Abstract
Betweenness centrality is an essential index for analysis of complex networks. However, the calculation of betweenness centrality is quite time-consuming and the fastest known algorithm uses 
time and 
space for weighted networks, where 
and 
are the number of nodes and edges in the network, respectively. By inserting virtual nodes into the weighted edges and transforming the shortest path problem into a breadth-first search (BFS) problem, we propose an algorithm that can compute the betweenness centrality in 
time for integer-weighted networks, where 
is the average weight of edges and 
is the average degree in the network. Considerable time can be saved with the proposed algorithm when 
, indicating that it is suitable for lightly weighted large sparse networks. A similar concept of virtual node transformation can be used to calculate other shortest path based indices such as closeness centrality, graph centrality, stress centrality, and so on. Numerical simulations on various randomly generated networks reveal that it is feasible to use the proposed algorithm in large network analysis.
Introduction
Networks, especially complex networks, have been extensively studied during the last decade [1]–[3]. Owing to the ability to gather and analyze large scale data using computers and communication networks, it is quite common to see studies on networks with millions of vertices (nodes) nowadays. The shift of studies from simple small graphs to large complex networks have increasingly contributed new findings of critical phenomena and development of theories in many fields, such as the scale-free distribution of network degrees [4], [5], burstness of human behaviors [6], vulnerability of internet networks [7], [8], and so on [1]–[3], [9].
However, the computation of several network properties, such as the shortest paths, betweenness centrality and closeness centrality, are hindered by the large computation complexity [3], [10]. As a result, many large-scale networks are regarded as unweighted when the above measures are reported [2], [3]. Large efforts have been made to improve the efficiency of algorithms for calculating those network properties [10], [11]. Take the betweenness centrality, for example [12], [13]: for a weighted network 
with 
nodes and 
edges, the naive algorithm requires 
time and 
storage, regardless of the algorithms implemented to find the shortest paths. A much faster algorithm proposed by Brandes [14], on the other hand, can calculate the betweenness centrality in 
time and 
space when the shortest paths are calculated by Dijkstra's algorithm implemented with a Fibonacci heap. Parallel algorithms are also proposed to improve the efficiency for the calculation of betweenness centrality [10], [11], [15]–[21]: for example, Bader and Madduri [10] proposed a betweenness centrality algorithm on a high-end shared memory symmetric multiprocessor and multithreaded architectures, with which is “possible” to achieve the computation in 
time with access conflicts, where 
is the number of processors used. However, the parallel algorithms requires much more complex programming and are highly dependent on the hardwares: for example, in Bader and Madduri's study [10], they used an IBM p5 570 on 16 processors and utilized 20GB RAM. These equipments are obviously not adaptable for general network researchers.
To circumvent the difficulties in calculating betweenness centrality with large time complexity, we propose a new algorithm for integer-weighted networks in this paper. By replacing the weighted edges with connected virtual nodes, the new algorithm computes the betweenness centrality in 
time and 
space, with 
and 
being the average edge weight and average degree of the network, respectively.
Methods
The Brandes' Algorithm
Given a network 
, with 
the number of nodes and 
the number of edges, for the purpose of this study, we consider strongly connected networks [22] with no self loops (acyclic). Let 
be the weight matrix of 
, where 
is the weight on edge 
. In real practice, 
can be distances between airports, information flows between computers, traffic loads between cities, etc.
Let 
denote the number of shortest paths from node 
to 
, and 
be the number of shortest paths from 
to 
that pass through 
, then the betweenness centrality of node 
is defined as [13], [14]:
 |
(1) |
From the definition we can see that betweenness centrality is the sum of the fraction of shortest paths over all pairs of nodes passing through the node, high betweenness centrality indicates that a node can reach others (or be reached by others) with relatively short paths, or the node lies on considerable fraction of shortest paths connecting others. In many fields, the betweenness centrality can be regarded as a measure of the extent to which the node has control over information flowing between others, and it is thus a core index for evaluating the importance of nodes in the network [13], [23]. For example, in the study of networks vulnerability to attacks, the removal of nodes with the highest betweenness centrality is shown to be one of the most harmful strategies that can break down the networks [8].
A straightforward way of calculating the betweenness centrality then use the following steps:
Step 1 Compute the length and number of shortest paths between all pairs of nodes;
Step 2 For each node 
, calculate 
(pair dependency) for each pair and sum them up.
