## 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 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!

## Network Theory I

2 March, 2014

Here’s a video of a talk I gave last Tuesday—part of a series. You can see the slides here:

One reason I’m glad I gave this talk is because afterwards Jamie Vicary pointed out some very interesting consequences of the relations among signal-flow diagrams listed in my talk. It turns out they imply equations familiar from the theory of complementarity in categorical quantum mechanics!

This is the kind of mathematical surprise that makes life worthwhile for me. It seemed utterly shocking at first, but I think I’ve figured out why it happens. Now is not the time to explain… but I’ll have to do it soon, both here and in the paper that Jason Eberle are writing about control theory.

• Brendan Fong, A compositional approach to control theory.

## Markov Models of Social Change (Part 1)

24 February, 2014

guest post by Alastair Jamieson-Lane

The world is complex, and making choices in a complex world is sometimes difficult.

As any leader knows, decisions must often be made with incomplete information. To make matters worse, the experts and scientists who are meant to advise on these important matters are also doing so with incomplete information—usually limited to only one or two specialist fields. When decisions need to be made that are dependent on multiple real-world systems, and your various advisors find it difficult to communicate, this can be problematic!

The generally accepted approach is to listen to whichever advisor tells you the things you want to hear.

When such an approach fails (for whatever mysterious and inexplicable reason) it might be prudent to consider such approaches as Bayesian inference, analysis of competing hypotheses or cross-impact balance analysis.

Because these methods require experts to formalize their opinions in an explicit, discipline neutral manner, we avoid many of the problems mentioned above. Also, if everything goes horribly wrong, you can blame the algorithm, and send the rioting public down to the local university to complain there.

In this blog article I will describe cross-impact balance analysis and a recent extension to this method, explaining its use, as well as some basic mathematical underpinnings. No familiarity with cross-impact balance analysis will be required.

### Wait—who is this guy?

Since this is my first time writing a blog post here, I hear introductions are in order.

Hi. I’m Alastair.

I am currently a Master’s student at the University of British Columbia, studying mathematics. In particular, I’m aiming to use evolutionary game theory to study academic publishing and hiring practices… and from there hopefully move on to studying governments (we’ll see how the PhD goes). I figure that both those systems seem important to solving the problems we’ve built for ourselves, and both may be under increasing pressure in coming years.

But that’s not what I’m here for today! Today I’m here to tell the story of cross-impact balance analysis, a tool I was introduced to at the complex systems summer school in Santa Fe.

### The story

Suppose (for example) that the local oracle has foretold that burning the forests will anger the nature gods

… and that if you do not put restrictions in place, your crops will wither and die.

Well, that doesn’t sound very good.

The merchant’s guild claims that such restrictions will cause all trade to grind to a halt.

Your most trusted generals point out that weakened trade will leave you vulnerable to invasion from all neighboring kingdoms.

The sailors guild adds that the wrath of Poseidon might make nautical trade more difficult.

The alchemists propose alternative sources of heat…

… while the druids propose special crops as a way of resisting the wrath of the gods…

… and so on.

Given this complex web of interaction, it might be a good time to consult the philosophers.

### Overview of CIB

This brings us to the question of what CIB (Cross-Impact Balance) analysis is, and how to use it.

At its heart, CIB analysis demands this: first, you must consider what aspects of the world you are interested in studying. This could be environmental or economic status, military expenditure, or the laws governing genetic modification. These we refer to as “descriptors”. For each “descriptor” we must create a list of possible “states”.

For example, if the descriptor we are interested in were “global temperature change” our states might be “+5 degree”, “+4 degrees” and so on down to “-2 degrees”.

The states of a descriptor are not meant to be all-encompassing, or offer complete detail, and they need not be numerical. For example, the descriptor “Agricultural policy” might have such states as “Permaculture subsidy”, “Genetic engineering”, “Intensive farming” or “No policy”.

For each of these states, we ask our panel of experts whether such a state would increase or decrease the tendency for some other descriptor to be in a particular state.

For example, we might ask: “On a scale from -3 to 3, how much does the agricultural policy of Intensive farming increase the probability that we will see global temperature increases of +2 degrees?”

By combining the opinions of a variety of experts in each field, and weighting based on certainty and expertise, we are able to construct matrices, much like the one below:

The above matrix is a description of my ant farm. The health of my colony is determined by the population, income, and education levels of my ants. For a less ant focused version of the above, please refer to:

• Elisabeth A. Lloyd and Vanessa J. Schweizer, Objectivity and a comparison of methodological scenario approaches for climate change research, Synthese (2013).

For any possible combination of descriptor states (referred to as a scenario) we can calculate the total impact on all possible descriptors. In the current scenario we have low population, high income and medium education (see highlighted rows).

