Hiding behind chaos and error in the double pendulum

If you want a visual intuition for just how unpredictable chaotic dynamics can be then the go-to toy model is the double pendulum. There are lots of great simulations (and some physical implementations) of the double pendulum online. Recently, /u/abraxasknister posted such a simulation on the /r/physics subreddit and quickly attracted a lot of attention.

In their simulation, /u/abraxasknister has a fixed center (block dot) that the first mass (red dot) is attached to (by an invisible rigid massless bar). The second mass (blue dot) is then attached to the first mass (also by an invisible rigid massless bar). They then release these two masses from rest at some initial height and watch what happens.

The resulting dynamics are at right.

It is certainly unpredictable and complicated. Chaotic? Most importantly, it is obviously wrong.

But because the double pendulum is a famous chaotic system, some people did not want to acknowledge that there is an obvious mistake. They wanted to hide behind chaos: they claimed that for a complex system, we cannot possibly have intuitions about how the system should behave.

In this post, I want to discuss the error of hiding behind chaos, and how the distinction between microdynamics and global properties lets us catch /u/abraxasknister’s mistake.

Four stages in the relationship of computer science to other fields

This weekend, Oliver Schneider — an old high-school friend — is visiting me in the UK. He is a computer scientist working on human-computer interaction and was recently appointed as an assistant professor at the Department of Management Sciences, University of Waterloo. Back in high-school, Oliver and I would occasionally sneak out of class and head to the University of Saskatchewan to play counter strike in the campus internet cafe. Now, Oliver builds haptic interfaces that can represent virtually worlds physically so vividly that a blind person can now play a first-person shooter like counter strike. Take a look:

Now, dear reader, can you draw a connecting link between this and the algorithmic biology that I typically blog about on TheEGG?

I would not be able to find such a link. And that is what makes computer science so wonderful. It is an extremely broad discipline that encompasses many areas. I might be reading a paper on evolutionary biology or fixed-point theorems, while Oliver reads a paper on i/o-psychology or how to cut 150 micron-thick glass. Yet we still bring a computational flavour to the fields that we interface with.

A few years ago, Karp’s (2011; Xu & Tu, 2011) wrote a nice piece about the myriad ways in which computer science can interact with other disciplines. He was coming at it from a theorist’s perspective — that is compatible with TheEGG but maybe not as much with Oliver’s work — and the bias shows. But I think that the stages he identified in the relationship between computer science and others fields is still enlightening.

In this post, I want to share how Xu & Tu (2011) summarize Karp’s (2011) four phases of the relationship between computer science and other fields: (1) numerical analysis, (2) computational science, (3) e-Science, and the (4) algorithmic lens. I’ll try to motivate and prototype these stages with some of my own examples.

Local maxima and the fallacy of jumping to fixed-points

An economist and a computer scientist are walking through the University of Chicago campus discussing the efficient markets hypothesis. The computer scientist spots something on the pavement and exclaims: “look at that $20 on the ground — seems we’ll be getting a free lunch today!”

The economist turns to her without looking down and replies: “Don’t be silly, that’s impossible. If there was a $20 bill there then it would have been picked up already.”

This is the fallacy of jumping to fixed-points.

In this post I want to discuss both the importance and power of local maxima, and the dangers of simply assuming that our system is at a local maximum.

So before we dismiss the economist’s remark with laughter, let’s look at a more convincing discussion of local maxima that falls prey to the same fallacy. I’ll pick on one of my favourite YouTubers, THUNK:

In his video, THUNK discusses a wide range of local maxima and contrasts them with the intended global maximum (or more desired local maxima). He first considers a Roomba vacuum cleaner that is trying to maximize the area that it cleans but gets stuck in the local maximum of his chair’s legs. And then he goes on to discuss similar cases in physics, chemisty, evolution, psychology, and culture.

It is a wonderful set of examples and a nice illustration of the power of fixed-points.

But given that I write so much about algorithmic biology, let’s focus on his discussion of evolution. THUNK describes evolution as follows:

Evolution is a sort of hill-climbing algorithm. One that has identified local maxima of survival and replication.

This is a common characterization of evolution. And it seems much less silly than the economist passing up $20. But it is still an example of the fallacy of jumping to fixed-points.

My goal in this post is to convince you that THUNK describing evolution and the economist passing up $20 are actually using the same kind of argument. Sometimes this is a very useful argument, but sometimes it is just a starting point that without further elaboration becomes a fallacy.

