A simple guide to chaos and complexity

Abstract

The concepts of complexity and chaos are being invoked with increasing frequency in the health sciences literature. However, the concepts underpinning these concepts are foreign to many health scientists and there is some looseness in how they have been translated from their origins in mathematics and physics, which is leading to confusion and error in their application. Nonetheless, used carefully, "complexity science" has the potential to invigorate many areas of health science and may lead to important practical outcomes; but if it is to do so, we need the discipline that comes from a proper and responsible usage of its concepts. Hopefully, this glossary will go some way towards achieving that objective.

Full text

GLOSSARY
A simple guide to chaos and complexity
Dean Rickles, Penelope Hawe, Alan Shiell
...................................................................................................................................
J Epidemiol Community Health 2007;61:933–937. doi: 10.1136/jech.2006.054254
The concepts of complexity and chaos are being invoked with
increasing frequency in the health sciences literature. However,
the concepts underpinning these concepts are foreign to many
health scientists and there is some looseness in how they have
been translated from their origins in mathematics and physics,
which is leading to confusion and error in their application.
Nonetheless, used carefully, ‘‘complexity science’’ has the
potential to invigorate many areas of health science and may
lead to important practical outcomes; but if it is to do so, we
need the discipline that comes from a proper and responsible
usage of its concepts. Hopefully, this glossary will go some way
towards achieving that objective.
.............................................................................
See end of article for
authors’ affiliations
........................
Correspondence to:
Dean Rickles, Department
of Philosophy, University of
Calgary, Calgary, Canada;
drickles@ucalgary.ca
Accepted 4 February 2007
........................
T
he concepts of complexity and chaos are being
increasingly invoked in the health sciences
literature (general treatments are outlined in
references
1–3
). Applications to date include (there
are many more):
N
epidemiology and infectious disease processes
4–14
N
healthcare organisation
15–26
N
general practice and ‘‘the clinical encounter’’
27–39
N
biomedicine
40–54
N
health social science
55 56
N
health geography.
57 58
However, despite their being so widespread, the
concepts underpinning complex systems science
and chaos theory are still foreign to many health
scientists and there is some looseness in how they
have been translated from their origins in mathe-
matics and physics, which is leading to much
confusion and error in their application.
36
This
glossary attempts to resolve these issues by
providing a simple (but not simplified) guide to
many central concepts in chaos theory and
complexity science. The many references provide
more detail.
Dynamical systems
System is simply the name given to an object
studied in some field and might be abstract or
concrete; elementary or composite; linear or non-
linear; simple or complicated; complex or chaotic.
Complex systems are highly composite ones, built
up from very large numbers of mutually interact-
ing subunits (that are often composites them-
selves) whose repeated interactions result in rich,
collective behaviour that feeds back into the
behaviour of the individual parts. Chaotic systems
can have very few interacting subunits, but they
interact in such a way as to produce very intricate
dynamics. Simple systems have very few parts that
behave according to very simple laws. Complicated
systems can have very many parts too, but they
play specific functional roles and are guided by
very simple rules. Complex systems can survive the
removal of parts by adapting to the change; to be
robust, other systems must build redundancy into
the system (eg by containing multiple copies of a
part). A large healthcare system will be robust to
the removal of a single nurse because the rest of
the members of the system will adapt to compen-
sate—however, adding more nurses does not
necessarily make the system more efficient.
23 24 37
On the other hand, a complicated piece of medical
technology, such as a positron emission tomogra-
phy scanner, will obviously not survive removal of
a major component. The behaviour of a chaotic
system appears random, but is generated by
simple, non-random, deterministic processes: the
complexity is in the dynamical evolution (the way
the system changes over time driven by numerous
iterations of some very simple rule), rather than
the system itself.
