## New IPCC Report (Part 1)

7 April, 2014

guest post by Steve Easterbrook

In October, I trawled through the final draft of this report, which was released at that time:

• Intergovernmental Panel on Climate Change (IPCC), Climate Change 2013: The Physical Science Basis.

Here’s what I think are its key messages:

I’ll talk about the first of these here, and the rest in future parts—click to get to any part you want. But before I start, a little preamble.

The IPCC was set up in 1988 as a UN intergovernmental body to provide an overview of the science. Its job is to assess what the peer-reviewed science says, in order to inform policymaking, but it is not tasked with making specific policy recommendations. The IPCC and its workings seem to be widely misunderstood in the media. The dwindling group of people who are still in denial about climate change particularly like to indulge in IPCC-bashing, which seems like a classic case of ‘blame the messenger’. The IPCC itself has a very small staff (no more than a dozen or so people). However, the assessment reports are written and reviewed by a very large team of scientists (several thousands), all of whom volunteer their time to work on the reports. The scientists are are organised into three working groups: WG1 focuses on the physical science basis, WG2 focuses on impacts and climate adaptation, and WG3 focuses on how climate mitigation can be achieved.

In October, the WG1 report was released as a final draft, although it was accompanied by bigger media event around the approval of the final wording on the WG1 “Summary for Policymakers”. The final version of the full WG1 report, plus the WG2 and WG3 reports, have come out since then.

I wrote about the WG1 draft in October, but John has solicited this post for Azimuth only now. By now, the draft I’m talking about here has undergone some minor editing/correcting, and some of the figures might have ended up re-drawn. Even so, most of the text is unlikely to have changed, and the major findings can be considered final.

In this post and the parts to come I’ll give my take on the most important findings, along with a key figure to illustrate each.

(1) The warming is unequivocal

The text of the summary for policymakers says:

Warming of the climate system is unequivocal, and since the 1950s, many of the observed changes are unprecedented over decades to millennia. The atmosphere and ocean have warmed, the amounts of snow and ice have diminished, sea level has risen, and the concentrations of greenhouse gases have increased.

(Fig SPM.1) Observed globally averaged combined land and ocean surface temperature anomaly 1850-2012. The top panel shows the annual values; the bottom panel shows decadal means. (Note: Anomalies are relative to the mean of 1961-1990).

Unfortunately, there has been much play in the press around a silly idea that the warming has “paused” in the last decade. If you squint at the last few years of the top graph, you might be able to convince yourself that the temperature has been nearly flat for a few years, but only if you cherry pick your starting date, and use a period that’s too short to count as climate. When you look at it in the context of an entire century and longer, such arguments are clearly just wishful thinking.

The other thing to point out here is that the rate of warming is unprecedented:

With very high confidence, the current rates of CO2, CH4 and N2O rise in atmospheric concentrations and the associated radiative forcing are unprecedented with respect to the highest resolution ice core records of the last 22,000 years

and there is

medium confidence that the rate of change of the observed greenhouse gas rise is also unprecedented compared with the lower resolution records of the past 800,000 years.

In other words, there is nothing in any of the ice core records that is comparable to what we have done to the atmosphere over the last century. The earth has warmed and cooled in the past due to natural cycles, but never anywhere near as fast as modern climate change.

You can download all of Climate Change 2013: The Physical Science Basis here. It’s also available chapter by chapter here:

## Civilizational Collapse (Part 1)

25 March, 2014

This story caught my attention, since a lot of people are passing it around:

• Nafeez Ahmed, NASA-funded study: industrial civilisation headed for ‘irreversible collapse’?, Earth Insight, blog on The Guardian, 14 March 2014.

Sounds dramatic! But notice the question mark in the title. The article says that “global industrial civilisation could collapse in coming decades due to unsustainable resource exploitation and increasingly unequal wealth distribution.” But with the word “could” in there, who could possibly argue? It’s certainly possible. What’s the actual news here?

It’s about a new paper that’s been accepted the Elsevier journal Ecological Economics. Since this paper has not been published, and I don’t even know the title, it’s hard to get details yet. According to Nafeez Ahmed,

The research project is based on a new cross-disciplinary ‘Human And Nature DYnamical’ (HANDY) model, led by applied mathematician Safa Motesharrei of the US National Science Foundation-supported National Socio-Environmental Synthesis Center, in association with a team of natural and social scientists.

So I went to Safa Motesharrei‘s webpage. It says he’s a grad student getting his PhD at the Socio-Environmental Synthesis Center, working with a team of people including:

Eugenia Kalnay (atmospheric science)
James Yorke (mathematics)
Matthias Ruth (public policy)
Victor Yakovenko (econophysics)
Klaus Hubacek (geography)
Ning Zeng (meteorology)
Fernando Miralles-Wilhelm (hydrology).

I was able to find this paper draft:

• Safa Motesharri, Jorge Rivas and Eugenia Kalnay, A minimal model for human and nature interaction, 13 November 2012.

I’m not sure how this is related to the paper discussed by Nafeez Ahmed, but it includes some (though not all) of the passages quoted by him, and it describes the HANDY model. It’s an extremely simple model, so I’ll explain it to you.

But first let me quote a bit more of the Guardian article, so you can see why it’s attracting attention:

By investigating the human-nature dynamics of these past cases of collapse, the project identifies the most salient interrelated factors which explain civilisational decline, and which may help determine the risk of collapse today: namely, Population, Climate, Water, Agriculture, and Energy.

These factors can lead to collapse when they converge to generate two crucial social features: “the stretching of resources due to the strain placed on the ecological carrying capacity”; and “the economic stratification of society into Elites [rich] and Masses (or “Commoners”) [poor]” These social phenomena have played “a central role in the character or in the process of the collapse,” in all such cases over “the last five thousand years.”

Currently, high levels of economic stratification are linked directly to overconsumption of resources, with “Elites” based largely in industrialised countries responsible for both:

“… accumulated surplus is not evenly distributed throughout society, but rather has been controlled by an elite. The mass of the population, while producing the wealth, is only allocated a small portion of it by elites, usually at or just above subsistence levels.”

The study challenges those who argue that technology will resolve these challenges by increasing efficiency:

“Technological change can raise the efficiency of resource use, but it also tends to raise both per capita resource consumption and the scale of resource extraction, so that, absent policy effects, the increases in consumption often compensate for the increased efficiency of resource use.”

Productivity increases in agriculture and industry over the last two centuries has come from “increased (rather than decreased) resource throughput,” despite dramatic efficiency gains over the same period.

