Self-organization, complexity, and networks.
Fritjof Capra, 1995. The Web of Life: A New Scientific Understanding of Living Systems. chapters 5-6.
Magoroh Maruyama. 1963. “The Second Cybernetics.” American Scientist. 51: 164-179. [link]
Albert-Laszlo Barabasi and Eric Bonabeau. 2003. “Scale-Free Networks.” Scientific American 288(May) [link]
David P. Reed. 2001. “The Law of the Pack.” Harvard Business Review (February): 23-24 [link]
Film: Mindwalk (screened April 11)
Please post comments below.
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April 18th, 2007 at 10:13 am
While the The Law of the Pack article may make the reader aware of mathematical rules used to calculate the number of potential connections in a network, this paper says little more of value. Instead, the author appears to have made the focus of this article self-glorification, as he names a law after himself, and then calls this law profound. While using networks of people to network with other people clearly strengthens the number of connections in the network, this discovery is hardly novel—nor worth naming a law after.
Meanwhile, the article on scale-free networks by Barabasi and Bonabeau addresses a series of related points about scale-free networks which itself can be connected to networked areas of study. The vast applications of scale-free networks and the understudied areas to which they could apply have potentially large implications for how we come to understand everything from brain function to cancer growth. While the concept of scale-free networks is discussed in introductory courses in psychology and sociology, it was interesting to consider the universal applicability of this concept.
Deviation-amplifying and deviation-counteracting loops would be an interesting concept to introduce to the scale-free network discussion, as network hubs are connected to their nodes in a loop, if you think about how the number of nodes influences the rate of growth (or lack thereof) of each node in the network itself. Likewise, the more popular the node, the more likely that node is to receive input from the network, either positive or negative. Maruyama’s article on the second cybernetics allows not only for network hubs but a directional causal relationship between a hub and its nodes. A hub, after all, can have positive or negative effects on its connected nodes. A person who is highly connected, for example, may foster relationships, thereby increasing her own network (an amplifying feedback loop) while this same person could just as easily break down relationships given the number of people she knows and the brevity in which a rumor would spread.
Connecting this discussion to Capra, how would one relate the sort of networks discussed by Barabasi and Maruyama to Eigen’s catalytic cycles or Mandelbrot’s Julia Sets? It appears to me, though I pose this question to hear other’s input, that the catalyst in catalytic cycles serves as hubs in a positive feedback loop. The hypercycles formed in these catalytic reactions are similar to scale-free networks, in that they are also relatively stable and self-correct for replication errors similar to the manner described in Barabasi’s article. Meanwhile, the Julia sets do not appear to mimic scale-free networks, on the surface, as some sets are disconnected, and they repeat a basic pattern, suggesting that the nodes might be connected to an equal number of other nodes. Rather than form scale-free network hubs, this would suggest a linear network is formed. Or could we consider the original pattern the hub from which other nodes stem? I am interested in other’s opinions, as I am fairly ignorant in this area but the articles got me thinking.
Lastly, a note about the film:
Mindwalk, though blatantly scripted from Capra’s book, The Web of Life, illustrates Capra’s concern for the interconnectedness of living organisms from the positions of both hard and social science. As his ecology theory heavily relies on physics as a lens through which to understand the world, one might consider his theory highly credible from all walks of life (especially given our discussion last week about the common disregard for social science among our discussion group). For those who didn’t watch the film, Mindwalk urges the viewer to relinquish Cartesian scientific methodologies, such as viewing living beings as solitary, mechanical units, and to instead adopt a view stressing the relationships between organisms. One issue I had with the film was that although on an atomic level we are interconnected, Capra did not explain how one distinguishes living organisms from the non-living, which makes difficult delineating what makes an organism alive. How does one make the connection, from the position of physics, between poverty and overpopulation, or between my interconnectedness with my pen and CO2 emissisions’ effects on the Earth’s average temperature? While I agree with Capra that a paradigm shift is necessary to solve global problems, I’m not sure that I agree with his approach.
