Presented at The Developing Group, 5 Jun 2004
(A second version, Thinking Networks II, was presented on 3 June 2006)
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 (2003)
Mark Buchanan, Nexus: Small worlds and the science of networks (2002)
Steven Strogatz, Sync: Rhythms of nature, rhythms of ourselves (2003)
Duncan Watts, Six Degrees: The Science of a connected age (2003)
Other books that have contributed to our thinking in terms of networks are:
Fritjof Capra, Hidden Connections: Integrating the biological, cognitive and social (2002)
Malcolm Gladwell, The Tipping Point: How little things can make a big difference (2000)
Steven Johnson, Emergence: The connected lives of ants, brains, cities and software (2001)
Mark Ward, Universality: The underlying theory behind life, the universe and everything (2002)
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 inMetaphors 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.
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
“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
Small-world architecture, Low degrees of separation, Short chains, Weak links, Ties, Long bridges, Short cuts, 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
“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-fee networks, Power laws, Fat-tail distributions, 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.
NETWORK DYNAMICS - I
Growth, Preferential attachment, Rich get richer, Stability, Robustness, Resilience, Tolerance, Vulnerability, Failure
“The goal before us is to understand complexity. To achieve that, we must move beyond structure and topology and start focusing on the dynamics that take place along the links.” Linked, p. 225
“Dynamics really has two meanings. The first meaning is what we might call dynamics of the network. In this sense of the word, dynamics refers to the evolving structure of the network itself, the making and breaking of network ties. … A dynamical view of networks, claims that existing structure can only be properly understood in terms of the nature of the processes that led to it.
The second meaning, is what we might call dynamics on the network. From this perspective, we can imagine the network as a fixed substrate linking a population of individuals, but now the individuals are doing something — the outcome of which is influenced by what their neighbors are doing and, therefore, the structure of the network. … In the real world, both kinds of dynamics are going on all the time. … The structure of the network could change, but so could the pattern of activity on the network.” Six Degrees, p. 54-55
“Equilibrium means nothing changes; stability means slight disturbances die out.” Sync, p. 60-63
“By viewing networks as dynamical systems that change continuously over time, the scale-free model embodies a new modelling philosophy. … Our goals have shifted from describing the topology to understanding the mechanisms that shape network evolution … understanding that structure and network evolution [can’t] be divorced from one another. … Networks are not en route from a random to an ordered state. neither are they at the edge of randomness and chaos. Rather, the scale-free topology is evidence of organising principles acting at each stage of the network formation process.” Linked, p. 90-91
“We find that real networks are governed by two laws: growth and preferential attachment.” Linked, p. 86
“The expansion of the network means that the early nodes have more time than the latecomers to acquire links. Thus growth offers a clear advantage to the senior nodes, making them the richest in links. Seniority, however, is not sufficient to explain the power laws. Hubs require the help of the second law, preferential attachment. Because new nodes prefer to link to the more connected nodes, early nodes with more links will be selected more often and will grow faster than their younger and less connected peers. Thus preferential attachment induces a rich-get-richer phenomenon that helps the more connected nodes grab a disproportionate large number of links at the expense of the latecomers.” Linked, p. 87-88
“Preferential attachment makes an additional statement about the way the world works: small differences in ability or even purely random fluctuations can get locked in and lead to very large inequalities over time.” Six Degrees, p. 109
“Even if every trace of racism were to vanish tomorrow, there may still be a natural tendency for races to separate, much like oil and water. Social realities are fashioned not only by the desires of people, but also by the action of blind and more or less mechanical forces — in this case forces that can amplify slight and seemingly harmless personal preferences into dramatic and troubling consequences.” Nexus, p. 186
“Whenever limitations or costs eventually come into play to impede the richest getting still richer, then a small-world network becomes more egalitarian. … On the one hand, the rich-get-richer mechanism leads inevitably to small-world networks, as if they were dictated by an architectural law of nature. Nevertheless, limitations and constraints sometimes get in the way and leave their telltale traces on the final form. Still, the similarities between the two kinds of networks are probably more important than the differences. The small-world character persists in either case.” Nexus, p. 125-126
“The model offers a general message: encouraging exchange between people, with other things being equal will tend to distribute wealth more equitably. [They] found greater equality whenever they boosted the flow of wealth along the links or increased the number of such links. Alternatively, stirring up the wildness and unpredictability of investment returns worked in the opposite direction, which is not surprising as it boosts the influence of the rich-get-richer phenomenon.” Nexus, p. 193
This suggests that so-called therapeutic approaches like ‘shaking the tree’ or ‘messing up a problem’ may be counter productive.
