Broken Symmetry

RIP Mandelbrot

A man after my own heart. Here's a short autobiographical summary of his intellectual odyssey. And here's a recent paper (via Marginal Revolution) applying some of his insights. I particularly liked this bit:

Economic systems tend to be around the critical point in general. Open markets are typical examples. When there are more buyers than sellers, the market price goes up causing decrease in the number of buyers and increase in sellers. Such well-known pulling back mechanism obviously keeps the system around the critical point. The important fact is that this criticality does not stabilize the absolute value of the market price but only stabilizes the statistics of market price fluctuations to follow the critical fluctuations. As the dealers in open markets tend to care about only the relative market price, whether it goes up or down, so the absolute value of the market price is nearly meaningless for the determination of the market price. Therefore, the market prices generally wander almost randomly having statistics characterized by the critical point.

What I picture is market price signals synchronizing the flux of supply with the flux of demand. See also the summary of the paper by Fretter, Krumov, Weihe, Müller-Hannemann, and Hütt.

The Internet as a Cricket Singing Burrows?

From a fascinating paper by Andy Clark:

Crickets sing to attract mates of their own species, and where these songs are concerned, louder is generally better. A louder song reaches more potential mates, and is typically deemed more attractive than any weaker-sounding competitor. The onboard mechanism responsible for producing the sound is a patch of flexible wing membrane, known as the harp, that is pulled taut and struck using a kind of tooth and peg arrangement. The overall process is termed ‘stridulation’.

...

Amazingly, several species of mole cricket (especially Scapteriscus acletus, and Gryllotalpa vinae ) exploit physically perfect Klipsch horns as a means of efficient sound production. These crickets tweak their own burrow-building activities so that one tunnel takes the form of a large bulb, while another flares like an exponential horn. The cricket sits in a narrow constriction between these two animal-built structures and stridulates (sings). In this way the burrow acts as a ‘tuned impedance transformer’ carefully fitted to the specific carrier wave of the species. Sometimes, a single cricket will sit at the intersection of two such tunnel/bulb arrangements. With the ground itself acting as a very large (near-infinite) baffle, and the boost of a double exponential horn, these crickets obtain an almost unimaginable increase in acoustic efficiency. In the case of Gryllotalpa vinae a full 34% of the muscle power used for stridulation is now turned into sound . Compare this to a maximum of 5% for an unaided field cricket, though even this is admirable since high end audio loudspeakers, as Turner points out, manage only a meager conversion efficiency of about 2% (of power into sound). The souped-up mole cricket, however, produces one of the loudest sounds made by any animal. The sound can be heard at 600 metres while the ground itself vibrates for a 20 cm radius.

Turner argues that the mole cricket’s burrows, like several other animal-built structures that he considers in the same mongraph, are functionally equivalent to the possession of a near-perfect physiological sound-emitting organ. Turner’s larger goal, in fact, is to argue that for many biological purposes such animal-built structures should be counted as proper parts of the animal’s physiology and thus to breach what he describes as “the essentially arbitrary boundary between organisms and the environment.”

Even more interesting are the lessons about human cognition that Clark urges could be drawn from this and similar observations:

The Mole cricket scenario is simply a comparative springboard: a dramatic device to bring home the quite staggering transformative potential of our own ‘Cognitive Singing Burrows’. The dramatic device is helpful because, despite our familiarity with the notion of human technologies as ‘cognition amplifiers’ [18, 14], the ubiquitous presence of these amplifiers (pen, paper, models, words, numbers, blueprints, compasses) often blinds us to the depth and importance of their role in ‘distinctively human thought’. By the close of the present treatment we will also see many fundamental respects in which our own ‘Cognitive Singing Burrows’ far exceed (in the cognitive realm) the transformative impact of the simple acoustic Singing Burrow. We thus better appreciate the deeply hybrid (biological and artifactual) nature of the human mind.

I haven't seen any articles where Andy Clark directly addresses the question of human sociality, especially non-kin sociality. That would be particularly interesting.

Do animals cooperate with non-kin?

From Nature:

Since Robert Trivers' seminal paper on reciprocal altruism in 1971, models of reciprocity and the Prisoner's Dilemma resource-sharing game have provided the dominant theoretical framework for explaining cooperation between unrelated individuals. Yet there is little evidence that, under natural conditions, animals other than man commonly exchange either resources or services. In a Review, Tim Clutton-Brock examines the case for reciprocal altruism in natural populations of non-human animals. The evidence is thin, he concludes, and many examples of apparent cooperation between non-kin are probably cases of mutualism or manipulation.

