Non-Linear Dynamics, Self-Organization and Cellular Automata Models - Environmental Modelling: Finding Simplicity in Complexity - page 49

more mathematically tractable. This assumption is at the
heart of much present-day mathematical modelling. The
assumption is a reasonable one for systems in which self-
organization does not take place, 10 but for those systems
that self-organize, to ignore the complex within-system
interactions that give rise to that self-organization is to
throw the baby out with the bathwater.

susceptible to local perturbations: thus any living thing
will be adversely affected by disturbance of even small
parts of its body. Local and global emergence presumably
form end-points in a continuum.

4.2.4 Self-organizedcriticality

Studies by Bak and co-workers (Bak et al ., 1988; Bak,
1996) on sand-pile models have provided a number
of insights into other generic aspects of complex sys-
tems. This work on so-called 'self-organized criticality'
(SOC) suggests that the presence of power law frequency-
magnitude relationships, 1/f properties of time-series
data, and spatial fractality form a kind of fingerprint
for SOC (Bak, 1996; Buchanan, 2000). This view sug-
gests that self-organization can be manifested not only
in emergent pattern formation, but also in terms of the
internal dynamics of systems (see also Chapter 16).
'Events' in SOC systems are of all magnitudes, from the
smallest to the largest that the system will support; there
is no 'most common' size of event. These events maintain
the whole system in a particular kind of dynamic equi-
librium (Bak, 1996; Phillips, 1999a: Sornette, 2006). The
canonical example of SOC is an idealized and frictionless
sandpile (Bak et al ., 1988). In such a sand pile, avalanches
of a wide range of sizes occur, from the movement of a
single grain up to a major avalanche, the maximum size
of which is only constrained by the size of the sandpile.
When the frequency-magnitude distribution of all these
avalanches is plotted on a graph, the result is a power-law
distribution. This power-law frequency-magnitude dis-
tribution is evidence for a critical state of the system, to
which the system readjusts after disturbance.
However this claim was challenged for real-world sand-
piles by Frette et al . (1996), who concluded that whereas
some models of sandpiles do show SOC, this behaviour
is not universal. Instead of using simulated frictionless
sand grains, Frette et al . used two different types of rice.
Only one of these displayed the expected SOC behaviour.
In general, laboratory experiments on real sandpiles have
not found consistent evidence of criticality. As well as
friction, this discrepancy may be related to inertial and
dilatational effects (Sornette, 2006). Thus any claim for
the universality of SOC, even in the rather limited domain
of real-world sandpiles, is questionable.
Further, even when power-law relationships are found
in real-world data, this is not unequivocal evidence for
SOC. Carlson and Doyle (1999) propose highly opti-
mized tolerance as a further mechanism to produce
power-laws; Sornette (2006) lists a further ten ways to

4.2.3.3 Emergence
The roots of the notion of emergence go back at least to
c.330 BCE with Aristotle's description of synergy: 'The
whole is more than the sum of its parts' ( Metaphysics ,
Book H 1045a 8-10). An emergent response is synergistic,
but 'more so' (in a qualitative sense) (see e.g. Corning,
2002; Bedau, 2009).
The continual flow of energy and matter through a
thermodynamically dissipative system maintains it in a
state far from equilibrium ( ¸ ambel, 1993; Ruelle, 1993).
Ordered structures 'emerge' as a result of interactions
between the system's subcomponents, such interactions
being driven by the flow of matter and energy which
characterizes such systems. As these structures grow
more common within the system, the system as a whole
'self-organizes'. This transition (see 'symmetry-breaking'
above) often occurs quite rapidly and abruptly, in the
manner of a phase change e.g. from water to ice. It is
crucial to note that this increase in systemic organization
is entirely a result of internal interactions, rather than
resulting from some externally imposed controlling fac-
tor (although a flow of energy and matter through the
system is essential). Bar-Yam (1997: 10) points out that
even in the scientific community there is still confusion
regarding the nature of emergence. One 'fingerprint' of
emergence is that the emergent properties of a system can-
not easily be derived from the properties of the system
components or subsystems (see Figure 4.2). Addition-
ally, Bar-Yam (1997: 10-12) distinguishes between local
and global emergence. In local emergence, the emer-
gent response of the system is relatively resistant to local
perturbations: for example, a standing wave on the lip
of a waterfall will remain, despite innumerable small
variations in flow conditions in its vicinity. In global
emergence, the emergent response of the system is more

10 Such as a idealized gas: see Bar-Yam (1997). It is also true (but
in a different way) for almost all systems, even quite complicated
ones, which have been constructed by some external agency, e.g.
by a human. A first question to ask when attempting to identify
self-organization is, 'Is this system more than the sum of its parts?'