Mike's Notes
Carlos Gershenson's paper about self-organising systems was recently published by Nature in npj complexity. Carlos is the editor of Complexity Digest.
I removed the references in the original paper because importing this into Google Blogger is messy. Please refer to the original paper for this missing information.
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11/04/2025
Self-organizing systems: what, how, and why?
By: Carlos Gershenson
Nature > npj complexity: 25/03/2025
Abstract
I present a personal account of self-organizing systems, framing relevant
questions to better understand self-organization, information, complexity,
and emergence. With this aim, I start with a notion and examples of
self-organizing systems (what?), continue with their properties and
related concepts (how?), and close with applications (why?) in physics,
chemistry, biology, collective behavior, ecology, communication networks,
robotics, artificial intelligence, linguistics, social science, urbanism,
philosophy, and engineering.
What are self-organizing systems?
"Being ill defined is a feature common to all important concepts.”
—Benoît Mandelbrot
I will not attempt to define a “self-organizing system”, as it involves the
cybernetic problem of defining “system”, the informational problem of
defining “organization”, and the ontological problem of defining “self”.
Still, there are plenty of examples of systems that we can usefully call
self-organizing: flocks of birds, schools of fish, swarms of insects, herds
of cattle, and some crowds of people. In these animal examples, collective
behavior is a product of the interactions of individuals, not determined by
a leader or an external signal. There are also several examples from
non-living systems, such as vortexes, crystallization, self-assembly, and
pattern formation in general. In these cases, elements of a system also
interact to achieve a global pattern.
Self-organization or similar concepts have been present since antiquity
(see Section 3.12), so the idea itself is not new. Nevertheless, we still
lack the proper conceptual framework to understand it properly. The term
“self-organizing system” was coined by W. Ross Ashby in the early days of
cybernetics. Ashby’s purpose was to describe deterministic machines that
could change their own organization. Ever since, the concept has been used
in a broad range of disciplines, including statistical mechanics,
supramolecular chemistry, computer science, and artificial life.
There is an unavoidable subjectivity when speaking about self-organizing
systems, as the same system can be described as self-organizing or not (see
Section 2.1). Stafford Beer gave the following example: an ice cream at room
temperature will thaw. This will increase its temperature and entropy, so it
would be “self-disorganizing”. However, if we focus on the function of an
ice cream for being eaten, it would be “self-organizing”, because it would
approach a pleasant temperature and consistency for degustating it,
improving its “function”. Ashby also mentioned that one just needs to call
the attractor of a dynamical system “organized”, and then almost any system
will be self-organizing.
So, the question should not be whether a system is self-organizing, but
rather (being pragmatic) when is it useful to describe a system as
self-organizing? The answer will slowly unfold along this paper, but in
short, it can be said that self-organization is a useful description when we
are interested on describing systems at multiple scales, and understanding
how these affect each other. For example, collective motion and
cyber-physical systems can benefit from such a description, compared to a
single-scale narrative/model. This is common with complexity, as
interactions can generate novel information that is not present in initial
nor boundary conditions, limiting predictability.
So rather than a definition, we can do with a notion: a system can be
described as self-organizing when its elements interact to produce a global
function or behavior. This is in contrast with centralized systems, where a
single or few elements “control” the rest, or with simply distributed
systems, where a global problem can be divided (reduced) and each element
does its part, but there is no need to interact nor integrate elementary
solutions. Thus, self-organizing systems are a useful description when we
want to relate individual behaviors and interactions to global patterns or
functions. If we can describe a system fully (for our particular purposes)
at a single scale, then self-organization could be perhaps identified, but
superfluos (not useful). And the “self” implies that the “control” comes
from within the system, rather than from an external signal/controller that
would explicitly indicate elements of what to do.
For example, we can decide to call a society “self-organizing” if we are
interested in how individual interactions lead to the formation of fashion,
ideologies, opinions, norms, and laws; but at the same time, how the
emerging global properties affect the behavior of the individuals. If we
were interested in an aggregate property of a population, e.g., its average
height, then calling the group of individuals “self-organizing” would not
give any extra information, and thus would not be useful.
It should be stressed that self-organization is not a property of systems
per se. It is a way of describing systems, i.e., a narrative.
