These notes were first presented at The Developing Group, 5 June 2004
(a second version, Thinking Networks II, was presented on 3 June 2006)
THINKING NETWORKS I
James Lawley
We have "a simple aim: To get you to
think networks.
It is about how networks emerge, what they look like, and how they evolve.
...
Networks are present everywhere. All we need is an eye for them."
Linked, p. 7
These preparatory notes consist of a series of quotations from four recently published books (with my comments interspersed between the quotes):
Albert-Laszlo Barabasi, Linked: How everything is connected (Plume, 2003)
Mark
Buchanan, Nexus: Small worlds and the science of
networks (Norton, 2002)
Steven
Strogatz, Sync: Rhythms of nature, rhythms of
ourselves (Allen Lane, 2003)
Duncan Watts, Six Degrees: The Science of a connected age (Norton, 2003)
Other books that have contributed to our thinking in terms of networks are:
Fritjof Capra, The Web of Life: A new synthesis of mind and matter
Fritjof Capra, Hidden Connections: Integrating the biological, cognitive and social
Malcolm
Gladwell, The Tipping Point: How little things can
make a big difference
Steven
Johnson, Emergence: The connected
lives of ants, brains, cities and software
Mark Ward, Universality: The underlying theory behind life, the universe and everything
A few points to remember:
1.
Time and again the authors of these books emphasise that in order to
think networks, a different way of thinking is required:
"Unfortunately,
our minds are bad at grasping these kinds of problems. We’re
accustomed to thinking in terms of centralized control, clear chains of
command, the straightforward logic of cause and effect. But in highly
interconnected systems, where every player ultimately affects every
other, our standard ways of thinking fall apart.” Sync p.
34-35
"The resulting small worlds are rather
different from the Euclidean world to which we are accustomed. ...
Navigating this non-Euclidean world repeatedly tricks our intuition and
reminds us that there is a new geometry out there that we need to
master in order to make sense of the complex world around us.” Linked,
p. 40
"When it comes to large-scale
coordinated social action, hindsight is not 20-20 — in fact it can be
actively misleading. ... The small-world phenomenon is so
counterintuitive — it is a global phenomenon, yet individuals are
capable only of local measurement.” Six Degrees p. 53 &
83
"What makes this paradoxical is that you
might think that strong social links would be the crucial ones holding
a network together. But they aren’t; in fact they are hardly important
at all. ... We are continually surprised [because] the long-distance
social shortcuts that make the world small are mostly invisible in our
ordinary social lives. We can only see as far as those to whom we are
directly linked.” Nexus, p. 41 & 55
2.
Because the study of networks is such a new field and so many people
from different backgrounds are contributing there are many terms for
similar phenomenon. We have tried to reflect this in the subheadings
by listing several of the most commonly occurring names. Also, the
terminology of networks can be mapped onto the terminology we use in
Metaphors in Mind:
COMPARING TERMINOLOGY: LEVELS
OF ORGANISATION
|
NETWORK THEORY
|
|
SYMBOLIC MODELLING
|
|
Network
|
<-->
|
Metaphor
Landscape
|
|
Cluster
|
<-->
|
Perception
|
|
Links
|
<-->
|
Relationships
|
|
Nodes
|
<-->
|
Components /
Symbols
|
3.
As most of the main ideas are interconnected (they would be wouldn't
they), our groupings reflect our personal preferences rather than
anything inherent in the information. We hope to reflect some of the
interconnections by
bolding the key concepts
wherever they appear. All
italics is in the original
text.
4. A few caveats:
This stuff is so new (all the popular books on network
theory have been published in the last five years) that there are bound
to be major revisions, it really is work in progress.
“Claiming that everything is a small-world network or a
scale-free network not only oversimplifies the truth but does so in a
way that can mislead one to think that the same set of characteristics
is relevant to every problem. If we want to understand the connected
age in any more than a superficial manner, we need to recognize that
different classes of networked systems require us to explore different
sorts of network properties.” p. 304 Six Degrees.
And how do we do
that? We use bottom-up modelling.
Except where there are physical entities that are
physically connected together, all talk of nodes, links, weak and
strong ties, hubs and connectors, etc.
is metaphor.