59–60
Systems possess properties that are represented
by variables or observables: quantities that have a
range of possible values such as the number of
people in a population, the blood pressure of an
individual, the length of time between consulta-
tions, and so on. The values taken by a system’s
variables at an instant of time describe the
system’s state. A state is often represented by a
point in a geometrical space (phase space), with
axes corresponding to the variables. The co-
ordinates then correspond to particular assign-
ments of values to each variable. The number of
variables defines the dimension of both the space
and the system. Each point in the phase space
represents a way in which the system could be at a
time, corresponding to an assignment of particular
values to the variables at an instant. The overall
health state of an individual, for example, might
include values for lung capacity, heart rate, blood-
sugar levels and so on. A path through the phase
space corresponds to a trajectory of the system, or a
way in which the system could evolve over time—
for example, the change in a person’s health state
over time (itself a function of many other
variables).
A dynamical system is a system whose state (and
variables) evolve over time, doing so according to
some rule. How a system evolves over time
depends both on this rule and on its initial
conditions—that is, the system’s state at some
initial time. Feeding this initial state into the rules
generates a solution (a trajectory through phase
space), which explains how the system will change
over time; chaos is generated by feeding solutions
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back into the rule as a new initial condition. In this way, it is
possible to say what state the system will be in at a particular
time in the future (Abraham and Shaw
61
offer an exceptionally
clear, graphical introduction to many aspects of dynamical
systems theory, including chaos).
Complex and chaotic systems are both examples of nonlinear
dynamical systems. A linear system is characterised by the
satisfaction of the superposition principle. The superposition
principle says that if A and B are both solutions for some
system (ways in which the system could evolve), then so is
their sum A + B—this implies that a linear system can be
decomposed into its parts and each part solved separately to
construct the full solution. For nonlinear systems, this is not
possible because of the appearance of nonlinear terms,
functions of the variables such as sin(x), x
3
and xy. In this
sense then, the whole here is more than the sum of its parts.
Given this, a nonlinear system is one for which inputs are not
proportional to outputs: a small (large) change in some variable
or family of variables will not necessarily result in a small
(large) change in the system. This kind of behaviour is well
known to those involved in intervention research: large
interventions, in some variable, do not necessarily have a large
effect on some outcome variable of interest. Likewise, a small
intervention can have large, unexpected outcomes.
24 25
A system (or process) is deterministic if it is possible to
uniquely determine its past and future trajectories (ie all points
it passed through and will pass through) from its initial
(present) state. A system is semi-deterministic if its future but not
its past trajectory can be uniquely determined. An indeterministic
system is one without a unique future trajectory, so that the
evolution is random. It is often possible to tell whether or not a
system is deterministic by inspecting the time series it generates,
plotting states at different times. The data points in the time
series might correspond to measurements of blood sugar levels,
population health indicators and so on. If closely matched
points at are found in different places in the series, then the
series is investigated to see if the points that follow are closely
matched too. Hence, attempts are made to infer from the spread
of points the kind of system or process that generated it.
The time-series can also be used to indicate whether or not a
system is chaotic or complex: a chaotic system’s time-series has
a fractal-like structure, meaning that it looks the same at
different scales (a property also known as self-similarity: take a
snapshot of the time series covering a certain interval of time
and then take another snapshot covering a much larger period
of time and the two will look the same). Fractals are really just
a spatial version of chaos; the interesting type of chaos is the
temporal kind. Inferring complexity from a time-series is more
difficult but can be done (this involves finding power law
correlations in the data; see below).
A related concept is that of the attractor.
45
Following an
intervention in a system (changing the value of some variable),
it takes a little while for it to settle down into its ‘‘normal’’
behaviour. The path traced out in phase space during this
period is known as a transient (or the start-up transient). The state
it reaches after this corresponds to the normal behaviour of the
system: the phase space points corresponding to this form the
system’s attractor. If the attractor is a point that does not move,
it is known as a fixed point. Such attractors often describe
dissipative systems (those that lose energy— for example, due to
friction). In general, however, it is possible to think of an
attractor as whatever the system behaves like after it has passed
the transient stage.