Thanks to Tyler Cowen for pointing out both this and the last post. We seem to be tuned into a similar wavelength today!

What do Ants and Neurons Have in Common?

From Seed Magazine:

According to Cornell University neurobiologist Thomas Seeley, the social insect metaphor can be scaled up even further, all the way to government. He is currently writing a book entitled Honeybee Democracy (Princeton University Press, 2010) about how we can improve our group decision making and government by imitating honeybees, which, like ants, are social insects. Honeybees have been making group decisions for tens of millions of years. And as Seeley says, “Democracy among humans traces back only 2,500 years or so to Athens, Greece, so it is perhaps not surprising that we are still fumbling around.”

Again via Omnivore

Acoustically driven programmable liquid motion using resonance cavities

From Sean M. Langelier, Dustin S. Chang, Ramsey I. Zeitoun and Mark A. Burns:

Performance and utility of microfluidic systems are often overshadowed by the difficulties and costs associated with operation and control. As a step toward the development of a more efficient platform for microfluidic control, we present a distributed pressure generation scheme whereby independently tunable pressure sources can be simultaneously controlled by using a single acoustic source. We demonstrate how this scheme can be used to perform precise droplet positioning as well as merging, splitting, and sorting within open microfluidic networks.

Via Technology Transfer Tactics

Organic Economics Revisited

Can you count how many levels there are in the hierarchy of networks? Consider the most recent issue of the Philosophical Transactions of the Royal Society (B), titled The network of life: genome beginnings and evolution. Here is an excerpt from the editor's intro:

For some 150 years, Darwin’s Origin of Species has been a cornerstone of the modern intellectual framework within which questions about the origin and evolution of organisms have been formulated. These questions themselves have evolved—some have been answered, others reformulated or rendered irrelevant, while new ones have arisen—as the framework has been enriched by the appearance and development of new disciplines (microbiology, genetics, biochemistry) and technologies (electron microscopy, DNA sequencing, scientific computing). The rapid growth of genome-sequence data since the mid-1990s is nowproviding unprecedented detail on the genetic basis of life, and not surprisingly is catalysing the most fundamental re-evaluation of origins and evolution since Darwin’s day. Several papers in this theme issue argue that Darwin’s tree of life is now best seen as an approximation—one quite adequate as a description of some parts of the living world (e.g. morphologically complex eukaryotes), but less helpful elsewhere (e.g. viruses and many prokaryotes); indeed, one of our authors goes farther, proclaiming the ‘demise’ of Darwin’s tree as a hypothesis on the diversity and seeming naturalness of hierarchical arrangements of groups of living organisms. It is now more complicated to ask—as do two papers in this issue—the location of the root of the tree of life, and whether there exist three, or four, major groups of organisms on this planet. Genomes are more dynamic than had been assumed, lineages less coherent through time, evolution less tree-like.

See an earlier post on Organic Economics here.

Network Theory of Bose-Einstein Condensates

This is pretty cool. I've been interested in how synchronicities in the models for BECs and Josephson Junctions might be useful in modeling markets since I read Steve Strogatz's Sync.

The mapping of a Bose gas predicts the existence of two distinct phases as a function of the energy distribution. In the fit-get-rich phase, describing the case of uniform fitness, the fitter nodes acquire edges at a higher rate than older but less fit nodes. In the end the fittest node will have the most edges, but the richest node is not the absolute winner, since its share of the edges (i.e the ratio of its edges to the total number of edges in the system) reduces to zero in the limit of large system sizes (Fig.2(b)). The unexpected outcome of this mapping is the possibility of Bose–Einstein Condensation for T < TBE, when the fittest node acquires a finite fraction of the edges and maintains this share of edges over time (Fig.2(c)).

e: The Fundamental Speed Limit on the Organic Growth of a Startup

Fred Wilson had a couple of good posts on the Venture Capital Math Problem last week. I made a few comments over there, but I wanted to offer a few more extended thoughts on this problem.

Although I haven't found anything on this in the literature I've read on finance, it is well known in other fields, including physics and biology, that there is a fundamental speed limit to the rate of growth of any quantity -- namely, e.

There are many places where one can read about e on the Internet, so I'm going to confine my remarks to the question of how this fundamental speed limit is relevant to venture capitalists and entrepreneurs. For one good introduction to e in finance, see here. But for one tantalizing hint at how interesting things get when one starts looking carefully at e, note that because most quantities we are interested in measuring vary in time (i.e., they have a relative-phase that is a coefficient to the complex quantity i), e together with pi and i provide a fundamental connection between algebra and calculus, or (equivalently) between geometry and dynamics.