How can self-organizing systems be measured?
"It is the function of science to discover the existence of a general
reign of order in nature and to find the causes governing this order. And
this refers in equal measure to the relations of man — social and
political — and to the entire universe as a whole.” —Dmitri Mendeleev
Even when self-organization had been described intuitively since antiquity
— the seeds of the narrative were present — the proper tools for studying it
became available only recently: computers. Since self-organizing systems
require the explicit description of elements and interactions, our brains,
blackboards, and notebooks are too limited to consider the number of
required variables to study the properties of self-organizing systems. It
was only through the relatively recent development of information technology
that we were able to study the richness of self-organization, just like we
were unable to study the microcosmos before microscopes and the macrocosmos
before telescopes.
Information
Computation can be generally described as the transformation of
information, although Alan Turing formally defined computable numbers with
the purpose of proving limits of formal systems (in particular, Hilbert’s
decision problem). In the same environment where the first digital computers
were built in the mid XXth century, Claude Shannon defined information to
quantify its transmission, showing that information could be reliably
transmitted through unreliable communication channels. As it turned out,
Shannon’s information H is mathematically equivalent to Boltzmann-Gibbs
entropy:
$$H=-K\mathop{\sum}\limits_{i=i}^{n}{p}_{i}\log {p}_{i},$$
(1)
where K is a positive constant and p is the probability of receiving symbol
i from a finite alphabet of size n. This dimensionless measure will be
maximal for a homogeneous probability distribution, and minimal when only
one symbol has a probability p = 1. In binary, we have only two symbols (n =
2), and information would be minimal with a string of only ones or only
zeroes (‘1111…’ or ‘0000…’). This implies that having more bits will not
tell us anything new, because we already know what the next bits will be
(assuming the probability distribution will not change). With a random-like
string, such as a sequence of coin flips (‘11010001011011001010…’),
information is maximal, because no matter how much previous information we
have (full knowledge of the probability distribution), we will not be able
to predict what the next bit might be better than chance.
In parallel, Norbert Wiener — one of the founders of cybernetics — proposed
an alternative measure of information, which was basically the same as
Shannon’s, but without the minus sign. Wiener’s information measured what
one knows already, so it is minimal when we have a random string
(homogeneous probability distribution) because all the information we
already have is “useless” (to predict the next symbol), and maximal when we
have a single symbol repeating (maximally biased probability distribution),
because the information we have allows us to predict exactly the next
symbol. Nevertheless, Shannon’s information is the one that everyone has
used, and we will do the same.
Shannon’s information is also known as Shannon’s entropy, which can be also
used as a measure of “disorder”. We already saw that it is maximal for
random strings, and thus minimal for particularly ordered strings. Then, we
can use the negative of Shannon’s information (which would be Wiener’s
information) as a measure of organization. If the organization is a result
of internal dynamics, then we can also use this measure for
self-organization.
Nevertheless, just like with many measures, the interpretation depends on
how the observer performs the measurement. Figure 1 shows how the same
system, divided into four microstates or two macrostates (with probabilities
represented as shades of gray) can increase its entropy/information (become
more homogeneous) or decrease it, depending on how it is observed.
Fig. 1: The same system, observed at different levels or with different
coarse grainings can be said to be disorganizing (entropy increasing) or
organizing (entropy decreasing), for arbitrary initial and final
states.
Probabilities of the system being in a state (a1, a2, b1, and b2 at the
lower level, which can be aggregated in different ways at a higher level)
are represented as shades of gray, so one can observe which configurations
are more homogeneous (i.e., with higher entropy): if there is a high
contrast in different states (such as between A' and B' in their initial
state), then this implies more organization (less entropy), while similar
shades (as between A' and B' in their final state) imply less organization
(more entropy).
Still, the fact that self-organization is partially subjective does not
mean that it cannot be useful. We just have to be aware that a shared
description and interpretation should be agreed upon.
Complexity
Self-organizing systems are intimately related to complex systems. Again,
the question is not so much whether a system is self-organizing or complex,
but when is it useful to describe it as such. This is because most systems
can be described as complex or not, depending on our context and
purposes.
