Information Technology Reference
In-Depth Information
Fig. 8.2
Scenarios that may increase or decrease individual terms in the modularity metric
by the dashed link
e
5,10
in the top diagram in Fig.
8.2
. The new link
e
ij
will increase
the degree of nodes
i
and
j
by one, i.e.
deg
(i) will become
deg
(i)
C
1. The total
number of edges
m
will increase to
m
C
1. A simple calculation at the bottom
of Fig.
8.2
shows that terms in the modularity formula involving blue links will
decrease from their previous values. However, if the network has clusters such as
C
A
with no changes in node degrees, then the corresponding values of terms of
lines in red will increase from their previous values as the denominator increases
from 2
m
to 2(
m
C
1). In summary, the updated modularity may increase as well
as decrease, depending on the structure of the network and where the new link
is added. With this particular definition of modularity, between-cluster links are
always associated with a zero valued term in the overall modularity formula due
to the Kronecker's delta. What we see in the change of modularity is a combination
of results from several scenarios that are indirectly affected by the newly added link.
We will introduce our next metric to reflect the changes in terms of between-cluster
links directly.
Search WWH ::
Custom Search