WWW 2012 – Session: Community Detection in Social Networks April 16–20, 2012, Lyon, France
QUBE: a Quick algorithm for Updating BEtweenness centrality
Min-Joong Lee Jungmin Lee Jaimie Y. Park Department of Computer Department of Computer Division of Web Science and Science, KAIST Science, KAIST Technology, KAIST 291 Daehak-ro, Yuseong-gu 291 Daehak-ro, Yuseong-gu 291 Daehak-ro, Yuseong-gu Daejeon, Korea Daejeon, Korea Daejeon, Korea [email protected] [email protected] [email protected] Ryan H. Choi Chin-Wan Chung Department of Computer Division of Web Science and Science, KAIST Technology & Department of 291 Daehak-ro, Yuseong-gu Computer Science, KAIST Daejeon, Korea 291 Daehak-ro, Yuseong-gu [email protected] Daejeon, Korea [email protected]
ABSTRACT General Terms The betweenness centrality of a vertex in a graph is a mea- Algorithms, Theory sure for the participation of the vertex in the shortest paths in the graph. The Betweenness centrality is widely used in network analyses. Especially in a social network, the re- Keywords cursive computation of the betweenness centralities of ver- Betweenness centrality, Update algorithm tices is performed for the community detection and finding the influential user in the network. Since a social network 1. INTRODUCTION graph is frequently updated, it is necessary to update the be- tweenness centrality efficiently. When a graph is changed, The betweenness centrality is a measure that computes the betweenness centralities of all the vertices should be re- the relative importance of a vertex in a graph, and it is computed from scratch using all the vertices in the graph. widely used in network analyses such as a social network To the best of our knowledge, this is the first work that analysis, biological graph analysis, and road network anal- proposes an efficient algorithm which handles the update ysis. For example, in the social network analysis, a vertex of the betweenness centralities of vertices in a graph. In with higher centrality can be viewed as a more important this paper, we propose a method that efficiently reduces the vertex than a vertex with lower centrality. The betweenness search space by finding a candidate set of vertices whose be- centrality of a vertex in a graph is a measure for the partic- tweenness centralities can be updated and computes their ipation of the vertex in the shortest paths in the graph. betweenness centeralities using candidate vertices only. As There are many previous works on the betweenness cen- the cost of calculating the betweenness centrality mainly de- trality problem. The concept of the betweenness central- pends on the number of vertices to be considered, the pro- ity is proposed in [1], but the definition proposed in [10] is posed algorithm significantly reduces the cost of calculation. more widely used. Recently, many variants of the definition The proposed algorithm allows the transformation of an ex- are proposed in [6]. [5] improves the computation time of isting algorithm which does not consider the graph update. the betweenness centrality based on a modified breadth-first Experimental results on large real datasets show that the search algorithm and the dependency of a vertex, and it is proposed algorithm speeds up the existing algorithm 2 to the fastest known algorithm that computes the exact be- 2418 times depending on the dataset. tweenness centralities of all the vertices in a graph. As the computation of shortest paths between all pairs of vertices are time consuming, [22] proposes another definition of be- tweenness centrality, which is based on a random walk. In [22], each vertex has a probability of visiting its neighbor Categories and Subject Descriptors vertices. Also, [7], [2] and [12] propose approximation algo- G.2.2 [Discrete Mathematics]: Graph Theory—Graph rithms for computing the betweenness centrality. [23] and algorithms, Path and circuit problems;E.1[Data]: Data [25] adopt the betweenness centrality for detecting commu- structures—Graphs and networks nities in a social network. Although many works on calculating the betweenness cen- trality exist and the betweenness centrality is one of the Copyright is held by the International World Wide Web Conference Com- major measures used in analyzing social network graphs, mittee (IW3C2). Distribution of these papers is limited to classroom use, and personal use by others. none of the works for computing the betweenness central- WWW 2012, April 16– 20, 2012, Lyon, France. ity address the problem of updating betweenness centrality. ACM 978-1-4503-1229-5/12/04.
