Sunday, July 24, 2016

Wednesday, May 11, 2016

Our new paper: Trenchard H., Perc, M. "Equivalences in biological and economical systems: Peloton dynamics and the rebound effect."



Our paper has just been published in PLOS ONE. This paper foreshadows what I anticipate will ultimately be a more important paper about energy savings mechanisms in biological systems, also co-authored with Matjaz Perc, currently under consideration for publication. When that paper comes out, I'll have more to say about some implications of what we discuss there in relation to what we discuss in this PLOS ONE paper. 

On a related side note, I am also hoping to hear definitive word soon on another paper prepared in collaboration with Andrew Renfree and Derek Peters from the University of Worcester, in which we apply our peloton model to groups of runners and test the effects of drafting on certain collective running dynamics. I have also now resumed work on a collaborative analysis of fish schooling dynamics that applies the peloton model.

So, in a sense this new PLOS ONE paper is a companion piece to the larger review paper in which we review literature on energy savings mechanisms in natural systems, and wherein we identify certain principles of the collective dynamics of pelotons that are common to other natural systems.

It is difficult to anticipate how our PLOS ONE paper may be viewed, if it gets much attention at all. I don't deny that we are somewhat perilously pushing the envelope of the peloton analogy into the realm of economics. Broadly speaking, some might argue that our attempts at identifying commonalities between certain economic parameters and biological ones are naive and audacious. There are risks to this interdisciplinary endeavor, but scientific breakthroughs are impossible without such risks and occasional failures. Regardless, clearly PLOS ONE and the reviewers have judged the analysis of sufficient merit to publish it for further appraisal in the wider cauldron of academic consideration.

A brief summary:
Our model of the rebound effect is premised on four main factors: 1. the price of the energy service, externally imposed upon the consumer; 2. the consumer’s maximum capacity (or budget) to pay for the energy service; 3. the reduction in cost to the consumer due to some energy service efficiency, as a percentage; 4. the rebound quantity, as a percentage. In our paper we identify the first three as the primary ones, but the fourth one is obviously of critical importance and discussed in detail in our paper.

These factors all have equivalences in peloton dynamics, respectively: 1. The speed set by a pacesetter in a non-drafting position (akin to the imposed cost of the service to the consumer); 2. The maximum sustainable output of a following cyclist in a drafting position (akin to the consumer’s available budget); 3. The energy savings quantity due to drafting, as a percentage (akin to the efficiency in the energy service that reduces the service cost to the consumer by some percentage); 4. Potentially some surplus energy facilitated by the energy savings mechanism of drafting that permits the cyclist to achieve higher speeds by drafting than she could without the drafting benefit (akin to the fraction of the consumer’s budget that has been freed for the consumer to purchase more of the energy service as a result of the reduction in cost of the energy service, than previously used).

By adapting a basic equation that describes the relationship of these factors in peloton dynamics, we have derived an analogous equation that describes the relationship between these four economic factors. Thus a ratio that models these factors allows us to identify two main thresholds that we have also identified in peloton dynamics.  The first one is clear: a decoupling threshold between the price of an energy service and a consumers ability to pay for it; the second is less clear but is based on upon the application of an analogous threshold in pelotons: the "protocooperative" threshold between one phase of collective behavior in which cyclists can share costly front positions, and a second phase of collective behavior in which cyclists can sustain the speed of a pacesetter only by exploiting the energy savings mechanism (drafting) but cannot pass the pacesetter in order to share the costly front position.

Our paper has just been published
Our paper has just been publishedI anticipate some criticisms of our approach and more particularly to our interpretation of the model. However, I feel it is prudent at this point first to hear what the criticisms may be before I try to anticipate and enumerate them here. My sense at this early stage is that the rebound ratio equation itself is defensible and, if there are problems in the paper, it isn't in the model itself, but in how best to interpret it. Our present interpretation of it and the effects of it may need to be modified in light of feedback and further analysis. 

If in the passage of time some or other aspects of our analysis turn out to be flawed, and if nothing else is achieved by our new paper in PLOS ONE, I hope that this singular point will stand the test of time: certain principles of peloton dynamics have application across a wide variety of systems, including not only a wide range of natural biological systems, but also among human-centric economic ones.  

Saturday, March 12, 2016

Come scientists and academics, publish while you still can; the times they are changing (with apologies to Bob Dylan).


It has been my intention to keep this blog focussed solely on peloton dynamics and its analogs, but I thought I would stray briefly across the boundaries and throw in my two cents worth about the historic third match yesterday between AlphaGo and master Go player Lee Sedol, summarized in this Wired article.

While I had not originally planned to watch the live game on the internet, serendipity led me to witness an unforgettable 3 1/2 hour game and move-by-move analysis by Michael Redmond, an American professional Go player, who has achieved the highest rank of this Asian dominated game. Certainly without Redmond's capable and animated analysis, the game would have been lost on me, although I could follow along roughly with my own crude evaluation of how the balance of advantage was unfolding. Redmond was accompanied by Go E-Journal Managing Editor, Chris Garlock, whose bias in favor of Sedol was palpable and contagious.

With Korean Lee Sedol down two games to nil in the best of 5 match, the third game was do or die for Sedol, and an historical milestone for the power of artificial intelligence, and for the AI and Go communities.