Entrepreneurs and the venture capitalists who finance them want their businesses to grow as fast as possible. At the nitty gritty level, this means building a product or service and creating a market for that product or service as fast as possible. In practice, this process of development, marketing, and sales, occurs in cycles. If things work right, with each iteration of the cycle, the product or service is adjusted to make a better fit to the existing or forecasted market, the market then expands either by word of mouth or by direct advertising, and the process repeats. Taking also into consideration the fact that as the market expands beyond a certain size it will begin eating into the money that used to be spent on other similar (but not identical) products or services, and you've got what Joseph Schumpeter called waves of "creative destruction."

Imagine you had the best product idea and the best team in the world. What is the fastest that you could possibly grow your startup? Would pumping more money into the startup help? Every venture capitalist would like to know the answers to these questions.

Some already know that the answer is that the very fastest that you can grow anything is by continuously compounding -- by feeding whatever new revenue you generate back into generating more revenue instantly. If you plot revenue as a function of time, and look at the base of the logarithm, the largest base results when you instaneously begin compounding every new bit of revenue as soon as it comes in. That base is e. There is no way to grow revenue faster than e.

In fact, you won't come close to compounding continuously most of the time. That's because there is always going to be some dissipative force in your organization that drains revenue off to processes other than the ones that are iterating and expanding your market share.

As a final thought, notice that if we assume a growth rate equal to f(x) = x^k (a power law), then log f(e) = k and log f(1) = 0. In other words, for our growth rate to scale up to new market sizes perfectly (that's what taking the logarithm requires) we have to be continuously compounding, which I just said we can't do. No startup's rate of growth is going to scale perfectly. The best one can hope for is to be closer to e than to 1.

UPDATE: It's worth pointing out that under the right conditions, change can happen faster than exponentially. Kurzweil has his double-exponential theory. Power-law fanatics have theirs. I can see how this might work by analogy to phase transitions in condensed matter physics. The organic growth described above would apply to a continuous transition. In some cases, however, there is a certain amount of energy that disappears from the system as local symmetries are broken -- like the latent heat of fusion as water goes from liquid to solid. In certain cases, one might be able to supercharge growth by adding a lot of resources into building out a basic structure that can scale with logarithmic increases in demand. But that would seem to be a risky proposition compared with organic growth in many scenarios.

Buried and a Bleg: Vector Analysis on a Lattice?

I'm buried for the next few days at work.

I have a few posts in the hopper for once I'm back online. In the meantime, here's a bleg for readers with some math and maybe physics or chemistry training:

Can you do vector calculus on a lattice? Suppose that instead of continuum in space, you can only translate between nodes in a network -- a lattice. Further suppose that directed edges between nodes can appear and disappear over some time interval associated with transfer of some quantity between nodes. Can we treat the scalar field of quantities at each node using the same vector calculus that would apply to a continuous scalar field? For example, can we define gradient, curl, and divergence?

The model I have in mind for a simple economy is of nodes representing actors. For each node, there is a vector representing that node's allocation of resources within the economy along with a time-dependent preference vector, which produces a scalar utility for each node when multiplied with the resource vector.

The set of exchanges that occur between nodes over time constitute a network, which evolves to maximize utility locally. Some exchanges, however, might increase utility locally at the same time that they decrease utility for the universe of nodes over all. We might call the difference between the gain in local utility and the loss in overall utility an "externality." Notice that if we cutoff our analysis of the impact on utility with the two nodes that are part of an exchange, then there can be no externalities.

Local asymmetries in preferences or resources will dictate what types of exchanges occur within this model.

Poincare on Inventing Mathematics

Poincare wrote an extraordinary book called The Foundations of Science, which is now in the public domain and available through Google Books here:

In fact, what is mathematical creation? It does not consist in making new combinations with mathematical entities already known. Any one could do that, but the combinations so made would be infinite in number and most of them absolutely without interest. To create consists precisely in not making useless combinations and in making those which are useful and which are only a small minority. Invention is discernment, choice.