John Maynard Smith on reductive vs effective thinking about evolution

“The logic of animal conflict” — a 1973 paper by Maynard Smith and Price — is usually taken as the starting for evolutionary game theory. And as far as I am an evolutionary game theorists, it influences my thinking. Most recently, this thinking has led me to the conclusion that there are two difference conceptions of evolutionary games possible: reductive vs. effective. However, I don’t think that this would have come as much of a surprise to Maynard Smith and Price. In fact, the two men embodied the two different ways of thinking that underlay my two interpretations.

I was recently reminded of this when Aakash Pandey shared a Web of Stories interview with John Maynard Smith. This is a 4 minute snippet of a long interview with Maynard Smith. In the snippet, he starts with a discussion of the Price equation (or Price’s theorem, if you want to have that debate) but quickly digresses to a discussion of the two kinds of mathematical theories that can be made in science. He identifies himself with the reductive view and Price with the effective. I recommend watching the whole video, although I’ll quote relavent passages below.

In this post, I’ll present Maynard Smith’s distinction on the two types of thinking in evolutionary models. But I will do this in my own terminology to stress the connections to my recent work on evolutionary games. However, I don’t think this distinction is limited to evolutionary game theory. As Maynard Smith suggests in the video, it extends to all of evolutionary biology and maybe scientific modelling more generally.

Drug holidays and losing resistance with replicator dynamics

A couple of weeks ago, before we all left Tampa, Pranav Warman, David Basanta and I frantically worked on refinements of our model of prostate cancer in the bone. One of the things that David and Pranav hoped to see from the model was conditions under which adaptive therapy (or just treatment interrupted with non-treatment holidays) performs better than solid blocks of treatment. As we struggled to find parameters that might achieve this result, my frustration drove me to embrace the advice of George Pólya: “If you can’t solve a problem, then there is an easier problem you can solve: find it.”

In this case, I opted to remove all mentions of the bone and cancer. Instead, I asked a simpler but more abstract question: what qualitative features must a minimal model of the evolution of resistance have in order for drug holidays to be superior to a single treatment block? In this post, I want to set up this question precisely, show why drug holidays are difficult in evolutionary models, and propose a feature that makes drug holidays viable. If you find this topic exciting then you should consider registering for the 6th annual Integrated Mathematical Oncology workshop at the Moffitt Cancer Center.^[1] This year’s theme is drug resistance.

Argument is the midwife of ideas (and other metaphors)

In their classic book Metaphors We Live By, George Lakoff and Mark Johnson argue — very convincingly, and as I’ve reviewed before — that “[m]etaphor is one of our most important tools for trying to comprehend partially what cannot be comprehended totally” and that these conceptual metaphors are central to shaping our understanding of and interaction with the world we are embedded in. Based on the authors’ grounding in linguistics, part of their case proceeds by offering examples of, by my count, over 58 different metaphors and metonymies in our everyday language; and given their book’s intentions, they chose a particularly pertinent first case: ARGUMENT is WAR.^[1]

They show this metaphor in action through some example of common usage (pg. 4):

Your claims are indefensible.
He attacked every weak point in my argument.
His criticisms were right on target.
I demolished his argument.
I’ve never won an argument with him.
You disagree? Okay, shoot!
If you use that strategy, he’ll wipe you out.
He shot down all my arguments.

Notice that the even the xkcd I borrowed for visual reinforcement is titled ‘Duty Calls’, an expression usually associated with a departure for war. With our awareness drawn to this militaristic structure, Lakoff and Johnson encourage the reader to ask themselves: how would discussions look if instead of structuring arguments adversarially, we structured them after a cooperative activity like dance?^[2]

Modeling influenza at ECMTB/SMB 2016

This week, I am at the University of Nottingham for the joint meeting of the Society of Mathematical Biology and the European Conference on Mathematical and Theoretical Biology — ECMTB/SMB 2016. It is a huge meeting, with over 800 delegates in attendance, 308 half-hour mini-symposium talks, 264 twenty-minute contributed talks, 190 posters, 7 prize talks, 7 plenary talks, and 1 public lecture. With seventeen to eighteen sessions running in parallel, it is impossible to see more than a tiny fraction of the content. And impossible for me to give you a comprehensive account of the event. However, I did want to share some moments from this week. If you are at ECMTB and want to share some of your highlights for TheEGG then let me know, and we can have you guest post.

I did not come to Nottingham alone. Above is a photo of current/recent Moffitteers that made their way to the meeting this year.

