A Small World Perspective on Urban Systems - Methods for Multilevel Analysis and Visualisation of Geographical Networks - page 20

explanations of the Zipf law proposed by Simon ( Simon , 1955 ), referring to the
“Yule process” ( Yu l e , 1925 ). This process thus induces the degree distribution of
the nodes to follow a power law; that is, the frequency of nodes of degree k is
approximately given by p

. In other words, a sharp inequality exists
between strongly central entities and others who cling to these centralities (the
interested reader is referred to Bornholdt & Schuster ( 2003 )orNewman( 2010 ),
for instance).
The scale-freeness of spatial networks makes sense, as the forthcoming sections
shall explain. However, this property makes their visual exploration and (automatic)
analysis much harder. Most often, networks show both characteristics simultane-
ously. That is, networks do contain small communities of interacting entities, while
the overall organization of the network is dominated by a scale-free process. The
high-degree nodes thus make it hard to identify communities or bridges and hide the
small-world structure. Hybrid approaches must be designed to reveal the structure
and properties of these complex networks.

( k ) ∼ c · ex p ( − k )

2.3 Visualizing Topological Proximities in Geographical

Networks

Visualizations are common tools for geographers, who use cartography to stress
geographical organizations and patterns. Nevertheless, because geographical dis-
tance is not the only factor constraining a system and its dynamics, a topological
representation can be much more useful to highlight organizational structures.
A basic geographical example is the worldwide container-traffic networks, for
which visualization allows a good overview of the overall structure (Fig. 2.1 )
( Ducruet, Rozenblat, & Zaidi , 2010 ). On one hand, the representation ignores the
geographical coordinates of ports. On the other hand, the ports' location in the graph
is determined by applying a layout based on topological proximities for maritime
exchanges. Therefore, the geographical situation is preserved due to the continuum
of movements from one port to another.
Fundamentally, two main sub-systems exist: Asia-Pacific and Europe-Atlantic.
These sub-systems are connected through a limited number of nodes: the Panama
and Suez canals. Removing those two global pivots would thus split the world
system in two parts, although one may notice some other links (e.g., Magellan Strait
and Cape of Good Hope), but those links remain very limited compared to the main
trunk lines concentrated at the two canals. Also, port ranges appear, such as the
Scandinavian range in the top left of the Fig. 2.1 , close to American ranges.
Another example maintaining some geographical position in a topological
network representation is the movement of commuters (Fig. 2.2 ). The example of
Switzerland shows especially the linguistic proximities among municipalities with
three visible isolated zones: French Swiss speaking at the left, Italian speaking at the
right, and Swiss German in the center, cut into two parts between a system around
Bern (the political federal capital) and another around Zürich (the economic capital).