“A significant fraction of nodes can be randomly removed from any scale-free network without its breaking apart. This resilience to errors is an inherent property of their topology. … In scale-free networks, failures predominantly affect the numerous small nodes. Thus, these networks do not break apart under failures. The accidental removal of a single hub will not be fatal either, since the continuous hierarchy of several large hubs will maintain the network’s integrity. Topological robustness is thus rooted in the structural unevenness of scale-free networks. … [However] the removal of a few hubs [can break a network] into tiny, hopelessly isolated pieces. … Hidden within their structure, scale-free networks harbor an unsuspected Achilles’ heel, coupling a robustness against failures with vulnerability to attack. … Several of the largest hubs must be simultaneously removed to crush them. This often requires taking out as many as 5 to 15 percent of all hubs at the same time.” Linked, p. 113-118
“Any hub or connector species has a huge number of links to other species. As a result, most of these links will be weak links; the two species interact infrequently. … The consequences of removing just one connector species can be especially dramatic, as a huge number of weak stabilizing links goes with it. Ecologists have long talked about ‘keystone’ species, crucial organisms the removal of which might bring the web of life tumbling down like a house of cards. … Ecologists have] found that the highly connected keystones were often inconspicuous organisms in the middle of the food chain or were sometimes basic plants at the very bottom of the web. In other cases, they were major predators. There appear to be no hard and fast rules for determining which kind of species are likely to be keystones. Identifying keystones means studying the network architecture and seeing which species are the connectors, the lynchpins of the living fabric.” Nexus, p. 151-154
“Weak links between species act to take the wind out of dangerous fluctuations. They are the natural pressure valves of ecological communities.” Nexus, p. 150
In small worlds, weak links are both change-propagating and change-restraining. They increase the chance of interacting with more ‘distant’ (not alike) nodes. This is particularly important at a time of crisis when, by definition, business-as-usual is not an option. Weak ties increase stability but this in turn works against radical change. At the same time, it is these same weak ties that propagate a change/failure/disease throughout the network.
NETWORK DYNAMICS - II
Cascades, Contagion, Epidemics, Domino effect
“Having an interconnected system really makes for a more efficient use of our natural resources and keeps the cost down, but it means when something goes wrong, it can cascadethrough the system A property of complex networks is their vulnerability due to interconnectivity. … In general, natural systems have a unique ability to survive in a wide range of conditions. Although internal failures can affect their behavior, they often sustain their basic functions under very high error rates. This is in stark contrast to most products of human design, in which the breakdown of a single component often handicaps the whole device.” Linked, p. 110-111
“[In] interacting systems ranging from forest fires to mass extinctions … the individual element is subjected to increasing pressure, builds up towards a threshold, then suddenly relieves its stress and spreads it to others, potentially triggering a domino effect.” Sync, p. 31
“The 1996 blackout is a typical example of a cascading failure. When a network acts as a transportation system, a local failure shifts loads or responsibilities to other nodes. If the extra load is negligible, it can be seamlessly absorbed by the rest of the system, and the failure remains effectively unnoticed. If the extra load is too much for the neighboring nodes to carry, they will either tip or again redistribute the load to their neighbors. Either way, we are faced with a cascading event. …
Simulations indicate that most cascades are not instantaneous: failures can go unnoticed for a long time before starting a landslide. Attempting to decrease the frequency of such cascades has inevitable consequences, however, as those cascades that do succeed are then more disruptive. …