What did Poincare mean by "useless" and "useful"? Poincare goes on to describe some of his experiences with mathematical discovery. Solutions often came to him while he was engaged in other activities, such as a geological expedition for the School of Mines: (my italics)

Most striking at first is this appearance of sudden illumination, a manifest sign of long, unconscious prior work. The role of this unconscious work in mathematical invention appears to me to be incontestable, and traces of it would be found in other cases where it is less evident. Often when one works at a hard question, nothing good is accomplished at the first attack. Then one takes a rest, longer or shorter, and sits down anew to the work. During the first half-hour, as before, nothing is found, and then all of a sudden the decisive idea presents itself to the mind. It might be said that the conscious work has been more fruitful because it has been interrupted and the rest has given back to the mind its force and freshness. But it is more probable that the rest has been filled out with unconscious work and that the result of this work has afterward revealed itself to the geometer just as in the cases I have cited; only the revelation, instead of coming during a walk or journey, has happened during a period of conscious work, but independently of this work which plays at most a role of excitant, as if it were the goad stimulating the results already reached during the rest, but remaining unconscious, to assume the conscious form.

For every successful new business model in an economy, there are many failed companies that packaged the wrong combination of capabilities. As an economy, we are less conscious of these failures than we are of the successes, which tend to explode on the scene when they are hit upon.

The similarities between the processes of mathematical creation and economic innovation were well-known to F.A. Hayek.

Comments on Julian Barbour's Essay The Nature of Time

Barbour won an essay contest on the subject of time. See here.

Barbour's clever argument is that our experience of time is always mediated through a comparison of the relative movement of two systems. A clock is simply a dynamical system that is phase-locked with another dynamical system, both of which satisfy a principle of least action.

In a key passage, Barbour takes issue with Einstein's definition of a clock, which is (basically) anything periodic that passes through an identical phase. Here are Barbour's criticisms:

First, no system ever runs through identical phases, so this is an idealization that hides the true nature of time; it lacks Poincare's insistence, which I have repeated, that only the universe and all that happens in it can tell perfect time. Second, Einstein's clock cannot measure time continuously. It can only indicate that a given interval has elapsed when identical phases recur. It can say nothing about the passage of time in intervals within phases.

Barbour's first point is true, but irrelevant to distinguishing between Einstein's definition of time and his own. Of course Poincare is right that the entire universe would provide the best frame of reference for time. But we cannot measure the entire universe, only a part. Is it possible, then, to know whether a given system obeys the principle of least action with perfect accuracy? Given these practical limitations, there is no obvious benefit that I can see to denying any reasonably periodic recurrence of phase the label of "clock." It is a matter of common knowledge that the things are not perfectly accurate.

Barbour's second point is partly true and partly false. It is true that Einstein's clocks could not measure time continuously. But this is just another way of making his first point. Again, until we become gods, we must content ourselves with approximations of whatever we seek to measure.

But it is plainly false that Einstein's clocks can say nothing about the passage of time in intervals within phases. They can say something about anything that happens within their period of oscillation at an interval of twice their period or longer.

The philosophical point here is that we ought to forget about deciding whether or not time exists in some metaphysical sense, and get down to the business of making use of it in a physical one.

Or to put it another way, here is Harald Weinrich in On Borrowed Time:

Considered in relation to the body, the sense of time can be either added to the canon of the five senses as a "sixth sense" with the same status as the others, or identified as superior "common sense" (sensus communis, bon sens) that is the psychic coordinator of all the other senses. In either case, the sense of time is a "real," that is, corporeal, sense, because it has its own bodily organ -- according to the older view, the temples, and according to the modern view, the circulation of the blood and its rhythmical pulse beat. Although we generally remain unaware of it, like the other senses, in crises it will be perceived beating on the surface of the body. However, since evenly flowing circulation of the blood is its condition, the sense of time takes on a special anthropological status insofar as it is at the same time an internal sense, as the only one that determines the beginning and the end of human life-time. To that extent we can say of the sense of time, and of it alone, that it creates or abolishes the presuppositions not only for itself, but also for all the activities of the other senses.

To put it still another way, since it is impossible for the future to be determined from the past, we should remain agnostic about applications of our theories that purport to say things about the universe or even about next week.

Hive Psychology?

From the Journal of Legal Studies, Hive Psychology, Happiness, and Public Policy by Jonathan Haidt, J. Patrick Seder, & Selin Kesebir:

Economists care about preference satisfaction, but where do these preferences come from? Evolutionary psychology offers what is arguably the most comprehensive explanation: we want things that helped our ancestors succeed at leaving surviving offspring in the environments in which the human mind was shaped by natural selection (Barkow, Cosmides, & Tooby, 1992). Our love of sweet and fatty foods, even when we know that we now eat too much of them; the desire for prestige and our concern for the opinions of others, even when we wish not to care; the desperate passion to protect our own children and the rapidly declining concern we show for more distantly related children; all of these human preferences flow readily from an analysis of the preferences that led early hunter-gatherers to succeed as individuals. David Buss (2000) has even offered a catalog of evolutionarily-informed methods for increasing human happiness in the modern environments we now inhabit. But nearly all analyses of happiness from evolutionary psychology, like those from economics, focus on individuals and their preferences. Might there be group-level preferences too? Might individuals be happiest when their groups are doing things that led, over eons of evolution, to group success?