I am not a Go player, but growing up I learned the rules and played a couple of games with my brothers, who played among themselves enough to become competent players. I certainly remember the Ko rule, a situation in which players can alternate taking a single surrounded stone, but who must play an intermediate move elsewhere before mirroring the Ko move. Michael Redmond remarked that an AI Go player version from a few years ago ran into difficulty around Ko moves, which the underlying computer algorithm, based on Monte Carlo simulation methods, was not well equipped to handle. Redmond remarked, however, that even in October of 2015, AlphaGo demonstrated competence around Ko moves. Despite this, Redmond noted there were rumors that colleagues had advised Lee Sedol to induce AlphaGo into error by drawing AlphaGo into Ko moves that might be difficult for the computer algorithms.

Each player was allotted a total of 2 hours for their moves, with three 1-minute overtime periods per player, which re-started if a move was made before the expiration of the 1-minute period. With about 40 minutes remaining on AlphaGo's clock, Sedol was down to his three overtime periods, and Redmond was predicting an AlphaGo victory of about 60 to 30 (total territory claimed).

With one 1-minute period remaining (again, re-started if a move was made before expiry), Sedol displayed remarkable brilliance under enormous pressure by deftly guiding AlphaGo into a sequence of Ko moves. Many times Sedol placed his stone with one or two seconds remaining. Meanwhile, in the commentator's box, Redmond's hands were flying across the working board, demonstrating variations in play, computing the relative "liberties" (viable options surrounding a critical point of play), and continually declaring that AlphaGo had the advantage. The game was becoming increasingly complex, and the likelihood of Sedol making a crucial mistake was frighteningly high, while AlphaGo still had several minutes of regular play time in hand.

What Sedol did was utterly brilliant, but what I witnessed was almost a kind of AlphaGo mockery. Twice Sedol and AlphaGo exchanged Kos, until AlphaGo surprisingly placed a stone in an uncontested opposite corner of the board. Redmond and Garland, scrutinizing variations, missed seeing AlphaGo's move, and when they looked back at the actual game board to the locus of Ko exchanges, they were momentarily confused as to where AlphaGo had placed its stone. Earlier, Redmond pointed out that once AlphaGo computes the probabilities of an advantage in a particular region of play, if it "feels" it is ahead, it will place a move elsewhere on the board. At an earlier stage of play, AlphaGo made another such move, which Redmond said actually allowed Sedol to recover from a losing position.

On one hand we see how amazingly brilliant is the mind of Sedol, a man with a human brain, knowing that he can apply human pattern recognition and intuition, against a massively powerful computer, programmed with learning algorithms that can play itself continuously and update optimal solutions over the course of millions of iterations. Yet, when AlphaGo made its seemingly casual move in the uncontested corner in the closing moves of the game, preceding Sedol's resignation a few moves later, for me it was a crushing recognition that artificial intelligence has advanced into a realm in which there are no longer problems or fields of human inquiry that cannot be solved by artificial intelligence.

For instance, Google has access to vast libraries of scientific and academic journals, in addition to massive quantities of data that are constantly, and at increasing rates, being uploaded into the Cloud. With such data and access to knowledge, algorithms need only be implemented to ask questions about what information or solutions are missing from the scientific literature, and then in turn to synthesize the vast store of available information in conjunction with enormous quantities of data, and to answer itself the questions that it poses.

This makes me anxious. I don't know how scientists generally feel, but in the context of my own miniscule contribution to human knowledge, whatever that may be, I sense a sudden and sky-high spike in urgency now for humanity to maximize its creative resources in the sciences and all of academia. Study, learn, be bold and creative and push the boundaries of knowledge now; discover, cogitate and publish while you can before your quests and thirst for knowledge are quenched far more rapidly and adroitly by machines with names like DeepBlue and DeepMind. Take the chance now, or it won't come again.

While there will always be ways for humans to satisfy their intellectual hunger, to justify their lives, to seek their own unique place amid the universal struggle to balance suffering and happiness, to me the defeat of Lee Sedol by AlphaGo represents a cross-roads for science and the human quest for discovery, as the form of the human contribution to science is bound to look very different in the coming years.



Sunday, October 25, 2015

Spider pelotons


How are pholcid spider collectives like a bicycle peloton?
Hugh Trenchard

Rebecca Boyle, for the New Scientist,  reported quite nicely one of the essential aspects of the peloton model discussed in my recent paper on protocooperative behavior [1] (free pre-print), which is that the model strips away the deliberate racing strategies and tactics of cyclists, and shows how certain collective peloton formations are dominated by basic physical/physiological principles.

Boyle also notes very well how these basic physical principles drive the changing shape of a peloton -- which was earlier presented in two collaborative papers [2,3] -- but she is circumspect in relation to other key points of my paper and only hints at something potentially more important in it. At the same time the headline emphasizes only the obvious point that weaker cyclists can gain advantage by drafting.

Key elements, not discussed in the New Scientist article, include that my peloton model shows: 1. a mechanism for how groups divide into smaller groups; 2. the threshold between cooperative behavior and predominantly free-riding behavior, and the features of these phases of behavior; and why these factors are significant in an evolutionary sense.

I have a lot of work ahead to show how the principles outlined in my paper may be applied to other biological systems, which is really the Holy Grail of my research. Rebecca Boyle intimates this in referencing fish schools and the comments of  James Herbert-Read and Shaun Killen. And while there is voluminous work ahead to show how peloton principles apply to schools of fish and flocks of birds, it is also instructive to consider how the principles apply in systems that are less obviously physically similar to a peloton.