And for everything that a metaphor illuminates, it hides
something else in shadow.
5.
And finally, a tribute to Fritjof Capra, who made an early and
significant contribution to bringing the importance of networks to our
attention when he wrote
The Web of Life in
1996:
“Having appreciated
the importance of pattern for the understanding of life, we can now
ask: Is there a common pattern of organization that can be identified
in all living systems? ... this is indeed the case. Its most important
property is that it is a network pattern. Whenever we encounter living
systems – organisms, parts of organisms, or communities of organisms —
we can observe that their components are arranged in network fashion.
Whenever we look at life, we look at networks.” p. 81-82
WHOLES, NONLINEARITY, COMPLEXITY, SELF-ORGANISATION,
CONTINGENCY
"Reductionism was the driving force behind much of the 20th century’s scientific
research. To comprehend nature, it tells us, we first must decipher
its components. The assumption is that once we understand the parts,
it will be easy to grasp the whole. ... Now we are close to knowing
just about everything there is to know about the pieces. But we are as
far as we have ever been from understanding nature as a
whole. Indeed, the reassembly turned out to be much harder
than scientists anticipated. The reason is simple: Riding
reductionism, we run into the hard wall of
complexity. We have learned that nature is not a
well-designed puzzle with only one way to put it back together. ...
Nature assembles the pieces with a grace and precision honed over
millions of years. It does so by exploiting the all encompassing laws
of self-organization, whose roots are still largely
a mystery to us." Linked, p. 6
“Virtually
all the major unsolved problems in science today have this intricate
character. Consider the cascade of biochemical reactions in a single
cell and their disruption when the cell turns cancerous; the booms and
crashes of the stock market; the emergence of consciousness from the
interplay of trillions of neurons in the brain; the origin of life from
a meshwork of chemical reactions in the primordial soup. All these
involve enormous numbers of players linked in complex webs. In every
case, astonishing patterns emerge spontaneously. The richness of the
world around us is due, in large part, to the miracle of
self-organization.” Sync p. 34-35
“The
study of nonlinear systems composed of enormous
number of parts [was] later christened as ‘complexity
theory’” Sync p. 209
“In
an abstract sense, any collection of interacting parts — from atoms and
molecules to bacteria, pedestrians, traders on a stock market floor,
and even nations — represent a kind of substance. Regardless of what
it is made of, that substance satisfies certain laws of form, the
discovery of which is the aim of complexity theory.”
Nexus, p. 18
“A big,
messy linear problem can always be broken into smaller, more manageable
parts. Then each part can be solved separately, and all the little
answers can be recombined to solve the bigger problem. So it is
literally true that in a linear problem, the whole is exactly equal to
the sum of the parts. The hitch, though, is that linear
systems are incapable of rich behavior.” Sync p.
50-1
“The word
linear refers to proportionality: If you
graph the deflection of a girder versus the force applies, the
relationship falls on a straight line. (Here, linear
does not mean sequential, as in “linear thinking”, plodding along, one
thing after another. That’s a different use of the same word.)
...
Most systems behave linearly only when they are close
to equilibrium, and only when we don’t push them too hard. When a
system goes nonlinear, driven out of its normal
operating range, all bets are off. ...
In any situation
where the whole is not equal to the sum of the parts, where things are
cooperating or competing, not just adding up their separate
contributions, you can be sure that nonlinearity is present. Our
nervous system is built from nonlinear components. The laws of ecology
are nonlinear. Combination therapy for AIDS patients — drug cocktails
— are effective precisely because the immune response and the viral
population dynamics are both nonlinear; three drugs taken in
combination are much more potent than the sum of the three of them
taken separately. Any human psychology is absolutely nonlinear. If you
listen to your two favorite songs at the same time you wont get double
the pleasure.
The synergistic character of
nonlinear systems is precisely what makes them so difficult to analyze.