Etymologically, complexity comes from the Latin plexus, which could be
translated as entwined. We can say that complex systems are those where
interactions make it difficult to separate the components and study them in
isolation, because of their interdependence. These interactions can generate
novel information that limit predictability in an inherent way, as it is not
present in initial nor boundary conditions. In other words, there is no
shortcut to the future, but we have to go through all intermediate steps, as
interactions partially determine the future states of the system.
For example, markets tend to be unpredictable because different agents make
decisions depending on what they think other agents will decide. But since
it is not possible to know what everyone will decide in advance, the
predictability of markets is rather limited.
Complex systems can be confused with complicated or chaotic systems.
Perhaps they will be easier to distinguish considering their opposites:
complicated are the opposite of easy, chaotic (sensitive to initial
conditions) are the opposite of robust, while complex systems are the
opposite of separable.
Given the above notion of self-organizing systems, then all of them would
also be complex systems, but not necessarily vice versa. This is because
interactions are an essential aspect of self-organizing systems, which would
make them complex by definition. However, we could have a description of a
complex system whose elements interact but do not produce a global pattern
or function we are interested in during the timeframe we are interested in.
So, the narrative of complexity would be useful, but not the one of
self-organization. Nevertheless, understanding complexity should be
essential for the study of self-organization.
Emergence
One of the most relevant and controversial properties of complex systems is
emergence. It could be seen as problematic because last century some people
described emergent properties as “surprising”. So then emergence would be a
measure of our ignorance, and then it would be reduced once we understood
the mechanisms behind emergent properties. Also, there are different flavors
of emergence, some easier to study and accept than others. But in general,
emergence can be described as information that is present at one scale and
not at another scale.
For example, we can have full knowledge of the properties of carbon atoms.
But if we focus only on the atoms, i.e. without interactions, we will not be
able to know whether they are part of a molecule of graphite, diamond,
graphene, buckyballs, etc. (all composed only of carbon atoms) which have
drastically different macroscopic properties. Thus, we cannot derive the
conductivity, transparency, or density of these materials by looking only at
the atomic properties of carbon. The difference lies precisely in how the
atoms are organized, i.e. how they interact.
If emergence can be described in terms of information, Shannon’s measure
can be used (understanding that we are measuring only the information that
is absent from another scale). Thus, emergence would be the opposite of
self-organization. This might seem contradictory, as usually emergence and
self-organization are both present in complex systems8. But if we take each
to its extreme, we can see that maximum emergence (information) occurs when
there is (quasi)randomness, so no organization. Maximum (self-)organization
occurs when entropy is minimal (no new information, and thus, no emergence).
Because of this, complexity can be seen as a balance between emergence and
self-organization.
Why should we use self-organizing systems?
"It is as though a puzzle could be put together simply by shaking its
pieces.” —Christian De Duve
Self-organization can be used to build adaptive systems. This is useful for
non-stationary problems, i.e., those that change in time. Since interactions
can generate novel information, complexity often leads to non-stationarity.
Thus, when a problem changes, the elements of a self-organizing system can
adapt through their interactions. Then, designers do not need to specify
precisely the problem beforehand, or how it will change, but just to
define/regulate interactions to achieve a desired goal.
For example, if we want to improve passenger flow in public transportations
systems, we cannot really change the elements of the system (passengers).
Still, we can change how they interact. In 2016, we successfully implemented
such a change to regulate boarding and alighting in Mexico City metro. In a
similar way, we cannot change teachers in an education system. But we can
change their interactions to improve learning. We cannot change politicians,
but we can regulate their interactions to reduce corruption and improve
efficiency. We cannot change businesspeople, but we can control their
interactions to promote sustainable economic growth.
There have been many other examples of applications of self-organization in
different field, and the following is only a partial enumeration.
Physics
The Industrial revolution led to the formalization of thermodynamics in the
XIXth century. The second law of thermodynamics states that an isolated
system will tend to thermal equilibrium. In other words, it loses
organization, as heterogeneities become homogeneous, and entropy is
eventually maximized. Still, non-equilibrium thermodynamics has studied how
open systems can self-organize.
Lasers can be seen as self-organized light, which Hermann Haken used as an
inspiration to propose the study of synergetics, which precisely studies
self-organization in open systems far from thermodynamic equilibrium and is
related to phase transitions, where criticality is found.
