ePubWU Institutional Repository
Angela Bohn and Norbert Walchhofer and Patrick Mair and Kurt Hornik
Social Network Analysis of Weighted Telecommunications Graphs
Paper
Original Citation:
Bohn, Angela and Walchhofer, Norbert and Mair, Patrick and Hornik, Kurt (2009) Social Network
Analysis of Weighted Telecommunications Graphs. Research Report Series / Department of
Statistics and Mathematics, 84. Department of Statistics and Mathematics, WU Vienna University
of Economics and Business, Vienna.
This version is available at: http://epub.wu.ac.at/708/
Available in ePubWU : March 2009
ePubWU , the institutional repository of the WU Vienna University of Economics and Business, is
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http://epub.wu.ac.at/
Social Network Analysis of Weighted
Telecommunications Graphs
Angela Bohn, Norbert Walchhofer, Patrick Mair, Kurt Hornik
Department of Statistics and Mathematics
Wirtschaftsuniversität Wien
Research Report Series
Report 84
March 2009
http://statmath.wu-wien.ac.at/
Social Network Analysis of Weighted
Telecommunications Graphs
Angela Bohn
Norbert Walchhofer
Kurt Hornik
Patrick Mair
Abstract
SNA provides a wide range of tools that allow examination of telecommunications graphs. Those graphs contain vertices representing cell phone
users and lines standing for established connections. Many sna tools do
not incorporate the intensity of interaction. This may lead to wrong conclusions because the difference between best friends and random contacts
can be defined by the accumulated duration of talks. To solve this problem, we propose a closeness centrality measure (ewc) that incorporates line
values and compare it to Freeman’s closeness. Small exemplary networks
will demonstrate the characteristics of the weighted closeness compared to
other centrality measures. Finally, the ewc will be tested on a real-world
telecommunications graph provided by a large Austrian mobile service
provider and the advantages of the ewc will be discussed.
1
Introduction
So far, the so called Freeman’s closeness centrality measure could only be calculated for unweighted graphs. As in many applications the line values are
important for the understanding of networks, we introduce an edge-weighted
closeness (ewc). Contrary to other versions of weighted closeness that can be
found in literature, the ewc incorporates not only line values but also distances.
The ewc code (Appendix A) for R (R Development Core Team, 2008) will be
provided. The ewc will be compared to Freeman’s closeness and other centrality
measures. Furthermore, it will be tested for a real-world cellphone calls graph.
We will show that the ewc detects nodes that have very close contacts to their
friends. On this note, the ewc is a counterpart of betweenness centrality which
identifies people with rather loose connections to many separated groups.
2
2.1
Literature Review
History of Closeness Centrality
A centrality measure based on distances was first introduced by Bavelas (1950).
He noticed that in different kinds of networks containing the same number of
vertices, the sum of distances differs. In addition, he showed on the basis of
examples that the calculation of the absolute distance between a vertex and
all other vertices does not suffice to describe a its role in the network. Thus
he suggested to use the proportion of the sum of all distances in the network
1
P
d(x, y) and the sum of thePdistances P
from a certain vertex i to all others
j to describe i’s importance:
d(x, y)/ d(i, j). Beauchamp (1965) had the
idea of rescaling Bavelas’ closeness to the 0-1 intervall. His “relative closeness”
RC(i) of vertex i is
n−1
,
(1)
RC(i) = P
j d(i, j)
where n is the network size (=number of vertices). This is the formula commonly used for closeness centrality, also called Freeman’s closeness. Moxley
and Moxley (1974) focused on the problem of disconnected graphs. While Flament (1963) defined the distance between two vertices that cannot reach each
other to be infinite and Beauchamp (1965) suggested to leave them undefined,
Moxley and Moxley (1974) assigned a penalty to each vertex that is not able
to reach every other vertex. This penalty has to be larger than the maximum
distance (=number of vertices-1). Freeman (1979) summarized and structured
the centrality measures existing so far like degree, betweenness and closeness.
Therefore, he is the most cited author when it comes to closeness centrality. He
also introduced a measure of (closeness) centralization, a measure of the extend
to which a network has a center and a periphery.
2.2
Freeman’s Closeness Centrality
Although Beauchamp (1965) created the closeness used today, it is usually referred to as Freeman’s closeness. The formula for closeness is
n−1
CC (i) = P
j d(i, j)
(2)
where i is the observed vertex, n is the network size (=number of vertices), j is
a vertex that is connected to i by a path and d(i, j) is the distance between i
and j (=length of the shortest path between i and j).