Topological robustness is a structural feature of networks. Cascading failures, however, are a dynamic property of complex systems. … The results of the research forced us to acknowledge that topology, robustness, and vulnerability cannot be fully separated from one another.” Linked, p. 119-122
“The [blackout] cascading failure that struck the West on August 10, 1996, was not a sequence of independent random events that simply aggregated to the point of a crisis. Rather, the initial failure made subsequent failures more likely, and once they occurred, that made further failures more likely still, and so on. … Perhaps the most perturbing aspect of cascading failures is that by installing protective relays on the power generators, by reducing, in effect, the possibility that individual elements of the system would suffer serious damage — the designers had inadvertently made the system as a whole more likely to suffer precisely the kind of global meltdown that occurred.” Six Degrees, p. 23-24
“There are three ways in which cascades can be forbidden. The first one is obvious: if everyone’s threshold is too high, no one will ever change and the system will remain stable regardless of how it is connected. Even when this is not the case, cascades can still be forbidden by the network itself, in two ways: either it is not well connected enough or (and this is the surprising part) it is too well connected.
Networks that are not connected enough, therefore, prohibit global cascades because the cascade has no way of jumping from one vulnerable cluster to another. And networks that are too highly connected prohibit cascades also, but for a different reason: they are locked into a kind of stasis, each node constraining the influence of any other and being constrained itself. In social contagion, a system will only experience global cascades if it strikes a trade-off between local stability and global connectivity.” Six Degrees, p. 237 & 241
The too connected scenario is a classic description of a binding pattern.
“Only when a disease reaches a shortcut does it start to display the worst-case, random mixing behavior. Epidemics in a small-world network have to survive first through a slow-growth phase, during which they are most vulnerable. And the lower the density of shortcuts, the longer this slow-growth phase will last.” Six Degrees, p. 181
While there is a good chance of preventing a full-scale epidemic during the slow-growth phase, when change is the intention, newness and difference will need to be nurtured through the slow-growth phase.
This finding may also support the notion of (i) Spending time at the beginning of a session to develop the links/relationships in a Metaphor Landscape as this will likely increase the density of shortcuts, thereby shortening the slow-growth phase and (ii) Taking your time at the beginning of the Maturing Changes phase to allow for the completion of the slow-growth phase.
“In scale-free networks even if a [computer] virus is not very contagious, it spreads and persists. Defying all wisdom accumulated during five decades of diffusion studies, viruses travelling in scale-free networks are practically unstoppable. The source of this unexpected behavior lies in the uneven topology. Scale-free networks are dominated by hubs. Because each hub is linked to a very large number of other [nodes], it has a high chance of being [re-]infected by one of them. Once infected, a hub can pass on the virus to all the other [nodes] it is linked to. Thus highly linked hubs offer a unique means by which viruses persist and spread. ” Linked, p. 135
This maybe one way to explain why ‘relapse’ after an apparently successful relief from depression or anxiety is not uncommon. If an unproductive thought (a ‘thought virus’ as Robert Dilts calls them) survives somewhere on the network it has a good chance of eventually re-infecting nodes that have become virus-free. Of course, the source of the thought virus may be outside the client.
This metaphor suggests that, rather than attempting to the eliminate all negative thoughts, it maybe wiser to establish a way of handling them when they occur, i.e. building up an immunity.
“It is not necessarily good ideas that spread — just infectious ones. … the infectious movement of desires and ideas from mind to mind is even the basis of a new theory of advertising known as ‘permission marketing’.” Nexus, p. 160
A more pleasant but less sticky name than ‘viral marketing’.
