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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 shortcutsbridges 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.



James Lawley

James LawleyJames Lawley is a UKCP registered psychotherapist, coach in business, and certified NLP trainer, and professional modeller. He is a co-developer of Symbolic Modelling and co-author (with Penny Tompkins) of Metaphors in Mind: Transformation through Symbolic Modelling. For a more detailed  biography see about us and his blog.

 
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