Because the current scenario has high ant income, this strongly influences us to have low population (+3) and prevents a jump to high population (-3). This combined with the non-influence from education (zeros) leads to low population being the most favoured state for our population descriptor. Thus we expect no change. We say this is “consistent”.

Education however sees a different story. Here we have a strong influence towards high education levels (summing the column gives a total of 13). Thus our current state (medium education) is inconsistent, and we would expect the abundance of ant wealth to lead to an improvements in the ant schooling system.

Classical CIB analysis acts as a way to classify which hypothetical situations are consistent, and which are not.

Now, it is all well and good to claim that some scenarios are stable, but the real use of such a tool is in predicting (and influencing) the future.

By applying a deterministic rule that determines how inconsistencies are resolved, we can produce a “succession rule”. The most straight-forward example is to replace all descriptor states with whichever state is most favoured by the current scenario. In the example above we would switch to “Low population, medium income, high education”. A generation later we would switch back to “Low population, High income, medium education”, soon finding ourselves trapped in a loop.

All such rules will always lead to either a loop or a “sink”: a self consistent scenario which is succeeded only by itself.

So, how can we use this? How will this help us deal with the wrath of the gods (or ant farms)?

Firstly: we can identify loops and consistent scenarios which we believe are most favourable. It’s all well and good imagining some future utopia, but if it is inconsistent with itself, and will immediately lead to a slide into less favourable scenarios then we should not aim for it, we should find that most favourable realistic scenario and aim for that one.

Secondly: We can examine all our consistent scenarios, and determine whose “basin of attraction” we find ourselves in: that is, which scenario are we likely to end up in.

Thirdly: Suppose we could change our influence matrix slightly? How would we change it to favour scenarios we most prefer? If you don’t like the rules, change the game—or at the very least find out WHAT we would need to change to have the best effect.

### Concerns and caveats

So… what are the problems we might encounter? What are the drawbacks?

Well, first of all, we note that the real world does not tend to reach any form of eternal static scenario or perfect cycle. The fact that our model does might be regarded as reason for suspicion.

Secondly, although the classical method contains succession analysis, this analysis is not necessarily intended as a completely literal “prediction” of events. It gives a rough idea of the basins of attraction of our cycles and consistent scenarios, but is also somewhat arbitrary. What succession rule is most appropriate? Do all descriptors update simultaneously? Or only the one with the most “pressure”? Are our descriptors given in order of malleability, and only the fastest changing descriptor will change?

Thirdly, in collapsing our description of the world down into a finite number of states we are ignoring many tiny details. Most of these details are not important, but in assuming that our succession rules are deterministic, we imply that these details have no impact whatsoever.

If we instead treat succession as a somewhat random process, the first two of these problems can be solved, and the third somewhat reduced.

### Stochastic succession

In the classical CIB succession analysis, some rule is selected which deterministically decides which scenario follows from the present. Stochastic succession analysis instead tells us the probability that a given scenario will lead to another.

The simplest example of a stochastic succession rule is to simply select a single descriptor at random each time step, and only consider updates that might happen to that descriptor. This we refer to as dice succession. This (in some ways) represents hidden information: two systems that might look identical on the surface from the point of view of our very blockish CIB analysis might be different enough underneath to lead to different outcomes. If we have a shaky agricultural system, but a large amount of up-and-coming research, then which of these two factors becomes important first is down to the luck of the draw. Rather than attempt to model this fine detail, we instead merely accept it and incorporate this uncertainty into our model.

Even this most simplistic change leads to dramatics effects on our system. Most importantly, almost all cycles vanish from our results, as forks in the road allow us to diverge from the path of the cycle.

We can take stochastic succession further and consider more exotic rules for our transitions, ones that allow any transition to take place, not merely those that are most favored. For example:

$P(x,y) = A e^{I_x(y)/T}$

Here $x$ is our current scenario, $y$ is some possible future scenario, and $I_x(y)$ is the total impact score of $y$ from the perspective of $x$. $A$ is a simple normalizing constant, and $T$ is our system’s temperature. High temperature systems are dominated by random noise, while low temperature systems are dominated by the influences described by our experts.

Impact score is calculated by summing the impact of each state of our current scenario, on each state of our target scenario. For example, for the above, suppose we want to find $I_x(y)$ when $x$ is the given scenario “Low population, High income, medium education” and $y$ was the scenario “Medium population, medium income, High education”. We consider all numbers that are in rows which were states of $x$ and in columns that are states of $y$. This would give:

$I_x(y)= (0+0+0) + (-2 +0 +10) +(6+7+0) = 21$

Here each bracket refers to the sum of a particular column.
More generically we can write the formula as:

$\displaystyle{ I_x(y)= \sum_{i \subset x, \;j \subset y} M_{i,j} }$

Here $M_{i,j}$ refers to an entry in our cross-impact balance matrix, $i$ and $j$ are both states, and $i \subset x$ reads as “$i$ is a state of $x$”.