They can’t be taken apart. The whole system has to be examined all at
once, as a coherent entity. This necessity for global
thinking is the greatest challenge to understanding how large
systems can spontaneously synchronize themselves. More generally, all
problems about self-organization are fundamentally
nonlinear. Sync p. 181-2
“Whenever
nonlinear elements are hooked together in gigantic
webs, the wiring diagram has to matter. It’s a basic principle:
Structure always affects function. The structure of
social networks affects the spread of information and disease; the
structure of the power grid affects the stability of power
transmission. The same must be true for species in an ecosystem,
companies in the global marketplace, cascades of enzyme reactions in
living cells.” Sync p. 237
"Nonlinear
dynamics [requires] an emphasis on geometry, visualisation
and global thinking." Sync p. 158
"The
evolutionary biologist Stephen Jay Gould argued, quite rightly, that
‘contingency’ lies at the very core of history.
Gould wrote, ‘A historical explanation does not rest on direct
deductions from the laws of nature, but on an unpredictable sequence of
antecedent states, where any major change in any step of the sequence
would have altered the final result. This final result is therefore
dependent, or contingent, upon everything that came before — the
unerasable and determining signature of history.’ " Nexus, p.
91
“In science, just as in life, one cannot
simply fast-forward the tape to see what the ending looks like, because
the ending is written only in the process of getting there.” Six
Degrees p. 161
“Ecologists estimate that
more than ten million chains of cause and effect link the seal to the
cod. In the face of this overwhelming complexity,
it is clearly not possible to foresee the ultimate effect of killing
seals on the numbers of some commercial fish.” Nexus, p. 142
“A
tiny difference in the character of just one person can have a dramatic
effect on the overall group.” Nexus, p. 108
"The
butterfly effect [Lorenz 1979 paper, Predictability:
Does the Flap of a Butterfly’s Wings in Brazil Set off a Tornado in
Texas?”] is the idea that in a chaotic system, small disturbances grow
exponentially fast, rendering long-term prediction impossible. Sync p.
183.
“The central idea of
The Tipping Point is that tiny
and apparently insignificant changes can often have consequences out of
all proportion to themselves.” Nexus, p. 158
“It’s
not as if the Millennium Bridge shakes for a little for a small number
of people and gradually builds up as the numbers increase. Either it
doesn’t shake at all, or it wobbles violently and without warning, once
the threshold is crossed. Like the straw that broke
the camel’s back, the onset of wobbling is a
nonlinear phenomenon.” Sync p. 174
"The
study of networks is part of the general idea of
science known as complexity
theory. ” Nexus, p. 18
Self-organisation, nonlinearity and
contingency mean there are no controllers, no root causes, no
predictable long-term effects, and the only way to find out what
happens is to find out what happens.
THE SCIENCE OF NETWORKS
“Just
as diverse humans share skeletons that are almost indistinguishable, we
have learned that these diverse maps follow a common blueprint. A
string of recent breathtaking discoveries has forced us to acknowledge
that amazingly simple and far-reaching natural laws govern the
structure and evolution of all the complex networks
that surround us.” Linked, p. 5-6
“The
interactions between the parts of a complex network often lead to
global patterns of organization that cannot be
traced to the particular parts. Network
architecture is not a property of parts but of the whole, as
is the existence of non existence of a tipping
point.” Nexus, p. 185
“Social
networks [such as] Web pages connected by hypertext
lines ... share deep structural properties with the food webs of any
nation’s economic activity. Incredibly, all these networks possess
precisely the same organization as the network of connected neurons in
the human brain and the network of interacting molecules that underlies
the living cell.” Nexus, p. 15
“Networks
have properties, hidden in their construction that limit or enhance our
ability to do things with them. ... A change in the layout, the
addition of only one extra link, [can] suddenly remove [a] constraint.
... The construction and structure of networks is the key to
understanding the complex world around us. Small changes in the
topology, affecting only a few of the nodes or links, can open up
hidden doors, allowing new possibilities to emerge.” Linked, p.
12
“A network, after
placing a critical number of links, drastically changes.
