An Efficient Algorithm for Approximate Betweenness
Centrality Computation
Mostafa Haghir Chehreghani
Department of Computer Science, KU Leuven
Celestijnenlaan 200a - box 2402
3001 Leuven, Belgium
mostafa.haghirchehreghani@cs.kuleuven.be
ABSTRACT
Betweenness centrality is an important centrality measure widely
used in social network analysis, route p lanning etc. However, even
for mid-size networks, it is practically intractable to compute ex-
act betweenness scores. In this paper, we propose a generic ran-
domized framework for unbiased approximation of betweenness
centrality. The proposed framework can be adapted with different
sampling techniques and giv e diverse methods. We discuss the con-
ditions a promising sampling technique should satisfy to minimize
the approximation error and present a sampling method partially
satisfying the conditions. We perform extensiv e experiments and
show the high efficiency and accuracy of the proposed method.
Categories and Subject Descriptors
G.2.2 [Discrete Mathematics]: Graph Theory—Graph algorithms,
Path and circuit problems;E.1[Data]: Data structures—Graphs
and networks
General Terms
Theory
Keywords
Centrality, betweenness centrality, social network analysis, approx-
imate algorithms.
1. INTRODUCTION
Betweenness centrality of a vertex, introduced by Linton Free-
man [6], is defined as the number of shortest paths (geodesic paths)
from all (source) vertices to all others that pass through that vertex.
He used it as a measure for quantifying the control of a human on
the communication b etween other humans in a social network [6].
Betweenness centrality is also used in some well-known algorithms
for clustering and community detection in social and information
networks [8].
Although betweenness centrality computation is tractable in the-
ory in the sense that there exist polynomial time and space algo-
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rithms, the most efficient existing exact method is Brandes’s algo-
rithm [3] which requires O(nm) time for unweighted graphs and
O(nm + n
2
log n) time for weighted graphs (n is the number of
vertices and m is the number of edges in the network). Therefore,
exact betweenness centrality computation is not practically appli-
cable, even for mid-size networks. The next bad news is that com-
puting exact betweenness centrality of a single vertex is not easier
than computing betweenness centrality of all vertices. Therefore,
the above mentioned worst case bounds hold if someone wants to
compute betweenness centrality of one or a few vertices. How-
ever, in many applications it might be required to compute be-
tweenness centrality of only one or a few vertices. For example,
the index might be computed only for core vertices of communi-
ties in social/information networks [12], or hubs in communication
networks.
To make betweenness centrality practically computable, in re-
cent years, several algorithms have been proposed for approximate
betweenness centrality computation [4], [1] and [7]. E xisting algo-
rithms fall into one of the following categories.
1. Some algorithms like [4] and [7] try to approximate between-
ness centrality of all vertices in the network. For such meth-
ods the value computed for every vertex is not of high im-
portance. Instead, the main goal is to correctly estimate the
relative rank of all vertices.
2. Some others, like the method presented in [1], aim to ap-
proximate betweenness centrality of a single vertex (or a few
vertices) in time faster than computing betweenness central-
ity of all vertices. For such methods, the accuracy of the
estimated betweenness centrality is important.
Our focus in this paper is the second cate gory of algorithms, i.e.
we aim at developing an efficient and accurate algorithm for be-
tweenness centrality computation of a single vertex. In this pa-
per, we propose a generic randomized framework for unbiased ap-
proximation of betweenness centrality. In the proposed framework,
a source vertex i is selected by some strategy, single-source be-
tweenness scores of all vertices on i are computed, and the scores
are scaled as estimations of betweenness centralities. While our
method might seem similar to the method of Brandes and Pich [4],
they have a key difference. In the method of [4], for a few source
vertices, single-source betweenness scores are computed and for
the rest, they are extrapolated. Betweenness centralities are the sum
of all single-source betweenness scores (which are either computed
or extrapolated). In our method, single-source betweenness scores
are computed for one single source chosen randomly, and the ob-
tained scores are scaled as estimations of betweenness centralities.
We discuss the condition a promising sampling technique should
satisfy to minimize the approximation error for a single vertex.
1489
Since it might be computationally expensive to find such a sam-
pling, we propose a sampling technique which partially satisfies
the condition. While the algorithm of [1] is intuitively presented
for high centrality vertices, in our method, the sampling technique
can be revised to optimize itself for both high centrality vertices and
low centrality vertices. The proposed method can be used to com-
pute similar centrality notions like stress centrality, which is also
based on counting shortest paths. We perform extensive experi-
ments on real-world netw orks from different domains, and show
that compared to existing exact and inexact algorithms, our method
works with higher accuracy or it gives significant speedups.
The rest of this paper is organized as follows. In Section 2, pre-
liminaries and definitions related to betweenness centrality com-
putation are given. In Section 3, we present a generic random-
ized algorithm for betweenness centrality computation. In Section
4, we discuss the sampling methods. We empirically evaluate the
proposed method in Section 5 and show its efficiency and high ac-
curacy. Finally, the paper is concluded in Section 6.
