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To what extent do you think of the climate models we have been considering at Azimuth as being network models, and to what extent is this stretching the term? In the latter case, we could fall back to the kindred notion of a complex system.

I realize that there may not be a clear-cut answer to this, and that the answer is model-dependent. I am putting this out here to get some perspective on the various models; to start a discussion. So any partial thoughts would be welcome!

## Comments

As we all know, many complex systems have evident network representations. Other times, networks are used as an approximate model. In biological sciences for example, they consider sometimes, not the Petri net, but the interaction graph (which is a projection of the Petri net where the transitions are replaced with directed edges and the species by nodes). They then try to infer properties of the system, from properties of this interaction graph. It does not always work. I want to try to fix this problem.

`As we all know, many complex systems have evident network representations. Other times, networks are used as an approximate model. In biological sciences for example, they consider sometimes, not the Petri net, but the interaction graph (which is a projection of the Petri net where the transitions are replaced with directed edges and the species by nodes). They then try to infer properties of the system, from properties of this interaction graph. It does not always work. I want to try to fix this problem.`

David wrote:

One climate model that's definitely a network is the box model used by Urban and Keller, which I discussed in week304 and the next week. It looks like this:

I think it's very good to think about a sequence of increasingly complicated box models starting with one like this! I'm very much hoping that network theory will (eventually) shed light on the qualitative behavior of models like this: tipping points, stability and the like. But so far I'm thinking mainly about

linearnetworked dynamical systems, which don't display as many interesting behaviors, but are a nice playground for developing the theory of networks.The stochastic energy balance models we've been programming would be networks of an extremely degenerate sort, scarcely worth thinking about as networks.

`David wrote: > To what extent do you think of the climate models we have been considering at Azimuth as being network models, and to what extent is this stretching the term? One climate model that's definitely a network is the box model used by Urban and Keller, which I discussed in [week304](http://math.ucr.edu/home/baez/week304.html) and the next week. It looks like this: <img width = "400" src = "http://math.ucr.edu/home/baez/nathan_urban_model.jpg" alt = ""/> I think it's very good to think about a sequence of increasingly complicated box models starting with one like this! I'm very much hoping that network theory will (eventually) shed light on the qualitative behavior of models like this: tipping points, stability and the like. But so far I'm thinking mainly about _linear_ networked dynamical systems, which don't display as many interesting behaviors, but are a nice playground for developing the theory of networks. The stochastic energy balance models we've been programming would be networks of an extremely degenerate sort, scarcely worth thinking about as networks.`

If more useful applications of network theory to climate science were found, I might pay more attention to the work being done about it in Azimuth. Box models are one thing, but can we build (useful) network representations of more complex systems? But just showing some interesting connections between network box models and nonlinear dynamics, as John suggests, would be a start.

The papers of Jonathan Donges on climate networks might be one starting point. Also, maybe some of the engineering perspective of input-output systems dynamics, where you can think of a dynamical system as being made up of interacting subsystems, each with their own feedbacks and characteristic dynamics, and then studying how the subsystem dynamics influence the whole system when combined. (I'm not really sure what exists out there, but that was my impression of what some of this engineering theory does.)

`If more useful applications of network theory to climate science were found, I might pay more attention to the work being done about it in Azimuth. Box models are one thing, but can we build (useful) network representations of more complex systems? But just showing some interesting connections between network box models and nonlinear dynamics, as John suggests, would be a start. The papers of [Jonathan Donges](http://www.pik-potsdam.de/members/donges/publications-1) on climate networks might be one starting point. Also, maybe some of the engineering perspective of input-output systems dynamics, where you can think of a dynamical system as being made up of interacting subsystems, each with their own feedbacks and characteristic dynamics, and then studying how the subsystem dynamics influence the whole system when combined. (I'm not really sure what exists out there, but that was my impression of what some of this engineering theory does.)`

Here's a paper that considers an application of reaction networks

Here's the abstract from the journals homepage.

