above by. Moreover, we have ˆ, because for ay u S, we have ad thus = max v,v V d(v, v ) max v,v V (d(u, v) + d(u, v )) = max d(u, v) + max d(u, v,v V v,v V v ) = max d(u, v). v V mi max u S v V d(u, v) = ˆ. Therefore, for all v T, â v âˆv + f(l) ˆ with probability at least (1 1 ) (because ). Hece, TOPRANK() icludes all top- vertices with exact average distaces i E i step 4, ad TOPRANK() fids these exact vertices i steps 5 ad 6, with high probability. This fiishes the proof of the lemma. We ow evaluate the complexity of algorithm TOPRANK(). The major computatio tass are l computatios of SSSP i step 1 ad E computatios of SSSP i step 5. We eed to choose a appropriate l to miimize the sum of these computatios. The umber of computatios of SSSP i step 5 depeds o the distributio of estimated average distaces of all vertices. The followig lemma provides a aswer whe this distributio is uiform.

7 Lemma. If the distributio of estimated average distaces is uiform with rage c (c is a costat umber), the TOPRANK() taes O(( + 3 log 1 3 )( log + m)) time, whe we choose l = Θ( 3 log 1 3 ). Proof. TOPRANK() taes O(l( log + m)) time at step 1 because SSSP algorithm taes O( log +m) time ad TOPRANK() iterates SSSP algorithm l times. Sice the distributio of estimated average distaces is uiform with rage f(l) ˆ f(l) ˆ c c, there are c vertices betwee âˆv ad âˆv + f(l) ˆ, ad is O(f(l)) because ˆ = mi u S max v V d(u, v) max u,v V d(u, v) =. So, the umber of vertices i E is +O(f(l)) ad TOPRANK() taes O(( + O(f(l)))( log + m)) time at step 5. Therefore, we select a l that could miimize the total ruig time at step 1 ad 5. I other words, we choose a l to miimize l + f(l), which implies l = Θ( 3 log 1 3 ). The TOPRANK() taes O( 3 log 1 3 ( log + m)) time at step 1, ad taes O(( + 3 log 1 3 )( log + m)) at step 5. Obviously TOPRANK() taes O( 3 log 1 3 ( log + m)) time at the other steps. So, TOPRANK() taes O(( + 3 log 1 3 )( log + m)) total ruig time. Combiig Lemmata 1 ad, we arrive at the followig theorem. Theorem 1. If the distributio of estimated average distaces is uiform with rage c (c is a costat umber), the algorithm TOPRANK() give i Figure 1 ras the top- vertices with the highest closeess cetralities i O(( + 3 log 1 3 )( log + m)) time w.h.p., whe we choose l = Θ( 3 log 1 3 ). Theorem 1 oly addresses the case whe the distributio of estimated average distaces is uiform. I this case, the complexity of TOPRANK() is better tha a brute-force algorithm that simply computes all average shortest distaces ad ras them, which taes O(( log + m)) time (assumig = o()). Eve though the theorem is oly for the case of uiform distributio, it could be applied to more geeral situatios, as explaied ow. Give a estimated average distace x, its desity d(x) is the umber of vertices whose estimate average distace is aroud x. The uiform distributio meas that the desity d(x) is the same aywhere i the rage of x. For ay other distributio, it has a average desity of d, which is the average of d(x) over all x s. Suppose that the distributio is such that whe x is sufficietly small, d(x) d (this property requires further ivestigatio but we believe it is reasoable for social etwors). Let x 0 be the largest value such that for all x x 0, d(x) d. The, i our algorithm, as log as âˆv + f(l) ˆ x0, the umber of vertices betwee âˆv ad âˆv +f(l) ˆ f(l) ˆ is at most c, as give i the proof of Lemma. Thus, Lemma uses a coservative upper boud for this umber, ad it will still hold for the distributios with the above property. Eve with the above geeralizatio, however, the savigs from O(( log + m)) to O(( + 3 log 1 3 )( log + m)) is still ot very sigificat. Ideally, we