Modelling a range of different scenarios, Motesharri and his colleagues conclude that under conditions “closely reflecting the reality of the world today… we find that collapse is difficult to avoid.” In the first of these scenarios, civilisation:

“…. appears to be on a sustainable path for quite a long time, but even using an optimal depletion rate and starting with a very small number of Elites, the Elites eventually consume too much, resulting in a famine among Commoners that eventually causes the collapse of society. It is important to note that this Type-L collapse is due to an inequality-induced famine that causes a loss of workers, rather than a collapse of Nature.”

Another scenario focuses on the role of continued resource exploitation, finding that “with a larger depletion rate, the decline of the Commoners occurs faster, while the Elites are still thriving, but eventually the Commoners collapse completely, followed by the Elites.”

In both scenarios, Elite wealth monopolies mean that they are buffered from the most “detrimental effects of the environmental collapse until much later than the Commoners”, allowing them to “continue ‘business as usual’ despite the impending catastrophe.” The same mechanism, they argue, could explain how “historical collapses were allowed to occur by elites who appear to be oblivious to the catastrophic trajectory (most clearly apparent in the Roman and Mayan cases).”

Applying this lesson to our contemporary predicament, the study warns that:

“While some members of society might raise the alarm that the system is moving towards an impending collapse and therefore advocate structural changes to society in order to avoid it, Elites and their supporters, who opposed making these changes, could point to the long sustainable trajectory ‘so far’ in support of doing nothing.”

However, the scientists point out that the worst-case scenarios are by no means inevitable, and suggest that appropriate policy and structural changes could avoid collapse, if not pave the way toward a more stable civilisation.

The two key solutions are to reduce economic inequality so as to ensure fairer distribution of resources, and to dramatically reduce resource consumption by relying on less intensive renewable resources and reducing population growth:

“Collapse can be avoided and population can reach equilibrium if the per capita rate of depletion of nature is reduced to a sustainable level, and if resources are distributed in a reasonably equitable fashion.”

### The HANDY model

So what’s the model?

It’s 4 ordinary differential equations:

$\dot{x}_C = \beta_C x_C - \alpha_C x_C$

$\dot{x}_E = \beta_E x_E - \alpha_E x_E$

$\dot{y} = \gamma y (\lambda - y) - \delta x_C y$

$\dot{w} = \delta x_C y - C_C - C_E$

where:

$x_C$ is the population of the commoners or masses

$x_E$ is the population of the elite

$y$ represents natural resources

$w$ represents wealth

The authors say that

Natural resources exist in three forms: nonrenewable stocks (fossil fuels, mineral deposits, etc), renewable stocks (forests, soils, aquifers), and flows (wind, solar radiation, rivers). In future versions of HANDY, we plan to disaggregate Nature into these three different forms, but for simpli cation in this version, we have adopted a single formulation intended to represent an amalgamation of the three forms.

So, it’s possible that the paper to be published in Ecological Economics treats natural resources using three variables instead of just one.

Now let’s look at the equations one by one:

$\dot{x}_C = \beta_C x_C - \alpha_C x_C$

This looks weird at first, but $\beta_C$ and $\alpha_C$ aren’t both constants, which would be redundant. $\beta_C$ is a constant birth rate for commoners, while $\alpha_C,$ the death rate for commoners, is a function of wealth.

Similarly, in

$\dot{x}_E = \beta_E x_E - \alpha_E x_E$

$\beta_E$ is a constant birth rate for the elite, while $\alpha_E,$ the death rate for the elite, is a function of wealth. The death rate is different for the elite and commoners:

For both the elite and commoners, the death rate drops linearly with increasing wealth from its maximum value $\alpha_M$ to its minimum values $\alpha_m$. But it drops faster for the elite, of course! For the commoners it reaches its minimum when the wealth $w$ reaches some value $w_{th},$ but for the elite it reaches its minimum earlier, when $w = w_{th}/\kappa$, where $\kappa$ is some number bigger than 1.

Next, how do natural resources change?

$\dot{y} = \gamma y (\lambda - y) - \delta x_C y$

The first part of this equation:

$\dot{y} = \gamma y (\lambda - y)$

describes how natural resources renew themselves if left alone. This is just the logistic equation, famous in models of population growth. Here $\lambda$ is the equilibrium level of natural resources, while $\gamma$ is another number that helps say how fast the resources renew themselves. Solutions of the logistic equation look like this:

But the whole equation

$\dot{y} = \gamma y (\lambda - y) - \delta x_C y$

has a term saying that natural resources get used up at a rate proportional to the population of commoners $x_C$ times the amount of natural resources $y.$ $\delta$ is just a constant of proportionality.

It’s curious that the population of elites doesn’t affect the depletion of natural resources, and also that doubling the amount of natural resources doubles the rate at which they get used up. Regarding the first issue, the authors offer this explanation:

The depletion term includes a rate of depletion per worker, $\delta,$ and is proportional to both Nature and the number of workers. However, the economic activity of Elites is modeled to represent executive, management, and supervisory functions, but not engagement in the direct extraction of resources, which is done by Commoners. Thus, only Commoners produce.

I didn’t notice a discussion of the second issue.

Finally, the change in the amount of wealth is described by this equation:

$\dot{w} = \delta x_C y - C_C - C_E$

The first term at right precisely matches the depletion of natural resources in the previous equation, but with the opposite sign: natural resources are getting turned into ‘wealth’. $C_C$ describes consumption by commoners and $C_E$ describes consumption by the elite. These are both functions of wealth, a bit like the death rates… but as you’d expect increasing wealth increases consumption:

For both the elite and commoners, consumption grows linearly with increasing wealth until wealth reaches the critical level $w_{th}.$ But it grows faster for the elites, and reaches a higher level.

So, that’s the model… at least in this preliminary version of the paper.

### Some solutions of the model

There are many parameters in this model, and many different things can happen depending on their values and the initial conditions. The paper investigates many different scenarios. I don’t have the energy to describe them all, so I urge you to skim it and look at the graphs.

I’ll just show you three. Here is one that Nafeez Ahmed mentioned, where civilization

appears to be on a sustainable path for quite a long time, but even using an optimal depletion rate and starting with a very small number of Elites, the Elites eventually consume too much, resulting in a famine among Commoners that eventually causes the collapse of society.

I can see why Ahmed would like to talk about this scenario: he’s written a book called A User’s Guide to the Crisis of Civilization and How to Save It. Clearly it’s worth putting some thought into risks of this sort. But how likely is this particular scenario compared to others? For that we’d need to think hard about how well this model matches reality.

It’s obviously a crude simplification of an immensely complex and unknowable system: the whole civilization on this planet. That doesn’t mean it’s fundamentally wrong! Its predictions could still be qualitatively correct. But to gain confidence in this, we’d need material that is not made in the draft paper I’ve seen. It says:

The scenarios most closely reflecting the reality of our world today are found in the third group of experiments (see section 5.3), where we introduced economic strati cation. Under such conditions,
we find that collapse is difficult to avoid.