So, here I explore, just a little, how peloton principles apply to a less obvious analogue: social spiders. Unlike the uniformity of movement we see in schools and flocks, spiders tend to move about in different directions and may be separated by considerable distances, and there are few obvious similarities in group shapes and formations, nor is there an obvious energy savings mechanism among spiders, like drafting.

So the question is, how is this:


Fig 1. South American social spiders feeding on katydid 

...like this?

Fig. 2. Tour de White Rock men's pro-1,2 criterium (2013)

There are some superficial similarities: they are both groups of organisms; individuals are assembled fairly near each other, and they both have cool shadows!

I suggest there are also deeper similarities. To see how, let's look at some research by  Elizabeth Jakob about  group dynamics of pholcid spiders [4]. I'm no spider expert, but note that pholcids are not the same spiders as the social spiders in Fig. 1; as I understand it, pholcids are more like the daddy long-legs we commonly see around B.C. I used Figure 1 because it shows a nice collective of spiders*.

For pholcids, groups can be as large as 15, but the most common composition is two sharing a web -- one small and one large spider [4].

Paraphrasing from the Abstract [4] Jakob conducted experiments to examine the role of individual variation in the dynamics of pholcid group formation. Pholcids (specifically H. pluchei) either share webs or live alone, and individuals shift frequently between these conditions. A spider's decision to move is influenced by its size and recent feeding success. Small hungry spiders joining a web already occupied by a larger one were more likely to abandon the web and build their own than small well-fed spiders. The reverse is true for large hungry spiders, which were more likely to stay in a web already occupied than a large well-fed spider.

Jakob explains why, at p. 18:
Hungry small spiders, closer to the bottom of their energetic reserves, may be more sensitive to the negative effects of competition, and thus more willing to take the risk of abandoning a web for potential gain...The behavior of the larger spiders is less obvious: why do well-fed spiders leave groups? One hypthesis is that H. pluchei judge each others' fighting ability by weight. Spiders that are very similar in size fight most intensely (Jakob 1994). Well fed instar 5 spiders have been close enough to the resident's weight that high-intensity potentially dangerous fights were likely.
For more background, Jakob reported in [5] that for H. pluchei (quoting from the Abstract):
The main benefit of group living is likely to be the reduction of the cost of web building when spiderlings take advantage of webs built by larger conspecifics. 
So this is where we can try some peloton analysis.

In [1-3] we apply an equation that I call the "peloton convergence ratio" (PCR), which is really just a description of the coupled, interactive, relationship between drafting and non-drafting cyclists.
PCR = (P_front * d) / MSO_follow
  • P_front is the power output of the leading, non-drafting, cyclist, who sets the pace for the group. 
  • d is the drafting coefficient (i.e. the ratio of output of the drafting cyclist to the output of the non-drafting cyclist; the actual energy savings is 1-d). It could just as easily be called the "power-output coefficient for coupled cyclists", and is a ratio that describes the relative energetic costs between them, given that a drafting cyclist's output is less than that of the pace-setter. This conceptual subtlety is important when we identify analogues in other systems. 
  • MSO_follow is the maximum sustainable power output of the following, drafting cyclist. 

The MSO_follow of the following rider can vary; it need not be static. In other words, MSO_follow can change according to fatigue or hunger. While we did mention the variability of MSO, in none of [1-3] did we actually model it. However, such a variable MSO is equivalent to changing the speed of the front rider: as the front rider accelerates, the closer that puts the follower to his/her MSO (assuming the follower is seeking to hold the pace of the pacesetter). Similarly, if you reduce the rider's MSO due to sudden fatigue, or the "bonks" due to hunger, he/or she is driven to his/her MSO at a lower speed.

When PCR > 1, coupled cyclists diverge. This simply means that the front cyclist is riding faster than the follower can sustain, even by drafting. 

For a small hungry pholcid who decides to abandon the web, we may describe the condition under which it leaves as SCR < 1 (where SCR is "spider convergence ratio").  This is the inverse of the cyclist situation and I will show why, but SCR, like PCR, is still a description of the energetic relationship between the spiders sharing the web, and SCR < 1 describes the threshold at which the smaller pholcid is forced to diverge from that energetic relationship. We can think of this relationship in terms of two pholcids sustaining their individual energetic requirements given that food is distributed between them; and given the smaller spiders' food intake is determined by the larger spider; and in view of energetic costs that are saved by the smaller spider.

In this case, the energy savings is enjoyed by the smaller spider because it saves the cost of constructing its own web by joining a web already constructed by the larger spider.  The food energy permitted for the smaller spider's consumption is set by the larger spider, who dominates the web. To understand how these factors might work for spiders and how we might model them, let's first outline some other pholcid behaviors that Jakob identified.

In [5] Jakob observed that (quoting from the Abstract): 
...when prey were introduced into group webs, the largest spider in the web that detected the prey won the prey approximately 77% of the time. In spite of this cost, spiderlings were found in groups more often than expected by chance.
Also, in [4] Jakob observes that the energy cost of generating silk to make a web is equivalent to about 9 flies, or an average of 4-5 days of foraging. Spiders normally capture 1-2 flies/day.  Young pholcids can survive up to 3 weeks without without food (21 days), and adults up to two months [4] (60 days).