There are other types of attractor. For example, an attractor
that describes a system that cycles periodically over the same
set of states, never coming to rest, is known as a limit cycle.A
system need not possess a single attractor either; the phase
space can have a number of attractors whose ‘‘attractiveness’’
depends upon the initial conditions of the system (ie the state
of the system at the outset).
The set of points that are ‘‘pulled’’ towards a particular
attractor are known as the basin of attraction. What is important
to note about the types of attractor discussed above (fixed
points and limit cycles) is that initial points that are close to
each other remain close as they each evolve according to the
same rules. This property is violated for so-called strange
attractors for which close points diverge exponentially over time.
Such attractors are also aperiodic; systems described by strange
attractors do not wind up in a steady state nor do they repeat
the same pattern of behaviour (an example that can be easily
observed in the household is a dripping water tap tuned to a
certain flow, not too high amd not too low). Chaotic systems
have strange attractors; complex systems have evolving phase
spaces and a range of possible attractors.
These concepts have been applied extensively, accurately and
successfully in the biomedical sciences.
42 47 48
The general
outcome of these investigations appears to be that chaos is
associated with ‘‘good health’’: pathologies (such as of the
brain, heart, lungs) occur when the dynamics becomes stable
and the attractor is a limit cycle.
42 46 49 50
Heart disease, epilepsy,
bipolarity and so on are considered to be dynamic diseases in that
they are not associated with something that can manifest itself
at an instant (like a broken arm), but only arise over time.
43
This is why chaos is so appropriate.
Chaosversuscomplexity
We can now consider further the similarities and differences
between chaotic systems and complex systems. Each shares
common features, but the two concepts are very different. Chaos
is the generation of complicated, aperiodic, seemingly random
behaviour from the iteration of a simple rule. This complicated-
ness is not complex in the sense of complex systems science,
but rather it is chaotic in a very precise mathematical sense.
Complexity is the generation of rich, collective dynamical
behaviour from simple interactions between large numbers of
subunits. Chaotic systems are not necessarily complex, and
complex systems are not necessarily chaotic (although they can
be for some values of the variables or control parameter; see
below).
62
The interactions between the subunits of a complex system
determine (or generate) properties in the unit system that cannot
be reduced to the subunits (and that cannot be readily deduced
from the subunits and their interactions). Such properties are
known as emergent properties. In this way it is possible to have an
upward (or generative) hierarchy of such levels, in which one
level of organisation determines the level above it, and that
level then determines the features of the level above it.
59
Emergent properties may also be universal or multiply realisable
in the sense that there are many diverse ways in which the
same emergent property can be generated. For example,
temperature is multiply realisable: many configurations of the
same substance can generate the same temperature, and many
different types of substance can generate the same temperature.
The properties of a complex system are multiply realisable since
they satisfy universal laws—that is, they have universal proper-
ties that are independent of the microscopic details of the
system. Emergent properties are neither identical with nor
reducible to the lower-level properties of the subunits because
there are many ways for emergent properties to be produced.
A necessary condition, owing to nonlinearity, of both chaos
and complexity is sensitivity to initial conditions. This means that
two states that are very close together initially and that operate
under the same simple rules will nevertheless follow very
different trajectories over time. This sensitivity makes it
difficult to predict the evolution of a system, as this requires
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the initial state of the system to be described with perfect
accuracy. There will always be some error in how this is
performed and it is this error that gets exponentially worse over
time. It is possible to see how this might pose problems for
replication of initial conditions in various types of trial and
intervention.
24 25
There are several less well-understood, but nonetheless
important properties that are characteristic features of complex
systems. Complex systems often exhibit self-organisation, which
happens when systems spontaneously order themselves (gen-
erally in an optimal or more stable way) without ‘‘external’’
tuning of a control parameter (see below). This feature is not
found in chaotic systems and is often called anti-chaos.
46
Such
systems also tend to be out of equilibrium, which means that the
system never settles in to a steady state of behaviour. This is
related to the concept of openness: a system is open if it is not or
cannot be screened off from its environment. In closed systems,
outside influences (exogenous variables) can be ignored. For
open systems, this is not the case. Most real-world systems are
open, thus this presents problems both for modelling and
experimenting on such systems, because the effect of exogen-
ous influences must be taken into account. Such influences can
be magnified over time by sensitivity to initial conditions.