But it would be nice to see a more careful approach to setting model parameters, justifying the simplifications built into the model, exploring what changes when some simplifications are reduced, and so on.

Here’s a happier scenario, where the parameters are chosen differently:

The main difference is that the depletion of resources per commoner, $\delta$, is smaller.

And here’s yet another, featuring cycles of prosperity, overshoot and collapse:

### Tentative conclusions

I hope you see that I’m neither trying to ‘shoot down’ this model nor defend it. I’m just trying to understand it.

I think it’s very important—and fun—to play around with models like this, keep refining them, comparing them against each other, and using them as tools to help our thinking. But I’m not very happy that Nafeez Ahmed called this piece of work a “highly credible wake-up call” without giving us any details about what was actually done.

I don’t expect blog articles on the Guardian to feature differential equations! But it would be great if journalists who wrote about new scientific results would provide a link to the actual work, so people who want to could dig deeper can do so. Don’t make us scour the internet looking for clues.

And scientists: if your results are potentially important, let everyone actually see them! If you think civilization could be heading for collapse, burying your evidence and your recommendations for avoiding this calamity in a closed-access Elsevier journal is not the optimal strategy to deal with the problem.

There’s been a whole side-battle over whether NASA actually funded this study:

• Keith Kloor, About that popular Guardian story on the collapse of industrial civilization, Collide-A-Scape, blog on Discover, March 21, 2014.

• Nafeez Ahmed, Did NASA fund ‘civilisation collapse’ study, or not?, Earth Insight, blog on The Guardian, 21 March 2014.

But that’s very boring compared to fun of thinking about the model used in this study… and the challenging, difficult business of trying to think clearly about the risks of civilizational collapse.

The paper is now freely available here:

• Safa Motesharri, Jorge Rivas and Eugenia Kalnay, Human and nature dynamics (HANDY): modeling inequality and use of resources in the collapse or sustainability of societies, Ecological Economics 101 (2014), 90–102.

## Programming with Chemical Reaction Networks

23 March, 2014

There will be a 5-day workshop on Programming with Chemical Reaction Networks: Mathematical Foundation at BIRS from Sunday, June 8 to Friday June 13, 2014 It’s being organized by

Anne Condon (University of British Columbia)
David Doty (California Institute of Technology)
Chris Thachuk (University of Oxford).

BIRS is the Banff International Research Station, in the mountains west of Calgary, in Alberta, Canada.

### Description

Here’s the workshop proposal on the BIRS website. It’s a pretty interesting proposal, especially if you’ve already read Luca Cardelli’s description of computing with chemical reaction networks, at the end of our series of posts on chemical reaction networks. The references include a lot of cool papers, so I’ve created links to those to help you get ahold of them.

This workshop will explore three of the most important research themes concerning stochastic chemical reaction networks (CRNs). Below we motivate each theme and highlight key questions that the workshop will address. Our main objective is to bring together distinct research communities in order to consider new problems that could not be fully appreciated in isolation. It is also our aim to determine commonalities between different disciplines and bodies of research. For example, research into population protocols, vector addition systems, and Petri networks provide a rich body of theoretical results that may already address contemporary problems arising in the study of CRNs.

#### Computational power of CRNs

Before designing robust and practical systems, it is useful to know the limits to computing with a chemical soup. Some interesting theoretical results are already known for stochastic chemical reaction networks. The computational power of CRNs depend upon a number of factors, including: (i) is the computation deterministic, or probabilistic, and (ii) does the CRN have an initial context — certain species, independent of the input, that are initially present in some exact, constant count.

In general, CRNs with a constant number of species (independent of the input length) are capable of Turing universal computation [17], if the input is represented by the exact (unary) count of one molecular species, some small probability of error is permitted and an initial context in the form of a single-copy leader molecule is used.

Could the same result hold in the absence of an initial context? In a surprising result based on the distributed computing model of population protocols, it has been shown that if a computation must be error-free, then deterministic computation with CRNs having an initial context is limited to computing semilinear predicates [1], later extended to functions outputting natural numbers encoded by molecular counts [5].

Furthermore, any semilinear predicate or function can be computed by that class of CRNs in expected time polylogarithmic in the input length. Building on this result, it was recently shown that by incurring an expected time linear in the input length, the same result holds for “leaderless” CRNs [8] — CRNs with no initial context. Can this result be improved to sub-linear expected time? Which class of functions can be computed deterministically by a CRN without an initial context in expected time polylogarithmic in the input length?

While (restricted) CRNs are Turing-universal, current results use space proportional to the computation time. Using a non-uniform construction, where the number of species is proportional to the input length and each initial species is present in some constant count, it is known that any S(n) space-bounded computation can be computed by a logically-reversible tagged CRN, within a reaction volume of size poly(S(n)) [18]. Tagged CRNs were introduced to model explicitly the fuel molecules in physical realizations of CRNs such as DNA strand displacement systems [6] that are necessary to supply matter and energy for implementing reactions such as X → X + Y that violate conservation of mass and/or energy.

Thus, for space-bounded computation, there exist CRNs that are time-efficient or are space-efficient. Does there exist time- and space-efficient CRNs to compute any space-bounded function?

#### Designing and verifying robust CRNs

While CRNs provide a concise model of chemistry, their physical realizations are often more complicated and more granular. How can one be sure they accurately implement the intended network behaviour? Probabilistic model checking has already been employed to find and correct inconsistencies between CRNs and their DNA strand displacement system (DSD) implementations [9]. However, at present, model checking of arbitrary CRNs is only capable of verifying the correctness of very small systems. Indeed, verification of these types of systems is a difficult problem: probabilistic state reachability is undecidable [17, 20] and general state reachability is EXPSPACE-hard [4].

How can larger systems be verified? A deeper understanding of CRN behaviour may simplify the process of model checking. As a motivating example, there has been recent progress towards verifying that certain DSD implementations correctly simulate underlying CRNs [16, 7, 10]. This is an important step to ensuring correctness, prior to experiments. However, DSDs can also suffer from other errors when implementing CRNs, such as spurious hybridization or strand displacement. Can DSDs and more generally CRNs be designed to be robust to such predictable errors? Can error correcting codes and redundant circuit designs used in traditional computing be leveraged in these chemical computers? Many other problems arise when implementing CRNs. Currently, unique types of fuel molecules must be designed for every reaction type. This complicates the engineering process significantly. Can a universal type of fuel be designed to smartly implement any reaction?

#### Energy efficient computing with CRNs

Rolf Landauer showed that logically irreversible computation — computation as modeled by a standard Turing machine — dissipates an amount of energy proportional to the number of bits of information lost, such as previous state information, and therefore cannot be energy efficient [11]. However, Charles Bennett showed that, in principle, energy efficient computation is possible, by proposing a universal Turing machine to perform logically-reversible computation and identified nucleic acids (RNA/DNA) as a potential medium to realize logically-reversible computation in a physical system [2].