Now we can further explore the notion of the energy savings quantity and what I call the "StarvationCoefficient". If we say that the StarvationCoeff is the equivalent of the drafting coefficient for cyclists, intuitively it doesn't seem to match: the drafting coefficient implies a reduction in output for the drafting cyclist relative to the lead cyclist, and how does this equate to a spider that saves the cost of building its own web by joining one already constructed?

Consider that the drafting coefficient is really a ratio of the relative costs of a leading and drafting cyclist. The cost is higher for the leading cyclist in terms of power output, and lower for the drafting cyclist. The actual energy savings for the drafting cyclist is 1 - d. For example, if the power output for a drafting cyclist is 300W and the power output for the pace-setter is 400W, then the drafting coefficient is 0.75 and the energy savings 25%.

For pholcids (or any organism for that matter), costs are daily energetic demands that prevent the ultimate cost of death by starvation. If we know how long a spider can go without food before death by starvation, and if we know the costs incurred in building a web, we can show the cost of web construction as a ratio of the ultimate cost of starvation.

I calculate the proportionate energy saving to a young pholcid by not building its own web is between 19% (4/21 days) and 23% (5/21 days), and between 6% (4/60 days) and 8% (5/60 days) for an adult. These values represent the energy saved by not building your own web, while the StarvationCoefficients -- the ratio of the reduced time to starvation due to time lost taken to build your own web, to the total cost of living to starvation (1 - energy savings) -- are 0.81, 0.77, 0.94, and 0.92 respectively.

There may be better ways of determining the energy savings benefit of not building a web, but it seems the time saved from imminent starvation is a reasonable quantity by which to develop a model that accounts for the energy savings of not building a web. There may be other factors too that contribute to energy savings or costs, but let's use this StarvationCoeff and the 19% and 8% energy savings and corresponding StarvationCoefficients to develop a model in principle.

Now let's consider the PCR equation and how it is equivalent to the pholcid situation. First, PCR describes the coupled relationship between a lead cyclist and a following drafting cyclist. The lead cyclist sets the pace for the follower, while that pace is some proportion of the maximum ability of the follower.  For pholcids, it makes sense to consider the larger spider who has constructed its own web as the "lead" spider who dominates the web and dictates what proportion of the total food available goes to the smaller spider. In this way, the larger spider "sets the pace" for the smaller spider in terms of food energy available. As noted, Jakob found the dominant spider won food 77% of the time [5]. So, roughly speaking, if the web catches 2 flies/day, the big spider will get 1.5 spiders, while the small spider gets 0.5 flies/day. This may or may not meet the needs of the small spider. But if 0.5 flies/day is insufficient for its needs, eventually the small spider will reach a threshold point of hunger and will be forced to abandon the shared web in order to make its own.

So for pholcids, 
SCR =  EnergySet_lead / (EnergyRequired_follower * StarvationCoefficient_follower)
  • SCR is "spider convergence ratio". 
  • EnergySet_lead is the number of flies/day allowed for the smaller spider as determined by the larger spider.
  • StarvationCoefficient_follower is the energy cost in days without food as a proportion of the total cost (starvation) in days to starvation, minus the number of days saved by not building a web (or some other energy saving not considered here); i.e. StarvationCoefficient = (1 - energy saved by not building a web).
  • EnergyCost_follower is the minimum number of flies required per day for the smaller spider to sustain life.
So for pholcids, let's use energy savings 19% for young ones (StarvationCoeff = 0.81), and 8% for adults (StarvationCoeff = 0.92). So let's say the small pholcid on a web gets 0.5 flies/day, as determined by the large spider, but it actually requires 1 fly/day for sustenance. 
SCR = (0.5 flies/day) / (1 fly/day * 0.81) 
SCR = 0.62
In this case, since SCR < 1, the small pholcid is below its daily energy requirement even as offset by the energy saved by not building its own web, and because SCR < 1 indicates spider divergence, the smaller spider will abandon the shared web to make its own.

Note the PCR and SCR thresholds are inverted. For cyclists the PCR equation denominator reflects the following cyclists' maximum output;  the numerator is the required output set by the front cyclist, and so the drafting coefficient reflects the reduction of the required output. For spiders, the denominator is the minimum energy requirement for living. Also, for SCR, the energy savings component implies that the spider's minimum daily food requirement is less than without the energy savings component, so we multiply the energy savings by the minimum requirement in the denominator, and not by the food quantity actually permitted it by the larger spider (in the numerator).

So despite that the PCR and SCR divergence thresholds are inverted, the parameters as between cyclists and spiders are essentially the same in that they both include: 1. an energy quantity determined by a leading agent; 2. an energy savings mechanism and quantity; 3. an energetic limitation for the follower (either a maximum output, as for cyclists, or a minimum food requirement, as for spiders).

Let's see how this works for an adult spider whose StarvationCoeff is 0.92, but which needs say 1.5 flies/ day, but is only allowed 0.5 flies/day by the larger spider.
SCR = (0.5 flies/day) / (1.5 flies/day * 0.92) 
SCR = 0.36
Here SCR is even smaller than for the example above and, according to this result, this spider will likely have chosen to leave well before it got to this state of hunger.

