Topological Clustering for Geographical Networks - Methods for Multilevel Analysis and Visualisation of Geographical Networks - page 44

mathematical definition, the parameter k corresponds to the tolerance (in terms
of lack of ties) inside the subgroup in comparison to the maximal cohesion
theoretically observed inside cliques. In the geographical application shown in
Fig. 3.12 , this criterion has been adjusted with the edge-density and two other rules
relating to the adjacent edges of each node inside the cluster.

3.4.4 Subgroups Built by Comparing Intra-

and Inter-Group Edges

The previous criteria are based on the properties of inner subgroup ties. Indeed,
members in cohesive social network groups are closer to each other than to outer
members.
Therefore, to identify a cohesive subgroup inside a graph, it is relevant to
consider:

￿ not only the density of edges inside the subgroup (as we did with exhaustiveness,

reachability and degree criteria),

￿ but also the intensities and the comparative frequencies of intra- and inter-

group ties.

Lambda-sets are subgroups built from this second criterion. For such subgroups,
cohesion is linked to the notion of connectivity: the identified subgroups must be
difficult to disconnect by edge removal. Thus, lambda sets are maximal subgraphs
for which it is harder to disconnect an inner node pair than a pair made up of an
inner node and an outer node.
In other words, the connectivity index of inner node pairs must be greater than
the same index computed for pairs composed of an inner node and an outer node
(cf. Fig. 3.13 ).
The computational methods used to identify lambda-sets are difficult to imple-
ment. However, the criterion based on the comparison between inner and outer ties
may be relevant for graphs whose clusters:

￿ may be made up of nodes that are not close to each other (actually, this criterion

does not depend on degree or reachability criteria),

￿ and need to be disjoint (as they are defined, lambda-sets do not overlap each other

unless they are included in each other).

Thus, Amiel, Melançon, and Rozenblat ( 2005 ) use a similar method to identify
clusters of international airports, given that the airports of a given cluster have to be
better connected to each other than to other airports. With this method, the authors
aim to explain how air relations are organized and to highlight their structures at
different scales (from global to local).
In practice, Amiel et al. build a graph of the air relations (on this graph, the nodes
represent the airports and the edges correspond to air relations between airports). On
this graph, they consider each edge and determine whether it can be part of a cluster.