Another important feature of a complex system is the idea of
feedback, in which the output of some process within the system
is ‘‘recycled’’ and becomes a new input for the system.
Feedback can be positive or negative: negative feedback works
by reversing the direction of change of some variable; positive
feedback increases the rate of change of the variable in a certain
direction. In complex systems, feedback occurs between levels
of organisation, micro and macro, so that the micro-level
interactions between the subunits generate some pattern in the
macro-level that then ‘‘back-reacts’’ onto the subunits, causing
them to generate a new pattern, which back-reacts again and so
on. This kind of ‘‘global to local’’ positive feedback is called
coevolution, a term originating in evolutionary biology to
describe the way organisms create their environment and are
in turn moulded by that environment.
If a system is stable under small changes in its variables, so
that it does not change radically when interventions occur then
it is said to be robust. Generally, complex systems increase in
robustness over time because of their ability to organise
themselves relative to their environment. However, it is possible
for single events to alter a complex system in a way that persists
for a long time (this is called path-dependence). For a complex
system, ‘‘history matters’’. Complex processes (processes
generated by complex systems) are, therefore, non-Markovian:
they have a long ‘‘memory’’. This non-Markovian aspect
(essentially due to feedback mechanisms) is often believed to
pose insuperable problems for describing causal events and
making causal predictions about complex systems. If this were
so, it would be a severe blow to the practical usefulness of
complex science; however, there has been some encouraging
preliminary work involving non-Markovian and modified
Markovian causal models and networks.
63 64
The key differences between chaotic systems and complex
ones lie, therefore, in the number of interacting parts and the
effect that this has on the properties and behaviour of the
system as a whole. As Nobel Laureate Phillip Anderson put it:
‘‘more is different’’.
65
Complex systems are coherent units in a
way that chaotic systems are not, involving instead interactions
between units. This simple difference concerning units and
subunits can be brought out using concepts from the theory of
critical phenomena, which is central to complexity science.
Critical phenomena
Some systems have a property known as criticality. A system is
critical if its state changes dramatically given some small input.
To make sense of this, we need to introduce several new
concepts.
An order parameter is a macroscopic (global or systemic)
feature of the system that tells one how the parts of the system
are competing or cooperating with one another. Hence, there is
a state of order among the parts when they act collectively (in
which case the order parameter is non-zero) and a state of
disorder when they fight against each other, doing the opposite
of their neighbours (in which case the order parameter is zero).
The control parameter is an external input to the system that
can be varied so as to change the order parameter and so the
macroscopic features of the system. This is termed tuning the
control parameter to shift the system between various phases or
regimes; it is possible to have ordered, chaotic and critical (edge of
chaos) phases. A common example concerns the phenomenon
of magnetism in a piece of metal. Here the order parameter is
the degree of magnetism and the control parameter is
temperature. As the magnet is heated up, the magnetism
decreases; increasing the control parameter decreases the order
parameter. The system undergoes a phase transition so that at a
critical temperature the magnetism vanishes. This is a general
feature of complex systems; tuning the system’s control
parameter to a certain critical point results in a phase transition
at which the system undergoes an instantaneous radical change
in its qualitative features (the phases of water, from gas to
liquid to solid, is another common example). The general study
of such behaviour is the theory of critical phenomena.
A system that is at a critical point has an extremely high
degree of connectivity between its subunits: everything
depends on everything else! Complex systems are said to be
poised at such a position, between order and chaos. The degree
of connectedness is encoded in the correlation function. This
function tells us by how much pairs of subunits are influencing
one another and how much this influence varies with distance.
