Information Technology Reference
In-Depth Information
from 0 to 1, the sparser is a graph, the lower is its value. When dealing with
weighted networks, a useful generalization of this quantity is represented by the
weighted-density
k
w
, which evaluates the intensities of the links composing the
network. The mathematical formulation of the network density is given by the
following:
∑
∈
k
(
A
)
=
w
w
ij
(10.13)
i
≠
j
V
Where
A
is the adjacency matrix and
w
ij
is the weight of the respective arc from
the point
j
to the point
i
.
V=1…N
is the set of nodes within the graph.
10.3.2.
Node strength
In the same way, the simplest attribute of a node is its connectivity degree, which
is the total number of connections with other vertices. In a weighted graph, the
natural generalization of the degree of a node
i
is the node strength or node
weight or weighted-degree. This quantity has to be split into in-strength
s
in
and
out-strength
s
out
, when directed relationships are being considered. The strength
index integrates the information of the links' number (degrees) with the
connections' weight, thus representing the total amount of outgoing intensity
from a node or incident intensity into it. The formulation of the in-strength index
s
in
can be introduced as follows:
∈
s
(
i
)
=
w
in
ij
(10.14)
j
V
It represents the whole functional flow incoming to the vertex
i
.
V
is the set of the
available nodes and
w
ij
is the weight of the particular arc from the point
j
to the
point
i
. In a similar way, for the out-strength:
∈
s
(
i
)
=
w
out
ji
(10.15)
j
V
It represents the whole functional flow outgoing from the vertex
i
.
10.3.3.
Strength distributions
For a weighted graph, the arithmetical average of all the nodes' strengths
<s>
only gives little information about the distributions of the links intensity within
the system. Hence, it is useful to introduce
R
(
s
) as the fraction of vertices in the
graph that have strength equal to
s
. In the same way,
R
(
s
) is the probability that a