What happens if either spider is fed before it wanders onto their neighbors' webs? Being fed means that current daily food requirement decreases. Let's say our adult spider is sufficiently well-fed so that for a few days its daily requirement is now only 0.1 flies/day. Say it's still permitted 0.5 flies/day by the dominant larger spider.
SCR = (0.5 flies/day) / (0.1 flies/day * 0.92) 
SCR = 5.43
Here SCR > 1, and the spider chooses to stay, since its needs are comfortably met. Eventually, as it burns off the food energy from its current state of satiety, the daily requirement will begin to increase again, and the SCR value will fall.

Now it becomes easy to find the rate of underfeeding the smaller spider will tolerate before it abandons the web. We simply apply the situation where SCR = 1, which is the point at which the equation states that the spiders will diverge, or the point at which the smaller spider's daily needs are not met, accounting for the energy saved by not building its own web. So with this we can show how hungry the smaller spider will become at its current rate (if 0.5 flies a day is insufficient to meet its daily caloric needs) before it leaves by determining the number of flies/day in the numerator when SCR=1. Let's say our adult smaller spider requires 1 flies/day for sustenance with the same StarvationCoeff of 0.92.  How many flies/day will it tolerate before it decides to abandon the web?
1 =  (x flies/day) / ( 1 * 0.92)
x= 0.92
Here, even though the spider requires 1 fly/day, it will tolerate 0.92 flies/day -- less than its daily requirement -- for some time because of the energy saved by not having to build its own web. Of course it can't sustain this indefinitely. As the spider becomes hungrier, its daily requirement also increases. At some point it will not tolerate only 0.92 flies/day.

In any event, this all seems somewhat promising, though there is more data in Jakob's paper [4] I can test against the framework here, and further work is required generally to develop a more complete model. Also, in terms of protocooperative behavior, a question arises as to how long a spider will go before it relocates, given its willingness to tradeoff the cost of building its own web for receiving less than daily required food needs.

To conclude, there is some promise in adopting an equation similar to the peloton PCR for pholcid coupling behavior. A spider coupling equation, SCR, involves a description of the energetic requirements of coupled spiders that is similar to that of the coupled energetic requirement of cyclists where there is an energy savings quantity. Namely, SCR incorporates: 1. an energy savings quantity; 2. an energetic output that is determined by a "lead" spider, and; 3. an energy quantity limitation for the following spider. These are the essential features of the peloton PCR coupling equation. The SCR equation may allow predictions of food quantities tolerated by pholcids before leaving webs and other related predictions. The apparent application of an SCR equation supports the universality of principles that determine the collective behavior of pelotons.

References

[1] Trenchard, H. The peloton superorganism and protocooperative behavior. Applied Mathematics and Computation Volume 270, 1 November 2015, pp. 179–192
[2] Trenchard, H., Ratamero, E., Richardson, A., Perc, M. A deceleration model for bicycle peloton dynamics and group sorting Applied Mathematics and Computation Volume 251, 15 January 2015, 24–34
[3] Trenchard, H.,Richardson, A.,Ratamero, E.,Perc, M., Collective behavior and the identification of phases in bicycle pelotons, Physica A: Statistical Mechanics and its Applications Volume 405, 1 July 2014, pp. 92–103
[4] Jakob, E. Individual decisions and group dynamics: why pholcid spiders join and leave groups, Animal Behaviour Volume 68, Issue 1, July 2004, pp.9–20
[5] Jakob, E. Costs and benefits of group living for pholcid spiderlings: losing food, saving silk, Animal Behaviour Volume 41, Issue 4, April 1991, pp. 711–722

* Figure 1 also comes with a creative commons licence.

Image credit (Fig. 1)
South American social spiders feeding on katydid
© Ken Preston-Mafham / Premaphotos Wildlife
Premaphotos Wildlife
Amberstone, 1 Kirland Road, Bodmin, Cornwall
PL30 5JQ United Kingdom
Tel: +44 (0) 1208 78 258
Fax: +44 (0) 1208 72 302
library@premaphotos.co.uk
http://www.premaphotos.com

Figure 2 is from video footage taken by either myself or Ash Richardson, I'm not sure which.

Saturday, August 29, 2015

The peloton superorganism and protocooperative behavior - a brief overview


My latest paper The peloton superorganism and protocooperative behavior is published at last. Obviously any published paper requires a lot of work, but for me this one has required the most so far. I began writing this in late 2014, and it was accepted for publication in early August 2015 after the usual process of review and revisions. I would like to think that while this one was more work than my previous papers, it will have the most impact.

I. Protocooperative behavior

Basically the paper argues for a new concept that I call "protocooperative behavior", or PB for short. While the evidence I present is in relation to bicycle pelotons, I suggest that it applies to any biological system in which there is some energy savings mechanism, such as may be found in bird flocks, or fish schools, among others.

Two phases of PB

PB is defined by two phases of behavior: a phase in which cyclists (or other organisms) proceed at a sufficiently low output or speed for cyclists to pass each other and to share the most costly front position(s); a second phase in which cyclists can maintain the speeds of stronger front riders, but cannot pass them. In the low speed phase, the peloton is high density, and in the higher speed phase, the peloton is stretched in single file. At a second threshold, cyclists decouple and diverge.

Phase 1: Low speed, passing and cooperative sharing of costly position

In the low-speed phase, cyclists can pass because to do so is well within their metabolic capacity. To simulate passing, simulated cyclists are programmed to accelerate randomly within a range of speeds up to their maximum capacities. In this low speed phase, cyclists naturally share the most costly high-drag front positions.