The furthest distance the influence extends is known as the
correlation length; beyond this distance, the subunits are
independent and are unaffected by one another. The farther
apart two subunits are, the less they influence each other (the
influence effect decreases exponentially with distance). The
correlation length can itself be influenced by the control
parameter. When the control parameter is tuned to the critical
point, the correlation length becomes infinite, and all the
subunits follow each other. The influence still decays exponen-
tially with distance, but in the critical regime, more pathways
are opened up between pairs of subunits so that the
connectivity of the system is massively amplified. As a
consequence, a small disturbance in the system (even to a
single subunit) can produce massive, systemic changes. Very
different kinds of critical system exhibit the same properties—
for example, crowds behaving like the atoms in a magnet. This
feature is known as universality.
14
Universality is connected to another concept: scaling. Scaling
laws,orpower laws have the following form: f(x) , x
2a
in other
words, the probability f(x) of an event of magnitude x occurring
is inversely proportional to x.
12
Sociologist Vilfred Pareto noticed
that the statistical distribution for the wealth of individuals in a
population followed such a law: few are rich, some are well-off
and many are poor. Whenever systems are described by scaling
laws that share the same exponents (the a term) then they will
exhibit similar behaviour in some way (hence, universality).
Such systems are said to belong to the same universality class.
66
For example, diverse countries follow Pareto’s law, as do cities.
This result could easily be extrapolated to the distribution of the
health of individuals within and across populations, with
significant implications for research on health inequalities.
If a system displays power law behaviour then it is scale-free and
its parts have scale-invariant correlations between them. What
A simple guide to chaos and complexity 935
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this means is that the system does not possess a characteristic scale
in the sense that events of all magnitudes can occur. To make
sense of this, take an example of a system that does have a
characteristic scale: human height. Most humans are of about the
same height. There are a few ‘‘outliers’’ who are either very tall or
very small, but most are around 59 80. Now consider earthquakes.
These can be both extremely tiny and extremely massive, but most
are negligible. This is an example of a system that does not have a
characteristic scale; it satisfies a scaling law (the Gutenberg–
Richter law). Note that self-similarity , scale invarianceandpower laws
are just equivalent ways of saying that there is no characteristic
scale. Various empirical studies have confirmed that healthcare
systems obey power laws for a number of quantities of interest,
such as hospital waiting lists.
416232437
These studies have clear
policy implications; if various healthcare delivery systems exhibit
power-law behaviour then we ought to reframe our intervention
and management theories in terms of complex systems science.
We should not be surprised if huge catastrophes occur for no
discernible reason, and we should not be surprised if our massive
intervention to reduce waiting times by employing more staff does
nothing for efficiency or even makes it worse. However, we should
show caution in using the power-law behaviour to infer an
underlying complex system, as power laws can be generated in
diverse ways.
67 68
CONCLUSION
Although complexity science is still a work in progress, having
neither a firm mathematical foundation nor the necessary and
sufficient conditions whose satisfaction entails that some
system is complex, it nonetheless has the potential to invigorate
many areas of health research as we hope to have indicated in
this brief guide. However, if it is to achieve its potential we need
the discipline that comes from careful and proper usage of its
concepts. Hopefully, this glossary will go some way towards
achieving that objective.
Authors’ affiliations
.......................
Dean Rickles, Department of Philosophy, University of Calgary, Calgary,
Canada
Penelope Hawe, Alan Shiell, Markin Institute, University of Calgary,
Calgary, Canada
Funding: This work was conducted as part of an International Collaboration
on Complex Interventions (ICCI) funded by the Canadian Institutes of Health
Research. DR is an ICCI Post Doctoral Fellow. AS and PH are Senior
Scholars of the Alberta Heritage Foundation for Medical Research. PH
holds the Markin Chair in Health and Society at the University of Calgary.
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What this paper adds
This glossary is intended to provide a ‘‘corrective’’ to what the
authors perceive to be a loose and misleading handling of
ideas stemming from the sciences of complexity. The aim is to
provide as simple an account as possible, without succumbing
to the ‘‘popular science’’-style accounts that appear to be fast
becoming the norm in the health-sciences literature. The going
is perhaps a little tough, but we feel the payoff (a responsible
guide) is worth the extra effort.
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