8 would lie a raig algorithm that is O(poly()( log +m)), where poly() is a polyomial of, which meas the umber of SSSP calculatios is oly related to, ot. This is possible for small whe the distributio of estimated average distaces is ot uiform but other distributios lie the ormal distributio. I this case, the umber of additioal SSSP computatios for vertices i the rage from âˆv to âˆv + f(l) ˆ could be small ad ot related to. We leave this as a future research wor (see more discussio i Sectio 4). 3. Improvig the algorithm with a heuristic Algorithm TOPRANK() 1 Use the approximatio algorithm RAND with a set S of l sampled vertices to obtai the estimated average distace â v for each vertex v. // Reame all vertices to ˆv 1, ˆv,..., ˆv such that âˆv1 âˆv âˆv. Fid ˆv. 3 Let ˆ = mi u S max v V d(u, v). 4 Compute cadidate set E as the set of vertices whose estimated average distaces are less tha or equal to âˆv + f(l) ˆ. 5 repeat 6 p E 7 Select additioal q vertices S + as ew samples uiformly at radom. 8 Update estimated average distaces of all vertices usig ew samples i S + (eed to compute SSSP for all ew sample vertices). 9 S S S + ; l l + q; ˆ mi( ˆ, mi u S + max v V d(u, v)) // Reame all vertices to ˆv 1, ˆv,..., ˆv such that âˆv1 âˆv âˆv. 10 Fid ˆv. 11 Compute cadidate set E as the set of vertices whose estimated average distaces are less tha or equal to âˆv + f(l) ˆ. 1 p E 13 util p p q 14 Calculate exact average shortest-path distaces of all vertices i E. 15 Sort the exact average distaces ad fid the top- vertices as the output. Fig.. Improved algorithm for raig vertices with top closeess cetralities The algorithm i Figure 1 speds its majority of computatio o the followig two steps: (1) step 1 computig SSSP for l samples, ad () step 5 computig SSSP for all cadidates i set E. The ey i reducig the ruig time of the algorithm is to fid the right sample size l to miimize l + E, the total umber of SSSP calculatios. However, this umber is difficult to obtai before ruig the algorithm, especially whe the distributio of average distaces is uow. I this sectio, we improve the algorithm by a heuristic to achieve the above goal. The idea of the heuristic is to icremetally add ew samples to compute more accurate average distaces of all vertices. I each iteratio, q ew sample vertices are added. After computig the ew average distaces with these q ew vertices, we obtai a ew cadidate set E. If the size of the cadidate set E decreases more tha q, the we ow that the savigs by reducig the umber of cadidates outweigh the cost of addig more samples. I this case, we

9 cotiue the ext iteratio of addig more samples. This procedure eds whe the cost of addig more samples outweighs the savigs obtaied by the reduced umber of cadidates. Figure provides this heuristic algorithm. Essetially, this is the dyamic way of fidig the optimal l to miimize l + E (or to mae l = E, where l is the small chage i l ad E is the correspodig chage i E ). The iitial value of the l i step 1 ca be obtaied based o Theorem 1 if we ow that the distributio of the estimated average distaces is uiform. Otherwise, we ca choose a basic value, for example, sice we eed to compute at least SSSP i step 14 i ay case. The icremetal uit q could be a small value, for example, log. However, we do ot ow yet if E strictly decreases whe the sample size l for estimatig average distaces icreases, ad if the rate of decrease of E slows dow whe addig more ad more sample vertices. Therefore, it is ot guarateed that the heuristic algorithm will always stop at the optimal sample size l. A ope problem is to study the coditios uder which the chage of E with respect to the chage of sample size l ideed has the above properties, ad thus the heuristic algorithm ideed provides the most efficiet solutio. 4 Coclusio ad discussios This paper ca be viewed as the first step towards the desig of more efficiet algorithms i obtaiig highest raed closeess cetrality vertices. By combiig the approximatio algorithm with the exact algorithm, we obtai a algorithm that has better complexity tha the brute-force exact algorithm. There are may directios to exted this study. First, as metioed i the previous sectio, we are iterested i more efficiet algorithms such that the umber of SSSP computatios is oly related to, ot to. This may be possible for some classes of social etwors with certai properties o their average distace distributios. Secod, the coditio uder which the heuristic algorithm results i the least umber of SSSP computatio is a ope problem ad would be quite iterestig to study. Third, we may be able to obtai faster algorithm if we ca relax the problem requiremet. Istead of fidig all top- vertices with high probability, we may allow the output to have a error boud t, which is the umber of vertices that should be raed withi top- vertices but are missed i the output. Geerally, give a algorithm for top- query of various cetralities, we defie the hit-ratio, deoted as η(a), of a output A = {v 1, v,, v κ } (ot ecessarily of size ) as A Q, where Q is the actual set of top- vertices. Let r(v i ) deote the the actual ra of v i. Here, we require that r(v i ) < r(v i+1 ), for i [1, κ 1]. Thus we have r(v i ) i. We defie the κ(κ+1) accuracy of A, deoted as α(a), as κ i=1 r(vi) (other defiitios of accuracy are possible). Clearly, η(a) [0, 1] ad α(a) [0, 1]. The hit-ratio ad accuracy of a algorithm are the its worst performace over all possible