2. PRELIMINARIES
Throughout the paper, G refers to a graph (network). For sim-
plicity, we suppose G is a connected and loop-free graph without
multi-edges. Throughout the paper, we assume G is an unweighted
graph, unless it is explicitly mentioned that G is weighted. V (G)
and E(G) refer to the set of vertices and the set of edges of G,re-
spectiv ely. Throughout the paper, n points to |V (G)| and m points
to |E(G)|. For an edge e =(u, v) ∈ E(G), u and v are two end-
points of e.Ashortest path (also called a geodesic path) between
two vertices u, v ∈ V (G) is a path whose size is minimum, among
all paths between u and v. For two vertices u, v ∈ V (G),we
use d(u, v), to denote the size (the number of edges) of a short-
est path connecting u and v.Bydefinition, d(u, v)=0and
d(u,
v)=d(v, u).Fors, t ∈ V (G), σ
st
denotes the number
of shortest paths between s and t,andσ
st
(v) denotes the number
of shortest paths between s and t that also pass through v.We
have σ
s
(v)=
t∈V (G)\{s,v}
σ
st
(v). Betweenness centrality of a
vertex v is defined as:
BC(v)=
s,t∈V (G)\{v}
σ
st
(v)
σ
st
(1)
A notion which is widely used for counting the number of shortest
paths in a graph is the directed acyclic graph (DAG) containing
all shortest paths starting from a vertex s (see e.g. [3]). In this
paper, we refer to it as the shortest-path-DAG,orSPD in short,
rooted at s. For every vertex s in a graph G,theSPD rooted at s
is unique, and it can be computed in O(m) time for unweighted
graphs and in O(m + n log n) time for weighted graphs [3]. In
[3], the authors introduced the notion of the dependency scor e of a
vertex s ∈ V (G) on a vertex x ∈ V (G) \{s}, which is defined
as:
δ
s•
(v)=
t∈V (G)\{v,s}
σ
st
(v)
σ
st
(2)
We have: BC(v)=
s∈V (G)\{v}
δ
s•
(v). As mentioned in [3],
given the SPD rooted at s, dependency scores of s on all other
vertices can be computed in O(m) time.
3. APPROXIMATE BETWEENNESS CEN-
TRALITY COMPUTATION
Algorithm 1 shows the high lev el pseudo code of the algorithm
proposed for approximate betweenness centrality computation. First
the following probabilities are computed
p
1
,p
2
,...,p
n
≥ 0 such that
n
i=1
p
i
=1 (3)
Then, at every iteration t of the loop in Lines 8-15 of Algorithm 1:
(I) an i ∈{1,...,n} is selected with probability p
i
, (II) the SPD
rooted at i is computed, (III) dependency score of vertex v on i,
δ
i•
(v), is computed, and (IV)
δ
i•
(v)
p
i
is the estimation of BC(v)
in iteration t. The average of betweenness centralities calculated
in different iterations is returned as the estimation of betweenness
centrality.
Algorithm 1 High level pseudo code of the algorithm of approxi-
mate betweenness centrality computation.
1: APPROXIMATEBE TWEENNESS
2: Require. A network (graph) G, the number of samples T .
3: Ensure. Betweenness centrality of vertices of G.
4: Compute probabilities p
1
,...,p
n
5: for all vertices v ∈ V (G) do
6: B[v] ← 0
7: end for
8: for all t =1to T do
9: Select a vertex i with probability p
i
10: Form the SPD D rooted at i
11: Compute dependency scores of e very vertex v on i
12: for all vertex v ∈ V (G) do
13: B[v] ← B[v]+
δ
i•
(v)
p
i
14: end for
15: end f or
16: for all i ∈{1,...,n} do
17: B[i] ←
B[i]
T
18: end f or
19: return B
LEMMA 1. In Algorithm 1, for every vertex v we have
E(B[v]) = BC(v) (4)
and
Va r (B[v]) =
1
T
n
i=1
δ
i•
(v)
2
p
i
−
BC(v)
2
T
(5)
P
ROOF.Wehave:
E(B[v]) =
T
n
i=1
p
i
δ
i•
(v)
p
i
T
= BC(v)
and
Va r (B
t
[v]) = E(B
t
[v]
2
) − E(B
t
[v])
2
=
n
i=1
δ
i•
(v)
2
p
i
− BC(v)
2
Since B[v] is the average of T independent copies of B
t
[v],there-
fore
Va r (B[v]) =
1
T
n
i=1
δ
i•
(v)
2
p
i
−
BC(v)
2
T
(6)
Some existing algorithms can be described as adaptations of Al-
gorithm 1 with specific sampling methods. For example, if vertices
i are selected uniformly at random (i.e. p
i
=
1
n
for 1 ≤ i ≤ n),
then it will give the randomized algorithm presented in [1]. We
1490
note that instead of taking T samples, we can defineacriteriafor
the termination of the loop in Lines 8-15 of Algorithm 1. For ex-
ample, similar to the algorithm of [1], we can stop when B[v] ≥ cn
for some constant c.
