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I’m very flattered to be invited to speak here. I was probably invited because of my abstract mathematical work on networks and category theory. But when I got the invitation, instead of talking about something I understood, I thought I’d learn about something a bit more practical and talk about that. That was a bad idea. But I’ll try to make the best of it.
I’ve been trying to learn climate science. There’s a subject called ‘complex networks’ where people do statistical analyses of large graphs like the worldwide web or Facebook and draw conclusions from it. People are trying to apply these ideas to climate science. So that’s what I’ll talk about. I’ll be reviewing a lot of other people’s work, but also describing some work by a project I’m involved in, the Azimuth Project.
The Azimuth Project is an all-volunteer project involving scientists and programmers, many outside academia, who are concerned about environmental issues and want to use their skills to help. This talk is based on the work of many people in the Azimuth Project, including Jan Galkowski, Graham Jones, Nadja Kutz, Daniel Mahler, Blake Pollard, Paul Pukite, Dara Shayda, David Tanzer, David Tweed, Steve Wenner and others. Needless to say, I’m to blame for all the mistakes.
Okay, let’s get started.
You’ve probably heard about the ‘global warming pause’. Is this a real thing? If so, is it due to ‘natural variability’, heat going into the deep oceans, some combination of both, a massive failure of our understanding of climate processes, or something else?
Here is chart of global average air temperatures at sea level, put together by NASA’s Goddard Institute of Space Science:
You can see a lot of fluctuations, including a big dip after 1940 and a tiny dip after 2000. That tiny dip is the so-called ‘global warming pause’. What causes these fluctuations? That’s a big, complicated question.
One cause of temperature fluctuations is a kind of cycle whose extremes are called El Niño and La Niña.
A lot of things happen during an El Niño. For example, in 1997 and 1998, a big El Niño, we saw all these events:
El Niño is part of an irregular cycle that happens every 3 to 7 years, called the El Niño Southern Oscillation or ENSO. Two strongly correlated signs of an El Niño are:
1) Increased sea surface temperatures in a patch of the Pacific called the Niño 3.4 region. The temperature anomaly in this region—how much warmer it is than usual for that time of year—is called the Niño 3.4 index.
2) A decrease in air pressures in the western side of the Pacific compared to those further east. This is measured by the Southern Oscillation Index or SOI.
You can see the correlation here:
El Niños are important because they can cause billions of dollars of economic damage. They also seem to bring heat stored in the deeper waters of the Pacific into the atmosphere. So, one reason for the ‘global warming pause’ may be that we haven’t had a strong El Niño since 1998. The global warming pause might end with the next El Niño. For a while it seemed we were due for a big one this fall, but that hasn’t happened.
The ENSO cycle is just one of many cycles involving teleconnections: strong correlations between weather at distant locations, typically thousands of kilometers. People have systematically looked for these teleconnections using principal component analysis of climate data, and also other techniques.
The ENSO cycle shows up automatically when you do this kind of study. It stands out as the biggest source of climate variability on time scales greater than a year and less than a decade. Some others include:
• The Pacific-North America Oscillation.
• The Pacific Decadal Oscillation.
• The North Atlantic Oscillation.
• The Arctic Oscillation.
For example, the Pacific Decadal Oscillation is a longer-period relative of the ENSO, centered in the north Pacific:
Recently people have begun to study teleconnections using ideas from ‘complex network theory’.
What’s that? In complex network theory, people often start with a weighted graph: that is, a set of nodes and for any pair of nodes a weight which can be any nonnegative real number.
Why is this called a weighted graph? It’s really just a matrix of nonnegative real numbers!
The reason is that we can turn any weighted graph into a graph by drawing an edge from node to node whenever This is a directed graph, meaning that we should draw an arrow pointing from to We could have an edge from to but not vice versa! Note that we can also have an edge from a node to itself.
Conversely, if we have any directed graph, we can turn it into a weighted graph by choosing the weight when there’s an edge from to and otherwise.
For example, we can make a weighted graph where the nodes are web pages and is the number of links from the web page to the web page
People in complex network theory like examples of this sort: large weighted graphs that describe connections between web pages, or people, or cities, or neurons, or other things. The goal, so far, is to compute numbers from weighted graphs in ways that describe interesting properties of these complex networks—and then formulate and test hypotheses about the complex networks we see in real life.