On the train ride to Nottingham, I needed to hear some success stories of mathematical biology. One of the ones that Dan Nichol volunteered was the SIR-model for controlling the spread of infectious disease. This is a simple system of ODEs with three compartments corresponding to the infection status of individuals in the population: susceptible (S), infectious (I), recovered (R). It is given by the following equations

where and are usually taken to be constants dependent on the pathogen, and the total number of individuals is an invariant of the dynamics.

As the replicator dynamics are to evolutionary game theory, the SIR-model is to epidemiology. And it was where Julia Gog opened the conference with her plenary on the challenges of modeling infectious disease. In this post, I will briefly touch on her extensions of the SIR-model and how she used it to look at the 2009 swine flu outbreak in the US.

Systemic change, effective altruism and philanthropy

The topics of effective altruism and social (in)justice have weighed heavy on my mind for several years. I’ve even touched on the latter occasionally on TheEGG, but usually in specific domains closer to my expertise, such as in my post on the ethics of big data. Recently, I started reading more thoroughly about effective altruism. I had known about the movement^[1] for some time, but had conflicting feelings towards it. My mind is still in disarray on the topic, but I thought I would share an analytic linkdex of some texts that have caught my attention. This is motivated by a hope to get some guidance from you, dear reader. Below are three videos, two articles, two book reviews and one paper alongside my summaries and comments. The methods range from philosophy to comedy and from critical theory to social psychology. I reach no conclusions.

Eukaryotes without Mitochondria and Aristotle’s Ladder of Life

In 348/7 BC, fearing anti-Macedonian sentiment or disappointed with the control of Plato’s Academy passing to Speusippus, Aristotle left Athens for Asian Minor across the Aegean sea. Based on his five years^[1] studying of the natural history of Lesbos, he wrote the pioneering work of zoology: The History of Animals. In it, he set out to catalog the what of biology before searching for the answers of why. He initiated a tradition of naturalists that continues to this day.

Aristotle classified his observations of the natural world into a hierarchical ladder of life: humans on top, above the other blooded animals, bloodless animals, and plants. Although we’ve excised Aristotle’s insistence on static species, this ladder remains for many. They consider species as more complex than their ancestors, and between the species a presence of a hierarchy of complexity with humans — as always — on top. A common example of this is the rationality fetish that views Bayesian learning as a fixed point of evolution, or ranks species based on intelligence or levels-of-consciousness. This is then coupled with an insistence on progress, and gives them the what to be explained: the arc of evolution is long, but it bends towards complexity.

In the early months of TheEGG, Julian Xue turned to explaining the why behind the evolution of complexity with ideas like irreversible evolution as the steps up the ladder of life.^[2] One of Julian’s strongest examples of such an irreversible step up has been the transition from prokaryotes to eukaryotes through the acquisition of membrane-bound organelles like mitochondria. But as an honest and dedicated scholar, Julian is always on the lookout for falsifications of his theories. This morning — with an optimistic “there goes my theory” — he shared the new Kamkowska et al. (2016) paper showing a surprising what to add to our natural history: a eukaryote without mitochondria. An apparent example of a eukaryote stepping down a rung in complexity by losing its membrane-bound ATP powerhouse.

Hadza hunter-gatherers, social networks, and models of cooperation

At the heart of the Great Lakes region of East Africa is Tanzania — a republic comprised of 30 mikoa, or provinces. Its border is marked off by the giant lakes Victoria, Tanganyika, and Malawi. But the lake that interests me the most is an internal one: 200 km from the border with Kenya at the junction of mikao Arusha, Manyara, Simiyu and Singed is Lake Eyasi. It is a temperamental lake that can dry up almost entirely — becoming crossable on foot — in some years and in others — like the El Nino years — flood its banks enough to attract hippos from the Serengeti.

For the Hadza, it is home.

The Hadza number around a thousand people, with around 300 living as traditional nomadic hunter-gatherers (Marlow, 2002; 2010). A life style that is believed to be a useful model of societies in our own evolutionary heritage. An empirical model of particular interest for the evolution of cooperation. But a model that requires much more effort to explore than running a few parameter settings on your computer. In the summer of 2010, Coren Apicella explored this model by traveling between Hadza camps throughout the Lake Eyasi region to gain insights into their social network and cooperative behavior.

Here is a video abstract where Coren describes her work:

The data she collected with her colleagues (Apicella et al., 2012) provides our best proxy for the social organization of early humans. In this post, I want to talk about the Hadza, the data set of their social network, and how it can inform other models of cooperation. In other words, I want to freeride on Apicella et al. (2012) and allow myself and other theorists to explore computational models informed by the empirical Hadza model without having to hike around Lake Eyasi for ourselves.