“[During] an information cascade individuals in populations essentially stop behaving like individuals and start to act more like a coherent mass. Sometimes information cascades occur rapidly [as when a market bubble burst]. And sometimes they happen slowly — new societal norms, like racial equality, woman’s suffrage, and tolerance of homosexuality, for example, can take generations to become [almost] universal. What all information cascades have in common, however, is that once one commences, it becomes self-perpetuating; that is, it picks up new adherents largely based on the strength of having attracted previous ones. Hence, an initial shock can propagate through a very large system, even if the shock itself is small.
Because they are often of a spectacular or consequential nature, cascades tend to make newsworthy events. This disguises the fact that cascades actually happen rather rarely.” Six Degrees, p. 205-65
“One of the most intriguing features of the cascade problem was how most of the time the system is completely stable even in the face of frequent external shocks. But once in a while, for reasons that are never obvious beforehand, one such shock gets blown out of all proportion in the form of a cascade.
And the key to a [social] cascade is that when making decisions about how to act or what to buy, individuals are influenced not only by their own pasts, perceptions, and prejudices but also but each other.
It seemed clear that contagion in a network was every bit as central to the outbreak of cooperation or the bursting of a market bubble as it is to an epidemic of disease. It just wasn’t the same kind of contagion. This is important because typically when we talk about social contagion problems, we use the language of disease. Thus we speak of ideas as infectious, crime waves as epidemics, and market safeguards as building immunity against financial distress, But the metaphors can be misleading because they suggest that ideas spread from person to person in the same way that diseases do — that all kinds of contagion are essentially the same. They are not. … Social contagion is a highly contingent process.” Six degrees, p. 220-224
“Social contagion is even more counterintuitive than biological contagion, because the impact of one person’s actions on another depends on what other influences the latter has been exposed to. The spread of ideas, unlike the spread of disease, requires a trade-off between cohesion within groups [clustering enables local reinforcement] and connectivity across them. A node can be vulnerable in one of two ways: either because it has a low threshold (thus, a predisposition to change); or because it possesses only a few neighbours, each of which thereby exert significant influence.” Six Degrees, pp. 231-3
Not only is timing of the introduction of an innovation important, so is where it is introduced. So when in the Maturing Changes phase you enquire if a change to one symbol has spread to another symbol (And when X, what happens to Y?) it may be prudent to start with the ‘closest’ and most similar symbols.
“The presence of a wide range of personal thresholds in a population tends to increase the chance of new ideas or products catching on considerably.
The term innovators can be used to refer not only to individuals who introduce new devices but also to advocates of new ideas, or more generally still, any small shock that disturbs a previously quiescent system. Early adopters are simply members of a population who are the first to be influenced by an external stimulus [innovator]. … Obviously the more early adopters there are in the population, the more likely a particular innovation is to spread. And the larger the connected cluster of early adopters in which the innovation lands, the farther it will spread.” Six Degrees, p. 227, p. 232-235
“The Pfizer study [‘How Physicians Adopt a New Drug’] demonstrated that innovations spread from innovators to hubs. The hubs in turn send the information out along their numerous links, reaching most people within a given social or professional network. … Conversion [of hubs] is the key to launching an idea or an innovation. If the hubs resist a product, they form such an impenetrable and influential wall that the innovation can only fail. If they accept it, they influence a very large number of people.” Linked, p. 129-130
Thus in a small-word network you don’t need to influence to a hub directly. Change in an early adopter connected to a hub may do just as well.
But how do you tell which symbols are the early adopters? And when a Metaphor Landscape is not changing, under what conditions would an innovation be above the personal threshold of an early adopter; and that early adopter (i) is able to influence a neighbour and (ii) is not overly influenced by its neighbours?
Of course, sometimes the problem is that some nodes change too easily.
For innovations to spread to early adopters and perhaps a few of the early majority:
“social contagion is largely equivalent to biological contagion because it undergoes the same phase transition that epidemics of disease do. And for the same reason — that network connectivity, rather than the resilience of individual decision makers, is the principal obstacle to a successful cascade … the cascade propagates until it occupies the vulnerable cluster and then it runs out of places to go.