Before, we have a bunch of tiny isolated clusters of
nodes, disparate groups of people that communicate only within the
clusters. After, we have a giant cluster, joined by
almost everybody. ... Networks around us are not just webs. They are
very dense networks from which nothing can escape and within which each
node is navigable.” Lines, p. 17-19
“Although
the structure of the relationships between a network’s components is
interesting, it is important principally because it
affects either their individual behavior or the behavior of
the system as a whole. Second, networks are dynamic
objects not just because things happen in networked systems,
but because the networks themselves are evolving and changing in time,
driven by the activities or decisions of those very components. In the
connected age, therefore, what happens and how it happens
depend on the network. And the network in turn depends on
what has happened previously. It is this view of a network — as an
integral part of a continuously evolving and self-constituting system —
that is truly new about the science of networks.”
Six Degrees p. 28-29
This is why, when we model a
behaviour, a state, a symbol, a relationship, we are always modelling
it firstly as part of a network of other behaviours, states, symbols
and relationships; and secondly as a dynamic system — one where the
parts are continually changing moment by moment and the whole is
continually evolving over time.
Note, it is
not possible to model ‘the whole’ because any element is always nested
in a network which in turn is nested in other networks. For practical
purposes, however, we don’t have to — thanks to modularity, metonymy
and metaphor:
Modularity
is a feature of self-organised systems. It means highly clustered nodes
have a relative autonomy, and can to some degree be studied
independently.
Metonymy
means a part can stand for the whole. This operates in at least two
ways: (i) The emergent behaviour of a system (part) is representative
of the overall interaction of the components (whole); and (ii) Because
of the fractal nature of many systems, any significant part contains
the essence of the whole.
Metaphor
enables us to model the complex and unknown in terms of the simpler and
better known. (As an aside, there is growing evidence that physical
weak links within the brain are what make the production and
comprehension of novel metaphors possible.)
“We
do what we do in part because of the position we occupy in our
surrounding social structure and in part because of our innate
preferences and characteristics. In sociology, these two forces are
called structure and agency, and
the evolution of a social network is driven by the
trade-off between the two. ... It will become clear that just a little
agency goes a long way.” Six Degrees p. 72 and 82
In this sense,
structure is equivalent to a symbol’s network of
relationships and agency is its attributes and
intention. Ken Wilber calls these communion and
agency.
“Unlike
networks of power generators or neurons, individuals in social
networks have their own ideas about what makes them who they
are. In other words each individual in a social network comes with a
social identity. And by driving both the creation
of the network and the notions of distance that
enable individuals to navigate through it, social identity is what
leads networks to be searchable.” Six Degrees p. 156
How might this apply to symbols in the
Present Configuration and their ‘distance’ from symbols in the Desired
Outcome? Distance could be perceived as space, time, speed,
difficulty, degree of change, control, probability of occurrence, etc.
And what happens when this perception does not match feedback from
actual events? e.g. when no magic wand/pill appears, or when a miracle
occurs. i.e. change happens slower/faster, easier/more difficult, etc.
than expected.
“A network might be a random network, an ordered network, a small-world network.” Nexus, p. 192
SMALL-WORLD ARCHITECTURE
LOW NUMBER of DEGREES OF SEPARATION, SHORT CHAINS
WEAK LINKS and TIES, LONG BRIDGES, SHORTCUTS
HIGH CLUSTERING, MODULARITY
“Small
worlds are a generic property of networks in general. Short
separation is not a mystery of our society or something peculiar about
the Web: most networks around us obey it. It is rooted in their
structure — it simply doesn’t take many links for me to reach a huge
number of web-pages or friends.” Linked, p. 40
“What
distinguishes a small-world network is not only that
it has a low number of degrees of separation but
also that it remains highly clustered. We might say
that the fabric of the network is densely weaved, so that any element
remains comfortable and tightly enmeshed within a local web of
connections. Consequently, the network overall can be viewed as a
collection of clusters, within which the elements are intimately
linked, as in a group of friends. A few ‘weak’
links between clusters serve to keep the whole world small.
... On the other hand, there are drawbacks to too
much clustering. ... At its core lies the idea that too much
order and familiarity is just as bad as too much disorder and novelty.
We instead need to strike some delicate balance between the two.”