There have been examples of logically-reversible DNA strand displacement systems — a physical realization of CRNs — that are, in theory, capable of complex computation [12, 19]. Are these systems energy efficient in a physical sense? How can this argument be made formally to satisfy both the computer science and the physics communities? Is a physical experiment feasible, or are these results merely theoretical footnotes?

#### References

[1] D. Angluin, J. Aspnes, and D. Eisenstat. Stably computable predicates are semilinear. In PODC, pages 292–299, 2006.

[2] C. H. Bennett. Logical reversibility of computation. IBM Journal of Research and Development, 17 (6):525–532, 1973.

[3] L. Cardelli and A. Csikasz-Nagy. The cell cycle switch computes approximate majority. Scientific Reports, 2, 2012.

[4] E. Cardoza, R. Lipton, A. R. Meyer. Exponential space complete problems for Petri nets and commutative semigroups (Preliminary Report). Proceedings of the Eighth Annual ACM Symposium on Theory of Computing, pages 507–54, 1976.

[5] H. L. Chen, D. Doty, and D. Soloveichik. Deterministic function computation with chemical reaction networks. DNA Computing and Molecular Programming, pages 25–42, 2012.

[6] A. Condon, A. J. Hu, J. Manuch, and C. Thachuk. Less haste, less waste: on recycling and its limits in strand displacement systems. Journal of the Royal Society: Interface Focus, 2 (4):512–521, 2012.

[7] Q. Dong. A bisimulation approach to verification of molecular implementations of formal chemical reaction network. Master’s thesis. SUNY Stony Brook, 2012.

[8] D. Doty and M. Hajiaghayi. Leaderless deterministic chemical reaction networks. In Proceedings of the 19th International Meeting on DNA Computing and Molecular Programming, 2013.

[9] M. R. Lakin, D. Parker, L. Cardelli, M. Kwiatkowska, and A. Phillips. Design and analysis of DNA strand displacement devices using probabilistic model checking. Journal of The Royal Society Interface, 2012.

[10] M. R. Lakin, D. Stefanovic and A. Phillips. Modular Verification of Two-domain DNA Strand Displacement Networks via Serializability Analysis. In Proceedings of the 19th Annual conference on DNA computing, 2013.

[11] R. Landauer. Irreversibility and heat generation in the computing process. IBM Journal of research and development, 5 (3):183–191, 1961.

[12] L. Qian, D. Soloveichik, and E. Winfree. Efficient Turing-universal computation with DNA polymers (extended abstract) . In Proceedings of the 16th Annual conference on DNA computing, pages 123–140, 2010.

[13] L. Qian and E. Winfree. Scaling up digital circuit computation with DNA strand displacement cascades. Science, 332 (6034):1196–1201, 2011.

[14] L. Qian, E. Winfree, and J. Bruck. Neural network computation with DNA strand displacement cascades. Nature, 475 (7356):368–372, 2011.

[15] G. Seelig, D. Soloveichik, D.Y. Zhang, and E. Winfree. Enzyme-free nucleic acid logic circuits. Science, 314 (5805):1585–1588, 2006.

[16] S. W. Shin. Compiling and verifying DNA-based chemical reaction network implementations. Master’s thesis. California Insitute of Technology, 2011.

[17] D. Soloveichik, M. Cook, E. Winfree, and J. Bruck. Computation with finite stochastic chemical reaction networks. Natural Computing, 7 (4):615–633, 2008.

[18] C. Thachuk. Space and energy efficient molecular programming. PhD thesis, University of British Columbia, 2012.

[19] C. Thachuk and A. Condon. Space and energy efficient computation with DNA strand displacement systems. In Proceedings of the 18th Annual International Conference on DNA computing and Molecular Programming, 2012.

[20] G. Zavattaro and L. Cardelli. Termination Problems in Chemical Kinetics. In Proceedings of the 2008 Conference on Concurrency Theory, pages 477–491, 2008.

## Networks of Dynamical Systems

18 March, 2014

guest post by Eugene Lerman

Hi, I’m Eugene Lerman. I met John back in the mid 1980s when John and I were grad students at MIT. John was doing mathematical physics and I was studying symplectic geometry. We never talked about networks. Now I teach in the math department at the University of Illinois at Urbana, and we occasionally talk about networks on his blog.

A few years ago a friend of mine who studies locomotion in humans and other primates asked me if I knew of any math that could be useful to him.

I remember coming across an expository paper on ‘coupled cell networks’:

• Martin Golubitsky and Ian Stewart, Nonlinear dynamics of networks: the groupoid formalism, Bull. Amer. Math. Soc. 43 (2006), 305–364.

In this paper, Golubitsky and Stewart used the study of animal gaits and models for the hypothetical neural networks called ‘central pattern generators’ that give rise to these gaits to motivate the study of networks of ordinary differential equations with symmetry. In particular they were interested in ‘synchrony’. When a horse trots, or canters, or gallops, its limbs move in an appropriate pattern, with different pairs of legs moving in synchrony:

They explained that synchrony (and the patterns) could arise when the differential equations have finite group symmetries. They also proposed several systems of symmetric ordinary differential equations that could generate the appropriate patterns.

Later on Golubitsky and Stewart noticed that there are systems of ODEs that have no group symmetries but whose solutions nonetheless exhibit certain synchrony. They found an explanation: these ODEs were ‘groupoid invariant’. I thought that it would be fun to understand what ‘groupoid invariant’ meant and why such invariance leads to synchrony.

I talked my colleague Lee DeVille into joining me on this adventure. At the time Lee had just arrived at Urbana after a postdoc at NYU. After a few years of thinking about these networks Lee and I realized that strictly speaking one doesn’t really need groupoids for these synchrony results and it’s better to think of the social life of networks instead. Here is what we figured out—a full and much too precise story is here:

• Eugene Lerman and Lee DeVille, Dynamics on networks of manifolds.

Let’s start with an example of a class of ODEs with a mysterious property:

Example. Consider this ordinary differential equation for a function $\vec{x} : \mathbb{R} \to {\mathbb{R}}^3$

$\begin{array}{rcl} \dot{x}_1&=& f(x_1,x_2)\\ \dot{x}_2&=& f(x_2,x_1)\\ \dot{x}_3&=& f(x_3, x_2) \end{array}$

for some function $f:{\mathbb{R}}^2 \to {\mathbb{R}}.$ It is easy to see that a function $x(t)$ solving

$\displaystyle{ \dot{x} = f(x,x) }$

gives a solution of these equations if we set

$\vec{x}(t) = (x(t),x(t),x(t))$

You can think of the differential equations in this example as describing the dynamics of a complex system built out of three interacting subsystems. Then any solution of the form

$\vec{x}(t) = (x(t),x(t),x(t))$

may be thought of as a synchronization: the three subsystems are evolving in lockstep.

