Use of Systems Theory, Directed Graphs, and Pulse Process Models in an Adaptive Approach to Agroecosystem Health and Sustainability - Integrated Assessment of Health and Sustainability of Agroecosystems - page 93

enable comparisons between graphs (Sorensen, 1978). There are several mathematical
approaches for analyzing signed digraphs based mostly on graph theory, matrix alge-
bra, and discrete and dynamic system models (Harary et al., 1965). The approaches
fall into two broad categories: arithmetic and geometric (Roberts, 1976b).
The aim of geometric analysis is usually to analyze the structure, shape, and pat-
terns that may impart important characteristics to the system. A typical geometric
conclusion is that some variable will grow exponentially or that some other variable
will oscillate in value. The numerical levels reached are not considered important in
such predictions (Roberts, 1976b). Geometric analysis of a signed digraph includes
(1) tracing out the different causal paths (Axelrod, 1976a), (2) identification of feed-
back loops (Roberts, 1976b), (3) detection of path imbalance (Nozicka et al., 1976),
(4) assessment of stability (Roberts, 1976a), (5) calculation of the strong components,
(6) assessment of connectedness (Roberts, 1976b), and (7) assessment of the effects
of different strategies (a change in the structure of the system) on system character-
istics (Roberts, 1976a).
Arithmetic analyses proceed from the perception of the signed digraph as a
dynamic system in which an element obtains a given value with each unit change
in time (or space) of another. The values obtained depend on previous changes in
other variables. The simplest assumption about how changes of value are propagated
through the system is the so-called pulse process (Roberts, 1971). By assuming that
change in values in the model follows a specified change-of-value process (such as
the pulse process), (1) stability can be assessed even for path-imbalanced digraphs,
(2) the effect of outside events on the system can be studied, and (3) forecasts can
be made. Roberts (1976a) cautioned that results from arithmetic analyses should be
regarded as suggestive and verified by further analysis since digraphs—as models of
a complex system—are not precisely correct due to oversimplifications made in the
modeling process.
This chapter describes the formulation of a problem-determined holon for an
agroecosystem and its analysis using graph theory and dynamic modeling tech-
niques. The overall objective was to gain an insight into the communities' definition
of health and to identify the factors they considered to be the most influential in
terms of the health and sustainability of their agroecosystems. This analytic frame-
work served as a basis for selecting indicators and in interpreting them. Specifically,
the objectives were (1) to assess how communities in the agroecosystem perceived the
interrelationships between problems, goals, values, and other factors; (2) to evaluate
what the communities perceived to be the overall benefits of various agroecosystem
management strategies; (3) to determine what would be the most relevant measures
of change in the problem situation; and (4) to find what would be the long-term
effects of various strategies and management policies, assuming that the communi-
ties' assertions were reasonably accurate depictions of the problem situation.
4.2 PRocess and metHods
Cognitive maps (also known as loop models, influence or spaghetti diagrams) were
defined as models that portrayed ideas, beliefs, and attitudes and their relationship to
one another in a form amenable to study and analysis (Eden et al., 1983; Puccia and