April 18th, 2007 at 4:50 pm
re:Siobhan
I agree that the Law of the Pack was the least interesting reading — the other two articles explored ideas that seemed widely applicable and were illustrated by very interesting phenomena, while the Law of the Pack simply states mathematical laws about the number of possible connections in different types of networks and then doesn’t do a good job of explaining why these laws are interesting. However, I do think that a similar connection could be made between the Law of the Pack and the Second Cybernetics as the connection you made between Scale-Free Networks and the Second Cybernetics. That is, mutual causal systems could be introduced into the discussion on Group-Forming Networks, because Reed’s Law assigns value to a network by adding up the number of possible two-, three-, four-person, etc. groups that can be formed. Each of these groups, however, could be viewed as links between nodes that have directional causal relationships between them. In short, Reed’s Law is saying that the value of a network is proportional to the number of causal loops that can be formed in the network.
However, Reed’s Law is unable to take into account several important factors when assigning value to Group-Forming Networks. First of all, it can’t take into account the different values of different groups. For example, are 5-person groups equally valuable as 4-person groups? Also, some groups are going to be more valuable than others due to factors besides the size of group-membership. Reed uses the example of chat-rooms and online games as examples of groups that add value to AOL. Some online games are going to add more value than others. Also, the same set of people can form multiple chat rooms — how do we measure the value this group of people adds? Something that might also be interesting to consider would be whether groups that have deviation-amplifying and deviation-counteracting causal loops add different value.
Furthermore, Reed’s Law fails to take into account one of the insights of Scale-Free Networks — we can’t just assume that every node has an equal opportunity to link up with every other node. An important factor to take into account when calculating the value of each new member of a network would be how many friends/acquaintances the member has that are already on the network, as this will affect the number of groups that actually get formed on the network and thus add value to the network. Also, it would be interesting to take into account how the existence of hubs affects the value of a network — very large groups will probably grow faster and get larger and larger, and in turn will also lead to lots of smaller groups being created from and splintering off from the large group. The existence of this very large group would probably be more valuable than the existence of any small group.
April 18th, 2007 at 5:38 pm
So, I’m going to mostly talk about the two “less important” articles, mostly because I like the mathematics. The Barabasi article makes a claim that the Internet is a scale-free network, and the various complex reasons behind it, can be approximated with a variation of a construction of a Zipfian distribution. Zipf’s law says that word frequency is inversely proportional to rank: f = k/r. This means that the most common words are extremely common and things fall out very quickly, with many many words being uncommon. It shows up in other places in corpus statistics. Here’s my model that is also something a feedback loop:
There are n pages on the internet. Thus for each page there are n possible links to consider, permitting self links for a while.
We construct pages in this fashion: Choose with some probability one of th n possible links l or a marker p, with p(p) = k*p(li), for all i and some k. If we run this algorithm we should get something like:
l1l2l3l4l5pl1pl1000l40p etc.
Take each p as a marker between each page, and the number of l’s to be the number of links into the page. Thus, we will be generating pages with an average length of E[links] = x, with x s.t. (1-p(p))^x = 0.5. However, the probability mass function for rank/frequency will still follow a power distribution: Pages with very few inbound links will occur very often, and those with many will occur much more rarely.
If you repeat this, but now differentiate between the p’s and assign them probabilities based on their number of inbound links, you’ll get a k/l^2 system, since more common nodes are more well known, and so they feed back into themselves.
Thus, you can actually create these seemingly complicated network structures from a rather simple algorithm that has nothing to do with networks at wall. You could call this an emergent property of this system, which I would be ok with, but I’m not sure there should be much surprise at their existence.
Continuing with the math: the Law of the Pack misses what I think is a very important point: not every connection is equally valuable. Some nodes are redundant, and some nodes are just useless. Thus, an article I’ve read a while back has a useful point:
http://www.spectrum.ieee.org/print/4109
In fact, in large networks, such as the Internet, with millions and millions of potential connections between individuals, most are not used at all. So assigning equal value to all of them is not justified. This is our basic objection to Metcalfe’s Law, and it’s not a new one: it has been noted by many observers, including Metcalfe himself.