The Three Laws of Performance by Zaffron and Logan

Back in October I read and blogged about Tribal Leadership by Logan, King, and Fischer-Wright. The thesis presented in Tribal Leadership is that the health of organizations correlates with growth in the average number of connections between members of the organization over time. If you haven't read Tribal Leadership, then you should check it out. That book, unlike this one, also includes a useful cheatsheet, which summarizes the most important points of the book. It's the better place to start, and should probably be read in conjunction with The Three Laws of Performance.

Both of these books promote what Mihalyi Csikszentmihalyi would call a systems model for explaining individual and group performance. According to the systems model, there is no particular habit or activity that will guarantee an individual who practices it better performance or more creativity. The reason for this is that whereas an individual produces the performance or creative work, the work must then be evaluated and accepted by the field of experts for the symbolic domain of the performance before it can become a part of that domain. In other words, transcendent performances (including creative works and the accomplishment of challenging business goals) require collaboration between individuals and groups. Growth (or decay) results as groups and the individuals who constitute them evolve to tackle successively more complex goals.

The Three Laws of Performance provides a list of rules (and stories explaining how they work in practice) that promise to help individuals in leadership roles facilitate the group coherence and cohesion that are necessary to bring about transcendent performance. The book is based on a wealth of experience from decades of applying the ideas it explains. If you're interested in this kind of theory, the book is worth a read. It was disappointing, however, that the authors didn't make more of an effort to connect their ideas up with similar work done by others in the past. I'll explain with reference to the three laws themselves.

The first law of performance holds that how people perform correlates to how situations "occur" to them. This odd choice of word raises many questions. Fortunately, the authors were merciful enough to address the confusion upfront:

So what exactly does occur mean? We mean something beyond perception and subjective experience. We mean the reality that arises within and from your perspective on the situation. In fact, your perspective is itself part of the way in which the world occurs to you. "How a situation occurs" includes your view of the past (why things are the way they are) and the future (where all this is going)."

Now without a systems model in mind, this is nearly incomprehensible to me. Again, the best way that I have of understanding it is in terms of a systems model that includes feedback loops between individuals, the field that includes those individuals, and the domain of symbolic knowledge that the field operates within. If you want a picture, consider this diagram from Roger Penrose's Road to Reality:

In this diagram we see the feedback loop that connects the physical world, our mental picture of it (our consciousness), and our models of it in a given symbolic domain (such as language or mathematics). We can associate each stage of the loop with different people who work within a particular symbolic domain. For a domain to be cohesive, each individual within the group has to a have a mental picture of the entire domain. For it to be coherent, those pictures have to be consistent with the rules of the domain set by a group of recognized experts. So the bottom left corner would correspond to individuals within the group. The top corner corresponds to the domain as it is defined by the group of people who are considered experts in the domain. Note that although the rules of the domain are determined by the group of experts, the domain exists independent of any one of the experts -- making it possible for individuals in the bottom left corner to contribute, and for the domain to outlive any particular expert.

Finally, in the bottom right corner we have the physical world in which everything is embedded. It is through this corner that new information (including experimental observations or discoveries) enters the loop.

Having a systems model in mind makes the first law more comprehensible, at least to me. With the systems model in mind, we can reinterpret the first law as "how individuals perform correlates to their memory and forecast for how the group and the world itself have and will evolve over time." (Or something pretty close to that.) So far this might seem fascinating, but also useless. So what? Should seeing things this way make any difference in how we act?

The answer is yes because as Werner Erhard knew well, it is possible to change the loop by convincing individuals to change how they understand their past and forecast their future, especially as it relates to others within the group. As the Author's Note makes clear, part of the intellectual history of this book is derived from the knowledge and experience of people who worked with Erhard in the est seminars popular several decades ago. Incidentally, that inheritance might also explain the willingness to leave some terminology inscrutable. Part of the ethics of est seems to have involved challenging the audience to engage with the teaching intellectually -- a practice best exemplified by Socrates and worst exemplified by Heidegger, both of whom might also be considered antecedents to this sort of philosophy.