351 WWW 2012 – Session: Community Detection in Social Networks April 16–20, 2012, Lyon, France
Applying the previous algorithms to find influential users or recomputation is necessary to compute the betweenness cen- detect communities over frequently updated graphs such as trality and the number of shortest paths that need to be a social network graph is inefficient. This is because, calcu- recomputed on G1 would be 12 · 11/2 = 66. On the other lating the betweenness centralities of all users in the graph hand, in our approach, only 5 · 4/2 = 10 shortest paths need involves computing the shortest paths between all pairs of to be recomputed. Clearly, the smaller the cardinality of users in the graph. In all previous works, the recomputa- the first set, the shorter the amount of time it would take in tion for all the vertices is inevitable whenever a new edge is computing the necessary shortest paths. inserted to the graph. This recomputation is clearly time- In order to recompute the betweenness centrality of a ver- consuming. As the number of edges in the social network tex in the reduced set, in addition to the betweenness cen- graph increases over time [19], the need for updating the trality in the reduced set, the number of shortest paths that betweenness centrality is evident. satisfy the following conditions needs to be considered. It is difficult to update the betweenness centrality, because even a single edge insertion or a single edge deletion leads 1. The shortest path goes through the vertex in the re- to the changes in many shortest paths in the graph. This duced set. change causes the updates of the betweenness centralities of 2. The shortest path’s source or target or both are not in many vertices in the graph. It is trivial to see that when the reduced set. an edge (vi, vj ) is inserted to a graph, the shortest path between vi and vj is changed. Also, the shortest paths that The number can be obtained based on the cardinality of include the original shortest path from vi to vj are changed. the set(s) of vertices whose betweenness centralities do not For example, in Figure 1, let G1 be a graph and G1 be an change without actually computing the shortest paths. For updated graph of G1. When an edge (v1, v5) is inserted, the example, the shortest paths from v12 to v6 always go through shortest path between v1 and v5 is changed. Also, there are vertices in the reduced set. Similarly, the shortest paths more shortest paths that are changed e.g., the shortest path from vi ∈{v12} to vj ∈{v6, v7, v8, v9, v10} always go between v12 and v5 and the shortest path between v10 and through vertices in the reduced set. Therefore, the number v11. of shortest paths from vi to vj is a product of the cardinal- However, we observe that there exist vertices whose be- ities of the two sets, which is 5. tweenness centralities do not change even when the graph The contributions of this paper are as follows. is updated. In Figure 1(b), the betweenness centralities of v1, v3, v4 and v5 change, while the betweenness centrali- 1. We propose a method that identifies a set of vertices ties of the other vertices do not change. The betweenness whose betweenness centralities can be updated and centralities of v2, v6, v7, v8, v9, v10, v11 and v12 do not sets of vertices whose betweenness centralities do not change, because the source-target pairs of original shortest change, based on the comprehensive analysis of changes paths that go through v2, v6, v7, v8, v9, v10, v11 or v12 do in the betweenness centrality when a graph is updated. not change even when G1 is updated. 2. We devise a Betweenness Centrality Update T heorem.
v9 v9 The theorem enables an efficient update of between- ness centrality without traversing the entire graph.
v v v v Based on the proposed theorem, we propose an effi- 10 8 10 8 cient algorithm for updating betweenness centrality. v v v v 1 7 1 7 3. We conduct experiments on various synthetic datasets
v v v v v v v v v v as well as large real datasets. The experimental results 2 3 4 5 6 2 3 4 5 6 show that the incorporation of our algorithm outper- forms an existing algorithm, in updating the between- v11 v12 v11 v12 ness centrality. In cases where the size of the reduced (a) G1 (b) G1 set of vertices is 1/10 of the number of vertices in the Figure 1: An example of a graph update synthetic graphs, the proposed algorithm speeds up the existing algorithm 577 times on the average. For Based on the above observation, we proposed a Quick al- real datasets, the proposed algorithm speeds up the ex- gorithm for Updating BEtweenness centrality (QUBE). The ist algorithm 2 to 2418 times depending on the dataset. key idea of the proposed algorithm is to perform the be- The rest of the paper is organized as follows. In Section 2, tweenness centrality computation on a reduced set of ver- related works on betweenness centrality are reviewed. In tices. We first find the set of vertices whose betweenness Section 3, we formally define betweenness centrality and centralities can be changed and the set(s) of vertices whose explain basic concepts. In Section 4, we devise a method betweenness centralities do not change. In Figure 1(b), which finds the reduced set of vertices whose betweenness {v1,v2,v3,v4,v5} is the set of vertices whose betweenness cen- centralities can be updated. Section 5 explains how to effi- tralities can be changed, and {v6,v7,v8,v9,v10}, {v11}, {v12} ciently update the betweenness centralities of vertices in the are the sets of vertices whose betweenness centralities do reduced set. In Section 6, we show experimental results, and not change. The method of finding these sets is explained in we conclude the paper in Section 7. Section 4, and it is easy to see that the latter three sets cor- respond to connected components after removing the first set from G1. 2. RELATED WORK We compute the betweenness centrality only on the first Computation of betweenness centrality has been gaining set of vertices. In the previous works, all pair shortest paths much importance in social network analyses, and is widely