They also cite Zipf’s Law, but in the context of network value they establish that the value of an network is actually n log n, which is a *negative* feedback loop. Another way to look at is that the more connection you have, the higher the chance of redundant information coming to a given node, and this should grow with the inverse of the number of nodes, (1/n), which is the derivative of log n.
On the movie/Capra reading:
I agree with Siobahn about the perhaps overzealous reach of Capra, but I like the direction he’s heading over all. His background on chaos theory and fractals was very well… integrated, and actually very informative. I’d heard of a lot of these theories, but I think he put it together in a way that makes the connectiosn more obvious. I’m not sure I follow Siobahn’s question, but I feel like the link between overpopulation and poverty is well-established: in an underdeveloped (poor) setting, human labor is key, and so you have to produce more human labor. In an age where food is almost plentiful enough (along with rapidly decreasing infant mortality rates and almost good enough medicine) , you can have a huge boom in population before societies can readily adapt, which reduces the amount of food and such over all, which reduces supply, which means you need more people to make more food, etc etc.
April 18th, 2007 at 5:52 pm
I will start my comment with a response to Siobhan. While I agree that naming a theory after one’s self is somewhat self-aggrandizing, I don’t think that the article is entirely without merit.
I do think the author glosses over some of the power of what he is saying. It’s entirely extraordinary that the value of a network should increase exponentially as more people join. According my basic intuitions, the first people to join a network are incredibly more valuable than the subsequent people who join. Once a network boasts thousands of members, one additional member seems almost superfluous.
Yet this intuition is based on a preference for objects over connection. I tend to think of people as basic components of a network. Additional people comprise an ever-dwindling proportion of the network as it gets larger. Yet these connections between people – and indeed the possible groups or interactions between them increase *exponentially.* The fact that these phenomena are almost diametrically opposed to each other (change in importance of objects vs relationships) makes our natural preferences about how we divide the world that much clearer. It becomes very apparent in the way we assess the value of a network. Perhaps now the merits of Capra’s divisive way of thinking – assertive vs. collaborative and object vs. relationship – are clearer now. Like Siobhan, I agree that the article doesn’t say much. I think it would be interesting to try to think of a network as a kind of undirected graph. That way we could identify features like cliques, bipartite groups, and cycles, and try to account for how relationships compound within a network and how we can foster new connections.
Perhaps this very mathematical part of me is the reason why I enjoyed this article in the first place – at least it’s trying to quantify the utility of the networks it’s talking about. I find systems theory fascinating, but I am rapidly becoming frustrated with the ineffectuality of quantifying observations about systems as a whole. For example, Maruyama’s illustration of the growth of a city identifies many causative factors – and loops and chains of interaction between these factors. However, without other knowledge about how the relative importance of these interactions, it’s impossible to predict the growth or decay of the city. Maruyama has identified this uncertainty in other aspects of the article. In fact, the difficulty that lies in tracing chains of self-reference is first mentioned in the context of human development. Maruyama asserts that we should look at the embryo rather than the adult when trying to piece together the nature of development, since complexity increases exponentially after several cycles of self-referential feedback.
But this strategy of looking at a system before it becomes a system is inherently flawed. By the time we realize that the city that a farmer has built is worthy of study, the farmer has already built the city. How should we think about systems study if we do not have access to the conditions that gave rise to a system? How do we look at the products and try to assess the relative importance of feedback loops that gave rise to such things? Maruyama seems dismissive of the feasibility an endeavor, but in my opinion, this should be an implicit goal of systems theorists in general. We need to be open to thinking about feedback systems in a retrospective sense in order to generate a full and meaningful account of past and present movements that have shaped our world.