Here’s a very simple example of what we can do with a weighted graph. For any node we can sum up the weights of edges going into
This is called the degree of the node For example, if lots of people have web pages with lots of links to yours, your webpage will have a high degree. If lots of people like you on Facebook, you will have a high degree.
So, the degree is some measure of how ‘important’ a node is.
People have constructed climate networks where the nodes are locations on the Earth’s surface, and the weight measures how correlated the weather is at the th and th location. Then, the degree says how ‘important’ a given location is for the Earth’s climate—in some vague sense.
For example, in Complex networks in climate dynamics, Donges et al take surface air temperature data on a grid and compute the correlation between grid points.
More precisely, let be the temperature at the th grid point at month after the average for that month in all years under consideration has been subtracted off, to eliminate some seasonal variations. They compute the Pearson correlation of and for each pair of grid points The Pearson correlation is the simplest measure of linear correlation, normalized to range between -1 and 1.
We could construct a weighted graph this way, and it would be symmetric, or undirected:
However, Donges et al prefer to work with a graph rather than a weighted graph. So, they create a graph where there is an edge from to (and also from to ) when exceeds a certain threshold, and no edge otherwise.
They can adjust this threshold so that any desired fraction of pairs actually have an edge between them. After some experimentation they chose this fraction to be 0.5%.
A certain patch dominates the world! This is the El Niño basin. The Indian Ocean comes in second.
(Some details, which I may not say:
The Pearson correlation is the covariance
normalized by dividing by the standard deviation of and the standard deviation of
The reddest shade of red in the above picture shows nodes that are connected to 5% or more of the other nodes. These nodes are connected to at least 10 times as many nodes as average.)
The Pearson correlation detects linear correlations. A more flexible measure is mutual information: how many bits of information knowing the temperature at time at grid point tells you about the temperature at the same time at grid point
Donges et al create a climate network this way as well, putting an edge between nodes if their mutual information exceeds a certain cutoff. They choose this cutoff so that 0.5% of node pairs have an edge between them, and get the following map:
The result is almost indistinguishable in the El Niño basin. So, this feature is not just an artifact of focusing on linear correlations.
We can also look at how climate networks change with time—and in particular, how they are affected by El Niños. This is the subject of a 2008 paper by Tsonis and Swanson, Topology and predictability of El Niño and La Niña networks.
They create a climate network in a way that’s similar to the one I just described. The main differences are that they:
create a link between grid points when their Pearson correlation has absolute value greater than $0.5;$
only use temperature data from November to March in each year, claiming that summertime introduces spurious links.
They get this map for La Niña conditions:
and this map for El Niño conditions:
They conclude that “El Niño breaks climate links”.
This may seem to contradict what I just said a minute ago. But it doesn’t! While the El Niño basin is a region where the surface air temperatures are highly correlated to temperatures at many other points, when an El Niño actually occurs it disrupts correlations between temperatures at different locations worldwide—and even in the El Niño basin!
For the rest of the talk I want to focus on a third claim: namely, that El Niños can be predicted by means of an increase in correlations between temperatures within the El Niño basin and temperatures outside this region. This claim was made in a recent paper by Ludescher et al. I want to examine it somewhat critically.
People really want to predict El Niños, because they have huge effects on agriculture, especially around the Pacific ocean. However, it’s generally regarded as very hard to predict El Niños more than 6 months in advance. There is also a spring barrier: it’s harder to predict El Niños through the spring of any year.
It’s controversial how much of the unpredictability in the ENSO cycle is due to chaos intrinsic to the Pacific ocean system, and how much is due to noise from outside the system. Both may be involved.
There are many teams trying to predict El Niños, some using physical models of the Earth’s climate, and others using machine learning techniques. There is a kind of competition going on, which you can see at a National Oceanic and Atmospheric Administration website.
The most recent predictions give a sense of how hard this job is:
When the 3-month running average of the Niño 3.4 index exceeds 0.5°C for 5 months, we officially declare that there is an El Niño.
As you can see, it’s hard to be sure if there will be an El Niño early next year! However, the consensus forecast is yes, a weak El Niño. This is the best we can do, now. Right now multi-model ensembles have better predictive skill than any one model.