352 WWW 2012 – Session: Community Detection in Social Networks April 16–20, 2012, Lyon, France used in many applications. The earliest work to define the works, intersections or junctions in transportation networks, measure which quantifies this idea of betweenness central- physical elements in computer networks, or documents in ity is introduced by Anthonisse et al. [1] and Freeman [10]. World Wide Web. For example, Leydesdorff [20] demon- Freeman’s original method of finding betweenness centrality strates in his research how betweenness centrality is shown is based on counting geodesic paths for all pairs of vertices to be an indicator of the interdisciplinarity of scientific jour- on a graph. Following Freeman’s work, variations of cen- nals, and del Sol et al. [8] use the betweenness centrality in trality measures are proposed. Everette et al. [17] propose identifying the most central residues in protein-protein com- a group betweenness measure which can be applied to groups plex structures. Jin et al. [15] demonstrate an application of and classes as well as individuals. Freeman et al. [11] extend parallel betweenness centrality to detect potentially harmful Freeman’s work [10] to introduce a new measure of centrality nodes in an electrical grid. The electrical grid is an inter- based on the concept of network flows, which considers both connected network for delivering electricity from suppliers to shortest and certain non-shortest paths. Newman [22] pro- consumers. Holme [13] studies the relationship between be- poses a measure of betweenness centrality based on random tweenness centrality and the density of a traffic model, and walks of any length instead of shortest paths. Brandes [6] Lammer et al. [18] use betweenness centrality in approximat- reviews a number of variants of betweenness centrality based ing the importance of a road or a junction and investigated on shortest paths including bounded-distance betweenness, the scaling laws associated with urban road networks in Ger- distance-scaled betweenness, edge betweenness, and group many. In many applications, the network structures are typ- betweenness, and discusses algorithms to compute each vari- ically not static. As the network evolves, the network graphs ant efficiently. As part of the discussion, Brandes points out constantly change over time, which implies that there is a that the efficient recomputation of betweenness centrality in strong need for an efficient algorithm to update betweenness dynamically changing networks on the algorithmic side is a centrality. remaining challenge. Betweenness centrality is also used in community detec- Currently, the fastest known algorithm to compute ex- tion. Newman et al. [23] propose a divisive community de- act betweenness centralities for all the vertices [5] requires tection technique which iteratively removes edges with the O(|V ||E|)andO(|V ||E| + |V |2log|V |)timeonweightedand highest betweenness centrality value from the network. Pin- unweighted graphs, respectively. Traditionally, betweenness ney et al. [25] suggest an alternative community detection centrality was determined by first computing the lengths and algorithm in which the network decomposition is based on number of shortest paths between all pairs, and then sum- vertex betweenness instead of edge betweenness. Newman et ming up pair-dependencies of all pairs [10]. Pair-dependency al. [23] discuss a weakness in the existing algorithms which of a pair s, t ∈ V on an intermediary vertex v ∈ V is defined is a high computation cost associated with iterative recalcu- as the ratio of shortest paths between s and t that v lies on lation of all-pair shortest paths when the edges are removed. to all shortest paths between s and t. Brandes [5] points out As observed in many applications, dynamic nature of many the weakness in this approach arguing it is computing more real-life networks is a clear evidence that efficiently updat- information than needed. The faster algorithm is presented ing betweenness centrality is an important issue. Yet no by Brandes [5], based on aggregating path counts from dif- literature dealing with the problem of efficiently updating ferent source vertices in the network. betweenness centrality in a dynamic network environment Although big improvement was made over the very ini- exists at present. tial betweenness centrality computation algorithm, many researchers argued that the Brandes’ algorithm is still too costly for large graphs. In order to overcome such limitation, 3. PRELIMINARY researchers propose approximation algorithms to compute In this section, we introduce the formal definition of be- the estimated betweenness centrality, claiming that good tweenness centrality of a vertex, and explain the basic con- approximation would be an acceptable alternative to exact cept of a minimum cycle basis. Also we briefly introduce the score as long as fast computation is possible. Brandes et overall process of the proposed algorithm. al. [7] propose a heuristic estimation method for between- ness centrality computation and conduct experiments with 3.1 Betweenness Centrality various selection strategies of the source vertices to assess Betweenness centrality is a measure that computes the the quality of the estimation. Bader et al. [3] present a relative importance of a vertex in a graph. The formal defi- parallel algorithm for computing betweenness centrality, op- nition is presented below. timized for scale-free sparse graphs. They [2] also suggest A graph is represented by G =(V,E), where V is the set an algorithm to compute the betweenness centrality of a sin- of vertices, and E ⊆ V × V is the set of edges. A path in gle vertex in time faster than computing the betweenness of a graph is represented by a sequence of vertices, (v1, ..., vn) all vertices. Geisberger et al. [12] suggest a bisection scal- where vi,vj ∈ V for 1 ≤ i, j ≤ n, i = j except possible ing algorithm for approximating a variant of betweenness 1=n. centrality. Makarychev [21] suggests a linear time approx- imation algorithm to find the ordering of the vertices that Definition 1 (Betweenness Centrality). The betweenness ∈ maximizes the number of satisfied betweenness constraints. centrality of a vertex vj G is: Betweenness centrality is used in diverse applications across σvi,vk (vj ) many different disciplines. Betweenness centrality allows an c(vj )= (1) σv ,v understanding of the extent to which a vertex contributes i,k i k in the flow of information. It is mainly used in finding the where vi,vj ,vk ∈ V , i = j = k, σv ,v (vj )isthenumberof most prominent vertices in complex networks, whether they i k shortest paths between vi and vk that include vj ,andσv ,v are individuals in social networks, elements in biological net- i k is the number of shortest paths between vi and vk.