10 iputs. We the may oly require that Pr (η(a) 1 ϱ, α(a) 1 ɛ) 1 δ, for sufficietly small ϱ, ɛ ad δ. Such relaxatios i the requiremet may allow much more efficiet algorithms, sice we observe that the complexity of our algorithm is maily because we eed to iclude all vertices i the extra rage from âˆv to âˆv + f(l) ˆ i order to iclude all top- vertices with high probability. Fourth, we would lie to study the stability of our algorithms for top- query usig radom samplig. Costebader et al. [CV03] studied the stability of various cetrality measures. Their study shows that, with a 50% sample of the origial etwor odes, average correlatios for the closeess measure raged from 0.54 to Fially, we ca loo ito other type of cetralities ad see how to ra them efficietly usig the techique i this paper. For example, Brades [Br00] presets a algorithm for betweeess cetrality of weighted graph with time-complexity (m + log ). Brades et al. [BP00] preset the first approximatio algorithm for betweeess cetrality. Improvemets over their method were preseted recetly i [GS08, BK07]. To coclude, we would lie to provide a brief compariso of our algorithms with the well ow PageRa algorithm [BP98]. PageRa is a algorithm used to ra the importace of the webpages, which are viewed as vertices coected by directed edges (hyperlis). As explaied i Footote 5, PageRa is a variat of the eigevector cetrality. Thus, it is a differet measure from closeess cetrality. More importatly, by defiitio the PageRa of a vertex (a webpage) depeds o the PageRas of other vertices liig to it, so the PageRa calculatio requires computig all PageRa values of all vertices, eve if oly the top- PageRa vertices are desired. However, for closeess cetrality measure, our algorithms do ot eed to compute closeess cetralities for all vertices. Istead, we may start with rough estimates of closeess cetralities of vertices, ad through refiig the estimates we reduce the cadidate set cotaiig the top- vertices. This results i reduced time complexity i our computatio. Refereces [BK07] Bader, D. A., Kitali, S., Madduri, K., Mihail, M.: Approximatig Betweeess Cetrality. The 5th Worshop o Algorithms ad Models for the Web-Graph (007) [Ba50] Bavelas, A.: Commuicatio patters i tas-orieted groups. The Joural of the Acoustical Society of America (6) (1950) [Be65] Beauchamp, M. A.: A improved idex of cetrality. Behavioral Sciece 10() (1965) [Br00] Brades, U.: Faster Evaluatio of Shortest-Path Based Cetrality Idices. Kostazer Schrifte i Mathemati ud Iformati 10 (000) [BP00] Brades, U., Pich, C.: Cetrality Estimatio i Large Networs. Itl. Joural of Bifurcatio ad Chaos i Applied Scieces ad Egieerig 17(7) [BP98] Bri, S., Page, L.: The aatomy of a large-scale hypertextual Web search egie. Computer Networs ad ISDN Systems 30 (1998)

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