Biomedical Engineering Reference
In-Depth Information
Figure 2
. The clustering coefficient
C
offers a measure of the degree of interconnectivity
in the neighborhood of a node. For example, the red node whose neighbors are all con-
nected to each other has
C
= 1 (left), whereas the red node with no links between its
neighbors has
C
= 0 (right).
the neighbors of a particular node are connected to each other (Figure 2). For
example, in a friendship network
C
reflects the degree to which friends of a par-
ticular person are friends with each other as well. Formally, the clustering coef-
ficient of node
i
is defined as
2
n
C
=
,
i
[2]
i
kk
(
)
i
i
where
n
i
denotes the number of links connecting the
k
i
neighbors of node
i
to
each other. Accordingly, we can define the average clustering coefficient as
1
N
=
.
C
C
[3]
i
N
i
=
1
An additional important measure of the network's structure is the function
C
(
k
),
defined as the average clustering coefficient of all nodes with
k
links. If
C
(
k
) is
independent of
k
, the network is either homogeneous or it is dominated by nu-
merous small tightly linked clusters. In contrast, if
C
(
k
) follows
C
(
k
) ~
k
-1
, the
network has a hierarchical architecture, meaning that sparsely connected nodes
are part of highly clustered areas (12,27,46,47). In such hierarchical networks
communication between the different highly clustered neighborhoods is main-
tained by a few hubs.
As we will see below, the degree distribution
P
(
k
) and the
k
dependence of
C
(
k
) can have generic features, allowing us to classify various networks. Pa-
rameters such as the average degree <
k
>, average path length <
l
>, and average
clustering coefficient <
C
> characterize the unique properties of the particular
network under consideration, and therefore are less generic.
