Spatially-embedded networks: Visualisation
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Current techniques for evaluating networks do not adequately meet the
needs of those trying to understand real-world spatial systems. Not
only are such systems vastly larger, more heterogeneous, and more
multi-variate than current techniques will cope with, but the scale of
interest has expanded in recent years from aggregate-level whole-system
understanding to a desire to understand how network embeddedness
effects individuals within a network across the whole range of scales
down to the individual level. As such, dynamics on networks are much
more important than blank network topology. Indeed, even when such
topology is of interest, it tends to be the evolution of such
structures that is interesting, and this tends to relate, at least in
part, to variables above and beyond simple topologies. As a simple
example; it is not enough to say that large cities tend to have a
larger number of connections to other cities, one also has say the
large cities attract more connections in a manner that isn’t just
related directly to the number of connections; the city size itself is
important. The development of such non-network properties under the
constraints of a network topology are a key area of research. Even for
a network that has reach topological stability or equilibrium, the
continuing dynamics associated with non-network properties, and their
relationship to network topology, are important. However, the tools we
have to analyse such systems are very limited. Most network-centred
tools concentrate on topology, and the evolution of topology as a
result of topological constraints.
As such, this section of the project looked at developing the initial
tools to visualise the properties associated with spatially embedded
networks.
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Characterising the evolution of networked properties
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The dynamic evolution of networked properties can be fruitfully explored through Rank-Size Clocks.
These clocks show the evolution of components within a network over time where variation is highlighted by a change in
the rank of a node within a network. As such they highlight the general system dynamics while also
concentrating attention on the importance of individual components within the system.
Such clocks can be combined with traditional graphs of aggregate metrics to show how aggregate
emergent properties do not demand individual-level consistency. An example is the power-law distribution of city sizes.
While at the aggregate level the size-distribution within a network of cities follows the
classic rank-size “Zipf” power-law, rank-size clocks of the data show that this constant
representation is actually the result of a wide variety of movements up and down the ranking
for individual cities. This suggests that the current explanatory model of this distribution,
which concentrates on growth proportional to the importance of nodes within the network, is
erroneous and the dynamics would be more appropriately characterised as emergent.
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Rank-clock of the top 100 cities in the USA, ranked by size. Cities
can be seen entering the clock as it progresses clockwise and the
number of cities increases. Once 100 cities exist, individual cities
can be seen rising and falling in rank with considerable variation. In
the situation where each city reached a rank (say, the top city) and
stayed there, the curves would be concentric.
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Batty, M. (2006) Rank Clocks. Nature, 444, 592-596.
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Characterising the evolution of interacting properties on networks
Recurrence plot of daily rainfall at a location contrasting each 25
day running average across the year with the other 25 day averages.
Running avergaes reveal broader patterns than day-on-day variation.
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Rank-size clocks show dynamic trends in networked properties, but
don't reveal the interaction between network nodes that may allow
us to understand the underlying behaviours generating these patterns.
Plainly this requires node-to-node comparison, with an inherent
rise in data complexity then needs addressing. Recurrence Plots provide
one methodology for such comparisons. While they simplify the presentation of individual
components, in doing so they reveal hidden structures in the data
and allow enhansed understanding of the co-evolution of nodes.
Recurrence plots can be used to reveal the nonstationarity of time series datasets and their degree of aperiodicity,
making them a valuable tools for characterising complex dynamics in the time domain. Such plots
can be used to reveal the structure in time series by comparing a single variable at a location
with itself over time. Running averages can then be used as part of this process to improve pattern clarity.
More importantly here, recurrence plots can also be
utilised to show variation over time of two linked nodes to show mutual periodicities.
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Heppenstall, A.J. and Ross, A.N. (2007) An Application of Agent-Based Modelling for Investigating
the Emergence of Structure and Dynamic Processes within a Complex Retail Market.
European Academy of Management Paris, May 16 - 19.
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Characterising the behaviour informing properties on networks
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Recurrence plots therefore present a useful tool for elucidating
network dyanmics over time. However, they don't provide any insight into
the underlying behaviour acting to create these dynamics. This final
challenge can be approached through the use of Genetic Algorithms, which
can be utilised to provide understanding of the underlying system dynamics.
Genetic Algorithms tend to be utilised as blank optimisation techniques,
however the optimisation process is clearly the point at which real data
about a system is squeeze into an abstract model of a system; stretching the model
to better fit reality. This process can therefore be used to generate information about how well
the model may map onto reality, the kinds of model elements that fit with
the data, and the kinds that don't, and, equally, what data is key to
fit in, and which isn't important for a specific model. For example, given a multi-variable
model of an economic network, a Genetic Algorithm may reject a given variable,
include it strongly, or flip between regarding it as important or swapping it with
another. As the algorithm progresses, careful interpretation of the optimisation
path allows researchers to reflect on the model as a framework for the real-world
data, and its quality as a representation of the behaviour presented on
a real-world network.
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Genetic Algorithm estimates of levels competing petrol stations are willing to overprice their
competitors within 3km on a road network.
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Heppenstall, A.J., Evans, A.J. and Birkin, M.H. (2007), Genetic
Algorithm Optimisation of a Multi-Agent System for Simulating a Retail Market. Environment and Planning B, 34 (4).
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In summary, then, the new or novelly applied techniques above do much to elucidate the dynamics
of complex real-world networks. They provide a suit of tools for examining the degree to which
global dynamics are played out at the local level, the manner in which nodes respond to
each other, and the underlying behaviour driving these changes. While they are by no means a
final toolkit for examining real spatial networks, with all their multi-variate complexity, they
take us a significant step forward in detailing their dynamics.
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