For the cascade to become global, and the innovation to spread to the early and late majority, the cascade has to cross the chasm, and that’s a different kind of phase transition.
Now, being simply well connected is less important than being connected to individuals who can be influenced easily … and whose neighbors have one or more vulnerable neighbors, and so on. So even if you can identify potential early adopters, unless you can view the network, you wont know whether or not they are all connected.
In other words, the structure of the network can have as great an influence on the success or failure of an innovation as the inherent appeal of the innovation itself. And even [when a cascade is possible] much of an innovation’s fate hangs on random chance. As much as we want to believe that it is the innate quality of an idea or product that determines its subsequent performance, or even the way it is presented, the model suggests that for any wild success, one could always find many deserving attempts that failed to receive more than a tiny fraction of the attention. And in general no one will know which one is which until all the action is over.” Six Degrees, pp. 239-244
Fitness
“Each node has a certain fitness … The introduction of fitness does not eliminate growth and preferential attachment, it changes, however, what is considered attractive in a competitive environment.” Linked, p. 95-96
“Independent of the nature of links and nodes, a network’s behavior and topology are determined by the shape of its fitness distribution. But even though each system, from the Web to Hollywood, has a unique fitness distribution, all networks fall into one of only two possible categories. The first category includes all networks in which, despite the fierce competition for links, the scale-free topology survives.
These networks display a fit-get-rich behavior, meaning that the fittest node will inevitably grow to become the biggest hub. The winner’s lead is never significant, however. The largest hub is closely followed by a smaller one, which acquires almost as many links as the fittest node. At any moment we have a hierarchy of nodes whose degree distribution follows a power law.
In networks belonging to the second category, the winner takes all, meaning that the fittest node grabs all links, leaving very little for the rest of the nodes. Such networks develop a star topology, in which all nodes are connected to a central hub. In such a hub-and-spokes network there is a huge gap between the lonely hub and everybody else in the system. A winner-takes-all network is not scale-free.” Linked, p. 102-103
An examples of a fit-get-rich distribution is that of Web search engines. Google is the fittest and the biggest, but there are plenty of others that are not far behind. An example of the winner-takes-all distribution is Microsoft Windows which runs on 86% of personal computers. The second most popular operating system, Mac OS by Apple, has only 5% of the market. Both Google and Apple are examples that the first innovator does not always have the advantage.
“The irregularity of investment return stirs up wealth differences, while transactions of all types between people tend to wipe them out. The competition between these two forces leads to Pareto’s Law, with a greater or lesser concentration of wealth falling into the hands of a small fraction of people.
The model, however, [suggests] that if the investment irregularities grows sufficiently strong, they can completely overwhelm the natural diffusion of wealth provided by transactions. In this case, an economy can pass through a sudden and dramatic transition in which the wealth disparities kicked up are simply too pronounced to be adequately tempered by flows between people.
The economy will tip and wealth, instead of being possessed by merely a small minority, will instead ‘condense’ into the pockets of a mere handful of super-rich ‘robber barons’ [cf. winner takes all]. … It has been estimated, for example, that the richest forty people in Mexico have nearly 30 percent of the money.” Nexus, p. 195
It seems to us that, in terms of networks, diversity is equivalent to the variety of nodes and choice to the number of links emanating from a node.
Scale-free networks have a greater range (diversity) of nodes than random, star or latticed networks. However, in social networks, too much diversity makes it impossible for a group or team to function effectively. The same is true of choice.
The bottom line? Gregory Bateson noted that attempting to maximise any part, characteristic or value of a system will ultimately become ‘pathological’ when the effects of the maximisation act against the interest of the system as a whole. In particular Bateson was keen to point out that this particularly applies to that part known as ‘conscious purpose’.
