L
E
the minimum value is 2, corresponding to the
index 4: in this case the center c
E
is K I.
Closeness. Next we consider the second type of lo-
cation problems - the minisum location problem, of-
ten also called the median problem or service facility
location problem. Suppose we want to place a ser-
vice facility, e.g., a shopping mall, such that the to-
tal distance to all customers in the region is minimal.
This would make traveling to the mall as convenient
as possible for most customers. We denote the sum
of the distances from a vertex u ∈ V to any other ver-
tex in a graph G as the total distance
∑
v∈V
d(u, v).
The problem of finding an appropriate location can be
solved by computing the set of vertices with minimum
total distance. In social network analysis a centrality
index based on this concept is called closeness. The
focus lies here, for example, on measuring the close-
ness of a person to all other people in the network.
People with a small total distance are considered as
more important as those with a high total distance.
Various closeness-based measures have been devel-
oped, see for example (Bavelas, 1950; Beauchamp,
1965; Moxley and Moxley, 1974; Sabidussi, 1966)
and (Valente and Foreman, 1998). The most com-
monly employed definition of closeness is the recip-
rocal of the total distance
c
C
(u) =
1
∑
v∈V
d(u, v)
In our sense this definition is a vertex centrality, since
c
C
(u) grows with decreasing total distance of u and
it is clearly a structural index. Before we discuss
the competitive location problem, we want to men-
tion the radiality measure and integration measure
proposed by Valente and Foreman (Valente and Fore-
man, 1998). These measures can also be viewed as
closeness-based indices. They were developed for di-
graphs but an undirected version is applicable to undi-
rected connected graphs, too. This variant is
c
R
(u) =
∑
v∈V
(4
G
+ 1 − d(u, v))
n − 1
where 4
G
and n denote the diameter of the graph and
the number of vertices, respectively. The index mea-
sures how well a vertex is integrated in a network. The
better a vertex is integrated the closer the vertex must
be to other vertices. The primary difference between
c
C
and c
R
is that c
R
reverses the distances to get a
closeness-based measure and then averages these val-
ues for each vertex.
Based on such method, we define a procedure to
compute the center of the graph as described in the
Algorithm 2. As for the eccentricity, we initialize the
closeness vector L
C
and calculate the matrix M. Then
for each node with index i we calculate the sum of
Algorithm 2: Center computation by closeness.
Input : The graph G
Output: The center c
C
1 n ← V.length;
2 L
C
← InitializeArray(n);
3 M ← FloydWarshall(G);
4 for i ← 0 to n do
5 L
C
[i] ←
∑
n−1
j=0
M[i][ j];
6 i
min
← MinIndex(L
C
);
7 c
C
← V [i
min
];
8 return c
C
;
distances from the other nodes (lines[4-5]). Finally,
as for the eccentricity, we calculate the index i
min
of
the minimum value in L
C
. Such index corresponds
to the center c
C
in G. Referring again to our exam-
ple, given the matrix M by the Floyd-Warshall algo-
rithm, we have the following closeness vector L
C
=
∞ ∞ ∞ 11 ∞ ∞ ∞ ∞ ∞
t
. Since the
minimum value is 11, i
min
is 4 (i.e. the center is K I).
Centroid Values. The last centrality index presented
here is used in competitive settings. Suppose each
vertex represents a customer in a graph. The service
location problem considered above assumes a single
store in a region. In reality, however, this is usually
not the case. There is often at least one competitor
offering the same products or services. Competitive
location problems deal with the planning of commer-
cial facilities which operate in such a competitive en-
vironment. For reasons of simplicity, we assume that
the competing facilities are equally attractive and that
customers prefer the facility closest to them. Con-
sider now the following situation: a salesman selects
a location for his store knowing that a competitor can
observe the selection process and decide afterwards
which location to select for her shop. Which vertex
should the salesman choose? Given a connected undi-
rected graph G of n vertices. For a pair of vertices
u and v, γ
u
(v) denotes the number of vertices which
are closer to u than to v, that is γ
u
(v) = |{w ∈ V :
d(u, w) < d(v, w)}|. If the salesman selects a vertex u
and his competitor selects a vertex v, then he will have
γ
u
(v) +
1
2
(n − γ
u
(v) − γ
v
(u)) =
1
2
n +
1
2
(γ
u
(v) − γ
v
(u))
customers. Thus, letting f (u, v) = γ
u
(v) − γ
v
(u), the
competitor will choose a vertex v which minimizes
f (u, v). The salesman knows this strategy and calcu-
lates for each vertex u the worst case, that is
c
F
(u) = min{ f (u, v) : v ∈ V − u}
c
F
(u) is called the centroid value and measures the
advantage of the location u compared to other loca-
tions, that is the minimal difference of the number of
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