353 WWW 2012 – Session: Community Detection in Social Networks April 16–20, 2012, Lyon, France
The betweenness centrality can be computed as follows: do and do not occur. Observed pattern is applicable for any type of connected graphs. Theoretical evidence on the 1. For each pair of vertices (vs and vt), compute the generalization is presented in Section 5.1. shortest paths between the two vertices. Next, we perform the betweenness centrality computation 2. For each pair of vertices, compute the ratio of the par- on the identified sets of vertices whose betweenness central- ticipation of each vertex in the shortest path(s). The ities can be changed. We refer to the computed values as ratio is the number of shortest paths between vs and vt the local betweenness centrality. On top of the local be- that go through vj divided by the number of shortest tweenness centrality, we perform additional calculations on paths between vs and vt. the vertices whose shortest paths are not yet considered. Details are presented in Section 5. Through simple addi- 3. Accumulate the ratio for all pairs of vertices. tional calculations, the exact betweenness centrality can be restored without performing an expensive computation on Let us consider updating the betweenness centrality caused all the vertices on a graph, such as the calculation of all pair by a graph update. Even a simple update, for example in- shortest paths. serting an edge to a graph, could change existing shortest paths for many pairs of vertices in the graph. One of the biggest drawbacks in updating the betweenness centrality 4. MINIMUM UNION CYCLE using the previous algorithms is that the shortest paths for In this section we introduce the concept of the minimum all pairs of vertices are recomputed whenever an update oc- union cycle (MUC) upon which our update algorithm is curs in a graph. built. As explained in the previous section, a set of ver- tices whose betweenness centralities can be changed is dis- 3.2 Minimum Cycle Basis tinguished from the set(s) of vertices whose betweenness cen- tralities do not change. Such sets are identified by using Definition 2 (Cycle Basis). Let a graph G =(V,E)bean MUCs obtained during the preprocessing time. The initial undirected graph. A cycle C is a subset of edges such that set of MUCs is found and stored during the preprocessing every vertex of V is incident to an even number of edges in time. As changes occur in a graph, stored MUCs also need C.EachcycleC can be represented by an edge incidence |E| to be changed. Changes in MUCs are managed during the vector in {0, 1} where a component is equal to 1 precisely runtime. In Sections 4.2 and 4.3, we explain how to find when e ∈ C. A maximal set of linearly independent cycles MUCs and how to update MUCs. is called a cycle basis. 4.1 Definition of MUC Definition 3 (Minimum Cycle Basis (MCB)). Let a graph G =(V,E) be an undirected connected graph with a non- Definition 4 (Minimum Union Cycle (MUC)). Given a negative weight we assigned to each edge e ∈ E. Minimum minimum cycle basis C and minimum cycles Ci ∈ C,let
Cycle Basis is a cycle basis C of minimum total weight, VCi be the set of vertices in Ci. Recursively union two VCi s v i.e., which minimizes w(C)= i=1 w(Ci), where w(Ci)= together if they share at least one common vertex. Then e e∈Ci w . each final set of vertices forms a MUC. Each vertex appears in only one MUC since MUCs are disjoint sets. We denote MUC(v) as MUC which contains v v1 3 c3 vertex v. c 1 Definition 5 (Connection vertex). Vertex v ∈ MUC is a c2 v 5 connection vertex, if v is an articulation vertex1 and v has an edge to a vertex w/∈ MUC(v).
v2 v4 In Figure 5, let us assume that an edge (v3, v4) is inserted. MUC(v3)is{v1, v2, v3, v4}, and the connection vertices of Figure 2: An example of a cycle basis and the min- MUC(v3)arev1,v2 and v3. imum cycle basis The deletion of a connection vertex makes the graph dis- connected since the connection vertex is also an articula- The example depicted in Figure 2 has three cycle basis sets tion vertex. We denote a graph that is disconnected from {C1,C2}, {C1,C3},and{C2,C3}. If every edge in the graph MUC(vi) as a result of the deletion of a connection vertex has the same weight (i.e., 1 for all edges), MCB is {C1,C2}. vi as a disconnected subgraph Gi.InFigure5,G1, G2,and The detailed definitions of cycle basis and minimum cycle G3 are disconnected subgraphs generated from the deletion basis can be found in [16]. of connection vertex v1, v2,andv3,respectively. 3.3 Overall Process 4.2 Finding MUCs The overall flow of the proposed betweenness centrality In this subsection, we present how to generate MUCs, a update algorithm is as follows. First, we identify the set set of connection vertices for each MUC and disconnected of vertices whose betweenness centralities can be changed, subgraphs derived from the deletion of connection vertices. and the set(s) of vertices whose betweenness centralities do 1Avertexa is called an articulation vertex if the deletion not change. Through the analysis of possible changes in the of a with its incident edges from G makes the graph discon- betweenness centrality that can occur as a result of graph nected. Equivalently, there must exist two vertices v and w updates, we discovered the characteristics of the sets of ver- such that every path from v to w goesthroughanarticula- tices in which the changes in the betweenness centralities tion vertex a.