AbstractHowever, as is the case with many studies, what is considered here is the

interaction graphcorresponding to the reaction network. This is a lossy projection from the reaction network. I think we can take this study further by going past the interaction graph approach, but it's not trivial. I think it will involve using ideas from John's paper. If others are interested, maybe we could start a new thread and read this paper and give it a try.Going past the interaction graph approach is not guaranteed to show us something new. Like many applications of network theory, many of the approaches use networks as an approximation tool. The empirical part is if such an approximation correlates with something we want to understand. I think it's worth a try though. More details later...

`Here's a paper that considers an application of reaction networks * [Returnability as a criterion of disequilibrium in atmospheric reactions networks](http://link.springer.com/article/10.1007%2Fs10910-012-9977-x), by Ernesto Estrada Here's the abstract from the journals homepage. **Abstract** > The concept of network returnability is reformulated as an equilibrium constant for a reaction network. Using this concept we study the atmospheric reaction networks of Earth, Mars, Venus and Titan. We found that the reaction network in the Earth’s atmosphere has the largest disequilibrium, followed by that of Titan which is still far from the most returnable atmospheres of Mars and Venus. We find that the chemical species with null or very low returnability are those in the highest disequilibrium in their respective atmospheres mainly due to physical, biogenic and/or anthropogenic mechanisms. Graphical Abstract > Highlights: A network theoretic approach to departure from equilibrium of reaction systems. Earth's atmosphere largely departed from equilibrium. Titan's atmosphere follows Earth in disequlibrium. Venus and Mars are the most in-equilibrium atmospheres. Biogenic species departs significantly from equilibrium. ![reaction network](http://link.springer.com/static-content/images/136/art%253A10.1007%252Fs10910-012-9977-x/MediaObjects/10910_2012_9977_Figa_HTML.gif) However, as is the case with many studies, what is considered here is the _interaction graph_ corresponding to the reaction network. This is a lossy projection from the reaction network. I think we can take this study further by going past the interaction graph approach, but it's not trivial. I think it will involve using ideas from John's [paper](http://arxiv.org/abs/1306.3451). If others are interested, maybe we could start a new thread and read this paper and give it a try. Going past the interaction graph approach is not guaranteed to show us something new. Like many applications of network theory, many of the approaches use networks as an approximation tool. The empirical part is if such an approximation correlates with something we want to understand. I think it's worth a try though. More details later...`

It may be more useful to thing of being a network model as a representation issue rather than an intrinsic property of a model. Existing models could be represented as various types of networks. This could provide new insights into existing models or make them more modular and easier to adapt for different purposes. Network formalisms like bond-graphs, circuit diagrams and various type of Petri nets are essentially languages or syntax (with semantics) while the models themselves are more like algorithms, semantic entities. The same algorithm can be written in many languages and different languages have different strengths. Translating some of the models that have been discussed here into circuits, bond graphs, or petri-nets or ... could also make them easier to compare.

`It may be more useful to thing of being a network model as a representation issue rather than an intrinsic property of a model. Existing models could be represented as various types of networks. This could provide new insights into existing models or make them more modular and easier to adapt for different purposes. Network formalisms like bond-graphs, circuit diagrams and various type of Petri nets are essentially languages or syntax (with semantics) while the models themselves are more like algorithms, semantic entities. The same algorithm can be written in many languages and different languages have different strengths. Translating some of the models that have been discussed here into circuits, bond graphs, or petri-nets or ... could also make them easier to compare.`

I spent an enjoyable couple of days trying to grasp Saedeleer and Crucifix's astronomical forcing paper, thanks to Nathan Urban, and noting the simulations which they said they hadn't done. I wrote some simple van der Pol oscillator codes which I meant to come back to when I'd relabelled the oscillator as a network. I like electrical circuit representations and thought it might be fun to see what happens when you replace a van der Pol oscillator with a Chua circuit because iiuc , unlike van der Pol oscillators which exhibit linear and bistable behaviour, they exhibit linear, bistable and chaotic behaviour.

