Introduction - Methods for Multilevel Analysis and Visualisation of Geographical Networks - page 4

explicitly reintroduced by Watts in the 1990s ( Watts , 1999 ; Watts & Strogatz ,
1998 ). Pred ( 1974 , 1977 ) identified the potential consequences of the structural
properties of these networks for geographical places, particularly for cities, which
accentuate “small world” structures in virtue of internal connectivity and interurban
channels. The organisation of the overall society into a “space of networks and
space of places” ( Castells , 1996 ) create complex systems of “multiterritories”
( Hassaert da Costa , 2004 ) with properties that exhibit multilevel, multiscale and
multidimensional characteristics.

1.2 Scale-Free Networks

A multi-territory is a system of interconnected territories. However, every territory
does not exhibit the same level of connection and integration within the system
because the hierarchy is based on past uneven diffusions and new diffusions of
innovations. Cities with established connections to multiple networks are more
likely to attract new connections, as demonstrated by Batty's simulation ( 2006 )of
the formation of urban systems. Batty found that preferential attachment provided a
better explanation for the formation of urban system hierarchies, which conform to
Zipf's law ( 1949 ) rather than to Gibrat's “law of proportional effect” ( Gibrat , 1931 ;
Guerin Pace , 1995 ; Pumain , 1982 ). Airline networks illustrate this process: cities
of secondary importance attempt to become airline hubs to more rapidly develop
links to the principal worldwide network ( Amiel, Melançon, & Rozenblat , 2005 ;
Guimerà, Mossa, Turtschi, & Amaral , 2005 ).
The scale-free model describes the distribution of networks as a type of power
law “because a power law is the only distribution that is the same whatever
scale we look at it on” ( Newman , 2005 , p. 334). In the same paper, Newman
noted that “'Zipf's law' and 'Pareto distribution' are effectively synonymous with
'power-law distribution”' (p. 327). Newman found that although the Yule “richer-
get-richer” effect continued to explain power-law distributions ( Simon , 1955 ; Willis
&Yule , 1922 ; Yu l e , 1925 ), it also could reflect processes of self-organising
systems and critical phenomena. The power law effect is estimated by two kinds
of adjustments: the log-log distribution of the cumulative frequencies of a variable,
which is commonly termed a “power law distribution”, or the interaction of the
log-distribution of a variable with the log-distribution of another variable describing
the general size of the system, which is termed a “scaling law distribution”, which
characterises certain functions or specific phenomena based on cities' populations
( Pumain , 2006 ).
Although physicists have proposed a single “universal” model, empirical studies
reveal many different cases and interpretations ( Newman , 2005 ). For example, three
classes of “small-world” networks have been identified based on their adjustments
of “power law distributions” ( Amaral, Buldyrev, Havlin, Salinger, & Stanley , 1998 ),
and “Apollonian networks” specifically describe networks as simultaneously scale-
free, ultra-small-world, Euclidean, matching and space-filling ( Andrade, Herrmann,
Andrade, & Da Silva , 2005 ).