integration of these models with diverse datasets that can be aligned to spatial and
temporal dimensions of the process model outputs. For example, this is currently being
done, to some extent, in the context of climate change modeling whereby dynamic,
spatially explicit climate models are coupled to ocean circulation models and terrestrial
biosphere models. However, this coupling imposes major computational challenges and
thus one model may simply serve as “boundary conditions” or “inputs” to another model
(e.g., climate model outputs are often treated as fixed input into terrestrial vegetation
models), and feedbacks between the models have been relatively difficult to
accommodate. At this time, computational methods do not exist for effectively integrating
such coupled models with the plethora of data available at the different temporal and
spatial scales, especially if placed within a probabilistic data-model integration
framework. Development of probabilistic methods for incorporating such feedbacks and
linking coupled process models to diverse datasets will be necessary for advancing
sustainability science, particularly in the context of forecasting future, multi-dimensional
states.
3.8 Complex Networks
The complexity in a sustainable society can be captured mathematically by
graphs (commonly called networks, which are used in a completely different context than
the aforementioned references to sampling or monitoring “networks”). There is an
embryonic discipline of “complex networks,” populated by physicists, statisticians,
computer scientists, epidemiologists, mathematicians, etc; see, for example, Kolaczyk
(2009) and Newman (2010). Complex networks are models of how “the world works,”
but they are extendable to allow for more complexity or more variables. That strength is
also a weakness; network sizes and complexities can clearly grow exponentially.
However, it does offer a paradigm to understand growth and its counterpart, recession.
In fact, by definition sustainability will require both growth and recession in different
sectors, in different regions, and at different times. To build, measure, and assimilate a
complex network is a worthy endeavor, but destined to fail. An analogy would be to try to
track every gas molecule in the atmosphere over time. To study complex networks, we
could move away from their mathematical building blocks (vertices and edges) and
consider instead identifiable “objects” made up of those building blocks and/or study
local densities (“fields”) within the network. The dependencies implied by the network are
expressed through conditional distributions. There is an important theoretical problem
that involves construction of the joint distribution from the conditional distributions
implied by the network. This will involve a generalization of the Hammersley-Clifford
Theorem (e.g., Cressie, 1993, Ch.6) for complex networks.
- Recommendations
Mathematical scientists should be encouraged to build and evaluate tools that
are not restricted to any particular field of application but that exploit the commonalities
among the different fields contributing to the sustainability science. There is a need to
develop methodologies for the iterative process of combining data and sampling designs
with models, leading to forecasting. Specific examples of problems to be solved include:
estimation of parameters in complex models based on diverse data sources, evaluating
the agreement between complex process models and observational data, and
integrating data from different sampling designs.