`I spent an enjoyable couple of days trying to grasp Saedeleer and Crucifix's astronomical forcing [paper](http://arxiv.org/abs/1109.6214), thanks to [[Nathan Urban]], and noting the simulations which they said they hadn't done. I wrote some simple van der Pol oscillator codes which I meant to come back to when I'd relabelled the oscillator as a network. I like electrical circuit representations and thought it might be fun to see what happens when you replace a van der Pol oscillator with a Chua circuit because iiuc , unlike van der Pol oscillators which exhibit linear and bistable behaviour, they exhibit linear, bistable and chaotic behaviour.`

Nathan wrote:

Yeah, I'll wake you up if and when that happens. Right now it's not really applications to climate science that I'm thinking about; it's more "foundational" stuff. I've got 5 grad students working to describe the math underlying

in a unified format that clarifies their relation. (The chemical reaction networks which Jacob and I were studying seem a bit distant from this cluster, which is something I should tackle.)

Thanks a lot, I'll look at those!

Yes, a lot of this falls under the heading of "control theory", and I've been working on that with a grad student named Jason Erbele. As usual, I'm starting with rather abstract work that seeks to clarify the mathematical foundations of control theory, like

what is the category in which signal flow diagrams are morphisms?Meanwhile, we're learning control theory, which traditionally focuses on the issues you mentioned: building up dynamical systems (described using signal flow diagrams) out of parts (described using smaller diagrams), studying how the behavior of the parts affect those of the whole, and designing parts that can "stabilize" or otherwise "control" the behavior of the whole.

Now that I'm on sabbatical I want to resume the network theory series on the blog, and start talking about what we've been up to.

`Nathan wrote: > If more useful applications of network theory to climate science were found, I might pay more attention to the work being done about it in Azimuth. Yeah, I'll wake you up if and when that happens. Right now it's not really applications to climate science that I'm thinking about; it's more "foundational" stuff. I've got 5 grad students working to describe the math underlying * electrical circuit diagrams * signal flow diagrams (from control theory) * Markov processes in a unified format that clarifies their relation. (The chemical reaction networks which Jacob and I were studying seem a bit distant from this cluster, which is something I should tackle.) > The papers of [Jonathan Donges](http://www.pik-potsdam.de/members/donges/publications-1) on climate networks might be one starting point. Thanks a lot, I'll look at those! > Also, maybe some of the engineering perspective of input-output systems dynamics, where you can think of a dynamical system as being made up of interacting subsystems, each with their own feedbacks and characteristic dynamics, and then studying how the subsystem dynamics influence the whole system when combined. Yes, a lot of this falls under the heading of "control theory", and I've been working on that with a grad student named Jason Erbele. As usual, I'm starting with rather abstract work that seeks to clarify the mathematical foundations of control theory, like _what is the category in which [signal flow diagrams](http://en.wikipedia.org/wiki/Signal-flow_graph) are morphisms?_ <img width = "500" src = "http://upload.wikimedia.org/wikipedia/commons/thumb/1/10/Position_servo_and_signal_flow_graph.png/800px-Position_servo_and_signal_flow_graph.png" alt = ""/> Meanwhile, we're learning control theory, which traditionally focuses on the issues you mentioned: building up dynamical systems (described using signal flow diagrams) out of parts (described using smaller diagrams), studying how the behavior of the parts affect those of the whole, and designing parts that can "stabilize" or otherwise "control" the behavior of the whole. Now that I'm on sabbatical I want to resume the network theory series on the blog, and start talking about what we've been up to.`

Correction: according to J. Rial van der Pol oscillators can demonstrate chaotic behaviour. http://goo.gl/OyKZTh

`Correction: according to J. Rial van der Pol oscillators can demonstrate chaotic behaviour. http://goo.gl/OyKZTh`

John,

As far as the control theory perspective is concerned, what I'm most interested in is systems of coupled PDEs, where the goal is to understand how coupling one PDE to another changes the behavior of both. (Or rather, coupling one system of PDEs to another system of PDEs.) i.e., coupling together atmospheric circulation, ocean circulation, atmospheric chemistry, terrestrial biogeochemistry, etc. Or even understanding how different terms in a single PDE contribute to the global dynamics. I realize you're far from doing this, but it's ultimately where my interest would lie. (And also in the idea of model reduction: what are the "essential" dynamics of these PDEs, and can we replace them by low-order ODE or SDE systems?).