The closeness is a point-centrality measure: it is used to measure the importance of a network member. It is based on the idea that, the farther a vertex j is
away from the observed vertex i, the less it should contribute to i’s importance.
Thus, the closeness of i is large if the sum of distances from i to all other nodes
is small. If i is located at the periphery of the network, the sum of distances is
large and the closeness is small.
If we had a network with the adjacency matrix


0 1 0
network1 =  1 0 10  ,
0 10 0
the two outer vertices had the same closeness, because only distances are taken
into account and weights are ignored.
0.667
1
0.667
Figure 1: Example of Freeman’s Closeness
2
2.3
Past Research Concerning “Weighted Closeness”
So far, a “weighted closeness” has been used in seven papers presenting five
different versions of it. Although all of them start with Freeman’s closeness
(Freeman, 1979), each version of weighted closeness comes out different because
it was taylored to the papers’ purposes:
Toivonen et al. (2006) analyzed the relationships between trustors and trustees
from a network perspective. They use a weighted closeness as a measure of
closeness between two people, not as a measure for a single person.
Cornwell (2005) suggested a different method to make Freeman’s closeness applicable to disconnected graphs. He called this new closeness “complementweighted closeness”, because he uses the complement of a graph in order to
calculate it. The complement GC of a graph G is a network consisting of the
same number of nodes as G and of those lines that do not exist in G. He can
use the complement GC to calculate the closenesses for a disconnected graph G,
because in an undirected or symmetric network, the complement GC is always
a connected graph.
Fowler (2005, 2006a,b) analyzed the co-sponsorship network of the US legislative
from 1973 to 2004. He used the formula for the normal unweighted closeness
and defined a weight w(i, j) indicating the intensity of the relationship between
a sponsor j and a co-sponsor i:
P
a(i, j, l)
,
(3)
w(i, j) = l
c(l)
where l is a co-sponsored bill, a(i, j, l) is a binary indicator taking the value 1
if legislator i co-sponsors bill l that is sponsored by legislator j and 0 otherwise
and c(l) is the number of co-sponsors of bill l. In a next step, Fowler set the
1
distance d(i, j) as w(i,j)
. Thus, he defined the weights as distances and this way,
he could use Freeman’s closeness as a “weighted closeness”.
Erath et al. (2007) used a weighted closeness measure in order to find important
points in the Swiss transport network. They used the importance of a destination as weight, such that the weight is attributed to a vertex and not to a line.
They defined the closeness CC (i) of a destination i as
P
X j6=i Wj T Tij
P
CC (i) =
,
(4)
,j6=i Wj
j6=i
where Wj is the importance of destination j and T Tij is the travel time between
i and j.
Newman (2001) analyzed scientific collaboration networks. He calculated “the
weighted version of the closeness centrality measure [. . . ] i.e., the average
weighted distance from a vertex to all others”. He did not state any formula,
but in the course of personal communication, he said that he used a distance
measure which was based on edge lengths. The edge lengths were the reciprocals
of their weights. Closeness was then the average of the lengths of these paths
over all vertices. From this description we assume that his formula was
P
1
X j w(i,j)
,
(5)
CN C (i) =
d(i, j)
j
3
where w(i, j) is the weight of the edges needed to go from i to j. In this formula, higher line values lead to smaller closeness. So, if we calculate Newman’s
weighted closeness for network1 we get:
1.05
0.6
1.1
Figure 2: Example of Newman’s Weighted Closeness
The R sna Package (Butts, 2007, 2008) is able to calculate a weighted closeness (use closeness([weighted adjacency
P matrix])). It uses Freeman’s formula
and
replaces
the
sum
of
distances
j d(i, j) by the sum of line weights
P
w(i,
j),
so
the
formula
is
j
n−1
CRC (i) = P
.
j w(i, j)
(6)
Consequently, high line values reduce the closeness, which is an undesired effect. Additionally, as the distances disappear from the formula, each vertex j’s
contribution to i’s closeness depends on the sum of line values on the path from
i to j, no matter how far j is away. Figure 3 shows the weighted closeness of
the sna Package calculated for network1:
0.167
0.182
0.095
Figure 3: Example of Weighted Closeness of R sna-Package
The right vertex has the smallest weighted closeness although his connection
to the middle vertex is strong, whereas the left vertex’ tie to the middle node
is weak. In order to improve the way the values are weighted, we present the
edge-weighted closeness (ewc).