Phase 2. High speed, no-passing, free-riding phase

As speeds increase, cyclists' capacity to pass diminishes until a threshold is reached when they cannot pass at all, yet they can still keep pace with stronger riders ahead (the stretched phase). This is possible due to the power output reductions afforded by drafting. In the stretched phase, cyclists are free-riders by physiological necessity, not by choice.


Individual and team strategies need not be modeled

These behaviors -- passing behavior/ sharing the most costly front position, and maintaining speed of stronger rider while being unable to pass -- need not be modeled by strategic probabilities or game theory. The behaviors self-organize from principles of energy savings, cyclists' maximal capacities, their current output, and a deceleration parameter that is triggered when cyclists are effectively driven over their maximal outputs. This is one of the main features that distinguishes this model from more standard models of cooperative behavior; i.e. no strategies are introduced into the model to generate cooperative behavior.

In that vein, team dynamics are not modeled, so circumstances in which teams might dominate the front at low speeds are not considered. This is important because, in natural biological collectives, we are more likely to see random passing based on inherent natural capacities, and less likely to see "teams" motivated by some tactical reason to dominate given positions. We may see something like that in nature, but I suggest such team-like domination would be observed among a comparatively advanced evolutionary stage than the primitive dynamics I am modeling.

The threshold between the phases

There is a clear threshold between a high-density passing phase, and a stretched phase. This threshold is demarcated by the equivalent of the coefficient of drafting. My paper sets out the details of how this works, but simply put, it is a function of the relationship between the coupled outputs of the cyclists, their maximum capacities, and the energy saved by drafting. I illustrate the transition here:

Figure 1. The protocooperative behavior threshold (pbt), approximately equivalent to the coefficient of drafting (d). See paper for details relating to "PCR" and Figure 2; Appl. Math and Computation Vol. 270, 1 Nov.  2015, Pages 179–192.


Cyclists sorting into ranges of output capacities

Additionally, cyclists (or any organism) exhibit a heterogeneous range of maximal outputs (i.e. they are not all the same). If the range of outputs is sufficiently broad, when the strongest riders drive peloton speeds to near maximums, the peloton tends to sort into subgroups in which the range of outputs is equivalent to the energy saved by drafting. My paper details why this is so.  Below I illustrate the group sorting hypothesis.

Figure 2.  Illustrating group sorting where, after speeds are driven to maximal speeds by the strongest riders, the range of maximal capacities within groups corresponds to the energy savings quantity.

II. Significance 

1. The two phases of PB and the threshold between them suggests a primitive form of cooperative behavior that does not rely on any evolutionary strategy (hence "proto" cooperative behavior) -- this proto-behavior may precede other evolutionary mechanisms for cooperation, such as kin selection, reciprocity and others; i.e. it is simply a function of changing speeds and organisms' outputs, coupled by an energy savings mechanism.

2. In its most primitive form, cooperation can't occur unless group members are below a critical threshold of outputs; i.e. they must have some spare energetic resources before they can cooperate, a la the quote from Roosevelt in the paper. So, if resources are strained such that group members are taxed to their physical limits, they will not physically be able to cooperate. This suggests that cooperation evolves in circumstances of "luxury".

As discussed,  the critical threshold for "luxury" corresponds to the variation range of maximal outputs that corresponds to the energy savings quantity (1 - d, where d is the energy savings coefficient): when organisms operate below this output threshold, they can cooperate; when above it they can free-ride, but cannot cooperate (Figs 1 and 2), up to a second de-coupling threshold.

So, as group member outputs are increased (and by similar process, resources become scarce), fewer and fewer among the group are within their limits of "luxury" and capable of cooperation; i.e. the strongest cooperate, while the weaker engage in free-riding behavior. Conversely, as outputs fall, or resources become more abundant, cooperation tends to be more widespread, even if it is less necessary for the survival of the group.

I have not researched the literature to find support for this, but it may be seen to be somewhat at odds with some current thinking -- for example, the "ecological constraints hypothesis" suggests that cooperative parenting occurs when resources are scarce [1] rather than when resources are more abundant, as I have suggested. While I need to study the literature a lot more, there may not be an inconsistency at all: what we may be seeing in circumstances of ecological constraint are situations when cooperation is narrowed to the stronger members of the group, while there is an increase in the number of free-riders. So, as a simplistic illustration, the weaker, young members may be fed by the cooperating stronger "parents" for an increased period of time (greater free-riding) before the young are permitted to feed on their own.

3.  In some models, such as that of Aviles [2], a cooperation parameter, y, is introduced which, when adjusted, generates different kinds of collective behavior.  However, at least insofar as [2] is set out, there is little or no explanation of what criteria are required to adjust the y parameter.  The peloton model suggests criteria for tuning that parameter as a function of individual maximal capacities, current outputs, and the energy savings quantity.

4. By modeling a mechanism for group sorting and the resulting range of differences among members of each group, we have a testable mechanism for niche formation, speciation, and group heterogeneity.

Although I didn't say this directly in the paper, here I go so far as to suggest that many branches in the evolutionary tree can be traced to this sorting mechanism. The model suggests an evolutionary mechanism that permits for the shedding of weak group members who are therefore readily susceptible to predators, or who become isolated and lose the opportunity to reproduce.  Where whole groups divide and separate, members of each may reproduce, but possibly in very different environments; e.g. one group might "make it over the Himalayas" (thank-you Ross Hooker for the illustration) into the rain-forest, while another may may end up in the high mountains. Indeed it may be possible to show mathematically how the entire range of differences among all existing species conforms to energy savings quantities at critical group sorting points. This would not be an easy task by any means, but it would have to begin with identifying the various energy savings mechanisms enjoyed by different organisms, and the energy savings quantities.