We refer to this function for computing transition probabilities as the Boltzmann succession law, due to its similarity to the Boltzmann distribution found in physics. We use it merely as an example, and by no means wish to imply that we expect the transitions for our true system to act in a precisely Boltzmann-like manner. Alternative functions can, and should, be experimented with. The Boltzmann succession law is however an effective example and has a number of nice properties: $P(x,y)$ is always positive, unchanged by adding a constant to every element of the cross-impact balance matrix, contains adjustable parameters, and unbounded above.

The Boltzmann succession rule is what I will refer to as fully stochastic: it allows transitions even against our experts’ judgement (with low probability). This is in contrast to dice succession which picks a direction at random, but still contains scenarios from which our system can not escape.

### Effects of stochastic succession

‘Partially stochastic’ processes such as the dice rule have very limited effect on the long term behavior of the model. Aside from removing most cycles, they behave almost exactly like our deterministic succession rules. So, let us instead discuss the more interesting fully stochastic succession rules.

In the fully stochastic system we can ask “after a very long time, what is the probability we will be in scenario $x$?”

By asking this question we can get some idea of the relative importance of all our future scenarios and states.

For example, if the scenario “high population, low education, low income” has a 40% probability in the long term, while most other scenarios have a probability of 0.2%, we can see that this scenario is crucial to the understanding of our system. Often scenarios already identified by deterministic succession analysis are the ones with the greatest long term probability—but by looking at long term probability we also gain information about the relative importance of each scenario.

In addition, we can encounter scenarios which are themselves inconsistent, but form cycles and/or clusters of interconnected scenarios. We can also notice scenarios that while technically ‘consistent’ in the deterministic rules are only barely so, and have limited weight due to a limited basin of attraction. We might identify scenarios that seem familiar in the real world, but are apparently highly unlikely in our analysis, indicating either that we should expect change… or perhaps suggesting a missing descriptor or a cross-impact in need of tweaking.

Armed with such a model, we can investigate what we can do to increase the short term and long term likelihood of desirable scenarios, and decrease the likelihood of undesirable scenarios.

As a last note, here are a few freely available resources that may prove useful. For a more formal introduction to CIB, try:

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

• Wolfgang Weimer-Jehle, Properties of cross-impact balance analysis.

You can find free software for doing a classical CIB analysis here:

• ZIRIUS, ScenarioWizard.

ZIRIUS is the Research Center for Interdisciplinary Risk and Innovation Studies of the University of Stuttgart.

Here are some examples of CIB in action:

• Gerhard Fuchs, Ulrich Fahl, Andreas Pyka, Udo Staber, Stefan Voegele and Wolfgang Weimer-Jehle, Generating innovation scenarios using the cross-impact methodology, Department of Economics, University of Bremen, Discussion-Papers Series No. 007-2008.

• Ortwin Renn, Alexander Jager, Jurgen Deuschle and Wolfgang Weimer-Jehle, A normative-functional concept of sustainability and its indicators, International Journal of Global Environmental Issues, 9 (2008), 291–317.

Finally, this page contains a more complete list of articles, both practical and theoretical:

## Network Theory Overview

22 February, 2014

Here’s a video of a talk I gave yesterday, made by Brendan Fong. You can see the slides here—and then click the items in blue, and the pictures, for more information!

The idea: nature and the world of human technology are full of networks! People like to draw diagrams of networks. Mathematical physicists know that in principle these diagrams can be understood using category theory. But why should physicists have all the fun? This is the century of understanding living systems and adapting to life on a finite planet. Math isn’t the main thing we need for this, but it’s got to be part of the solution… so one thing we should do is develop a unified and powerful theory of networks.

We are still far from doing this. In this overview, I briefly described three parts of the jigsaw puzzle, and invited everyone to help fit them together:

• electrical circuits and signal-flow graphs.

• stochastic Petri nets, chemical reaction networks and Feynman diagrams.

• Bayesian networks, information and entropy.

In my talks coming up, I’ll go into more detail on each of these.﻿ With luck, you’ll be able to see videos here.

But if you’re near Oxford, you might as well actually attend! You can see dates, times, locations, my slides, and the talks themselves as they show up by going here.

## Relative Entropy (Part 4)

16 February, 2014

In recent posts by Manoj Gopalkrishnan and Marc Harper we’ve seen how not just entropy but relative entropy—the entropy of a probability distribution relative to the equilibrium distribution—is a driving force in chemistry and evolution. Now Tobias Fritz and I have finally finished our paper on this subject:

Very roughly, we proved that any reasonable measure of the information you gain when you to update your assumptions about the world based on discovering what a system is really doing must be some constant $c$ times the relative entropy.