One can also view the result geometrically: the diagonal

$\displaystyle{ \Delta = \{(x_1,x_2, x_3)\in {\mathbb{R}}^3 \mid x_1 =x_2 = x_3\} }$

is an invariant subsystem of the continuous-time dynamical system defined by the differential equations. Remarkably enough, such a subsystem exists for any choice of a function $f.$

Where does such a synchronization or invariant subsystem come from? There is no apparent symmetry of ${\mathbb{R}}^3$ that preserves the differential equations and fixes the diagonal $\Delta,$ and thus could account for this invariant subsystem. It turns out that what matters is the structure of the mutual dependencies of the three subsystems making up the big system. That is, the evolution of $x_1$ depends only on $x_1$ and $x_2,$ the evolution of $x_2$ depends only on $x_2$ and $x_3,$ and the evolution of $x_3$ depends only on $x_3$ and $x_2.$

These dependencies can be conveniently pictured as a directed graph:

The graph $G$ has no symmetries. Nonetheless, the existence of the invariant subsystem living on the diagonal $\Delta$ can be deduced from certain properties of the graph $G.$ The key is the existence of a surjective map of graphs

$\varphi :G\to G'$

from our graph $G$ to a graph $G'$ with exactly one node, call it $a,$ and one arrow. It is also crucial that $\varphi$ has the following lifting property: there is a unique way to lift the one and only arrow of $G'$ to an arrow of $G$ once we specify the target node of the lift.

We now formally define the notion of a ‘network of phase spaces’ and of a continuous-time dynamical system on such a network. Equivalently, we define a ‘network of continuous-time dynamical systems’. We start with a directed graph

$G=\{G_1\rightrightarrows G_0\}$

Here $G_1$ is the set of edges, $G_0$ is the set of nodes, and the two arrows assign to an edge its source and target, respectively. To each node we attach a phase space (or more formally a manifold, perhaps with boundary or corners). Here ‘attach’ means that we choose a function ${\mathcal P}:G_0 \to {\mathsf{PhaseSpace}};$ it assigns to each node $a\in G_0$ a phase space ${\mathcal P}(a).$

In our running example, to each node of the graph $G$ we attach the real line ${\mathbb{R}}.$ (If we think of the set $G_0$ as a discrete category and ${\mathsf{PhaseSpace}}$ as a category of manifolds and smooth maps, then ${\mathcal P}$ is simply a functor.)

Thus a network of phase spaces is a pair $(G,{\mathcal P}),$ where $G$ is a directed graph and ${\mathcal P}$ is an assignment of phase spaces to the nodes of $G.$

We think of the collection $\{{\mathcal P}(a)\}_{a\in G_0}$ as the collection of phase spaces of the subsystems constituting the network $(G, {\mathcal P}).$ We refer to ${\mathcal P}$ as a phase space function. Since the state of the network should be determined completely and uniquely by the states of its subsystems, it is reasonable to take the total phase space of the network to be the product

$\displaystyle{ {\mathbb{P}}(G, {\mathcal P}):= \bigsqcap_{a\in G_0} {\mathcal P}(a). }$

In the example the total phase space of the network $(G,{\mathcal P})$ is ${\mathbb{R}}^3,$ while the phase space of the network $(G', {\mathcal P}')$ is the real line ${\mathbb{R}}.$

Finally we need to interpret the arrows. An arrow $b\xrightarrow{\gamma}a$ in a graph $G$ should encode the fact that the dynamics of the subsystem associated to the node $a$ depends on the states of the subsystem associated to the node $b.$ To make this precise requires the notion of an ‘open system’, or ‘control system’, which we define below. It also requires a way to associate an open system to the set of arrows coming into a node/vertex $a.$

To a first approximation an open system (or control system, I use the two terms interchangeably) is a system of ODEs depending on parameters. I like to think of a control system geometrically: a control system on a phase space $M$ controlled by the the phase space $U$ is a map

$F: U\times M \to TM$

where $TM$ is the tangent bundle of the space $M,$ so that for all $(u,m)\in U\times M,$ $F(u,m)$ is a vector tangent to $M$ at the point $m.$ Given phase spaces $U$ and $M$ the set of all corresponding control systems forms a vector space which we denote by

$\displaystyle{ \mathsf{Ctrl}(U\times M \to M)}$

(More generally one can talk about the space of control systems associated with a surjective submersion $Q\to M.$ For us, submersions of the form $U\times M \to M$ are general enough.)

To encode the incoming arrows, we introduce the input tree $I(a)$ (this is a very short tree, a corolla if you like). The input tree of a node $a$ of a graph $G$ is a directed graph whose arrows are precisely the arrows of $G$ coming into the vertex $a,$ but any two parallel arrows of $G$ with target $a$ will have disjoint sources in $I(a).$ In the example the input tree $I$ of the one node of $a$ of $G'$ is the tree

There is always a map of graphs $\xi:I(a) \to G.$ For instance for the input tree in the example we just discussed, $\xi$ is the map

Consequently if $(G,{\mathcal P})$ is a network and $I(a)$ is an input tree of a node of $G,$ then $(I(a), {\mathcal P}\circ \xi)$ is also a network. This allows us to talk about the phase space ${\mathbb{P}} I(a)$ of an input tree. In our running example,

${\mathbb{P}} I(a) = {\mathbb{R}}^2$

Given a network $(G,{\mathcal P}),$ there is a vector space $\mathsf{Ctrl}({\mathbb{P}} I(a)\to {\mathbb{P}} a)$ of open systems associated with every node $a$ of $G.$

In our running example, the vector space associated to the one node $a$ of $(G',{\mathcal P}')$ is

$\mathsf{Ctrl}({\mathbb{R}}^2, {\mathbb{R}}) \simeq C^\infty({\mathbb{R}}^2, {\mathbb{R}})$

In the same example, the network $(G,{\mathcal P})$ has three nodes and we associate the same vector space $C^\infty({\mathbb{R}}^2, {\mathbb{R}})$ to each one of them.