April 18th, 2007 at 6:20 pm
The movie reminded me a lot of My Dinner With Andre. To make an SAT analogy, Mindwalk was to My Dinner With Andre what Systems Theory is to LSD-induced naked romps through the woods. I’ll try to abstain from commenting on its characteristics as a film, and instead try to focus on the ideas it presents and how they relate to what we’ve seen in class.
Replying to David and Siobahn: One issue I had with the movie and with the Capra reading is what last week I called “sweeping unsupported claims”. To this concern add the a priori assumptions that Capra makes. For example, consider Capra’s claim that “stabilizing world population will only be possible when poverty is reduced worldwide”. Even conceding to David that the link between overpopulation and poverty is well-established, underlying Capra’s argument is the a priori assumption that the world has an overpopulation problem. Now, I’m not trying to claim the opposite (that there is no overpopulation problem), but this tendency to make sweeping assumptions gets in the way of productive argument. If (as I assume Capra’s argument might go) overpopulation is bad because of its effects on the environment, then why is it that the least poor countries (thus presumably those that don’t have problems of overpopulation) have the greatest environmental footprint (i.e., are most harmful to the environment)? I guess the link between overpopulation and poverty concerns me because of the undeniable strands of racist/eugenicist subtexts it carries.
Another thing caught my attention in the movie, and it brought into focus something that had been bothering me about the Capra reading. In the scene where they’re in the torture room, Sonia (the physicist) explains how there are two underlying themes in everything they’ve talked about, the male and the female. The male half represents aggression, power, growth, while the female half represents nurturing, care, protection. To reduce the entirety of a philosophical discussion into such a constricted gender binary seems… well, very pre-Queer Theory, at the very least. For those unfamiliar, since about the 1990’s, Queer Theory has (among other things) been deconstructing the notion of the gender binary and finding that there is not really as clear a line between male and female as most people think there is. To limit oneself to a sharp divide between male and female is to overlook a lot of possibilities in the spectrum of gender.
Anyway, back to more substance on systems. I found the article on Reed’s Law very stimulating. I was, of course, familiar with Metcalfe’s law, but I had never heard of Reed’s law. It got me thinking about modal logic and the concept of ‘common knowledge’. To say that a fact x is common knowledge is to say that everyone knows x, that everyone knows that everyone knows x, that everyone knows that everyone knows that everyone knows x, and so on and so forth. Therefore, establishing that x is common knowledge is a very tricky business (as the recursion of ‘everyone knows that everyone knows…’ goes infinitely deep). One way to do it is for an authoritative and credible source to publicly announce “x” in front of everyone. Then everyone knows that x is true, furthermore, everyone knows that everyone else is present when x’s truth is revealed, so everyone knows that everyone knows that x is true, and everyone knows that everyone else can figure all this out, so everyone knows… You get the idea. However, if you are limited to one-on-one interactions, establishing that x is common knowledge becomes a lot trickier. In a two-person network it’s not so bad. If A knows that x, A can tell B that x is true, and then x is common knowledge between them. In a three-person network, it becomes trickier. A knows that x, so A tells B that x. A can also tell C that x. Then A knows x, and A knows that B and C know x. However, B doesn’t know whether C knows x, and C doesn’t know whether B knows x. B can tell C that B knows x, and C can tell B that C knows x. Then both B and C know that B and C know x. But now A doesn’t know that B knows that C knows x. I believe (dammit I lost my lecture notes) that establishing common knowledge in this manner is impossible.
As David said, a lot of the connections in (for example) the Internet are only potential connections. To the extent that they are unrealized connections it seems unfair to count them. However, the upper bound on the number of connections does give us some idea of the potential of the network. Furthermore, it might be interesting to try to model how much new information (’x is common knowledge’ is new information, if before the use of the network only A knew x) a network can generate. In the case of a one-to-many network, anything that the ‘one’ node knows can be established as common knowledge, whereas in the one-to-one network it is impossible to establish anything as common knowledge. In the many-to-many network, anything that ANYONE knows can be established as common knowledge.