`John, As far as the control theory perspective is concerned, what I'm most interested in is systems of coupled PDEs, where the goal is to understand how coupling one PDE to another changes the behavior of both. (Or rather, coupling one system of PDEs to another system of PDEs.) i.e., coupling together atmospheric circulation, ocean circulation, atmospheric chemistry, terrestrial biogeochemistry, etc. Or even understanding how different terms in a single PDE contribute to the global dynamics. I realize you're far from doing this, but it's ultimately where my interest would lie. (And also in the idea of model reduction: what are the "essential" dynamics of these PDEs, and can we replace them by low-order ODE or SDE systems?).`

Hi Nathan, what you're mentioning seems interesting and, in the general case difficult I reckon. What about the equations of motion for Petri nets? All of your questions could maybe lead to a partial solution inside of this framework.

Also, what about the equations of motion for electrical networks stuff?

`Hi Nathan, what you're mentioning seems interesting and, in the general case difficult I reckon. What about the equations of motion for Petri nets? All of your questions could maybe lead to a partial solution inside of this framework. Also, what about the equations of motion for electrical networks stuff?`

I'm not sure if Petri nets are the best framework within which to study continuous dynamical systems, nor am I sure they're a useful tool to think about discrete numerical approximations to continuous systems, but perhaps there is something useful to be gained. I guess it would come down to identifying a "minimal useful example". It's hard for me to elicit these from numerical modelers because their models are so far from minimal.

I know modelers frequently encounter problems where they will tune model subcomponents using prescribed boundary conditions as stand-ins for other subcomponents ... but when they truly couple together the two components together, the whole system behavior changes. It would help to understand this better. And otherwise find ways to get a handle on model complexity by studying components in isolation, yet being able to reason about the fully coupled dynamics.

`I'm not sure if Petri nets are the best framework within which to study continuous dynamical systems, nor am I sure they're a useful tool to think about discrete numerical approximations to continuous systems, but perhaps there is something useful to be gained. I guess it would come down to identifying a "minimal useful example". It's hard for me to elicit these from numerical modelers because their models are so far from minimal. I know modelers frequently encounter problems where they will tune model subcomponents using prescribed boundary conditions as stand-ins for other subcomponents ... but when they truly couple together the two components together, the whole system behavior changes. It would help to understand this better. And otherwise find ways to get a handle on model complexity by studying components in isolation, yet being able to reason about the fully coupled dynamics.`

I should probably keep emphasizing that Petri nets are just one aspect of network theory. I've written the most about them not because I think they're the most important kind of network — just because they were the first I studied and they turned out to be pretty interesting!

The real fun will start when we figure out how various kinds of network theories are related. Many of these amount to ways to describe the interaction between systems modeled by multivariable ODE's. When we get to PDE's things get higher-dimensional and we may need to replace categories by n-categories; in highly theoretical physics this is almost "fashionable", but I'm not sure the rest of the world is ready for it (or needs it).

`I should probably keep emphasizing that Petri nets are just one aspect of network theory. I've written the most about them not because I think they're the most important kind of network — just because they were the first I studied and they turned out to be pretty interesting! The real fun will start when we figure out how various kinds of network theories are related. Many of these amount to ways to describe the interaction between systems modeled by multivariable ODE's. When we get to PDE's things get higher-dimensional and we may need to replace categories by n-categories; in highly theoretical physics this is almost "fashionable", but I'm not sure the rest of the world is ready for it (or needs it).`