Back to the book. The remaining two laws basically spell out a prescription for how to implement transformational change within an organization given that the systems model applies. So the second law holds that "how a situation occurs arises in language." This is the Sapir-Whorf hypothesis, which is now an accepted fact at least among cognitive psychologists. The Sapir-Whorf hypothesis explains how corners of the loop are linked.

The third law holds that "[f]uture based language transforms how situations occur to people." In other words, when enough individuals working within a domain decide to change the domain, then the domain and hence the activities of the indviduals and groups that constitute it will shift into a different mode of operating. What is perhaps less intuitive is the fact that such shifts may occur voluntarily when enough individuals agree to make them happen. That's another reason why this book should be read in conjunction with Tribal Leadership.

I like the vision that the authors have in this book, and believe their prescriptions are sound advice for carrying out transformational change within an organization. There is so much theory compacted into a short book, however, that it is a bit difficult to digest, and is probably best used in conjunction with training by somebody familiar with the theory. (The authors are also professional consultants with an impressive track record of helping organizations achieve transformational change.)

As a final note, I was reminded by this book of this post on how Thomas Schelling's concept of focal points is useful in facilitating institutional change, and this book review on how the culture of reformed Protestantism may have influenced the Founding Fathers. There is more to be said about insitutional design than is captured by the static mental models of economic equilibrium taught in law and business schools, and this book is a good contribution to the evolution of these mental models.

UPDATE: If you're interested in systems theory in business, then you might also like some of the books and papers published by Prof. John Sterman at MIT.

FULL DISCLOSURE: Per new FTC rules, I want to disclose that I received a free copy of this book before providing the review here. I guess they found me because of my earlier review of Tribal Leadership and thought I'd be interested. I probably would have written a nice review if I didn't feel myself pulled in that direction by my desire to get more free books in the future!

On Why Economists and Physicists Need to Take Time Seriously

Lee Smolin nails this in an Edge symposium:

I used to think that the goal of physics was the discovery of a timeless mathematical equation that was isomorphic to the history of the universe. But if there is no Platonic realm of timeless mathematical object, this is just a fantasy. Science is then only about what we can discover is true in the one real universe we find ourselves in.

More specifically, this view challenges how we think about cosmology. It opens up new ways to approach the deepest questions, such as why the laws we observe are true, and not others, and what determined the initial conditions of the universe. The philosopher Charles Sanders Pierce wrote in 1893 that the only way of accounting for which laws were true would be through a mechanics of evolution, and I believe this remains true today. But the evolution of laws requires time to be real. Furthermore, there is, I believe, evidence on technical grounds that the correct formulations of quantum gravity and cosmology will require the postulate that time is real and fundamental.

But the implications of this view will be far broader. For example, in neoclassical, economic theory, which is anchored in the study of equilibria of markets and games, time is largely abstracted away. The fundamental results on equilibria by Arrow and Debreu assume that there are fixed and specifiable lists of goods, and strategies, and that each consumer’s tastes and preferences are unchanging.

But can this be completely correct, if growth is driven by opportunities that suddenly appear from unpredictable discoveries of new products, new strategies, and new modes of organization? Getting economic theory right has implications for a wide range of policy decisions, and how time is treated is a key issue. An economics that assumes that we cannot predict key innovations must be very different from one that assumes all is knowable at any time.

Bleg: Geometry of Algorithms?

On a walk with a mathematician friend today we fell to talking about the value that geometric insights have had in solving difficult problems. The general thought is that great analysis is useful in solving difficult problems in well-understood systems. But geometric understanding has often been necessary to setting up the analysis properly in the first place. When I think of great physics, for example, an extraordinary ability to visualize geometry seems to run as a common thread through the best theory.

The other topic of conversation was algorithmic theory. We've been putting together algorithms for a long time -- every mathematical proof (whether true, false, or undecidable) -- is an algorithm. Algorithms are more general than equations.

So naturally we were talking about how one might go about setting up a geometry of algorithms. The connection between geometry and analysis (both real and complex) would be a subset of a more general geometry of algorithms -- viz., the subset in which real and complex functions are associated with a geometric manifold.

But let's say I wanted to start visualizing algorithms in two or three dimensions. Is there a theory for this? I remember from years ago that there are many different sorting algorithms, with quicksort being one of the fastest. Is there a systematic way to represent these algorithms in two or three dimensions and bring geometric intuition to bear on their similarities and differences?

