Mapping Science - Mapping Scientific Frontiers: The Quest for Knowledge Visualization - page 261

8.1.3.1 Modularity Change Rate (MCR)

Given a partition of a network, i.e. a configuration of clusters, the modularity of
the network measures the degree of interconnectivity among the groups of nodes
identified by the partition. If different clusters are loosely connected, then the
overall modularity would be high. In contrast, if clusters are interwoven, then
the modularity would be low. We follow Newman's algorithm (Newman 2006 )
to calculate the modularity with reference to a cluster configuration generated by
spectral clustering (Chen et al. 2010 ; von Luxburg 2006 ). Suppose the network G is
partitioned by a partition C into k clusters such that G D c 1 C c 2 C ::: C c k , Q ( G )
is defined as follows, where m is the total number of edges in the network G. n is
the number of nodes in G. •(c i ,c j ) is known as the Kronecker's delta. It is 1 if nodes
n i and n j belong to the same cluster and 0 otherwise. deg ( n i ) is the degree of node
n i . The range of Q ( G ) is between 1 and 1.

A ij !

deg .n i / deg n j
2m

X

n

ı c i ;c j

1
2m

Q.G; C/ D

i;j D 0

The modularity of a network is a measure of the overall structure of the network.
Its range is between 1 and 1. The Modularity Change Rate of a scientific paper
measures the relative structural change due to the information from the published
paper with reference to a baseline network. For each article a , and a baseline network
G baseline , we define the Modularity Change Rate (MCR) as follows:

Q.G baseline ;C/ Q.G baseline ˚ G a ;C/
Q.G baseline ;C/

MCR.a/ D 100

where G baseline ˚ G a is the updated baseline network by information from the article
a . For example, suppose reference nodes n i and n j are not connected in a baseline
network of co-cited references but they are co-cited by article a , a new link between
n i and n j will be added to the baseline network. In this way, the article changes the
structure of the baseline network.
Intuitively, adding a new link anywhere in a network should not increase the
modularity of the network. It should either reduce it or leave it intact. However, the
change of modularity is not a monotonic function as we initially expect. In fact, it
depends on where the new link is added and how the network is structured. Adding
a link may reduce the proportion of the modularity in some clusters, but it may
increase the modularity in other clusters in the network. Thus, the overall modularity
change is not monotonic.
Without losing any generality, assume that an article adds one link at a time
to a given baseline network. If the new link connects two distinct clusters, then
it has no effect on the corresponding term in the updated modularity because by
definition • ij D 0 and the corresponding term becomes 0. Such a link is illustrated