3
Definition of Edge-Weighted Closeness
The edge-weighted closeness is based on Freeman’s closeness. In contrast to
the versions of weighted closeness stated above, it incorporates distances as
well as line weights. The distances are weighted according to the last line value
appearing on the shortest path (geodesic) from i and j. Thus, the edge-weighted
closeness of vertex i is
P llv(i,j)
Cewc (i) =
j d(i,j)
max(lv)(n − 1)
,
(7)
where max(lv) is the maximum of the line values occurring in the observed network, n is the network size, j are all vertices that can be reached by i, llv(i, j)
is the average1 last line value occurring on the path from i to j and d(i, j) is
1 There can be several shortest paths. In this case we take the average of the last line
values.
4
the distance between i and j. Read Wasserman and Faust (1997) to see how to
obtain the distance matrix.
The denominator max(lv)(n − 1) standardizes the values, so an ewc close to
1 is associated with a high centrality, whereas an ewc close to 0 indicates low
centrality.
If we calculate the ewc for network1, we get:
0.3
0.55
0.525
Figure 4: Example of Edge-Weighted Closeness
Now the right vertex has a higher centrality because it has a larger line value
than the left node.
Line values and degree contribute in equal measure to the ewc. One degree
weights the same as a line value of one. (This can be changed by transforming
the line values.) Example:

0
network2 =  2
0
Before
standardization:
2
0
3

0
3 
0
network3
=
0 5
5 0
5
5
0.83
After
standardization:
Tabular 1: ewc for different line values and degrees in comparison
1
A vertex with two adjacency lines with values two and three has the same nonstandardized ewc as a vertex with one adjacency line with value five. However,
the standardization causes the ewc values to be not the same because max(lv)
and n change.
The code for R is provided in Appendix A.
4
EWC Compared to Other Centrality Measures
The graphs show the mailing list conversations of R-devel2 in January 2008,
visualizing six different centrality measures. In each plot, the node having the
largest centrality is marked with 1, the vertex with the second highest centrality
is labeled with 2, and so on. The vertex size corresponds to the actual centrality.
Broad dark lines indicate high communication intensity while narrow light lines
2 R-devel is a mailing list for R developers. http://stat.ethz.ch/mailman/listinfo/
r-devel. The network can be downloaded from http://www.angela-bohn.de/data.html
5
mark weak ties.
Edge−Weighted Closeness
13
Freeman's Closeness
16
18
11
18
12
17
17
11
12
4
4
3
7
8
5
3
2
14
6
1
13
5
2
6
16
10
15
10
8
14
15
1
9
Weighted Closeness of R−sna−Package
18
Edge−Weighted Degree
11
10
11
17
18
9
16
2
10
6
15
8
4
12
7
16
1
12
9
7
3
1
13
3
9
5
6
14
15
5
8
13
14
17
4
Betweenness
Edge−Weighted PageRank
6
11
18
8
17
18
17
16
3
10
3
4
6
5
7
15
12
1
9
2
7
4
2
12
2
5
9
16
11
14
14
10
8
13
13
15
7
1
Figure 5: EWC compared to five other centrality measures
The blue graph in the top left-hand corner shows the ranked results of ewc. The
large vertex at the bottom right corner has the highest ewc. Therefore so it is
marked with 1. The node labeled with 2 has the second highest ewc because
6
the large number of direct neighbors contributes to its ewc. The vertex with the
third highest ewc has a slightly smaller line values than Node 1 and a smaller
degree than Node 2. If we look at the results of Freeman’s closeness by
comparison, we see that the large vertex in the center is the number 1 vertex.
This is due to the fact that line values are not taken into account. The vertex
with the hightest ewc has only the 9th highest closeness. In contrast, the second
highest closeness node is not very important according to the 3, where it is on
the fifth position.
If we compare the ewc to the weighted closeness of the R sna package, we
see that the latter does not yield the desired results. While the fact that the
node in the center is number 1 can be discussed, the vertex on the bottom right
corner having high line values should not be on the 17th position.
The main difference between weighted degree and ewc is the fact that the
degree considers direct neighbors only. Thus, the results are similar for networks
with small maximal distance like this one.
The betweenness looks for a different type of centrality (brokerage) than the
closeness (proximity). Therefore, it gives very different results. The betweenness
looks for vertices connecting separated subgroups. Granovetter (1973) states
that such ties can be loose and yet very useful: Nodes with high betweenness
have easy access to new information and they can benefit from the power of
brokerage. In contrast, the ewc detects nodes having very close friends who
often know each-other as well. Therefore, the vertices with the highest and
third highest ewc have a betweenness of zero, because they have no brokerage
power.
The edge-weighted PageRank (Brin and Page, 1998; Kiss and Bichler, 2007)
gives similar results as the ewc, but it emphasizes the line values implicitly.
Therefore, the ranks of the third and fourth highest ewc nodes is switched by
the weighted PageRank.