The naive method that we talked about would involve taking a turing tape of 0s and 1s, breaking it into finite segments (of, say, 16 bits), and then mapping each segment into a two dimensional discrete grid of 256 points. An algorithm, then, would be a trajectory through that discrete space. This is unlikely to produce geometrical insights, however.

My friend insists that this is what graph theory is all about. I need to learn more graph theory. But if there are any readers that can point me towards a web page, paper, or book that provides an introduction to this type of theory, I'd be much obliged.

More on the Application of Non-equilibrium Statistical Mechanics to Economics

Edge.org recently published a manifesto on how science can help economists and politicians improve forecasts and regulations:

Instead of being in equilibrium, markets can be understood as self-organized critical systems. This is a theoretical construct that can be usefully applied to real markets. It leads one to expect that in steady states markets are approximately scale invariant. This implies a prediction for how certain quantities will be distributed in an economy, which includes wealth, income, sizes of firms, populations of cities, and total values of currencies. The prediction is that these distributions are power laws and the prediction is observed in the real economy.

Note that the theory is that the entire market displays self-organized criticality, not individual firms within the market.

Time scales matter. Different functional cash to cash cycles have very different time scales. Instabilities can be easily introduced by too strongly coupling processes on different time scales.

What this means is that if in a population of 300 million people, only 3 million or so are going to be eligible to borrow under subprime terms in a given year, then you're going to create a bubble by coupling enough money to fund 30 million a year into the market through things like CDOs and CDO^2s.

Read the whole thing. There's also mention of periodicity in cycles of economic activity.

Miscellaneous Sunday Links

* Fourier transforms can be useful for analyzing many time-dependent signals, including that cool "PRAANG" at the beginning of A Hard Day's Night.

* I've been looking for a natural experiment that demonstrates how consumption patterns are poisson-distributed in time for months. Then I found this post by shmula on how diggs are poisson-distributed. The distribution at the top is a cumulative distribution function that represents the summation of clicks over a period of time. Unfortunately, this isn't quite the poisson distribution I had in mind. I expect individual acts of consumption to be poisson-distributed in frequency, which would mean that we should see the poisson distribution over a finite interval reproduced when measured over a different same-length interval. And there's not enough data here to know whether that would be true also. (And I'm guessing it probably wouldn't.)

* The transactions costs of keeping risks uncorrelated = management fees?

* Disabling circadian rhythms in hamsters disrupts memory.

* "At the core of the recent financial market crisis has been the discovery that these securities are actually far riskier than originally advertised." You'd think these guys could have gotten their paper out a little earlier.

* A while back I got after Lawrence Summers for his vacillation on regulation. He says it's just oscillation.

* Lessons from the Japanese attitude toward shareholders. (The pros and cons to owner-oriented management.)

How much would you pay for good math?

I was both happy and sad to see Applied Hyperfunction Theory selling used for over $700 on Amazon.  That's a one-volume treatise on hyperfunctions, not an encyclopedia.  Happy because I'm very intrigued by hyperfunctions, and I'm glad to see there's strong demand for a treatise about them.  Sad because how much would it cost a publisher to print another thousand copies and clear the market?  Seems like it shouldn't be so hard to clear the market for highly technical books.

Etymology of the day: "nature"

From an interesting article by Leo Marx:

But it also should be said that the word nature is a notorious semantic and metaphysical trap. As used in ordinary discourse nowadays, it is an inherently ambiguous word. We cannot always tell whether references to nature are meant to include or exclude people. Besides, the word also carries the sense of essence : of the ultimate, irreducible character or quality of something, as for example, 'the nature of femininity' or, for that matter, 'the nature of nature.' When this meaning is in play, the word tacitly imputes an idealist or essentialist hence ahistorical - character to the particular subject at hand, whether it be femaleness or nature itself. The word's multiple meanings testify to its age : its roots go back (by way of Latin and Old French) to the concept of origination of being born. As Raymond Williams famously noted, nature is probably the most complex word in the English language.

A dramatic example of gravitational symmetry breaking

Did you know that if you boil water in zero gravity you get a sphere of vapor attached at a tangent to the burner?  Thanks to Backreaction for the excellent summary.

The force of gravity is necessary to achieving the density gradient that generates flow within a pot as water is heated at the surface of the earth.  Gravity is a source of spontaneous symmetry breaking in water in its transition from liquid to vapor.

If there were a larger philosophical point to be drawn from this (and I'm not sure there is), then how about this one:  Weak forces (like gravity) don't seem to matter much most of the time.  But weak forces matter greatly (and have system-wide consequences) during phase transitions when stronger forces cancel out dynamically.