Of course, it is not necessarily true that all biological collectives enjoy energy savings mechanisms. In fact a recent study that shows pigeon flocks involve increased energetic costs due to positional adjustments especially during turning motions [3]*.  Still, it is apparent that such mechanisms do appear in many species. Even collections of bacteria tend to move faster than individuals [4] which suggests an energy savings mechanism.

In terms of other applications, there are other types of energy savings mechanisms. For example, any sort of leader-follower situation involves some sort of energy saving for the follower. The person who tramps snow first saves energy for the follower; the person who cuts through the forest makes it easier for those who come after.  A teacher saves a student energy because the student does not have to reinvent the wheel, as it were. These are but a few examples. While PB and group sorting for such situations may be more complicated, it is open to consider how the principles I present may be broadly applied.

[1] B.J. Hatchwell and J. Komdeur  Ecological constraints, life history traits and the evolution of cooperative breeding. (2000) 59(6):1079-1086.
[2] Aviles, L. "Cooperation and non-linear dynamics: An ecological perspective on the evolution of sociality." Evolutionary Ecology Research (1999), 1: 459-477
[3] Usherwood, James R., et al. "Flying in a flock comes at a cost in pigeons."Nature 474.7352 (2011): 494-497. 

[4] Cisneros, Luis H., et al. "Dynamics of swimming bacteria: Transition to directional order at high concentration." Physical Review E 83.6 (2011): 061907.  http://www.physics.arizona.edu/~kessler/micro/PhysRevE.83.061907.pdf


* Usherwood et al. seem to focus on energetic costs incurred in banking and turning, as they demonstrate in this video: https://www.youtube.com/watch?v=ssUslfD47l0. I've observed pigeons to fly several km at a stretch in roughly mean straight trajectories, and there are pigeon races over hundreds of km, meaning they are capable of long flight: https://en.wikipedia.org/wiki/Pigeon_racing.  I would be interested to see if the same principles of increased costs in flock formations apply in these circumstances. 



Thursday, July 30, 2015

Friday Harbor laboratories presentation


I was privileged to receive an invitation from Dr. Paolo Domenici to give a peloton dynamics presentation
at the Friday Harbor Laboratories (University of Washington) on San Juan island. The opportunity was serendipitous given how near San Juan Island is to Victoria.

Monday, December 29, 2014

Peloton hysteresis revisited



In analyzing peloton dynamics, an area ripe for exploration involves the mechanics of peloton density as it stretches and compresses. Speed and power output changes are obvious underlying mechanisms for these oscillations, but there are other more subtle dynamics at play. Among these subtleties are lags or delays in the transitions between density oscillations which are characteristic of hysteresis effects.

There is more than one type of hysteresis, but the general category of hysteresis I consider involves a lag between system input and output [1], meaning there is an asymmetry between a system trajectory, such as acceleration, and its complementary trajectory, such as deceleration. The following description, made in the context of vehicle traffic [2], well describes hysteretic behavior in pelotons:

"The dynamics of traffic flow result in the hysteresis phenomenon. This consists of a generally retarded behavior of vehicle platoons after emerging from a disturbance compared to the behavior of the same vehicles approaching the disturbance."

In a 2010 conference paper "Hysteresis in Competitive Bicycle Pelotons" [3] I proposed three different types of hysteresis in pelotons. Below I re-state and re-categorize them, hopefully with better clarity. In the 2010 paper I referred to "flow" to describe changing peloton density to be consistent with vehicle traffic models, but it is more intuitive to refer to compression (increase in peloton density) and stretching (peloton lengthening).

Types of peloton hysteresis

I.    Where a peloton decelerates relatively rapidly with corresponding rapid compression, followed by a proportionately longer acceleration time and peloton stretching. There are two sub-types:
  • A: occurs with corresponding changes in speed and power output preceding the disturbance and after it, such as slowing before a corner and accelerating out of it;
  • B: occurs with corresponding changes in speed but where power output may be approximately retained, such as when peloton compression and deceleration precede a climb and stretching occurs during the climb, but power output is roughly constant before and during the climb.
II. Where a peloton accelerates rapidly with corresponding stretching effect, followed by increased stretching  even as speeds decrease.

III. Inversely to II, where a peloton decelerates rapidly with corresponding compression, following by an increase in speed where compression either continues to increase for a short period or is temporarily retained even as the peloton accelerates.

For the purpose of this post, I am primarily concerned with types II and III.

Figures 1 & 2 below are re-scaled images from my 2010 hysteresis paper. Figures 3 & 4 show data from indoor velodrome races I videoed a couple of years ago.  Figures 1 - 4 involve speed data that is generated by the front rider in the group, which speed is applied to the entire group.