Obviously, the complexity of the naive algorithm is dominated by the second step which requires 
time summation and 
storage of pair dependencies. To introduce Brandes' algorithm, we first define the set of predecessors of node 
on the shortest paths from 
:
 |
(2) |
where 
is the distance of the shortest path from 
to 
. Then the number of shortest paths from 
to 
can be calculated as:
 |
(3) |
To eliminate the need for explicit summation of all pair dependencies, Brandes [14] defines the dependency of node 
as:
 |
(4) |
has the recursive property that
 |
(5) |
Note that 
is merely a partial sum of Eq. (1), then the betweenness centrality can be expressed by:
 |
(6) |
The summation of pair dependencies is then reduced to accumulation of dependencies defined by Eq. (5). Specifically, given the shortest paths from 
in 
, the array storing 
for all nodes can be recursively calculated according to Eq. (5), by traversing the nodes in non-increasing order of their distances from 
. An illustrative algorithm is shown in Algorithm 1. We can see that the calculation for Step 2 is now in 
time and 
space, then the calculation complexity of betweenness centrality is determined by the shortest path algorithms used in Step 1. Using Dijkstra's algorithm implemented with Fibonacci heap [24], which requires 
time for the single source shortest path problem [25], the betweenness centrality can be computed by Brandes' algorithm in 
time and 
space on weighted networks [14].
Computing Betweenness Centrality with Virtual Nodes
Brandes' algorithm has greatly reduced the computation burden for betweenness centrality, however, the time complexity is still too high for networks with millions of nodes since the shortest path algorithm would cost a lot of computation time anyway. In this section, we propose a new algorithm that can reduce the time complexity in Step 1, such that the betweenness centrality can be calculated within reasonable time under certain conditions.
Replacement of Weighted Edges
Our new algorithm originates from the idea that an integer-weighted network can be broken down into a simple unweighted network with virtual nodes, such that the calculation of shortest paths in Step 1 can be solved as a breadth-first search (BFS) problem.
Algorithm 1: Brandes' algorithm [14] .
1 
2 for 
do
3 [
] = single source shortest path algorithm()
/*
set of predecessors for shortest paths from 
to 
;*/
/*
array storing the number of shortest paths from 
passing through 
; */
/*
stack storing the distances of nodes from 
in non-increasing order; */
/*accumulate dependency from the most distant nodes */
4 
5 while 
not empty do
6 pop 
7 for 
do 
8 if 
then 
9 end
10 end
Figure 1 illustrates the representation of an undirected weighted network by an undirected unweighted network with three additional virtual nodes. We can see that edge 
and 
are replaced by 3 and 2 unit edge segments with two and one virtual nodes inserted, respectively. The number of virtual nodes to be inserted on a weighted edge 
, is then 
.
Figure 1. Illustration of representing the weighted network (a) by an unweighted network with virtual nodes (b).
Let 
be the unweighted representation of 
with virtual nodes, where 
with 
the set of virtual nodes, then the number of virtual nodes in 
, is 
, where 
is the average edge weight.
Virtual Node Based Algorithm for Betweenness Centrality. Obviously, the insertion of virtual nodes does not change the distances between pairs of nodes in 
and consequently the number of shortest paths between nodes is the same as in 
. The calculation of shortest paths on 
can then be solved by the BFS algorithm, instead of the traditional Dijkstra's algorithm.
However, before applying the BFS on 
to calculate the betweenness centrality for nodes in 
, there is at least one problem to be solved: to use the existing theories on summation of pair dependency in Algorithm 1, the predecessors of nodes in 
recorded during the shortest path calculation in 
, should be kept as the same as if they were calculated by any shortest path algorithm in 
. This can be achieved as follows: suppose the BFS finds a shortest path from 
to 
: 
, where 
, 
are two virtual nodes inserted on edge 
, then the predecessor of 
, which is 
, can be passed through 
to the next non-virtual node 
:
 |
An implementation of the above process is presented in Algorithm 2, the steps for accumulation of dependency are identical as the Brandes' algorithm and thereby are omitted.
Algorithm 2: Virtual node algorithm for betweenness centrality
1 
2 for 
do
3 
empty stack;
4 
empty list, 
;
5 
, 
; 
;
6 
, 
; 
;
7 
empty queue;
8 enqueue 
;
9 while 
not empty do
10 dequeue 
;
11 push 
;
12 foreach neighbor 
of 
do
13 if 
then /*visit 
the first time*/
14 enqueue 
;
15 
;
16 end
17 if 
then
18 
;
19 if 
then
20 append 
;
21 else /*if 
is a virtual node, retrieve the latest non-virtual node as predecessor*/
22 append 
;
23 end
24 end
25 end
26 end
27 accumulate dependency()/*as shown in Algorithm 1
28 end
Note that in Algorithm 2, we don't need to calculate shortest paths between virtual nodes. The BFS then requires 
time. For the sake of clarity, let 
be the average degree of nodes in 
such that 
, then we have 
. The computation of betweenness centrality with virtual nodes (the VN algorithm), is dominated by the BFS and has a time complexity of 
, and needs 
space.