Relative Entropy (Part 1): how various structures important in probability theory arise naturally when you do linear algebra using only the nonnegative real numbers.

Relative Entropy (Part 2): a category related to statistical inference, $\mathrm{FinStat},$ and how relative entropy defines a functor on this category.

Relative Entropy (Part 3): statement of our main theorem, which characterizes relative entropy up to a constant multiple as the only functor $F : \mathrm{FinStat} \to [0,\infty)$ with a few nice properties.

Now that the paper is done, we’re having a nice conversation about it on the n-Category Café. Since I don’t want to splinter the conversation, I won’t enable comments here—please go there and join the fun!

One thing is that our conversation is getting more deeply into the category-theoretic aspects. Read the long parenthetical remarks in my post on the n-Café to get up to speed on that aspect.

Another is that by looking at our paper, you can actually see the proof of our result. As I mention on the n-Café.

The proof is surprisingly hard. Or maybe we’re just surprisingly bad at proving things. But the interesting thing is this: the proof is swift and effective in the ‘generic’ case—the case where the support of the probability measures involved is the whole set they’re living on, and the constant $c$ is finite.

It takes some more work to handle the case where the probability measures have smaller support.

But the really hard work starts when we handle the case that, in the end, has $c = \infty.$ Then the proof becomes more like analysis than what you normally expect in category theory. We slowly corner the result, blocking off all avenues of escape. Then we close in, grab its neck, and strangle it, crushing its larynx ever tighter, as it loses the will to fight back and finally expires… still twitching.

We haven’t gotten into discussing this much yet, perhaps because the mathematicians on the n-Café are too dainty and civilized. But someone into analysis might be able to find a more efficient proof.

That would make me a bit sad—since why didn’t we find it?—but mainly happy—since this subject deserves to be clean and elegant. We really need a category-theoretic formulation of the second law of thermodynamics that’s suitable for studying complex networks: that’s the long-term goal here.

## Triangular Numbers

12 February, 2014

This post is just for fun. I’ll start with Pascal’s triangle and show you the number e hiding inside it. Using this, we’ll see how the product of all numbers in the nth row of Pascal’s triangle is related to the nth triangular number.

That’s cute, because Pascal’s triangle looks so… triangular!

But then, with a massive amount of help from Greg Egan, we’ll dig a lot deeper, and meet strange things like superfactorials, the magic properties of the number 1/12, and the Glaisher–Kinkelin constant.

First let’s get warmed up.

### Warmup

Pascal’s triangle is a famous and much-studied thing. It was discovered long before Pascal. It looks like this:

We write 1′s down the edges. We get every other number in the triangle by adding the two numbers above it to its left and right. People call these numbers binomial coefficients, since you can also get them like this:

$(x + y)^5 = x^5 + 5 x^4 y + 10 x^3 y^2 + 10 x^2 y^3 + 5 x y^4 + y^5$

We get the 10 in $10 x^3 y^2$ here because there are 10 ways to take 5 things and choose 3 to call x’s and 2 to call y’s. In general we have

$\displaystyle{ (x + y)^n = \sum_{k=0}^n \binom{n}{k} x^k y^{n-k} }$

where the binomial coefficient $\binom{n}{k}$ is the kth number on the nth row of Pascal’s triangle. Here count both rows and the numbers in a row starting from zero.

Since we can permute n things in

$n! = 1 \cdot 2 \cdot 3 \cdot \cdots \cdot n$

ways, there are

$\displaystyle{ \frac{n!}{k! (n-k)!} }$

ways to take n things and choose k of them to be x’s and (n-k) of them to be y’s. You see, permuting the x’s or the y’s doesn’t change our choice, so we have to divide by $k!$ and $(n-k)!.$

So, the kth number in the nth row of Pascal’s triangle is:

$\displaystyle{\binom{n}{k} = \frac{n!}{k! (n-k)!} }$

All this will be painfully familiar to many of you. But I want everyone to have an fair chance at understanding the next section, where we’ll see something nice about Pascal’s triangle and the number

$e \approx 2.718281828459045...$

### The hidden e in Pascal’s triangle

In 2012, a guy named Harlan Brothers found the number e hiding in Pascal’s triangle… in a very simple way!

If we add up all the numbers in the nth row of Pascal’s triangle we get $2^n$. But what if we take the product of all these numbers? Let’s call it $t_n.$ Then here’s what Brothers discovered:

$\displaystyle{ \lim_{n \to \infty} \frac{t_n t_{n+2}}{t^2_{n+1}} = e }$

This may seem mysterious, but in fact it’s rather simple once you see the trick. I’ll use a nice argument given by Greg Egan.