Self-organized criticality (SOC) was proposed to explain why power laws and
scale-free-like distributions and fractals are so prevalent in nature. SOC
was illustrated with the sandpile model, where grains accumulate and lead to
avalanches with a scale-free (critical) distribution. Similarly,
self-organization has been used to describe granular media.
Generalizing principles of granular media, self-organization can be used to
describe and design “optimal” configurations in biological, social, and
economic systems.
Chemistry
Around 1950, Boris P. Belousov was interested in a simplified version of
the Krebs cycle. He found that a solution of citric acid in water with
acidified bromate and yellow ceric ions produced an oscillating reaction.
His attempts to publish his findings were rejected, arguing that it violated
the second law of thermodynamics (which only applies to systems at
equilibrium, and this system is far from equilibrium). In the 1960s, Anatol
M. Zhabotinsky began working on this reaction, and only in the late 1960s
and 1970s the Belousov-Zhabotinsky reaction became known outside the Soviet
Union. Since then, many chemical systems far from equilibrium have been
studied. Some have been characterized as self-organizing, because they are
able to use free energy to increase their organization.
More generally, self-organization has been used to describe pattern
formation, which includes self-assembly.
Molecules are basically atoms joined by covalent bonds. Supramolecular
chemistry studies chemical structures formed by weaker forces (Van Der
Waals, hydrogen bonds, electrostatic charges), and can also be described in
terms of self-organization.
Biology
The study of form in biology (morphogenesis) is far from new, but far from
complete.
Alan Turing was one of the first to describe morphogenesis with
differential equations. Morphogenesis can be seen as pattern formation with
local stimulation and long-range inhibition (skins, scales), or as fractals
(capillaries, neurons). These processes are more or less well understood.
Still, it becomes more sophisticated for embryogenesis and regeneration,
where many open questions remain.
Humberto Maturana and Francisco Varela proposed autopoiesis
(self-production) to describe the emergence of living systems from complex
chemistry. Autopoiesis can be seen as a special case of self-organization
(to the disagreement of Maturana), because molecules self-organize to
produce membranes and metabolism. Moreover, it can be argued that living
systems also need information handling, self-replication, and
evolvability.
There are further examples of self-organization in biology, that include
firefly synchronization, ant foraging, and collective behavior.
Collective Behavior
Groups of agents can produce global patterns or behavior through local
interactions. Craig Reynolds presented a simple model of boids, where agents
followed three simple rules: separation (don’t crash), alignment (head to
average heading of neighbors), and cohesion (go towards average position of
neighbors). Varying its parameters, this simple model produces dynamic
patterns similar to those of flocks, schools, herds, and swarms. It was used
to animate bats and penguins in the 1992 Batman Returns film and contributed
to earning Reynolds an Oscar in 1998.
A flock of boids self-organize even only with the alignment rule and added
noise. It has been shown that when the number of boids increases, novel
properties emerge.
Slightly more sophisticated models have been used to describe more
precisely animal collective behavior.
Furthermore, similar models and rules have been used to study the
self-organization of active matterand robots (see below).
Ecology
Species self-organize to produce ecological patterns. These include trophic
networks (who eats who), mutualistic networks (cooperating species), and
host-parasite networks.
At the biosphere level, ecosystems also self-organize. This is a central
aspect of the Gaia hypothesis, which defends that our planet self-regulates
its own conditions that allow life to thrive.
Self-organization can be useful to study how ecosystems can be robust,
resilient, or antifragile
Communication networks
Self-organization has been useful in telecommunication networks106, as it
is desirable to have the ability to self-reconfigure based on changing
demands. Also, having local rules to define global functions makes them
robust to potential failures or attacks of central nodes: if there is a path
that is not responsive, then an alternative is sought. These principles have
been used in Internet protocols, peer-to-peer networks, cellular networks,
and more.
Robotics
There has been a broad variety of self-organizing robots, terrestrial,
aerial, aquatic, and/or hybrid (for a review see ref. 26).
A common aspect of self-organizing robots is that there is no leader, and
the collective function or pattern is the result of local interactions. Some
have been inspired in the collective behavior of animals, and their
potential applications are vast.