Nexus, p. 199-207
“As long as we have a way
of generating clustering and a way of
allowing shortcuts, we will always get a
small-world network. Small-world networks arise
from a very simple compromise between very basic forces — order and
disorder — and not from the specific mechanisms by which that
compromise is brokered.” Six Degrees p. 91
“The
‘average pathlength,’ formalizes the intuitive idea of degrees
of separation. To calculate it, take any pair of nodes and
count the number of links in the shortest chain between them; then
repeat for all other pairs of nodes, and average the resulting chain
length. ... The average amount of overlap in a network is quantified by
a second statistic, the ‘clustering,’ defined as the
probability that two nodes linked to a common node will also be linked
to each other. ... Average pathlength reflects the global structure; it
depends on the way the entire network is connected, and cannot be
inferred from any local measurement. Clustering reflects the local
structure; it depends only on the interconnectedness of a typical
neighborhood, the inbreeding among nodes tied to a common center.
Roughly speaking pathlength measures how big the network is, clustering
measures how incestuous it is.” Sync p.
239-241
“There are, it seems, two
flavors of small: egalitarian networks in
which all of the elements have roughly the same number of links, and
aristocratic networks characterized by a spectacular
disparity. The Internet and the WWW, the networks of sexual contacts
between people, of scientific papers linked by citations, and of
scientists linked by having coauthored papers, and of words linked by
appearing next to one another in English sentences, are aristocratic
networks with hubs or connectors, presumably the
consequence of the rich getting richer.
But for other small-world networks, this is not the case. The neural
network of the nematode worm, for example, has no connectors. As each
neuron is linked to roughly fourteen others. The same egalitarian
character seems to describe the neural network of the human brain, as
well as transportation networks of many kinds, including the webs of
roads and railways that cover the continents. In the case of the US
electrical power grid, each generator, transformer or substation links
up with roughly three others, and again there is a conspicuous lack of
highly linked connectors.” Nexus, p. 119-120
“In
the cat brain, for example, the number of degrees of
separation turns out to be between only two and three. The
number is identical in the macaque brain. ... The small-world
architecture not only makes the brain efficient and quick,
but it also gives it the ability to stand up in the face of faults.”
Nexus, p. 65-66
“Cells are small worlds with
three degrees of separation.
That is, most pairs of molecules can be linked by a path of
three reactions. Perturbations, therefore, are never localised: any
change in the concentration of a molecule will shortly reach most other
molecules. ... Surprisingly the measurements indicate that whether we
are navigating the tiny network of a small parasite bacterium or the
highly developed highway system of a multi-cellular organism, such as a
flower, the separation is the same. ... For the vast majority of
organisms, the ten most-connected molecules are the same. ... The
most-connected molecules have an early evolutionary history as well.”
Linked, p. 186-187
“Researchers put to the
test seven distinct food webs sampled from ecosystems globally. Each
of these studies found exactly the same thing: small worlds
with only two or three degrees of separation ... most species
within a food web can be thought of as ‘local’ to each other and exist
in surprisingly ‘small worlds’ where species can potentially interact
with other species through at least one short trophic chain. ... This
suggests that the effect of adding, removing or altering species will
propagate both widely and rapidly throughout large complex
communities.” Nexus, p. 150-152
“What we
found amazed us. The slightest bit of randomness contracted the
network tremendously. The average pathlength plummeted at first — with
only one percent rewiring the graph dropped by 85 percent from its
original level. Further rewiring had only minimal effect; indicating
that the network had already gotten about as small as it could possibly
get. Meanwhile, the clustering barely budged. ...
The first few random links act as shortcuts —
bridges between parts of the network that would
otherwise be remote. Their disproportionate impact comes from a
powerful nonlinear effect: not only do they pull two
nodes together; they pull entire worlds together. ... The first few
shortcuts drastically reduced the size of the world, but had far less
effect on the clustering. The implication is that the transition to a
small world is essentially undetectable at a low level. ... The most
important result of the simulations was that over a broad intermediate
range of rewiring, the model networks were very clustered and very
small at the same time. ... [In an organism] the short pathlength
facilitates rapid communication throughout the creature’s body, while
the high clustering probably reflects the presence of feedback loops
and modular structure in its nervous system.” Sync
p. 241-244
Note, the extra links do not result in an 80:20 Pareto
Effect, but an 85:1. The implies that, if a network is
fragmented, only a few extra links are needed to produce high
interconnectivity — and it really doesn’t matter what is connected to
what as long as new connections are produced. However, if a network is
already highly connected then extra links don’t make much
difference.