We then construct an interconnection map

$\displaystyle{ {\mathcal{I}}: \bigsqcap_{a\in G_0} \mathsf{Ctrl}({\mathbb{P}} I(a)\to {\mathbb{P}} a) \to \Gamma (T{\mathbb{P}}(G, {\mathcal P})) }$

from the product of spaces of all control systems to the space of vector fields

$\Gamma (T{\mathbb{P}} (G, {\mathcal P}))$

on the total phase space of the network. (We use the standard notation to denote the tangent bundle of a manifold $R$ by $TR$ and the space of vector fields on $R$ by $\Gamma (TR)$). In our running example the interconnection map for the network $(G',{\mathcal P}')$ is the map

$\displaystyle{ {\mathcal{I}}: C^\infty({\mathbb{R}}^2, {\mathbb{R}}) \to C^\infty({\mathbb{R}}, {\mathbb{R}}), \quad f(x,u) \mapsto f(x,x). }$

The interconnection map for the network $(G,{\mathcal P})$ is the map

$\displaystyle{ {\mathcal{I}}: C^\infty({\mathbb{R}}^2, {\mathbb{R}})^3 \to C^\infty({\mathbb{R}}^3, {\mathbb{R}}^3)}$

given by

$\displaystyle{ ({\mathscr{I}}(f_1,f_2, f_3))\,(x_1,x_2, x_3) = (f_1(x_1,x_2), f_2(x_2,x_1), f_3(x_3,x_2)). }$

To summarize: a dynamical system on a network of phase spaces is the data $(G, {\mathcal P}, (w_a)_{a\in G_0} )$ where $G=\{G_1\rightrightarrows G_0\}$ is a directed graph, ${\mathcal P}:{\mathcal P}:G_0\to {\mathsf{PhaseSpace}}$ is a phase space function and $(w_a)_{a\in G_0}$ is a collection of open systems compatible with the graph and the phase space function. The corresponding vector field on the total space of the network is obtained by interconnecting the open systems.

Dynamical systems on networks can be made into the objects of a category. Carrying this out gives us a way to associate maps of dynamical systems to combinatorial data.

The first step is to form the category of networks of phase spaces, which we call ${\mathsf{Graph}}/{\mathsf{PhaseSpace}}.$ In this category, by definition, a morphism from a network $(G,{\mathcal P})$ to a network $(G', {\mathcal P}')$ is a map of directed graphs $\varphi:G\to G'$ which is compatible with the phase space functions:

$\displaystyle{ {\mathcal P}'\circ \varphi = {\mathcal P}. }$

Using the universal properties of products it is easy to show that a map of networks $\varphi: (G,{\mathcal P})\to (G',{\mathcal P}')$ defines a map ${\mathbb{P}}\varphi$ of total phase spaces in the opposite direction:

$\displaystyle{ {\mathbb{P}} \varphi: {\mathbb{P}} G' \to {\mathbb{P}} G. }$

In the category theory language the total phase space assignment extends to a contravariant functor

$\displaystyle{ {\mathbb{P}}: {({\mathsf{Graph}}/{\mathsf{Region}})}^{\mbox{\sf {\tiny {op}}}} \to {\mathsf{Region}}. }$

We call this functor the total phase space functor. In our running example, the map

${\mathbb{P}}\varphi:{\mathbb{R}} = {\mathbb{P}}(G',{\mathcal P}') \to {\mathbb{R}}^3 = {\mathbb{P}} (G,{\mathcal P})$

is given by

$\displaystyle{ {\mathbb{P}} \varphi (x) = (x,x,x). }$

Continuous-time dynamical systems also form a category, which we denote by $\mathsf{DS}.$ The objects of this category are pairs consisting of a phase space and a vector field on this phase space. A morphism in this category is a smooth map of phase spaces that intertwines the two vector fields. That is:

$\displaystyle{ \mathrm{Hom}_\mathsf{DS} ((M,X), (N,Y)) = \{f:M\to N \mid Df \circ X = Y\circ f\} }$

for any pair of objects $(M,X), (N,Y)$ in $\mathsf{DS}.$

In general, morphisms in this category are difficult to determine explicitly. For example if $(M, X) = ((a,b), \frac{d}{dt})$ then a morphism from $(M,X)$ to some dynamical system $(N,Y)$ is simply a piece of an integral curve of the vector field $Y$ defined on an interval $(a,b).$ And if $(M, X) = (S^1, \frac{d}{d\theta})$ is the constant vector field on the circle then a morphism from $(M,X)$ to $(N,Y)$ is a periodic orbit of $Y.$ Proving that a given dynamical system has a periodic orbit is usually hard.

Consequently, given a map of networks

$\varphi:(G,{\mathcal P})\to (G',{\mathcal P}')$

and a collection of open systems

$\{w'_{a'}\}_{a'\in G'_0}$

on $(G',{\mathcal P}')$ we expect it to be very difficult if not impossible to find a collection of open systems $\{w_a\}_{a\in G_0}$ so that

$\displaystyle{ {\mathbb{P}} \varphi: ({\mathbb{P}} G', {\mathscr{I}}' (w'))\to ({\mathbb{P}} G, {\mathscr{I}} (w)) }$

is a map of dynamical systems.

It is therefore somewhat surprising that there is a class of maps of graphs for which the above problem has an easy solution! The graph maps of this class are known by several different names. Following Boldi and Vigna we refer to them as graph fibrations. Note that despite what the name suggests, graph fibrations in general are not required to be surjective on nodes or edges. For example, the inclusion

is a graph fibration. We say that a map of networks

$\varphi:(G,{\mathcal P})\to (G',{\mathcal P}')$

is a fibration of networks if $\varphi:G\to G'$ is a graph fibration. With some work one can show that a fibration of networks induces a pullback map

$\displaystyle{ \varphi^*: \bigsqcap_{b\in G_0'} \mathsf{Ctrl}({\mathbb{P}} I(b)\to {\mathbb{P} b) \to \bigsqcap_{a\in G_0} \mathsf{Ctrl}({\mathbb{P}}} I(a)\to {\mathbb{P}} a) }$

on the sets of tuples of the associated open systems. The main result of Dynamics on networks of manifolds is a proof that for a fibration of networks $\varphi:(G,{\mathcal P})\to (G',{\mathcal P}')$ the maps $\varphi^*,$ ${\mathbb{P}} \varphi$ and the two interconnection maps ${\mathcal{I}}$ and ${\mathcal{I}}'$ are compatible:

Theorem. Let $\varphi:(G,{\mathcal P})\to (G',{\mathcal P}')$ be a fibration of networks of manifolds. Then the pullback map

$\displaystyle{ \varphi^*: \mathsf{Ctrl}(G',{\mathcal P}')\to \mathsf{Ctrl}(G,{\mathcal P}) }$

is compatible with the interconnection maps

$\displaystyle{ {\mathcal{I}}': \mathsf{Ctrl}(G',{\mathcal P}')) \to \Gamma (T{\mathbb{P}} G') }$

and

$\displaystyle{ {\mathcal{I}}: (\mathsf{Ctrl}(G,{\mathcal P})) \to \Gamma (T{\mathbb{P}} G) }$

Namely for any collection $w'\in \mathsf{Ctrl}(G',{\mathcal P}')$ of open systems on the network $(G', {\mathcal P}')$ the diagram

commutes. In other words,

$\displaystyle{ {\mathbb{P}} \varphi: ({\mathbb{P}} (G',{\mathcal P}'), {\mathscr{I}}' (w'))\to ({\mathbb{P}} (G, {\mathcal P}), {\mathscr{I}} (\varphi^* w')) }$

is a map of continuous-time dynamical systems, a morphism in $\mathsf{DS}.$

At this point the pure mathematician in me is quite happy: I have a theorem! Better yet, the theorem solves the puzzle at the beginning of this post. But if you have read this far, you may well be wondering: “Ok, you told us about your theorem. Very nice. Now what?”