We’ve said $t_n$ is the product of all numbers in the nth row of Pascal’s triangle. Just for fun, let’s divide this by the product of all numbers in the next row:

$\displaystyle{ u_n = \frac{t_n}{t_{n+1}} }$

And since this is so much fun, let’s do it again! Divide this quantity by its next value:

$\displaystyle{ v_n = \frac{u_n}{u_{n+1}} }$

Now look:

$\displaystyle{v_n = \frac{t_n/t_{n+1}}{t_{n+1}/t_{n+2}} = \frac{t_n t_{n+2}}{t_{n+1}^2} }$

So, this is the thing that should approach e.

But why does it approach e? To see this, first take Pascal’s triangle and divide each number by the number to its lower left. We get a second triangle of numbers. Then take each number in this second triangle and divide it by the number to its lower right! We get a third triangle, like this:

If you think a bit, you’ll see:

• In the first triangle, the product of all numbers in the nth row is $t_n.$

• In the second, the product of all numbers in the nth row is $u_n.$

• In the third, the product of all numbers in the nth row is $v_n.$

And look—there’s a cool pattern! In the third triangle, all the numbers in any given row are equal. In the row with n numbers, all those numbers equal

$\displaystyle{ (n+1)/n = 1 + \frac{1}{n} }$

So, the product of all these numbers is

$\displaystyle{ \left(1 + \frac{1}{n}\right)^n }$

But it’s a famous fact that

$\displaystyle{ \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e}$

Some people even use this as a definition of e. So,

$\displaystyle{ \lim_{n \to \infty} v_n = e}$

which is just what we wanted!

Puzzle 1. Use the formula I mentioned:

$\displaystyle{\binom{n}{k} = \frac{n!}{k! (n-k)!} }$

to show all the numbers in the same row of the third triangle are equal.

### Triangular numbers

The number of balls in a triangular stack with n balls at the bottom is called the nth triangular number,

$T_n = 1 + 2 + \cdots + n$

If you chop a square of balls in half along a diagonal you get a triangle, so $T_n$ is approximately half the nth square number:

$\displaystyle{ T_n \approx \frac{n^2}{2} }$

But if you chop the square in half this way, the balls on the diagonal get cut in half, so to get the nth triangle number we need to include their other halves:

$T_n = \displaystyle{ \frac{n^2}{2} + \frac{n}{2} = \frac{n(n+1)}{2} }$

These numbers go like

$1, 3, 6, 10, \dots$

and they are one of the simple pleasures of life, like sunshine and good bread. Spotting them in a knight-move zigzag pattern in the multiplication table was one of my first truly mathematical experiences. You can also see them in Pascal’s triangle:

because

$T_n = \displaystyle{ \binom{n+1}{2} }$

But today it’s time to see how $T_n$ is related to $t_n$, the product of all the numbers in the nth row of Pascal’s triangle!

If we subtract each triangular number from the the one before it we get the numbers $-n$, and if we subtract each of these numbers from the one before it we get the number 1. This should remind you of something we’ve seen:

$\displaystyle{ u_n = \frac{t_n}{t_{n+1}} }$

$\displaystyle{ v_n = \frac{u_n}{u_{n+1}} }$

$\displaystyle{ \lim_{n \to \infty} v_n = e }$

Here we’re dividing instead of subtracting. But if take logarithms, we’ll be subtracting, and we get

$\ln(u_n) = \ln(t_n) - \ln(t_{n+1})$

$\ln(v_n) = \ln(u_n) - \ln(u_{n+1})$

$\displaystyle{ \lim_{n \to \infty} \ln(v_n) = 1 }$

What can we do with this? Well, suppose $\ln(v_n)$ were equal to 1, instead of approaching it. Then $\ln(t_n)$ could be the nth triangular number… and we’d get a cool formula for the product of all numbers in the nth row of Pascal’s triangle!

But since $\ln(v_n)$ is just approximately equal to 1, we should only expect an approximate formula for the product of all numbers in the nth row of Pascal’s triangle:

$\ln(t_n) \approx T_n$

or

$\displaystyle{ t_n \approx e^{T_n} = e^{n(n+1)/2} }$

This was my hope. But how good are these approximations? I left this as a puzzle on Google+, and then Greg Egan stepped in and solved it.

For starters, he graphed the ratio $\ln(t_n)/T_n,$ and got this:

That looks pretty good: it looks like it’s approaching 1. But he also graphed the ratio $t_n/e^{T_n},$ and got this:

Not so good. Taking exponentials amplifies the errors. If we want a good asymptotic formula $t_n,$ we have to work harder. And this is what Egan did.