“Modularity
is a defining feature of most complex systems. ... The cell is not
only modular, but its modularity has a strict architecture: Numerous
small but highly interlinked modules combine in a hierarchical fashion
into a few larger, less interlinked modules. There are not “typical”
or “characteristic” modules in the cell. Rather, the metabolism can be
equally well deconstructed into many small, highly interlinked modules
or into a few larger but less cohesive ones. ...
Language, viewed as a network of synonyms, is
hierarchical as well, a few highly connected words like “turn,” “take,”
or “go,” each with over one hundred synonyms hold the various lexical
modules together. ...
"Hierarchical
modularity is a generic property of most real networks,
accompanying the scale-free architecture. It is
this hierarchical modularity that makes multitasking possible: While
the dense interconnections within each module help the efficient
accomplishments of specific tasks, the hubs
coordinate the communication between the many parallel functions. ...
Hierarchical modularity has significant design advantages: It permits
parts of the system to evolve separately.
Bottlenecks and
slowdowns are inevitable if the same module is simultaneously
confronted with several tasks. The computer’s dependence on a single
central processing unit is its main bottleneck, and when our cerebral
cortex is taxed with too many tasks, we slow down too.” Linked, p.
236-7
“In a series of computer-simulations,
Watts and Strogatz found that fireflies were able to manage the
synchronization almost as readily as if everyone were talking to
everyone else. By itself, the small-world
architecture offered a reduction in the required number of links by a
factor of thousands. There is a profound message lurking here — the
message is not about biology, but about computation. ... Whatever the
setting, computation requires information to be moved about between
different places. And since the number of degrees of
separation reflects the typical time needed to shuttle
information from place to place, the small-world architecture makes for
computational power and speed.” Nexus, p. 58
“Suppose
somehow we could remove a strong link from the social network. What
effect would this have on the number of degrees of
separation? Hardly any. Since strong links almost always
appear in special triangles, you would still be able to go from one end
of the missing link to the other in just two steps, by moving along the
remaining two edges of the triangle. ... The crucial links are the weak
links between people, especially those that he called social
‘bridges’. ... Bridges are almost always
formed from weak links. Granovetter was able to reach a
surprisingly conclusion: weak links are often of greater importance
than strong links because they act as the crucial ties that sew the
social network together. These are the social
‘shortcuts’ that if eliminated, would cause the
network to fall to pieces. [Hence] ‘The Strength of Weak
Ties’ was the elegant title of Granovetter’s classic paper
from 1973.” Nexus, p. 41-43
It is my guess that symbols in a
Metaphor Landscape will likely be as interconnected as any physical
ecosystem. This means that any two symbols or ideas will be connected
by a maximum of two or three links. In other words you can get from
anywhere in your mind to anywhere else is just a few hops. Of course,
just because there is a pathway doesn’t mean you can always find it
when you need it!
“At an
anatomical level — the level of pure, abstract connectivity — we seem
to have stumbled on a universal pattern of
complexity. Disparate networks show the same three
tendencies: short chains, high clustering, and scale-free link
distributions.” Sync p.256
SCALE-FREE NETWORKS
POWER LAWS, FAT-TAIL DISTRIBUTION
HUBS, CONNECTORS
“Real
networks are not random, [but] chance and randomness do play an
important role in their construction. ... A scale-free
network is a web without a spider. Real networks
are self-organized. ... The robustness of the laws governing
the emergence of complex networks is the explanation for the ubiquity
of the scale-free topology, describing such diverse systems as the
network behind language, the links between the proteins in the cell,
sexual relationships between people, the wiring diagram of a computer
chip, the metabolism of a cell, the Internet, Hollywood, the World Wide
Web, the web of scientists linked by co-authorships, and the intricate
collaborative web behind the economy, to name only a few.” Linked, p.