5
Analysis of a Telecommunication Network
Figure 6 shows an Ego’s 3-step calls graph from an Austrian telecommunications
network. The line values represent call durations of two nodes within one month.
The broader and darker the lines, the longer two people talked to each-other.
The size of the red vertices symbolize Freeman’s closeness and the size of the
blue nodes indicate the ewc. In order to compare the two measures, the vertex
size represents the rank, not the actual centrality score. So the largest blue
vertex has the highest ewc and the second largest red node has the second
highest closeness, for example.
Vertex A has many weak ties. This causes its closeness and its ewc to be
approximately equally high, because the ewc allows a large number of loose
friends to compensate close friendships. In contrast, Vertices B and C have
relatively few friends, but they have a very strong relationship to each-other.
Consequently, their ewc is higher than their closeness. The group of blue nodes
in the bottom right corner is in periphery of the network. Therefore their
closeness is low. However, their line values and ewc3 shows that they are well
connected locally. In contrast, the group of red vertices at the bottom left corner
are closely connected to Vertex A, that is very central. Thus, their closeness is
high while the small line values causes their ewc to be low.
7
C
A
B
Ego
EWC
Freeman's Closeness
Line Values (accumulated call durations):
1s−106s
107s−3.92min
>3.92min−8.76min
>8.76min−24.73min
>24.73min−19.84h
Figure 6: 3-step ego-network-extraction of a mobile calls graph
6
Discussion
In order to create a weighted version of closeness that yields the intuitively
correct results, we introduced the “edge-weighted closeness” (ewc), which incorporates line values as well as distances. We found out, that the ewc is somewhat
the counterpiece of betweenness, because it identifies nodes having a circle of
very good friends (which is not the idea of Freeman’s closeness). The ewc has
similarities with the weighted degree and with the edge-weighted PageRank.
However, the interpretation of the graphs clearly shows the differences. The
telecommunications graph exemplary shows the differences between Freeman’s
closeness and ewc and the derivable insights into the social behavior of nodes.
The R code (Appendix A) code allows everyone to calculate the ewc.
The current definition and the R code allows the calculation of the ewc for
connected graphs only. The question of how to calculate the closeness for disconnected graphs is as old as the closeness itself and not yet finally solved. As
the line values increase the complexity of the problem, some solutions found for
the closeness cannot be carried over to the ewc. This also means that the ewc
only works for directed graphs if the network is still connected when only one
8
direction is considered.
In some applications, the line values do not correspond to a high proximity but
to great distance, like in traffic networks. A corresponding ewc version is in
progress.
Due to the large number of matrix multiplications necessary in the current code
(Appendix A), the calculation might take long for large networks. We tested
the ewc code for a graph with 3432 nodes and 9172 undirected lines on a dual
core 2.2 GHz machine with 2GB RAM. It took around 4.4 minutes on average.
As we have two variables determining the ewc, line values and distances, the
question of weighting these variables automatically arises. We are working on a
way to emphasize either the values, so that strong ties contribute more to closeness than many loose ties, or the distances, so that a large k-step neighborhood
becomes important.
Finally, we discovered that it can be interesting to see at which distance the
vertices gain most of their closeness. The most central node always gets most of
its closeness in the first distance. This does not have to be true for peripheral
vertices. Depending on the line values and the degree distribution, they might
get most of the closeness in the first distance or in higher distances. This insight
allows to group vertices according to their social roles and it is a topic a further
research.
A
EWC Code for R
### Edge-Weighted Closeness for R by Angela Bohn & Norbert Walchhofer ###
ewc <- function(vnetwork){
network <- vnetwork
network[network>0] <- 1
W <- network
Sigma <- network
wd <- colSums(vnetwork, na.rm=TRUE)
i <- 2
while(min(Sigma)==0){
Y <- vnetwork %*% W
X <- network %*% W
TM <- X
TM[TM>0] <- 1
TM <- TM-Sigma
TM[TM<0] <- 0
X <- TM * X
diag(Y) <- 0
Z <- (Y/X)/i
Z[Z == "Inf" | Z == "-Inf"] <- 0
wd <- rbind(wd,colSums(Z, na.rm=TRUE))
Sigma <- Sigma + TM
W <- W %*% network
i <- i + 1
}
wd <- wd/(max(vnetwork)*(n-1))
print(colSums(wd))
9
}
Example:
vnetwork <- cbind(c(0,1,0),c(1,0,10),c(0,10,0))
ewc(vnetwork)
[1] 0.300 0.550 0.525
The code can be downloaded from http://www.angela-bohn.de/publications/
ewc_code.html.
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