4. SAMPLING METHODS
Suppose that we want to approximate betweenness centrality of
avertexv. The following Lemma defines the probabilities mini-
mizing the approximation error.
L
EMMA 2. If in Algorithm 1 source vertices i are selected with
probabilities
p
i
=
δ
•i
(v)
n
j=1
δ
•j
(v)
(7)
the approximation error (i.e. variance of B[v]) is minimized. In
this case, variance of B[v] will be 0
1
.
P
ROOF. Omitted due to lack of space.
Therefore, using probabilities p
i
defined in Equation 7, gives an
exact method in the sense that it makes the approximation error 0.
However, time complexity of computing optimal p
i
’s is the same
as exact betweenness centrality computation. Although it is not
practically efficient to use probabilities p
i
defined in Equation 7,
they can help us to define properties of a good sampling. From
Equation 7, we can conclude that in a good sampling, for every two
vertices i and i
, the following must hold:
p
i
≥ p
i
⇔ δ
•i
(v) ≥ δ
•i
(v) (8)
which means vertices with higher dependency scores on v,mustbe
selected with a higher probability.
However, finding probabilities p
1
,p
2
,...,p
n
which satisfy Equa-
tion 8 might be computationally expensive, since it needs to com-
pute dependency scores of all vertices on v which is as bad as com-
puting dependency scores of every source vertex on all vertices. In
order to design practically efficient sampling techniques, we con-
sider relaxations of Equation 8. Consider two vertices i and i
such
that d(i
,v) >d(i, v). If in the SPD rooted at v there exists an
ancestor-descendant relationship between i and i
and i is the only
ancestor of i
at the level d(i, v), then, it can be shown that for
k ∈{i, i
}, probability p
k
defined as
p
k
=
1
d(k,v)
n
j=1
1
d(j,v)
(9)
satisfies Equation 8.
The positive aspect of the sampling technique presented in Equa-
tion 9 is that it only needs to compute the distance between vertex
v and every vertex in the graph: the single-source shortest path, or
SSSP in short, problem. For unweighted graphs, this problem can
be solved in O( m) time and for weighted graphs, using Fibonacci
heap, it is solvable in O(m + n log n) time[5]. Itmeansthatthe
sampling method presented in E quation 9 is practically efficient.
1
What this lemma suggests, somehow contradicts the source ver-
tex selection procedure presented in [7]. In the method of [7] the
scheme for aggregating dependency scores changes so that vertices
do not profit from being near the selected source vertices. How-
ever, Lemma 2 says it is better to select source vertices based on
their dependency scores on v, and as we will see later, it might
result in preferring source vertices which are closer to v.
The reason of this contradiction is that while here we aim at pre-
cisely approximating betweenness centrality of some specific ver-
tex v, the method of [7] aims to rank all vertices based on their
betweenness scores.
Therefore, with probabilities p
i
defined in Equation 9, a vertex
i is selected and dependency score of i on v is computed, and the
result is scaled. For unweighted graphs, it gives an O(Tm) time al-
gorithm for approximate betweenness centrality computation. For
weighted graphs (with positive weights), time complexity of the
approximation algorithm will be O(Tm+ Tnlog n).
5. EXPERIMENTAL RESULTS
Our experiments were done on one core of a single AMD Proces-
sor 270 clocked at 2.0 GHz with 8 GB main memory and 2×1 MB
L2 cache, running Ubuntu Linux 12.0. The program was compiled
by the GNU C++ compiler 4.0.2 using optimization level 3. We
compare our proposed method with the algorithm presented in [1].
As mentioned earlier, methods lik e [4] and [7] aim to rank vertices
based on betweenness scores (and the betweenness score of an in-
dividual vertex is not very important for them). Therefore, they are
not suitable for our comparisons. We refer to the algorithm of [1]
as uniform sampling, since it chooses source vertices uniformly at
random. We refer to our proposed method as distance-based sam-
pling. We also compare the methods with Brandes’s algorithm for
exact betweenness centrality computation [3].
We performed extensive experiments on different real-world net-
works to assess the quantitative and qualitative behavior of the pro-
posed algorithm. We used two DBLP citation networks dblp0305
and dblp0507 [2], the Wiki-Vote social network [9], the Enron-
Email communication network [10], and the CA-CondMat collabo-
ration network [11]. Table 1 summarizes specifications of our real-
world networks.
Table 1: Summary of real-world networks.
