A Small World Perspective on Urban Systems - Methods for Multilevel Analysis and Visualisation of Geographical Networks - page 27

Fig. 2.7 Identification of hubs based on contribution indexes

Whatever method of classification is used, these groups imply some relative closure
due in part to the property of transitivity, as previously discussed. In other cases,
the network organizes into a star-shaped structure built around a central node that
acts as a hub connecting the whole group to the outside world. This phenomenon
can be tracked by measuring the relative density or “cohesion” of a group. As in
classical classifications, some nodes (like cities) contribute more than other nodes
to the cohesion of their groups, which may be seen as the “contribution” of each
point to its group. Conversely, each city can also have links going out to the group,
calling for a measure of the “participation” of a node in other groups in the network,
as explored by ( Guimerà et al. , 2005 ). Contribution and participation indexes can be
measured as individual shares in one node to one group or the opposite (details are
discussed in the next chapter).
These indexes can help identify hubs in networks The role of hubs is often
concentrated in or between particular communities. Guimerà et al. ( 2005 )presented
a z -score and “participation coefficient” of nodes that can be extended to take edge
weights into account. The z -score of a node measures how many of the node's con-
nections are devoted to its own cluster, while a node's participation coefficient mea-
sures how much its connections cover all of the other clusters in the network. Be-
cause we cluster the graph into a hierarchy of subgraphs, we can moreover measure
these indices at every level and study how the indices vary down the hierarchy of
clusters. Again, the full definition of these measures is delayed until the next chapter.
The figure below provides a good illustration (Fig. 2.7 ). The nodes have been
sized according to their z -score (internal connections) and colored according to
their participation coefficient (external connections). For example, Hong-Kong has
a greater role at the worldwide level than at the Asian level.
Looking at node contribution and node participation as measures of interactions
between nodes and groups and computing entropy help to provide a global view of