354 WWW 2012 – Session: Community Detection in Social Networks April 16–20, 2012, Lyon, France
Algorithm 1: FindMUC(C) Algorithm 2: UpdateMUC(vi,vj ,MUCSet,G) input : C - minimum cycle basis input : vi - a vertex in inserted/deleted edge 1 begin vj - a vertex in inserted/deleted edge 2 MUCSet := A minimum cycle basis C; MUCSet -asetofMUCs 3 while ∃ ci,cj ∈ MUCSet,whereci and cj share at G - an original graph least one common vertex do 1 begin 4 ci := ci union cj ; // if vi is not contained any MUC, MUC(vi) 5 Remove cj from MUCSet ; returns vi only 2 6 for each MUC ∈ MUCSet do if Insertion Operation then 3 7 Conn(MUC) := a set of connection vertices in MUC if MUC(vi) = MUC(vj ) then 4 ; // Do Nothing 8 for each connection vertex vi ∈ Conn(MUC) do 5 else 9 Gi := disconnected subgraphs originated from 6 NewMUC := Let be an empty set; the deletion of a connection vertex vi ; 7 for each vertices v in ςρ(vi,vj ) in G do 8 NewMUC := NewMUC ∪ MUC(v); 9 remove MUC(v)fromMUCSet ; 10 add NewMUC to MUCSet ; 11 Algorithm 1 uses a minimum cycle basis C as an input, add edge (vi,vj )tographG; and it finds a set of MUCs (MUCSet), and a set of connec- 12 else 13 tion vertices with corresponding subgraphs. delete edge (vi,vj ) from graph G; 14 if 1=|Path(vi,vj )| in G then The calculation of a minimum cycle basis is well studied in 15 remove MUC(vi)fromMUCSet ; the field of graph theory, and many efficient algorithms, such 16 else as Horton’s algorithms [14] and Kavitha’s algorithm [16], ex- 17 if ∃v in all Path(vi,vj ) then ist. In line 2, we calculate a minimum cycle basis using an 18 split MUC(v)intoMUCs; existing algorithm. In line 3-line 5, the algorithm finds a set 19 else of MUCs (MUCSet) by unioning the cycles in a minimum 20 // Do Nothing cycle basis until the unioned cycles are disjoint from each other. A set of connection vertices for each MUC and dis- connected subgraphs derived from the deletion of connection vertices are found in line 7-line 9. Note that Algorithm 1 is performed during the prepro- tion vertex. We explain each case of updating MUCs accord- cessing time. However, the MUC updating algorithm (Al- ing to the insertion or deletion of an edge as follow (Initial gorithm 2) needs to be processed during the runtime. MUC MUCs in Figure 3(a) are {v1,v2,v3,v4,v5} and {v8,v9,v10}. updating algorithm for an insertion and deletion of an edge Initial MUCs in Figure 3(b) are {v1,v2,v3,v4,v5 ,v6,v7} is presented in the following subsection. and {v8,v9,v10}.): 4.3 Updating MUCs 1. When an edge is inserted (a) No change, if the new edge connects two ver- b v tices in one MUC. In Figure 3(a), the insertion v v 6 v 2 12 9 of edge(a) does not affect any MUCs (Line 4 in a Algorithm 2). v v4 1 v7 v8 v10 (b) A new MUC is created, if vertices in an exist- ing shortest path between two vertices in the new c edge are not included in any MUC. In Figure 3(a), v3 v5 v11 the insertion of edge(b) induces a creation of a new MUC consisting of {v6,v7,v12} (Line 6-Line (a) Insertion 10 in Algorithm 2. ςρ(vi,vj )isthesetofvertices in shortest paths between vi and vj ). v (c) MUC is merged with the vertices and other MUCs v2 b 6 v9 c to create a new MUC, if vertices in existing short- est paths between two vertices of the new edge v v4 a 1 v7 v8 v10 are included in some MUCs. In Figure 3(a), the insertion of edge(c) induces MUC(v10)tomerge with v11 (Line 6-Line 10 in Algorithm 2). v3 v5 v11 2. When an edge is deleted (b) Deletion (a) MUC is destroyed, if there exists only one path Figure 3: An example of updating MUC between two vertices in the deleted edge as a re- sult of the deletion. In Figure 3(b), the deletion We now present our technique on maintaining a set of of edge(a) causes the destruction of MUC(v10) MUCs, a set of connection vertices for each MUC and dis- (Line 14-Line 15 in Algorithm 2. Path(vi,vj )is connected subgraphs derived from the deletion of a connec- the set of paths between vi and vj .).
355 WWW 2012 – Session: Community Detection in Social Networks April 16–20, 2012, Lyon, France
(b) No change, after the deletion, if there still exists lated using the entire vertices in the graph as the global more than one path between the two vertices and betweenness centrality. does not exist a vertex appearing in all the paths between the two vertices. In Figure 3(b), the dele- 5.1 Betweenness Centrality Update Theorem tion of edge(b) does not affect any MUCs(Line Before we introduce our technique, we define some ter- 20 in Algorithm 2). minologies for a better understanding. cMUC (vi) denotes (c) An existing MUC is split into MUC(s) and ver- the local betweenness centrality of a vertex vi calculated us- tex(s), after the deletion, if there still exists more ing the vertices in MUCU only. ci represents a connection than one path between two vertices and exists vertex. ςρ(vi,vj ) is the set of vertices in the shortest paths a vertex appearing in all paths between the two between vi and vj ,andSP(vi,vj )isthesetofshortestpaths | | vertices. In Figure 3(b), the deletion of edge(c) between vi and vj . Therefore, SP(vi,vj ) is the number of induces the separation of MUC into two MUCs the shortest paths between vi and vj . For example, in Fig- | | | | (Line 18 in Algorithm 2). ure 4, SP(v1,v5) =1, SP(v1,v2) =2,andςρ(v1,v2)is {v1,v2,v3,v4}. Gj represents a disconnected subgraph originated from a l 5. UPDATING BETWEENNESS deletion of the connection vertex, cj . Gj represents the lth CENTRALITY connected component of Gj . VGj is the set of vertices of In this section, we describe how to compute the between- Gj .InFigure5,G1, G2 and G3 represent disconnected sub- graphs originated from the deletions of connection vertices, ness centrality values. As mentioned in Section 4, after an 1 2 v1, v2,andv3,respectively. G2 and G2 are connected com- insertion or deletion of the edge e(vi,vj ), we guarantee that ponents of G2. If the dotted edge is inserted, MUCU is the betweenness centralities of vertices in MUC(vi)canbe { } changed. Therefore, after we find the reduced set of vertices, v1,v2,v3,v4 and connection vertices of MUCU to G1, G2, which we refer to as MUC, we need to efficiently calculate and G3 are v1, v2,andv3, respectively. and update the betweenness centralities of the vertices in the MUC to which the updated vertices belong. From now G1 on, we simply denote such MUC as MUCU . MUC |VG |=5 U G 1 v1 G’ GG’ v1 c(v1)1 0.5 G3 v3 v4 c(v2)1 0.5 v3 v4 v5 c(v3)0.50.5 |V |=6 v2 G3
v2 c(v4)3.50.5
c(v5)0 G2 |V |=4 v Figure 4: An example of the dependency of the be- G2 6 tweenness centrality
Recomputing betweenness centralities of the vertices in a v8 v5 v7 graph every time an update occurs is expensive, because in general, the recomputation involves computation of all pair 1 1 2 G2 |VG |=3 G |V 2|=1 shortest paths in the graph. In the previous section, we 2 2 G2 proposed a way to find a set of vertices whose betweenness Figure 5: An example of updating the betweenness centralities can be changed. This set of vertices is referred centrality (vertices in G1 and G3 are omitted.) to as MUCU . Yet, calculating the betweenness centrality Lemma 1. Let vs ∈ VGj , vt ∈ MUCU and cj be a con- using only the vertices in MUCU is insufficient. In fact, the nection vertex which connects MUCU with Gj .Theneach betweenness centralities calculated using only the vertices vertex in ςρ(cj ,vt) must be included in a ςρ(vs,vt). in MUCU are always smaller than the betweenness central- Proof: Since a connection vertex in MUCU is also an ar- ities calculated using all the vertices in a graph. This is ticulation vertex, all paths from vs ∈ VGj to vt ∈ MUCU go because, (1) the shortest paths whose source or target is through a connection vertex cj . Therefore ςρ(vs,vt)always not in MUCU , and (2) the shortest paths that pass though includes ςρ(cj ,vt).2 MUCU and both the source and the target of the shortest Lemma 1 allows us to calculate the increase of the be- paths are not in MUCU , are not yet considered. For ex- ample, Figure 4 shows the betweenness centralities of the tweenness centrality due to the shortest paths whose source vertices in G, and a subgraph of G, G . The betweenness or target is not in MUCU (the shortest paths between the centralities of vertices calculated using only the vertices in vertices in MUCU and the vertices not in MUCU ). Such G are smaller than the values calculated using the vertices increase of the betweenness centrality for vi is denoted as cb (vi). in G. This is because, the paths from v5 to each vertex in j ⎧ |V | G are missing. ⎪ Gj ⎨ if vi in ςρ(cj ,vt) −{vt} Based on this idea, we now explain how to restore the |SP(vs,vt)| cbj (vi)= (2) exact betweenness centrality by considering the vertices in ⎩⎪ 0 otherwise MUCU only. Let us refer to the betweenness centrality cal- culated using only the vertices in MUCU as the local be- where vs in VGj , vt ∈ MUCU , cj is a connection vertex to tweenness centrality and the betweenness centrality calcu- Gj .
356 WWW 2012 – Session: Community Detection in Social Networks April 16–20, 2012, Lyon, France
∈ ∈ Lemma 2. Let vs VGj , vt VGk ,andcj and ck be con- Algorithm 3: QUBE(MUCU ) nection vertices which connect MUCU with Gj ,andMUCU input : MUCU - Minimum Union Cycle that updated with Gk, respectively. Then each vertex in ςρ(cj ,ck)must vertices belong to be included in a ςρ(vs,vt). output : C[vi] - Updated Betweenness Centrality Array Proof: Since cj and ck are articulation vertices, all paths 1 begin 2 U from vs ∈ VGj to vt ∈ VGk go through connection vertices Let SP be the set of all pair shortest paths in MUC ; 3 ∈ cj and ck. Therefore ςρ(vs,vt) always includes ςρ(cj ,ck).2 Let C[vi] be an empty array, vi MUCU ; 4 SP, C[vi] ← Betweenness() ; 5 for each shortest path
G2,whichis‘|VG1 |·|VG2 |/2=5· 4/2’ for each path. The 5.2 Betweenness Centrality Update Algorithm path v2-v3 connects G2 and G3 via v2 and v3. Similar to the | |·| | · (QUBE) above case, we add ‘ VG2 VG3 =4 6’ to v2’s betweenness centrality. G2 is a disconnected graph and v2 is a connection Algorithm 3 shows how to update betweenness centrality vertex to G2. Therefore, we add the product of the numbers only using vertices in MUCU that updated vertices belong 1 2 of vertices in G2 and G2,whichis3· 1. Finally, we get 59 to. Algorithm 3 uses MUCU as an input and calculates the updated betweenness centrality (C[vi]) as an output. The 2Algorithm 3 can use any existing betweenness centrality set of all pair shortest paths in MUCU and the local be- algorithm. For the implementation of Algorithm 3, we use tweenness centralities of vertices in MUCU are calculated the Brandes’ algorithm which is the fastest known algorithm using the existing betweenness centrality algorithms (Line so far. Our implementation is explained in Section 6.1