There is plenty to do. On the practical side of things, the continuous-time dynamical systems are much too limited for contemporary engineers. Most of the engineers I know care a lot more about hybrid systems. These kinds of systems exhibit both continuous time behavior and abrupt transitions, hence the name. For example, anti-lock breaks on a car is just that kind of a system — if a sensor detects that the car is skidding and the foot break is pressed, it starts pulsing the breaks. This is not your father’s continuous time dynamical system! Hybrid dynamical systems are very hard to understand. They have been all the rage in the engineering literature for the last 10-15 years. Sadly, most pure mathematicians never heard of them. It would be interesting to extend the theorem I am bragging about to hybrid systems.

Here is another problem that may be worth thinking about: how much of the theorem holds up to numerical simulation? My feeling is that any explicit numerical integration method will behave well. Implicit methods I am not sure about.

And finally a more general issue. John has been talking about networks quite a bit on this blog. But his networks and my networks look like very different mathematical structures. What do they have in common besides nodes and arrows? What mathematical structure are they glimpses of?

## Network Theory III

16 March, 2014

In the last of my Oxford talks I explain how entropy and relative entropy can be understood using certain categories related to probability theory… and how these categories also let us understand Bayesian networks!

The first two parts are explanations of these papers:

• John Baez, Tobias Fritz and Tom Leinster, A characterization of entropy in terms of information loss

• John Baez and Tobias Fritz, A Bayesian characterization of relative entropy.

Somewhere around here the talk was interrupted by a fire drill, waking up the entire audience!

By the way, in my talk I mistakenly said that relative entropy is a continuous functor; in fact it’s just lower semicontinuous. I’ve fixed this in my slides.

The third part of my talk was my own interpretation of Brendan Fong’s master’s thesis:

• Brendan Fong, Causal Theories: a Categorical Perspective on Bayesian Networks.

I took a slightly different approach, by saying that a causal theory $\mathcal{C}_G$ is the free category with products on certain objects and morphisms coming from a directed acyclic graph $G$. In his thesis he said $\mathcal{C}_G$ was the free symmetric monoidal category where each generating object is equipped with a cocommutative comonoid structure. This is close to a category with finite products, though perhaps not quite the same: a symmetric monoidal category where every object is equipped with a cocommutative comonoid structure in a natural way (i.e., making a bunch of squares commute) is a category with finite products. It would be interesting to see if this difference hurts or helps.

By making this slight change, I am claiming that causal theories can be seen as algebraic theories in the sense of Lawvere. This would be a very good thing, since we know a lot about those.

You can also see the slides of this talk. Click on any picture in the slides, or any text in blue, and get more information!

## Network Theory II

12 March, 2014

Chemists are secretly doing applied category theory! When chemists list a bunch of chemical reactions like

C + O₂ → CO₂

they are secretly describing a ‘category’.

That shouldn’t be surprising. A category is simply a collection of things called objects together with things called morphisms going from one object to another, often written

f: x → y

The rules of a category say:

1) we can compose a morphism f: x → y and another morphism g: y → z to get an arrow gf: x → z,

2) (hg)f = h(gf), so we don’t need to bother with parentheses when composing arrows,

3) every object x has an identity morphism 1ₓ: x → x that obeys 1ₓ f = f and f 1ₓ = f.

Whenever we have a bunch of things (objects) and processes (arrows) that take one thing to another, we’re likely to have a category. In chemistry, the objects are bunches of molecules and the arrows are chemical reactions. But we can ‘add’ bunches of molecules and also add reactions, so we have something more than a mere category: we have something called a symmetric monoidal category.

My talk here, part of a series, is an explanation of this viewpoint and how we can use it to take ideas from elementary particle physics and apply them to chemistry! ﻿For more details try this free book:

• John Baez and Jacob Biamonte, A Course on Quantum Techniques for Stochastic Mechanics.

as well as this paper on the Anderson–Craciun–Kurtz theorem (discussed in my talk):

• John Baez and Brendan Fong, Quantum techniques for studying equilibrium in reaction networks.

You can also see the slides of this talk. Click on any picture in the slides, or any text in blue, and get more information!

## Markov Models of Social Change (Part 2)

5 March, 2014

guest post by Vanessa Schweizer

This is my first post to Azimuth. It’s a companion to the one by Alaistair Jamieson-Lane. I’m an assistant professor at the University of Waterloo in Canada with the Centre for Knowledge Integration, or CKI. Through our teaching and research, the CKI focuses on integrating what appears, at first blush, to be drastically different fields in order to make the world a better place. The very topics I would like to cover today, which are mathematics and policy design, are an example of our flavour of knowledge integration. However, before getting into that, perhaps some background on how I got here would be helpful.

### The conundrum of complex systems

For about eight years, I have focused on various problems related to long-term forecasting of social and technological change (long-term meaning in excess of 10 years). I became interested in these problems because they are particularly relevant to how we understand and respond to global environmental changes such as climate change.

In case you don’t know much about global warming or what the fuss is about, part of what makes the problem particularly difficult is that the feedback from the physical climate system to human political and economic systems is exceedingly slow. It is so slow, that under traditional economic and political analyses, an optimal policy strategy may appear to be to wait before making any major decisions – that is, wait for scientific knowledge and technologies to improve, or at least wait until the next election [1]. Let somebody else make the tough (and potentially politically unpopular) decisions!

The problem with waiting is that the greenhouse gases that scientists are most concerned about stay in the atmosphere for decades or centuries. They are also churned out by the gigatonne each year. Thus the warming trends that we have experienced for the past 30 years, for instance, are the cumulative result of emissions that happened not only recently but also long ago—in the case of carbon dioxide, as far back as the turn of the 20th century. The world in the 1910s was quainter than it is now, and as more economies around the globe industrialize and modernize, it is natural to wonder: how will we manage to power it all? Will we still rely so heavily on fossil fuels, which are the primary source of our carbon dioxide emissions?

Such questions are part of what makes climate change a controversial topic. Present-day policy decisions about energy use will influence the climatic conditions of the future, so what kind of future (both near-term and long-term) do we want?