Artificial Intelligence
As mentioned in the first section of this paper, the study of
self-organizing systems originated in cybernetics, which had a strong
influence and overlap in the early days of artificial intelligence. Claude
Shannon, William Grey Walter, Warren McCulloch and Walter Pitts contributed
to both fields in their early days.
If brains can be described as self-organizing, it is no surprise that
certain flavors of artificial neural networks have also been described as
self-organizing. Independently on the terminology, adjustments to local
weights between artificial neurons lead to an error reduction in the task of
the network.
Even when their interpretation is still controversial, large language
models have been useful in multiple domains. Whether describing them as
self-organizing would bring any benefit or not, still remains to be
seen.
Linguistics
The statistical study of linguistics became popular after Zipf. Different
explanations have been put forward to try to explain statistical
regularities found across languages, and in even more general
contexts.
Naturally, some of these explanations focus on the evolution of language.
It has been shown that a shared vocabulary and grammar140 can evolve using
self-organization: individuals interact locally leading to a population
converging to a shared language. This is useful not only for understanding
language evolution, but also to build adaptive artificial systems. Similar
mechanisms can be used in other social systems, e.g. to reach
consensus.
Social Science
Individuals in a society interact in different ways. These interactions can
lead to social properties, such as norms, fashions, and expectations. In
turn, these social properties can guide, constrain, and promote behaviors
and states of individuals.
Computers have allowed the simulation of social systems, including
systematic explorations of abstract models. Combined with an increase in
data availability, computational social science has been increasingly
adopted by social scientists. The understanding and implications of
self-organization are naturally relevant to this field.
Urbanism
It is similar to the scientific study of cities.
For example, city growth can be modeled as a self-organizing process152.
Similar to the metro case study mentioned above, self-organization has been
shown to efficiently and adaptively coordinate traffic lights, and is
promising for regulating interactions among autonomous vehicles.
More generally, urban systems tend to be non-stationary, as conditions are
changing constantly. Thus, self-organization offers a proven alternative to
design urban systems that adapt as fast as their conditions change.
Philosophy
Concepts similar to self-organization can be traced to ancient Greece in
Heraclitus and Aristotle and also to Buddhist philosophy.
There has been a long debate about the relationship between mechanistic
principles and the purpose of systems. This question was at the origins of
cybernetics. It has been argued12 that self-organization can be used to
explain teleology, in accordance with Kant’s attempt from the late XVIIIth
century, as purpose can also be described in terms of organization.
Also, self-organization is related to downward causation: when higher-level
properties cause changes in lower-level elements. This is still debated,
along with other philosophical questions related to self-organization.
Engineering
There have been several examples of self-organization applied to different
areas of engineering apart from those already mentioned, such as power
grids, computing, sensor networks, supply networks and production systems,
bureaucracies, and more.
In general, self-organization has been a promising approach to build
adaptive systems, as mentioned above. It might seem counterintuitive to
speak about controlling self-organization, since we might think that
self-organizing systems are difficult to regulate because of a certain
autonomy of their components. Still, we can speak about a balance between
control and independence, in what has been called “guided
self-organization”.
Conclusions
"We can never be right, we can only be sure when we are wrong” —Richard
Feynman
There are many open questions related to the scientific study of
self-organizing systems. Even when their potential has been promising, they
are far from being commonly used to address non-stationary problems. Could
it be because of a lack of literacy in concepts related to complex systems?
Might there be any conceptual or technical obstacle? Do we need further
theories? Independently of the answers, these questions are worth
exploring.
For example, we have yet to explore the relationship between
self-organization and antifragility: the property of systems that benefit
from perturbations. Self-organization seems to be correlated with
antifragility, but why or how still has to be investigated. In a similar
vein, a systematic exploration of the “slower is faster” effect might be
useful to better understand self-organizing systems and vice versa.
Many problems and challenges we are facing — climate change, migration,
urban growth, social polarization, etc. — are clearly non-stationary. It is
not certain that with self-organization we will be able to improve the
situation in all of them. But it is almost certain that with the current
tools we have, we will not be able to make much more progress (otherwise we
would have made it already). It would be imprudent not to make efforts to
use the narrative of self-organization, even if for slightly improving
situations related to only one of these challenges.