357 WWW 2012 – Session: Community Detection in Social Networks April 16–20, 2012, Lyon, France
As we mentioned in Section 5, in addition to the local be- Table 1: An example of updating the betweenness tweenness centrality, we need to calculate (1) the increase centrality of the betweenness centrality (Equation 2) due to the short- est paths whose source or target is a connection vertex, and SP(cj, •) Paths v1 v2 v3 v4 (2) the increase of the betweenness centrality (Equation 3
cMUC (vi) 0 0 0.5 0.5 and Equation 4) due to the shortest paths whose source and SP(c1,v2) v1-v3-v2 5/2 5/2 target are both connection vertices. Since the Brandes’ al- cb1 (vi) v1-v4-v2 5/2 5/2 gorithm does not explicitly calculate the all pair shortest SP(c1,v3) v1-v3 5 paths, the necessary values for the calculation of (1) and (2) SP(c1,v4) v1-v4 5 are obtained during the computation of the one-sided pair SP(c2,v1) v2-v3-v1 4/2 4/2 dependencies on the Brandes’ algorithm. The detailed im- cb2 (vi) v2-v4-v1 4/2 4/2 SP(c2,v3) v2-v3 4 plementation of the updatable version of the Brandes’ algo- SP(c2,v4) v2-v4 4 rithm using QUBE is shown in Algorithm 4. The additional SP(c3,v1) v3-v1 6 increases in the betweenness centrality explained through cb3 (vi) SP(c3,v2) v3-v2 6 Equation 2, Equation 3, and Equation 4 are computed in SP(c3,v4) v3-v4 6 (Line 34), (Line 23-Line 26 and Line 29-Line 31), and (Line ct2 (vi) SP(c1,c2) v1-v3-v2 5·4/2 5·4/2 5·4/2 1 38), respectively. The additional lines apart from the origi- v1-v4-v2 5·4/2 5·4/2 5·4/2 nal Brandes’ algorithm are underlined. c 3 (vi) SP(c1,c3) v1-v3 5·6 5·6 t1 ct3 (vi) SP(c2,c3) v2-v3 4·6 4·6 2 6.2 Experiment Results ct2 (vi) v2 3·1 c(vi) 65 59 87 15 To evaluate the proposed algorithm, we measure the be- Actual 130 118 174 30 tweenness centrality update time using synthetic datasets value and real datasets. We synthetically generate connected, undirected, and unweighted graphs of varying numbers of vertices and edges in order to observe the performance with respect to the graph size and proportions. The proportion as the global betweenness centrality which is the same value is computed as (|MUCU |/|V |)·100, and it indicates the per- resulting from the calculation of the betweenness centrality centage of vertices whose betweenness centralities should be of v2 using all the vertices in the original graph. The be- recalculated due to the update of the graph. Therefore, it tweenness centrality values for the other vertices v1, v3 and mainly affects the performance of QUBE. The Erd¨os-R´enyi v4 can also be calculated in the same way as the case of v2. model [9], the most widely used random graph model, is used to generate synthetic graphs. Each edge in the graph is generated independently of existing edges, with an equal 6. EXPERIMENTS probability of being generated. We explain how we implement an updatable version of the Figure 6 shows the running time for updating the between- Brandes’ algorithm using QUBE based on the Brandes’ al- ness centrality on synthetic graphs of size 1000, 3000, and gorithm in Section 6.1. We compare this updatable version 5000, respectively. For each graph, we randomly insert 100 of the Brandes’ algorithm with the original Brandes’ algo- edges and take the average value. QUBE significantly re- rithm and we show how much improvement is achieved with duces the betweenness centrality update time as the pro- the help of QUBE in Section 6.2. Recall that all previous portion decreases. In Figure 6(c), when the proportions are betweenness centrality calculation algorithms which do not 80, 40, and 10, QUBE enables the original Brandes’ algo- consider the graph update, such as the Brandes’ algorithm, rithm to perform about 2, 13, and 623 times faster, respec- inevitably require the computation of the betweenness cen- tively. These results provide a clear evidence that finding trality from scratch whenever a graph changes. We conduct MUCU dramatically improves the performance of updating experiments on Intel Xeon CPU with 2.53GHz and 20GB the betweenness centrality. Regardless of the size of a graph, main memory. QUBE makes the original Brandes’ algorithm perform much faster. 6.1 Implementation Besides the update time of the betweenness centrality, we As we explained in Algorithm 3, QUBE can be applied to measure the update time of MUCs. Since it is negligible any betweenness centrality calculation algorithm since it re- compared to the overall processing time, we do not explicitly duces the search space by identifying the candidate vertices present the update time of MUCs. and restores the global betweenness centrality using the local In order to estimate how QUBE performs in real world betweenness centrality. Since the Brandes’ algorithm known graphs, we select various real datasets which are prone to to be the fastest algorithm so far for computing the exact frequent changes. For each real graph, we extract the maxi- betweenness centrality, we implement QUBE based on the mally connected subgraph. In cases of directed real graphs, Brandes’ algorithm. The Brandes’ algorithm computes one- we convert directed edges into undirected edges. We com- sided pair dependencies of all the vertices in a graph for a pare the betweenness centrality update time over 8 different given source vertex by solving a single source shortest path real datasets. The results are shown in Table 2 and Fig- problem. The one-sided pair dependency of a vertex vt is the ure 7. Table 2 shows the speed-up achieved by QUBE and number of all shortest paths that go through vt,wherethe the overall statistics of each real dataset. Recall that the start vertex of the paths is fixed to a specific vertex vs.A proportion is the percentage of vertices in MUC. The low detailed description of the Brandes’ algorithm can be found proportion means that there exists a small number of ver- in [5]. tices whose betweenness centralities can be changed. Speed-