Global phenomena, Collective behaviour, Swarm logic, Mob rule
“The trouble with systems like the power grid is that they are built up of many components whose individual behavior is reasonably understood, but whose collective behavior, like that of football crowds and stock market investors, can be sometimes orderly and sometimes chaotic, confusing, and even destructive.
How does individual behavior aggregate to collective behavior? This is one of the most fundamental and pervasive questions in all of science. … What makes complex systems complex, is that the parts making up the whole don’t sum up in any simple fashion. Rather they interact with each other, and in interacting, even quite simple components can generate quite bewildering behavior. …
The flip side of complex systems [is that] while knowing the rules that govern the behavior of individuals does not necessarily predict the behavior of the mob, we may be able to predict the same mob behavior without knowing very much at all about the unique personalities and characteristics of the individuals that make it up.
Sometimes, the interactions of individuals in a large system can generate greater complexity than the individuals themselves display, and sometimes much less. Either way, the particular manner in which they interact can have profound consequences for the sorts of new phenomena that can emerge at the level of groups, systems, and populations. In particular, what is it about the patterns of interactions between individuals in a large system that we would pay attention to? No one has the answer yet, but in recent years a group of researchers has been chasing a promising new lead, the science of networks.” Six Degrees, p. 23-27
“Individuals have severe limitations imposed on what they can deduce about the world based on what they can observe. A well-known aphorism contends that all politics is local, but really we should say all experience is local — we only know what we know, and the rest of the world, by definition, lies beyond our radar screen. That’s why the small-world phenomenon is so counterintuitive — it is a global phenomenon, yet individuals are capable only of local measurement.” Six Degrees, p. 83
“Weak ties can be thought of as a link between individual- and group-level analysis in that they are created by individuals, but their presence affects the status and performance not just of the individuals who own them, but of the entire group to which they belong.” Six Degrees, p. 49
“The real issue is that there is a big difference between two people being connected by a short path (which is all the small-world network models claim) and their being able to find it. … The fundamental difficulty being that you are trying to solve a global problem using only local information about the network. … Finding paths to the right information becomes particularly important in times of crisis or rapid change. … Cleinberg’s deep insight was that mere shortcuts are not enough for the small-world phenomenon to be of any actual use to locally informed individuals. In order for social conditions to be useful — in the sense of finding anything deliberately — they have to encode information about the underlying social structure.” Six Degrees, p. 136-145
“[When a] system is decentralized, no one has global knowledge. And that’s what makes the puzzle so challenging: How can the system using a local rule, solve a problem that is fundamentally global in character? This puzzle captures the essence of what’s called collective computation. Think of a colony of ants building a nest. Individually, no ant knows what the colony is supposed to be doing, but together, they act like they have a mind.” Sync, p. 250
Just for fun, Penny Tompkins and I have summarised some of the above into ‘The Eight Laws of Small-world, Scale-free Marketing’ soon to be available in all good bookshops:
| Law 1 | The fit get rich so get very good at what you do. |
| Law 2 | The rich get richer because of preferential attachment, so give people a small extra reason to attach to you. |
| Law 3 | Early nodes acquire more links so be one of the first in some area of specialism and keep doing what you’re doing — the early birds eventually get the worms. |
| Law 4 | Fitness has to be noticed so work in contexts where your particular skills are visible and keep putting your talent out there — persistence pays. (J K Rowling’s first manuscript was turned down by several publishers.) |
| Law 5 | Connectors influence the most so be linked to hubs in your chosen field and be recommendable. |
| Law 6 | Six degrees of separation means you can quiet easily get to anyone — providing you find out who you know knows. |
| Law 7 | Establish weak ties, you never know where they will lead and when they might come in handy. According to Robert Cialdini in Influence, one good way to do this is to do stuff for other people as it activates the Law of Reciprocity. |
| Law 8 | The network decides which cascades go global and which peter out. So don’t take it personally if your idea fails to capture the minds of millions, and even more importantly, don’t take it personally it is succeeds. |