April 18th, 2007 at 6:32 pm
Sorry that I slid in past the new deadline! I almost forgot to post my response. Two topics interested me the most out of this group of readings:
(1) That the “ordered” or “self-organizing” systems that we are concerned with — from living organisms to lasers — require consistent outside stimulation. This is obvious on some level, because organisms need to eat, and lasers need to be plugged in to a wall. Now take, for example, Maturana’s theory that perception/cognition is “dissipative phenomenon” not unlike other self-organizing life structures or the chemical “honeycombs”. This is a quite reasonable conjecture, and widely accepted now. The brain reacts to outside stimuli in pattern of brain activity governed by neural relationships. If you’ve ever seen a PET scan or fMRI of a person reacting to a single novel stimulus, you can trace the wave pass through. Now a fun question: where do dreams come from? If you watch a live feed of a sleeper’s brain, there is usually very little activity. The brain is dark, with a little sporatic noise here and there. It literally sleeps. Now a person starts to dream; they hit REM. The brain wakes in a blaze of activity, almost identical to the waves of activity seen when awake. It quite looks very, very much similar to the same process that happens in lasers: a sudden, seemingly spontaneous ignition of ordered splendor that sustains itself until the stimulus is gone. But the million dollar question (still) is, where does this stimulation come from? It should be reasonable to deduce (whether you read the systems readings or not) that the energy cannot come from completely within the brain. Because where would it come from? Indeed, this energy usually does not come from within the brain. Usually it comes from the environment. Our eyes and ears constantly feed our brain with the stimuli (energy) that it needs to sustain it’s fire. Just as Maruana postulated. But as we dream there are no perceptions. We are (literally) blind, deaf, and paralyzed. So how can our brains “see” anything? How can they be awake? Scientists don’t have an answer to this yet that I know of, only theories. I just thought it would be an interesting thing to throw out there. Nonetheless, it seems probably, by anology, that whatever energy is stimulating our brains (while we wake or sleep) crosses a threshold not unlike the threshold in Benard’s chemistry experiement, and this point ignites our brain into a structure of perceptions and congnitions. When we are awake, we are constantly bombarded enough that our brain is always on. But when we sleep certain parts get bored enough that they do turn off, until those crucial tipping points when something, ’causes a perception, a dream, to suddenly emerge.
(2) All the readings thus far have made passing comments about the self-correcting or resilient nature of these feedback loops and self-organized systems. I’ve always known this, but it wasn’t until watching the movie that I started to think about it more. At one point, what’s his name says, “I certainly doubt that we’re harming the earth any more than the glaciers did during the ice age, and the world was fine after that.” Now, I know my jump is a bit of a
disanalogy, but I am very curious how much disturbance a given system can sustain, given a stable flow of energy through it. Perhaps there is a mathmatical way to answer this problem, and likely it is beyond my grasp. That said, I know that some systems and subsystems can be incredibly stable. Take, for example, the design of a biped or x-ped walker (e.g. almost every animal). Most people don’t realize that the system of bones and muscles in an
x-ped absorb and rebound energy in a way that is fundamentally self-correcting. That is, if I am walking straight and someone tries to push me off course as I walk, my gait will naturally rebound to their force withot my thinking. This “design” makes us much better at moving around than we would be if everything was up to our brain. Before or after class, I’ll find everyone a link or two to the following… Some guy whose seminar I sat in on did an
experiment where they put cocaroaches on a treadmill with (small) gunpowder cannons strapped to their backs and electrical sensors on their heads. They made the cocaroaches run, and once they were running they shot the cannon, which jerked them straight sideways about a centimeter. The roaches corrected their course and began walking straight again before their brain actually responded to any stimuli. The stabalizing feedback loops that help them (and others) walk straight operate independent of the will; they can walk
straight (literally) without having to think about it. I think that’s cool. Anyway, there’s an anecdote into how pervasive feedback is in the living world. Again, knowing how stable feedback loops can be, in systems from cocaroaches to earths, I wonder if there is a way to quantify their stability. Understanding just how resilent a system can be, to what, and why, would be fascinating
and also quite applicable.