358 WWW 2012 – Session: Community Detection in Social Networks April 16–20, 2012, Lyon, France
3500 100000 400000 3000 350000 80000 2500 300000 250000 2000 60000 200000 1500 40000 QUBE+Brandes QUBE+Brandes Time(ms) QUBE+Brandes Time(ms) Time(ms) 150000 1000 Brandes 100000 Brandes Brandes 20000 500 50000 0 0 0 10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80
Proportion Proportion Proportion
(a) |V|=1000 (b) |V|=3000 (c) |V|=5000 Figure 6: The betweenness centrality update time on the synthetic data
Table 2: The speed-up on real data | | | | Algorithm 4: QUBE-BRANDES(MUCU ) Name Type V E Avg. Speed- Prop. up input : MUCU - Minimum Union Cycle that an updated Eva[24] Ownership 4457 4562 6.41 2418.17 vertices belong to Erdos02a Collaboration 5534 8472 28.66 39.57 SGs - A set of disconnected subgraphs connected Erdos972a Collaboration 4680 7030 30.00 34.39 by each connection vertices in MUCU Pgp[4] Social 4680 24316 42.14 13.09 output : C[vi] - Updated Betweenness Centrality Array b 1 Epa Web link 4253 8897 52.25 6.67 begin c 2 for vs ∈ MUCU do Contact Social 11604 65441 62.60 4.00 3 S ← empty stack ; Wikivote[19] Trust 7066 100736 67.73 3.00 4 P [vi] ← empty list, for all vi ∈ MUCU ; CAGrQc[19] Collaboration 4158 13422 77.92 2.06 5 σ[vi]:=0,forallvi ∈ MUCU ; σ[vs]:=1; a 6 σt[vi]:=0forallvi ∈ MUCU ; d[vs]:=0; http://vlado.fmf.uni-lj.si/pub/networks/data b 7 d[vi]:=-1,forallvi ∈ MUCU ; d[vs]:=0; http://www.cs.cornell.edu/courses/cs685/2002fa/ 8 Q ← empty queue ; chttp://stuff.metafilter.com/infodump/ 9 enqueue vs → Q ; 10 while Q not empty do 10000000 11 dequeue vi ← Q ; 12 push vi → S ; 1000000 13 for each neighbor vn of vi do 100000 14 if d[vn] < 0 then 10000 15 enqueue vn → Q ; 16 d[vn]:=d[vi]+1; 1000 100 17 if d[vn]=d[vi]+1then (ms, log Time scale) 18 σ[vn]:=σ[vn]+σ[vi]; 10 19 i → n append v P [v ]; 1 Eva Erdos02 Erdos972 Pgp Epa Contact Wikivote CAGrQc 20 δ[vi]:=0,forallvi ∈ MUCU ; QUBE+Brandes 106 12289 8640 270419 34056 1150801 361362 101895 21 while S not empty do Brandes 256326 486267 297100 3538417 227158 4600805 1082843 210831 22 pop vn ← S ; 23 if vs, vn are connection vertices and vn = vs Figure 7: The betweenness centrality update time then on real data 24 ct := |VGs |·|VGn | ; 25 σt[vn]:=σt[vn]+ct ; 26 C[vn]:=C[vn]+ct ; up in Table 2 shows how fast the updatable version of the Brandes’ algorithm is compared to the original Brandes’ al- 27 for vp in P [vn] do σ[vp] gorithm. Table 2 clearly shows that the performance of the 28 δ[vp]:=δ[vp]+ · (1 + δ[vn]) ; σ[vn] updatable version of the Brandes’ algorithm increases as the 29 if vs is connection vertex then proportion decreases. Figure 7 shows the average between- σ[vp] 30 t p t p t n · σ [v ]:=σ [v ]+σ [v ] σ[vn] ; ness centrality update times measured on real graphs. Note σ[vp] that we use a log scale for the y-axis in Figure 7. To rep- 31 C[vp]:=C[vp]+σt[vn] · ; σ[vn] resent the precise update time, a table is included in Figure 7. QUBE makes the original Brandes’ algorithm perform 32 if vn = vs then 33 C[vn]:=C[vn]+δ[vn]; about 2 times faster on ‘CAGrQc’ dataset whose proportion is about 77 and perform about 2418 times faster on ‘Eva’ 34 if vs is connection vertex then dataset whose proportion is about 6. When the proportion 35 C[vn]:=C[vn]+δ[vn] ·|VGs |·2; is about 30, QUBE makes the original Brandes’ algorithm
36 for Gi ∈ SGs do perform about 37 times faster and when the proportion is 37 if Gi is disconnected then about 70, QUBE makes the original Brandes’ algorithm per- 38 | |2 − n | |2 C[vi]:=C[vi]+ VGi l=1( V l ) ; form about 3 times faster. Gi
359 WWW 2012 – Session: Community Detection in Social Networks April 16–20, 2012, Lyon, France
7. CONCLUSIONS Publicationes Mathematicae (Debrecen), 6:290–297, In this paper, we devise a betweenness centrality update 1959. theorem and propose an efficient algorithm (QUBE) based [10] L. C. Freeman. A set of measures of centrality based on this theorem. QUBE identifies a set of vertices whose on betweenness. Sociometry, 40(1):35–41, 1977. betweenness centralities can be changed. QUBE efficiently [11] L. C. Freeman, S. P. Borgatti, and D. R. White. updates the betweenness centralities based on the between- Centrality in valued graphs: A measure of ness centrality calculated using the vertices in the set only betweenness based on network flow. Social Networks, and the number of vertices not in the set. Any existing be- 13(2):141 – 154, 1991. tweenness centrality algorithm which does not consider the [12] R. Geisberger, P. Sanders, and D. Schultes. Better graph update can be changed to an efficient updatable be- approximation of betweenness centrality. In J. 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