### Digging deeper

So far we’ve talked a lot about $t_n,$ the product of all numbers in the nth row in Pascal’s triangle… but we haven’t actually computed it. Let’s try:

$\begin{array}{ccl} t_n &=& \displaystyle{ \binom{n}{0} \cdot \binom{n}{1} \cdot \cdots \cdot \binom{n}{n} } \\ \\ &=& \displaystyle{ \frac{n!}{0! \cdot n!} \cdot \frac{n!}{1! \cdot (n-1)!} \cdot \cdots \cdot \frac{n!}{n! \cdot 0!} } \\ \\ &=& \displaystyle{ \frac{(n!)^{n+1}}{\left(0! \cdot 1! \cdot \cdots \cdot n!\right)^2} } \end{array}$

So, we’re seeing the superfactorial

$\displaystyle{ 0! \cdot 1! \cdot \cdots \cdot n! }$

raise its pretty head. This is also called the Barnes G-function, presumably by people who want to make it sound more boring.

Actually that’s not fair: the Barnes G-function generalizes superfactorials to complex numbers, just as Euler’s gamma function generalizes factorials. Unfortunately, Euler made the mistake of defining his gamma function so that

$\Gamma(n) = (n-1)!$

when $n = 1, 2, 3, \cdots.$ Everyone I trust assures me this was a bad idea, not some deep insight… but Barnes doubled down on Euler’s mistake and defined his G-function so that

$G(n) = 0! \cdot 1! \cdot \cdots \cdot (n-2)!$

So, we’ll have to be careful when looking up formulas on Wikipedia: the superfactorial of n is $G(n+2).$ Thus, we have

$\displaystyle{ t_n = \frac{(n!)^{n+1}}{G(n+2)^2} }$

so

$\displaystyle{ \ln(t_n) = (n+1) \ln (n!) - 2 \ln G(n+2) }$

Now, there’s a great approximate formula for the logarithm of a factorial. It’s worth its weight in gold… or at least silver, so it’s called Stirling’s formula:

$\displaystyle{ \ln(n!) = n \ln (n) \; - \; n \; + \;\tfrac{1}{2} \ln(2 \pi n) \; + \; \frac{1}{12n} \cdots }$

where the dots mean stuff that goes to zero when divided by the last term, in the limit $n \to \infty.$

There’s also an approximate formula for the logarithm of the superfactorial! It’s a bit scary:

$\begin{array}{ccl} \ln G(n+2) &=& \displaystyle{ \left(\tfrac{1}{2} \ln(n+1) - \tfrac{3}{4}\right) (n+1)^2 \; + } \\ \\ && \tfrac{\ln(2\pi)}{2} \, (n+1) \; - \\ \\ && \tfrac{1}{12} \ln(n+1) \; + \\ \\ && \tfrac{1}{12} - \ln A \; + \cdots \end{array}$

where the dots mean stuff that actually goes to zero as $n \to \infty.$

What’s A? It’s the Glaisher–Kinkelin constant! If you’re tired of memorizing digits of pi and need a change of pace, you can look up the first 20,000 digits of the Glaisher–Kinkelin constant here, but roughly we have

$A \approx 1.282427129100622636875342...$

Of course most mathematicians don’t care much about the digits; what we really want to know is what this constant means!

Euler, who did some wacky nonrigorous calculations, once argued that

$\displaystyle{ 1 + 2 + 3 + \cdots = -\tfrac{1}{12} }$

Riemann made this rigorous by defining

$\displaystyle{ \zeta(s) = 1^{-s} + 2^{-s} + 3^{-s} + \cdots }$

which converges when $\mathrm{Re}(s) > 1,$ and then analytically continuing this to other values of $s.$ He found that indeed

$\displaystyle{ \zeta(-1) = -\tfrac{1}{12} }$

This fact has many marvelous ramifications: for example, it’s why bosonic string theory works best in 24 dimensions! It’s also connected to the $\tfrac{1}{12n}$ term in Stirling’s formula.

But anyway, we might wonder what happens if we compute $\zeta(s)$ for $s$ near -1. This is where the Glaisher–Kinkelin constant shows up, because if we try a Taylor series we get

$\displaystyle{ \zeta'(-1) = \tfrac{1}{12} - \ln A }$

To me, this means that $\tfrac{1}{12} - \ln A$ is more fundamental than A itself, and indeed you’ll see it’s this combination that shows up in the approximate formula for superfactorials. So, we can simplify that formula a bit:

$\begin{array}{ccl} \ln G(n+2) &=& \displaystyle{ \left(\tfrac{1}{2} \ln(n+1) - \tfrac{3}{4}\right) (n+1)^2 \; + } \\ \\ && \tfrac{\ln(2\pi)}{2} \, (n+1) \; - \\ \\ && \tfrac{1}{12} \ln(n+1) \; + \\ \\ && \zeta'(-1) \; + \cdots \end{array}$