### Futures studies and trying to learn from the past

Many approaches can be taken to answer the question of what kind of future we want. An approach familiar to the political world is for a leader to espouse his or her particular hopes and concerns for the future, then work to convince others that those ideas are more relevant than someone else’s. Alternatively, economists do better by developing and investigating different simulations of economic developments over time; however, the predictive power of even these tools drops off precipitously beyond the 10-year time horizon.

The limitations of these approaches should not be too surprising, since any stockbroker will say that when making financial investments, past performance is not necessarily indicative of future results. We can expect the same problem with rhetorical appeals, or economic models, that are based on past performances or empirical (which also implies historical) relationships.

### A different take on foresight

A different approach avoids the frustration of proving history to be a fickle tutor for the future. By setting aside the supposition that we must be able to explain why the future might play out a particular way (that is, to know the ‘history’ of a possible future outcome), alternative futures 20, 50, or 100 years hence can be conceptualized as different sets of conditions that may substantially diverge from what we see today and have seen before. This perspective is employed in cross-impact balance analysis, an algorithm that searches for conditions that can be demonstrated to be self-consistent [3].

Findings from cross-impact balance analyses have been informative for scientific assessments produced by the Intergovernmental Panel on Climate Change Research, or IPCC. To present a coherent picture of the climate change problem, the IPCC has coordinated scenario studies across economic and policy analysts as well as climate scientists since the 1990s. Prior to the development of the cross-impact balance method, these researchers had to do their best to identify appropriate ranges for rates of population growth, economic growth, energy efficiency improvements, etc. through their best judgment.

A retrospective using cross-impact balances on the first Special Report on Emissions Scenarios found that the researchers did a good job in many respects. However, they underrepresented the large number of alternative futures that would result in high greenhouse gas emissions in the absence of climate policy [4].

As part of the latest update to these coordinated scenarios, climate change researchers decided it would be useful to organize alternative futures according socio-economic conditions that pose greater or fewer challenges to mitigation and adaptation. Mitigation refers to policy actions that decrease greenhouse gas emissions, while adaptation refers to reducing harms due to climate change or to taking advantage of benefits. Some climate change researchers argued that it would be sufficient to consider alternative futures where challenges to mitigation and adaptation co-varied, e.g. three families of futures where mitigation and adaptation challenges would be low, medium, or high.

Instead, cross-impact balances revealed that mixed-outcome futures—such as socio-economic conditions simultaneously producing fewer challenges to mitigation but greater challenges to adaptation—could not be completely ignored. This counter-intuitive finding, among others, brought the importance of quality of governance to the fore [5].

Although it is generally recognized that quality of governance—e.g. control of corruption and the rule of law—affects quality of life [6], many in the climate change research community have focused on technological improvements, such as drought-resistant crops, or economic incentives, such as carbon prices, for mitigation and adaptation. The cross-impact balance results underscored that should global patterns of quality of governance across nations take a turn for the worse, poor governance could stymie these efforts. This is because the influence of quality of governance is pervasive; where corruption is permitted at the highest levels of power, it may be permitted at other levels as well—including levels that are responsible for building schools, teaching literacy, maintaining roads, enforcing public order, and so forth.

The cross-impact balance study revealed this in the abstract, as summarized in the example matrices below. Alastair included a matrix like these in his post, where he explained that numerical judgments in such a matrix can be used to calculate the net impact of simultaneous influences on system factors. My purpose in presenting these matrices is a bit different, as the matrix structure can also explain why particular outcomes behave as system attractors.

In this example, a solid light gray square means that the row factor directly influences the column factor some amount, while white space means that there is no direct influence:

Dark gray squares along the diagonal have no meaning, since everything is perfectly correlated to itself. The pink squares highlight the rows for the factors “quality of governance” and “economy.” The importance of these rows is more apparent here; the matrix above is a truncated version of this more detailed one:

(Click to enlarge.)

The pink rows are highlighted because of a striking property of these factors. They are the two most influential factors of the system, as you can see from how many solid squares appear in their rows. The direct influence of quality of governance is second only to the economy. (Careful observers will note that the economy directly influences quality of governance, while quality of governance directly influences the economy). Other scholars have meticulously documented similar findings through observations [7].

As a method for climate policy analysis, cross-impact balances fill an important gap between genius forecasting (i.e., ideas about the far-off future espoused by one person) and scientific judgments that, in the face of deep uncertainty, are overconfident (i.e. neglecting the ‘fat’ or ‘long’ tails of a distribution).

### Wanted: intrepid explorers of future possibilities

However, alternative visions of the future are only part of the information that’s needed to create the future that is desired. Descriptions of courses of action that are likely to get us there are also helpful. In this regard, the post by Jamieson-Lane describes early work on modifying cross-impact balances for studying transition scenarios rather than searching primarily for system attractors.

This is where you, as the mathematician or physicist, come in! I have been working with cross-impact balances as a policy analyst, and I can see the potential of this method to revolutionize policy discussions—not only for climate change but also for policy design in general. However, as pointed out by entrepreneurship professor Karl T. Ulrich, design problems are NP-complete. Those of us with lesser math skills can be easily intimidated by the scope of such search problems. For this reason, many analysts have resigned themselves to ad hoc explorations of the vast space of future possibilities. However, some analysts like me think it is important to develop methods that do better. I hope that some of you Azimuth readers may be up for collaborating with like-minded individuals on the challenge!

### References

The graph of carbon emissions is from reference [2]; the pictures of the matrices are adapted from reference [5]:

[1] M. Granger Morgan, Milind Kandlikar, James Risbey and Hadi Dowlatabadi, Why conventional tools for policy analysis are often inadequate for problems of global change, Climatic Change 41 (1999), 271–281.

[2] T.F. Stocker et al., Technical Summary, in Climate Change 2013: The Physical Science Basis. Contribution of Working Group I to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change (2013), T.F. Stocker, D. Qin, G.-K. Plattner, M. Tignor, S.K. Allen, J. Boschung, A. Nauels, Y. Xia, V. Bex, and P.M. Midgley (eds.) Cambridge University Press, New York.

[3] Wolfgang Weimer-Jehle, Cross-impact balances: a system-theoretical approach to cross-impact analysis, Technological Forecasting & Social Change 73 (2006), 334–361.

[4] Vanessa J. Schweizer and Elmar Kriegler, Improving environmental change research with systematic techniques for qualitative scenarios, Environmental Research Letters 7 (2012), 044011.

[5] Vanessa J. Schweizer and Brian C. O’Neill, Systematic construction of global socioeconomic pathways using internally consistent element combinations, Climatic Change 122 (2014), 431–445.

[6] Daniel Kaufman, Aart Kray and Massimo Mastruzzi, Worldwide Governance Indicators (2013), The World Bank Group.

[7] Daron Acemoglu and James Robinson, The Origins of Power, Prosperity, and Poverty: Why Nations Fail. Website.