What's interesting and useful about transitions like the one our world economy is going through right now is what they reveal about the weak forces that are at work in the background.

Mathematical Models for Synchronized Flow

Through a friend, I just heard about the work of Steve Strogatz.  Here is a link to a paper on the Kuramoto Model for phase synchronization of an ensemble of oscillators.

Collective synchronization occurs throughout the living world, from the rhythmic firing of thousands of pacemaker cells in our hearts, to the chorusing of crickets on a warm summer evening [Winfree 1967, 1980; Pikovsky et al. 2001; Strogatz 2003]. What is remarkable is that these and many other biological populations somehow manage to synchronize themselves spontaneously, without any external cue, despite the inevitable diversity in the natural frequencies of their constituent oscillators.

My conjecture on this blog has been that market price signals perform a similar function by synchronizing periodic oscillations in supply with periodic oscillations in demand.

Here is a much easier to read paper about how the unconscious synchronization of the crowd on the Millenium Bridge in London led to its instability and required its reengineering.

According to their models, when a population of weakly coupled oscillators grows past a certain critical number, the oscillators will spontaneously synchronize.  In the case of the Millenium Bridge, the synchronization began to drive the bridge into an oscillation that could ultimately have resulted in its collapse.  In the case of coupling of supply and demand in a market, such synchronization could lead to a bubble and crash.

Here is an introductory article.  This part is particularly interesting:

While modelling populations of biological oscillators, Winfree discovered a new kind of cooperative phenomenon, the temporal analogue of a phase transition.  He proposed a mean-field model of nearly identical, weakly coupled limit oscillators and showed that when the coupling is small compared to the spread of natural frequencies, the system behaves incoherently, with each oscillator running at its natural frequency.  As the coupling increased, the incoherence persists until a certain threshold is crossed -- then a small cluster of oscillators suddenly 'freezes' into synchrony.  For still greater coupling, all the oscillators become locked in phase and amplitude. (Fig. 2)

To an approximation, I believe this describes what firms do with people.  An increase in radius in Fig. 2 corresponds to increased efficiency through divisions of labor.  The mean field corresponds to a measure of profit.  Notice that it moves in time along with the people, and increases in amplitude as the people synchronize.  These are very tentative interpretations; but not inconsistent with what is observed in practice.

The Kuramoto model relies on a lorentzian distribution in frequency of oscillation.  I wonder how fatal it would be to the model to substitute in a poisson distribution, which seems a better fit for most frequency distributions in consumption and production.  According to the Kuramoto model, above the threshold frequency:

[T]he population splits into a partially synchronized state consisting of two groups of oscillators: a synchronized group ..., and a desynchronized group whose natural frequencies lie in the tails of the distribution ... and are too extreme to be contained.

There are definitely people within our economy at any given moment whose frequencies of consumption or production are either too fast or too slow to be accommodated.

Hayek would have been pleased to see progress in the mathematical model.  (But not with slow or off-beat people within an economy being left out.)

The Aesthetics of Entrepreneurship

Via Peter Klein at the Organizations and Markets blog, I find this quote from Evan Osborne:

What is seldom appreciated is that successfully reconciling the interests of and coordinating scientists, financiers, doctors, patients and other constituencies is a complex and subtle human achievement.  It is similar to other acts of creation and coordination that we admire as the highest forms of such achievement – acts such as the writing or performance of great music and the creation of great paintings.  The highest praise for such works of art is that they are beautiful, and commerce too should be seen not in purely utilitarian terms but also as an act of human creativity and expression.

Worth checking out here.

More on the Neuroscience of Synchronized Flow

"Long-term potentiation" is what the neurobiologists call it when two neurons that have not fired together before show an increased frequency of firing after being stimulated together at high-frequency once.

Neurons "learn" to cooperate over a longer time period by being stimulated together at high-frequency for a short time period.

See my earlier post on the connection between this research and Hayek's Sensory Order.  The mechanism for learning at the level of institutional design has similarity to the mechanism for learning at the cellular level.

In Rainbows

Seems to be the best Radiohead album since Okay Computer (1997).  I'm very into Reckoner and House of Cards.  Don't miss this post on the making of the video for House of Cards.  They reconstructed 3D images of the band playing in pitch darkness using spinning lasers and detectors.

Etymology of the Day: "Rhythm"

Two branches:

Branch one is from the Latin words rime and rhyme, demonstrating the connection between language and perception.

Branch two is from the Greek rein (which meant literally "flow") demonstrating the connection between nature and language.