Now, let’s use this together with Stirling’s formula to estimate the logarithm of the product of all numbers in the nth row of Pascal’s triangle. Remember, that’s

$\displaystyle{ \ln(t_n) = (n+1) \ln (n!) - 2 \ln G(n+2) }$

So, we get

$\begin{array}{ccl} \ln(t_n) &\approx& \displaystyle{(n+1)\Big[ n \ln(n) - n + \tfrac{1}{2} \ln(2 \pi n) + \frac{1}{12n} \Big] } \\ \\ && - \displaystyle{ \Big[ \left( \ln(n+1) - \tfrac{3}{2}\right) (n+1)^2 + \ln(2\pi) \cdot (n+1) -} \\ \\ && \quad \displaystyle{ \tfrac{1}{6} \ln(n+1) + 2\zeta'(-1) \Big] } \end{array}$

and exponentiating this gives a good approximation to $t_n.$

Here is a graph of $t_n$ divided by this approximation, created by Egan:

As you can see, the ratio goes to 1 quite nicely!

So, we’ve seem some nice relationships between these things:

$1 + 2 + \cdots + n = T_n$

$1 \cdot 2 \cdot \cdots \cdot n = n!$

$\binom{n}{1} \cdot \binom{n}{2} \cdot \cdots \cdot \binom{n}{n} = t_n$

$1! \cdot 2! \cdot \cdots \cdot n! = G(n+2)$

$\frac{1}{0!} +\frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \cdots = e$

and Euler’s wacky formula

$\displaystyle{ 1 + 2 + 3 + \cdots = -\tfrac{1}{12} }$

Puzzle 2. Can you take the formula

$\begin{array}{ccl} \ln(t_n) &\approx& \displaystyle{(n+1)\Big[ n \ln(n) - n + \tfrac{1}{2} \ln(2 \pi n) + \frac{1}{12n} \Big] } \\ \\ && - \displaystyle{ \Big[ \left( \ln(n+1) - \tfrac{3}{2}\right) (n+1)^2 + \ln(2\pi) \cdot (n+1) -} \\ \\ && \quad \displaystyle{ \tfrac{1}{6} \ln(n+1) + 2\zeta'(-1) \Big] } \end{array}$

and massage it until it looks like $n(n+1)/2$ plus ‘correction terms’? How big are these correction terms?

### References

Any errors in the formulas above are my fault. Here are the papers that first squeezed e out of Pascal’s triangle:

• Harlan J. Brothers, Pascal’s triangle: The hidden stor-e, The Mathematical Gazette, March 2012, 145.

• Harlan J. Brothers, Finding e in Pascal’s triangle, Mathematics Magazine 85 (2012), 51﻿.

I learned the idea from here, thanks to Richard Elwes:

• Alexander Bogomolny, e in the Pascal Triangle, Interactive Mathematics Miscellany and Puzzles.

For how Euler derived his crazy identity

$\displaystyle{ 1 + 2 + 3 + \cdots = -\tfrac{1}{12} }$

and how it’s relevant to string theory, try:

• John Baez, My favorite numbers: 24.

## Network Theory Talks at Oxford

7 February, 2014

I’m giving some talks at Oxford:

### Network Theory

Nature and the world of human technology are full of networks. People like to draw diagrams of networks: flow charts, electrical circuit diagrams, signal-flow graphs, Bayesian networks, Feynman diagrams and the like. Mathematically minded people know that in principle these diagrams fit into a common framework: category theory. But we are still far from a unified theory of networks. After an overview, we will look at three portions of the jigsaw puzzle in three separate talks:

I. Electrical circuits and signal-flow graphs.

II. Stochastic Petri nets, chemical reaction networks and Feynman diagrams.

III. Bayesian networks, information and entropy.

If you’re nearby I hope you can come! All these talks will take place in Lecture Theatre B in the Computer Science Department—see the map below. Here are the times:

• Friday 21 February 2014, 2 pm: Network Theory: overview. See the slides or watch a video.

• Tuesday 25 February, 3:30 pm: Network Theory I: electrical circuits and signal-flow graphs. See the slides or watch a video.

• Tuesday 4 March, 3:30 pm: Network Theory II: stochastic Petri nets, chemical reaction networks and Feynman diagrams. See the slides or watch a video.

• Tuesday 11 March, 3:30 pm: Network Theory III: Bayesian networks, information and entropy. See the slides.

The first talk will be part of the OASIS series, meaning the “Oxford Advanced Seminar on Informatic Structures”.

I thank Samson Abramsky, Bob Coecke and Jamie Vicary of the Computer Science Department for inviting me, and Ulrike Tillmann and Minhyong Kim of the Mathematical Institute for helping me get set up. I also thank all the people who helped do the work I’ll be talking about, most notably Jacob Biamonte, Jason Erbele, Brendan Fong, Tobias Fritz, Tom Leinster, Tu Pham, and Franciscus Rebro.