221
“The striking visual and structural
differences between a random network and one described by a
power law degree distribution are best seen by
comparing a US road map with an airline routing map. On the road map
cities are the nodes and the highways connecting them the links. This
is a fairly uniform network: each major city has at least one link to
the highway system, and there are no cities served by hundreds of
highways. Thus most nodes are fairly similar, with roughly the same
number of links. The airline routing map differs drastically from the
road map. The nodes of this network are airports connected by direct
flights between them. ... A few hubs ... from which
flights depart to almost all other US airports. The vast majority of
airports are tiny, appearing as nodes with at most a few links
connecting them to one or several hubs.” Linked, p. 69
“Power-law
curves have what are called ‘fat tails’.
That is compared to the bell curve, the power-law curve tails off
towards zero much more slowly. The fat tail implies that you are far
more likely to find a node with a very high number of links than you
would be if these networks followed normal statistics.” Nexus, p.
84
“If the heights of an imaginary planet’s
inhabitants followed a power law distribution, most
creatures would be really short. But nobody would be surprised to see
occasionally a hundred-feet-tall monster walking down the street. In
fact, among six billion inhabitants, there would be at least one over
8,000 feet tall. So the distinguishing feature of a power law is not
only that there are many small events, but that the numerous tiny
events coexist with a few very larger ones. These extraordinary large
events are simply forbidden in a bell curve.” Linked, p.
67-68
“The absence of a peak in a
power law distribution implies that in a real
network there is no such thing as a characteristic node. We see a
continuous hierarchy of nodes, spanning from rare
hubs to numerous tiny nodes. The power law
distribution thus forces us to abandon the idea of a scale, or a
characteristic node. There is no intrinsic scale in these networks.
[JL- Hence they are described as scale-free.]”
Linked, p. 70
“Each
scale-free network will have several large
hubs that will fundamentally define the network’s
topology. The finding that most networks of conceptual importance,
ranging from the World Wide Web to the network within the cell, are
scale-free gave legitimacy to hubs. They determine
the structural stability, dynamic
behavior, robustness, and error and attack
tolerance of real networks. They stand as proof of
the highly important organizing principles that govern network
evolution.” Linked, p. 71-72
“Connectors
— nodes with an anomalously large number of links — are present in very
diverse complex systems, ranging from the economy to the cell. They
are a fundamental property of most networks. Their discovery has
turned everything we thought we knew about networks on its head. ...
[e.g.] 90% of all documents on the web have 10 or fewer links pointing
to them, while a few, about 3, are referenced by close to a million
other pages.” Linked, p. 56-58
“For every
organism [studied], the distribution of nodes according to their number
of links — the number of chemical reactions in which the molecule
participates — followed a power law. Cellular
metabolism involves hubs. In the bacterium E. coli,
for example, one or two specific molecules take part in several-hundred
different chemical reactions involved in the bacterium’s metabolism,
whereas many thousands of other molecules take part in only one or two
reactions. The biochemical network of cellular metabolism is also a
small world, and the diameter is just about the same
for all forty-three species: in every one no more than about four
reactions link any two molecules.” Nexus, p. 87
“Power
laws hint that a system may be organizing
itself. They arise at phase transitions,
when a system is poised at the brink, teetering between order and
chaos. They arise in fractals, when an arbitrarily
small piece of a complex shape is a microcosm of the whole. They arise
in the statistics of natural hazards — avalanches and earthquakes,
floods and forest fires — whose sizes fluctuate so erratically from one
event to the next that the average cannot adequately stand in for the
distribution as a whole.” Sync p.255
“The
power law implies that if you magnify any small
portion of a river network, you will get a pattern that looks much like
the whole. In other worlds, the network is not nearly as complex as it
appears. Innumerable accidents may make every river network unique,
and yet what goes on at one scale is in every case intimately connected
to what goes on at another. This feature, which reveals a hidden
simplicity in the structure of all river networks, is known as
self-similarity,
and structures of this sort are sometimes called
fractals. The real importance of
the power law is that it reveals how, even in a historical process
influenced by random chance, lawlike patterns can still emerge. ... If
history were run over again, the storm and its water might have gone
elsewhere and the entire river network in its details would be
different. Nevertheless, the network as a whole would still have the
very same fractal character and would satisfy the same power law that
reflects its globally organized self-similar
architecture.” Nexus, p. 102-3
If mental links
follow a power-law, scale-free distribution, it has tremendous implications for modelling and change work. For example: Hubs
and weak links both help to keep the network stable
and propagate any changes. As all paths will very
quickly lead to a hub, hubs should be fairly easy to find. As should
the strong and most used links. Weak links, on the other hand, will not
be so obvious as they are rarely used. When a weak link is brought into
operation, it may be accompanied by surprise, confusion or an a-ha
experience. Or it may be sign-posted by that little
something-out-of-the-ordinary that almost goes unnoticed. (What David
Grove refers to as a ‘non sequitur’ and Caroline Myss alludes to when
she says “The Gods prefer to enter by the backdoor.”)
