Geography Reference
In-Depth Information
for node connectivity. They propose a simple growth model in which at each time step a
new node is added and preferentially attaches itself to the node with the highest degree.
More precisely, the probability that a new node attaches itself to an existing node is
exactly proportional to the latter's degree. The specii cation of this mechanism rel ects
the benei ts of linking to nodes with high degrees, as such 'hubs' provide new nodes with
short pathways to many other nodes in the network. Assuming each node attaches to
only one existing node, this mechanism leads to a power law distribution with an expo-
nent equal to three.
In reality, most networks have degree distributions that are dif erent from the pure
Barabasi-Albert model. In particular, the degree of the best connected nodes is gener-
ally less than the model predicts. Indeed, the tendency of i rms to connect to highly
connected i rms is found to be not as strong (Powell et al., 2005; Ter Wal, 2009). One
explanation holds that i rms are limited in the number of network relations they can
meaningfully maintain. In the case of inter-i rm networks, it is obvious there are limits
to the number of partners a i rm can maintain (Holme et al., 2004). This implies that
well-connected nodes typically refuse proposals for networking and will select only the
most benei cial partners (Giuliani 2007). A second reason why the degree distribution
is less skewed than one would predict from preferential attachment is that proximity
matters. This means that new nodes - even though attracted by the ones with highest
connectivity - often connect to nodes with lower degrees if these are more proximate
in any of the i ve dimensions we outlined before. Consider, for example, geographical
proximity. A company may opt to collaborate locally to save on travel time and trans-
portation costs, even though companies with the highest connectivity are located in
other countries. The preferential attachment model can be easily adapted to incorporate
this ef ect of proximity by assuming that the probability of a node linking to an existing
node is not only dependent on the latter's degree but also on the geographical proximity
between them (GuimerĂ  and Amaral, 2004). The same reasoning holds for other forms
of proximity. Depending on the benei ts of proximity, such a constraint yields dif erent
network structures, ranging from very skewed degree distributions and low clustering as
in the original Barabasi-Albert model when overcoming distance (in whatever dimen-
sion) is cheap, to networks with a normal degree distribution and a high clustering as in
small worlds (Watts and Strogatz, 1998; Zhang et al., 2004) when overcoming distance
is rather expensive, to an empty network where any relation is just too expensive to
establish.
Closure
Another driving force of network formation is closure. In many instances, new network
relations follow from existing relations as two actors are introduced to one another by
a third actor with whom both already have a relation. The probability of the two actors
forming a relation, who already relate to a common third, is expected to be much higher
than the probability of two actors forming a relation who do not relate to a common
third. The establishment of such triangle relationships is called 'closure' and such a
closure mechanism will increase the degree of clustering (in a network sense) over time.
The reason for closure to be common is twofold: (1) each actor can be informed by the
common third about the properties of the other (what knowledge it possesses) and trust-
worthiness of the other, and (2) once the relationship is formed each actor has less incen-
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