Dataset # vertices # edges Avg. degree
dblp-0305 109,044 233,961 4.29
dblp-0507 135,116 290,363 4.28
Enron-Email 36,692 367,662 20.04
W iki-Vote 7,115 103,689 29.14
CA-CondMat 23,133 93,497 8.08
For a vertex v, the empirical betweenness approximation error
(which is reported in the experiments) is defined as:
err(v)=
|App(v) − BC(v)|
BC(v)
× 100 (10)
where App(v) is the computed approximate score.
In our experiments, we consider several vertices of a dataset
and for every vertex, we compute distance-based probabilities, ex-
act betweenness scores, and approximate betweenness scores using
distance-based and uniform samplings. Table 2 summarizes the av-
erage results (i.e. the sum of results of all vertices divided by their
number) obtained for different datasets. In all experiments, for
both uniform and distance-based samplings, the number of sam-
ples (the number of source vertices selected) is 10% of the number
of vertices in the network. Therefore, the running time of the ap-
proximate methods is around 10% of the running time of the exact
method.
The first dataset is Wiki-Vote which is dense (its average de-
gree is 29.14). For most of vertices of Wiki-Vote, distance-based
sampling gives a better approximation. The extra time needed by
the distance-based sampling to compute required shortest path dis-
tances is quite tin y and ignorable compared to running time of the
whole process. Since for all datasets it is a tiny value varying in
different runs, we report an upper bound for it. For the Wiki-Vote
1491
Table 2: Comparing average approximation error and average running time of uniform sampling, distance-based sampling, and
exact method, for single vertices in different datasets.
Database
Exact BC Approximate BC
Avg. BC score Avg. time (sec)
Distance-based sampling Uniform sampling
Avg. time (sec) # iterations
Avg. error (%) Avg. dist.
comp. time
(sec)
Avg. error (%)
Wiki Vote 76056.85 515.09 37.0% < 1 41.13% 46.05 10%
Email-Enron 2775100.8 9033.11 15.75% < 1 25.28% 925.80 10%
dblp0305 564246.41 19149.8 7.59% < 2 64.73% 1747.15 10%
dblp0507 798125.00 35140 7.19% < 2 50.17% 2863.82 10%
CA-CondMat 691667 3026.9 10.8% < 1 20.81% 315.3 10%
dataset, it is always smaller than 1 second. The next dataset is
Email-Enron. Compared to Wiki-Vote, it is less dense (but still
dense) and larger. Over this dataset, the approximation error of
distance-based sampling is better than the uniform sampling.
Figure 1: In sparse
graphs, distance-based
sampling is closer to opti-
mal sampling. The graph
in the left side shows a
SPD in a dense graph, and
the graph in the right side
shows a SPD in a sparse
graph.
Dblp0305 and dblp0507 are
large and relatively sparse datasets.
As reflected in Table 2, over
these datasets, distance-based
sampling works much better
than uniform sampling. This
means that on sparse datasets,
the difference between the ap-
proximation quality of two meth-
ods is more considerable. It
has several reasons. The first
reason is that in very dense
datasets, many vertices have the
same (and small) distance from
v (v is the vertex whose be-
tweenness centrality is approx-
imated). Therefore, distance-
based sampling becomes closer
to the uniform sampling. The
second reason is that in sparse networks, in the SPD rooted at
v, the probability that a vertex i has only one ancestor at some
level k is lower than this probability in dense graphs. Figure 1
compares these two situations. It means that in sparse networks,
distance-based sampling is closer to the optimal sampling, because
by distance-based sampling, a larger number of vertices will satisfy
the condition expressed in Equation 7. As a result, on sparse net-
works, distance-based sampling becomes much more effective than
uniform sampling.
Finally, the methods were compared on the CA-CondMat dataset
which contains scientific collaborations between authors of papers
submitted to Condense Matter category [11]. The average degree
in this dataset is 8.08 which means it is denser than dblp0305 and
dblp0507, but less dense than Wiki-Vote and Email-Enron. Over
this dataset, the approximation error of uniform sampling i s almost
twice of the approximation error of distance-based sampling.
6. CONCLUSION
In this paper , we presented a generic randomized framework for
unbiased approximation of betweenness centrality. In the proposed
framework, a source vertex i is selected by some strategy, single-
source betweenness scores of all vertices on i are computed, and
the scores are scaled as estimations of betweenness centralities.
Our proposed framework can be adapted with different sampling
techniques to give diverse methods for approximating betweenness
centrality. We discussed the conditions a promising sampling tech-
nique should satisfy to minimize the approximation error, and pro-
posed a sampling technique which partially satisfies the conditions.
Our experiments show the high efficiency and quality of the pro-
posed method.
7. ACKNOWLEDGMENTS
This work was partially supported by ERC Starting Grant 240186
"MiGraNT: Mining Graphs and Netwo rks: a Theory-based approach".
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