The
power law says there are no typical nodes in
scale-free network. Hence groups can be categorized easily but
individuals cannot. Yet much of psychology is related to categorising
and diagnosing ‘the typical’, e.g. Psychometric tests,and the
Diagnostic and Statistical Manual, DSM IV. In Symbolic Modelling while
we recognise archetypical patterns, we are most interested in modelling
the idiosyncratic and the unique — as identity is a function of the
individual.
PHASE TRANSITIONS, THRESHOLDS, TIPPING and CRITICAL
POINTS
“Normally
nature hates power laws. In ordinary systems all quantities follow
bell curves, and correlations decay rapidly, obeying exponential laws.
But all that changes if the system is forced to undergo a
phase transition. Then power
laws emerge — nature’s unmistakable sign that chaos is
departing in favor of order. The theory of phase transitions told us
loud and clear that the road from disorder to order is maintained by
the powerful forces of self-organization and is
paved by power laws. It told us that power laws are not just another
way of characterizing a system’s behavior. They are the patent
signatures of self-organization in complex systems.” Linked, p.
77
“[A phase transition] is a crisp
transition between two utterly distinct regimes. ... When gasoline
evaporates to vapor or a hot copper wire melts, or when any of a
thousand other substances suddenly change from one form to another, the
atoms or molecules remain the same. In every case, it is only the
overall, collective organization of the atoms or molecules that
changes. ... In ordinary life, details usually matter. At
phase transitions most details simply do not matter.
... There is not just one, unique kind of phase transition. Instead,
there are a handful of several different kinds. ... There is a
universal theory of organizational transformation. ... The
critical state is the knife’s
edge between two utterly different conditions. The word
critical arises in connection with the peculiar
condition that matter gets itself into when poised exactly between two
kinds of organizations. Water held under those conditions, for
example, is neither a vapor or a liquid.” Nexus, p. 163-166
“The
freezing of a liquid and the emergence of a magnet are both transitions
from disorder to order. ... Right
at the transition point the system is poised to choose between the two
phases, just like a climber on a crest choosing which side to go down
the mountain. Undecided which way to go, the system frequently goes
back and forth, and its vacillations increase near the
critical point. ... In the vicinity of the critical
points we need to stop viewing atoms separately. Rather they should be
considered communities that act in unison. Atoms must be replaced by
boxes of atoms such that within each box all atoms behave as one.”
Linked, p. 74
This is like a binding pattern where
the system acts as one, and none of the actions of components make
sense without knowledge of the whole pattern.
“At
the critical point of transition, all parts of the
system act as if they can communicate with each
other, despite their interactions being purely local. In this
condition, known as criticality,
tiny perturbations, which in any other state would be felt only
locally, can propagate without bound throughout even an infinitely
large system.” Six Degrees p. 63-64
“Given
the unusual richness of our complex world [you may be surprised to
know], everything that physicists have discovered indicates that no
matter how you bend the rules, there is always a sharp tipping
point. ... Consequently, even though we know very little,
perhaps even next to nothing at all about the psychology and sociology
of ideas, mathematical physics guarantees that there is a tipping
point. All the details that we do not know about are irrelevant to
this question.” Nexus, p. 168
This means there are
always conditions under which an individual, a group
or a Metaphor Landscape will change. When a system goes beyond a
threshold changes occur regardless of individual nodes or links. Of
course, whether the change ends up being a breakthrough or a breakdown
is another matter.