In Fig 1, the circled area and arrows show a period of decreasing speed following a rapid acceleration with a corresponding increase in stretching. In Fig 2, the arrows show the direction of the hysteresis curve as speed drops even as the peloton continues to stretch. The short arrow pointing down and slightly to the right (Fig 2 speed-stretch curve) is the hysteretic period of deceleration accompanies by increasing stretch. This direction in the speed-stretch trajectory characterizes Type II hysteresis. This indicates a lag or delay in the system's return to a compressed state as deceleration occurs. If the system dynamics were symmetrical, one would expect compression to accompany deceleration immediately, but we can see that is not necessarily the case. This delay is explainable by cyclists' collective fatigue and a recovery period following high acceleration. I refer to this fatigue induced delay in density oscillation as competitive hysteresis. This is the defining feature of type II peloton hysteresis.


Figure 1 independent speed and stretch curves
Figure 2 speed-stretch curve


Figures 3 & 4 below show another example of Type II hysteresis from a 3km indoor mass-start race (red arrows). Increasing stretch follows a double-whammy acceleration (the two sharp spikes preceding the arrows). Following the first of these accelerations to about 48km/h, there is deceleration and corresponding compression, as one would expect. After the second acceleration "whammy" there is the characteristic Type II down-and-to-the-right trajectory in system dynamics, as shown in Figure 4 (red arrow), indicating deceleration accompanied by increased stretching.

Figures 3& 4 also show an example of Type III hysteresis (green). Following soon after the Type II period, there is a period of increasing speed and increasing compression (Fig 3, green arrows), characterized by an up-and-to-the-left trajectory (Fig 4, green arrow) in the speed-stretch curve. This may occur when there is an acceleration from cyclists toward the rear of the peloton who pass riders ahead. In this case the hysteretic lag is in the return to a stretched state.

Type III hysteresis is also a precursor dynamic to the convective phase discussed in earlier papers [e.g. 5].


Figure 3 speed of front rider, and stretch 

Figure 4 speed-stretch

Using the model in [4], I've sought to identify occurrences of Types II and III hysteresis in simulated peloton dynamics.

Similar to the speed data in Figures 1 - 4, Figures 5 & 6 show the speed of the front rider versus peloton density. Speed of the front rider was coded to randomly fluctuate every 60 ticks (equivalent to 60s) within a narrow range, hence the "square" appearance of the speed data, as well as in the speed-stretch curve.  In Fig 6, generally wherever the curve proceeds at an angle it indicates a hysteretic delay. Using Fig 6, it is difficult to trace the direction of the curve which would tell us the hysteretic Type, although we can roughly eyeball regions of both Types II and III in Fig 5.  I've noted two periods of stable front-rider speed versus increased stretching, which appear to be Type II periods (although difficult to know given the speed curve is stable in those regions).

In contrast, Figures 7 & 8 show the mean speeds of all the riders versus peloton density, which is a level of detail that provides a clearer indication of whether there is acceleration or deceleration occurring. These aggregate speeds are easy to trace in simulations, but are a level of detail I have not obtained for any empirical data. This level of detail could be obtained empirically if speed data was obtained individually for each rider and averaged over the course of the race. This sort of data is desirable.

In Fig 7 there are arrows in about the same locations as for Fig 5, and the opposing direction of the aggregate speed curve do suggest that these regions are indeed hysteretic periods, of Type II. While I have not noted a Type III occurrence, the Fig 8 speed-stretch curve winds in all directions rather like the trajectory of a fruit fly, so it seems the simulation captures a fairly continuous but non-linear hysteresis trajectory. A speed-stretch curve of the moving averages of the mean speeds and the stretch might produce visually more apparent looping patterns.

Figure 5 speed of front rider, and stretch

Figure 6 speed-stretch

Figure 7 mean speeds of all riders, and stretch 

Figure 8 speed-stretch 

Where else might this be applied? 

Regarding other types of flocking systems, similar system lags could be tested by accelerating groups to maximal sustainable outputs for specific periods, followed by deceleration periods, in much the same way as demonstrated here for pelotons.

Here I get highly speculative, but I wonder if there are similar kinds of competitive hysteresis processes in bacterial colonies, or perhaps cancer growths, that might reveal vulnerabilities in those systems. I do not know enough about either of those systems even to propose an experimental protocol to test such a hypothesis. It does seem worth looking into further, however, if it has not already been done.  

It further occurs to me that, having mentioned fruitflies, when we look at 3-dimensional fruit fly trajectories, perhaps there is a way to decompose those trajectories into 2-dimensional speed and stretch data. I don't have a good sense of how this would be done at this point, however, and am simply putting the idea out there.

If nothing else, we gain insight into peloton dynamics specifically, and may gain understanding of general evolutionary mechanisms for group formation in a variety of systems and how these systems behave.  

References
1. http://en.wikipedia.org/wiki/Hysteresis
2. Treiterer, J., Myers, J.A. 1974. The Hysteresis Phenomenon in Traffic Flow. In: Buckley, D.J. (ed.), Proceedings of the 6th International Symposium on Transportation and Traffic Theory: 13-38
3. Complex Adaptive Systems —Resilience, Robustness, and Evolvability: Papers from the AAAI Fall Symposium (FS-10-03)
4. Trenchard, H., Ratamero, E., Richardson, A., Perc, M. 2015. A deceleration model for peloton dynamics and group sorting. Applied Mathematics and Computation 251, 24–34
5. Trenchard, H.Richardson, A.Ratamero, E.  Perc, M. 2014. Collective behavior and the identification of phases in bicycle pelotons. Physica A: Statistical Mechanics and its Applications, 405 . pp. 92-103. ISSN 0378-4371