Compared with Brandes' algorithm, we can see that the VN algorithm will perform better when 
, that is, 
. We henceforth denote 
as the critical threshold for the average edge weight on a network; if 
, the VN algorithm will be able to calculate the betweenness centrality faster than Brandes' algorithm. Figure 2 shows the distribution of 
over the domain of combinations of different network sizes and average degrees. We can see that the advantage of the VN algorithm becomes evident when the network is large and sparse, for example, when the network size is 1 million (
), and the average degree is 5, the VN algorithm would be faster for those with 
; for the same average degree, 
increases to 7 when the network size reaches 1 billion (
). For an average degree of 10, 
lies beyond 3 for networks larger than 1 million. Note that many large-scale networks are reported to have rather small average degrees; for example, the mobile communication network reported in [26], contains 4.6 million nodes and an average of 3.04 edges. The Internet network [27], math co-authorship network [28], and power grid [29] reported in [1], are found to have average degrees of 3.5–4.1, 3.9 and 2.7, respectively. Networks with low integer weights are also reported in the literature; for example, the neural network of the Caenorhabditis elegans worm [29], the communication network of the online community [30], and the political support network of the US Senate [31], have average edge weights of 3.74, 2.95 and 3.74, respectively.
Figure 2. Critical threshold for average weights (
) on networks with specified network size (
) and average degree (
).
Results and Discussion
Numerical Experiments
To evaluate the algorithms, we generate scale-free networks [32] with different network sizes and edge weights, and the execution time of VN algorithm and Brandes' algorithm are then tested on these networks. Algorithms are coded in C and run on a PC with an Intel Core 2 Quad CPU (2.66 GHz, 6 Mb) and 6 Gb of RAM, all the following reported running times are the average of 100 simulations.
It is intuitive that when seldom edges in the network are weighted, the VN algorithm will calculate the betweenness centrality approximately as fast as the BFS, which is much faster than the Brandes' algorithm. For example, when the network size is 100,000 and we set the average degree as 2 and take 1000 edges to be weighted with random numbers generated from 1 to 10, the execution time for Brandes' algorithm is 8460 seconds, while the VN algorithm needs only 3830 seconds, which is around 1.3 hours faster than the Brandes' algorithm. Since when 
becomes large, we have 
, more time can be expected to be saved in larger networks with fixed number of weighted edges. We calculated the VN and Brandes' algorithm on networks with 1% of edges being weighted as 2, and the execution times are presented in Figure 3(a). We can see that the difference in execution time become larger when the network size increases. When the network size is 50,000, the VN algorithm is 3 and 1.5 times faster than the Brandes' algorithm, for average network degrees of 2 and 10, respectively.
Figure 3. Running time of the VN algorithm and Brandes' algorithm.
(a) Networks with average degree 
and 
, 1% of the network edges are weighted with 
; (b) Networks with average degree 
, all edges are weighted with 
.
The above results reveal that the VN algorithm is much faster on large sparse networks with limited number of weights. However, we should note that the VN algorithm is quite sensitive to the average degree and weight sum of the network, for any network with 
, the VN algorithm will not outperform Brandes' algorithm as long as 
. To illustrate the sensitivity of the VN algorithm, we run algorithms on networks with 
and 
, and the difference in running times between the two algorithms decreases quickly as expected (Figure 3(b)).
Discussion
By replacing the weighted edges with connected virtual nodes, we propose the VN algorithm to calculate the betweenness centrality in weighted networks with the BFS rather than shortest path algorithms. The VN algorithm uses 
time and 
space. Theoretically, the VN algorithm outperforms the Brandes' algorithm when 
, indicating that when the average edge weight is low, considerable time can be saved on large sparse networks. The simulation study confirms that when 
, more time can be saved when the network grows large.
We should note that the VN algorithm is quite sensitive to the density and weight of the networks, it can hardly outperform the Brandes' algorithm when the network is dense and weighted with large values. What's more, the theoretical threshold value 
, could be even lower in practice since the VN algorithm requires more space. Despite these limitations, given the evidences that large-scale networks in real life are mostly sparse, and the BFS is much easier to implement than the Fibonacci heap based shortest path algorithms, the VN algorithm is expected to be able to save analysis time in many scenarios. Moreover, the VN algorithm can easily be generalized to calculate other shortest path based network properties, such as closeness centrality [33], graph centrality [34], stress centrality [35], and so on. We henceforth recommend that network researchers to use the VN algorithm when the studied network is large, sparse, and lightly weighted, but continue to use the Brandes' algorithm otherwise.
Supporting Information
Both the Brandes' algorithm and the VN algorithm are written in C and are available upon request from the author.
Footnotes
Competing Interests: The authors have declared that no competing interests exist.
Funding: This work is funded by the National Science Foundation (no. 70971131). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
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