Ulrike Tillmann has also kindly invited me to give a topology seminar:

### Operads and the Tree of Life

Trees are not just combinatorial structures: they are also biological structures, both in the obvious way but also in the study of evolution. Starting from DNA samples from living species, biologists use increasingly sophisticated mathematical techniques to reconstruct the most likely “phylogenetic tree” describing how these species evolved from earlier ones. In their work on this subject, they have encountered an interesting example of an operad, which is obtained by applying a variant of the Boardmann–Vogt “W construction” to the operad for commutative monoids. The operations in this operad are labelled trees of a certain sort, and it plays a universal role in the study of stochastic processes that involve branching. It also shows up in tropical algebra. This talk is based on work in progress with Nina Otter.

I’m not sure exactly where this will take place, but surely somewhere in the Mathematical Institute building:

• Monday 24 February, 3:30 pm, Operads and the Tree of Life. See the slides.

The Computer Science Department is shown in the map here:

The Mathematical Institute is a bit to the west:

## Categories in Control

6 February, 2014

I’m visiting Erlangen from now until the end of May, since my wife got a grant to do research here. I’m trying to get a lot of papers finished. But today I’m giving a talk in the math department of the university here, which with Germanic brevity is called the Friedrich-Alexander-Universität Erlangen-Nürnberg.

You can see my slides here, or maybe even come to my talk:

Categories in control, Thursday 6 February 2014, 16:15–18:00, Mathematics Department of the FAU, in Übungsraum 1.

The title is a pun. It’s about categories in control theory, the branch of engineering that studies dynamical systems with inputs and outputs, and how to optimize their behavior.

Control theorists often describe these systems using signal-flow graphs. Here is a very rough schematic signal-flow graph, describing the all-important concept of a ‘feedback loop’:

Here is a detailed one, describing a specific device called a servo:

The device is shown on top, and the signal-flow graph describing its behavior is at bottom. For details, click on the picture.

Now, if you have a drop of category-theorist’s blood in your veins, you’ll look at this signal-flow graph and think my god, that’s a string diagram for a morphism in a monoidal category!

And you’d be right. But if you want to learn what that means, and why it matters, read my talk slides!

The slides should make sense if you’re a mathematician, but maybe not otherwise. So, here’s the executive summary. The same sort of super-abstract math that handles things like Feynman diagrams:

also handles signal-flow graphs. The details are different in important and fascinating ways, and this is what I’m mainly concerned with. But we now understand how signal-flow graphs fit into the general theory of networks. This means we can proceed to use modern math to study them—and their relation to other kinds of networks, like electrical circuit diagrams:

### More talks

Thanks to the Azimuth Project team, my graduate students and many other folks, the dream of network theory as a step toward ‘green mathematics’ seems to be coming true! There’s a vast amount left to be done, so I’d have trouble convincing a skeptic, but I feel the project has turned a corner. I now feel in my bones that it’s going to work: we’ll eventually develop a language for biology and ecology based in part on category theory.

So, I think it’s a good time to explain all the various aspects of this project that have been cooking away—some quite visibly, but others on secret back burners:

• Jacob Biamonte and I have written a book on Petri nets and chemical reaction networks. You may have seen parts of this on the blog. We started this project at the Centre for Quantum Technologies, but now he’s working at the Institute for Scientific Interchange, in Turin—and collaborating with people there on various aspects of network theory.

• Brendan Fong is working with me on electrical circuits. You may know him for his posts here on chemical reaction networks. He’s now a grad student in computer science at Oxford.

• Jason Erbele, a math grad student at U.C. Riverside, is working with me on control theory. This work is the main topic of my talk—but I also sketch how it ties together with what Brendan is doing. There’s a lot more to say here.

• Tobias Fritz, a postdoc at the Perimeter Institute, is working with me on category-theoretic aspects of information theory. We published a paper on entropy with Tom Leinster, and we’ve got a followup on relative entropy that’s almost done. I should be working on it right this instant! But for now, read the series of posts here on Azimuth: Relative Entropy Part 1, Part 2 and Part 3.

• Brendan Fong has also done some great work on Bayesian networks, using ideas that connect nicely to what Tobias and I are doing.

• Tu Pham and Franciscus Rebro are working on the math that underlies all these projects: bicategories of spans.

The computer science department at Oxford is a great place for category theory and diagrammatic reasoning, thanks to the presence of Samson Abramsky, Bob Coecke and others. I’m going to visit them from February 21 to March 14. It seems like a good time to give a series of talks on this stuff. So, stay tuned! I’ll try to make slides available here.