The Azimuth Project has carefully examined a 2013 paper by Ludescher et al called Very early warning of next El Niño, which uses a climate network for El Niño prediction.
They build their climate network using correlations between daily surface air temperature data between points inside the El Niño basin and certain points outside this region, as shown here:
The red dots are the points in their version of the El Niño basin.
(Next I will describe Ludescher’s procedure. I may omit some details in the actual talk, but let me include them here.)
The main idea of Ludescher et al is to construct a climate network that is a weighted graph, and to say an El Niño will occur if the average weight of edges between points in the El Niño basin and points outside this basin exceeds a certain threshold.
As in the other papers I mentioned, Ludescher et al let be the surface air temperature at the th grid point at time minus the average temperature at that location at that time of year in all years under consideration, to eliminate the most obvious seasonal effects.
They consider a time-delayed covariance between temperatures at different grid points:
where is a time delay, and the angle brackets denote a running average over the last year, that is:
where is the time in days.
They normalize this to define a correlation that ranges from -1 to 1.
Next, for any pair of nodes and and for each time they determine the maximum, the mean and the standard deviation of as the delay ranges from -200 to 200 days.
They define the link strength as the difference between the maximum and the mean value of divided by its standard deviation.
Finally, they let be the average link strength, calculated by averaging over all pairs where is a grid point inside their El Niño basin and is a grid point outside this basin, but still in their larger rectangle.
Here is what they get:
The blue peaks are El Niños: episodes where the Niño 3.4 index is over 0.5°C for at least 5 months.
The red line is their ‘average link strength’. Whenever this exceeds a certain threshold and the Niño 3.4 index is not already over 0.5°C, they predict an El Niño will start in the following calendar year.
Ludescher et al chose their threshold for El Niño prediction by training their algorithm on climate data from 1948 to 1980, and tested it on data from 1981 to 2013. They claim that with this threshold, their El Niño predictions were correct 76% of the time, and their predictions of no El Niño were correct in 86% of all cases.
On this basis they claimed—when their paper was published in February 2014—that the Niño 3.4 index would exceed 0.5 by the end of 2014 with probability 3/4.
The latest data as of 1 December 2014 seems to say: yes, it happened!
Graham Jones of the Azimuth Project wrote code implementing Ludescher et al’s algorithm, as best as we could understand it, and got results close to theirs, though not identical. The code is open-source; one goal of the Azimuth Project is to do science ‘in the open’.
More interesting than the small discrepancies between our calculation and theirs is the question of whether ‘average link strengths’ between points in the El Niño basin and points outside are really helpful in predicting El Niños.
Steve Wenner, a statistician helping the Azimuth Project, noted some ambiguities in Ludescher et al‘s El Niño prediction rules and disambiguated them in a number of ways. For each way he used Fischer’s exact test to compute the -value of the null hypothesis that Ludescher et al‘s El Niño prediction does not improve the odds that what they predict will occur.
The best he got (that is, the lowest -value) was 0.03. This is just a bit more significant than the conventional 0.05 threshold for rejecting a null hypothesis.
Do high average link strengths between points in the El Niño basin and points elsewhere in the Pacific really increase the chance that an El Niño is coming? It is hard to tell from the work of Ludescher et al.
One reason is that they treat El Niño as a binary condition, either on or off depending on whether the Niño 3.4 index for a given month exceeds 0.5 or not. This is not the usual definition of El Niño, but the real problem is that they are only making a single yes-or-no prediction each year for 65 years: does an El Niño occur during this year, or not? 31 of these years (1950-1980) are used for training their algorithm, leaving just 34 retrodictions and one actual prediction (1981-2013, and 2014).
So, there is a serious problem with small sample size.
We can learn a bit by taking a different approach, and simply running some linear regressions between the average link strength and the Niño 3.4 index for each month. There are 766 months from 1950 to 2013, so this gives us more data to look at. Of course, it’s possible that the relation between average link strength and Niño is highly nonlinear, so a linear regression may not be appropriate. But it is at least worth looking at!
Daniel Mahler and Dara Shayda of the Azimuth Project did this and found the following interesting results.
Here is a scatter plot showing the Niño 3.4 index as a function of the average link strength on the same month:
(Click on these scatter plots for more information.)
The coefficient of determination, is 0.0175. In simple terms, this means that the average link strength in a given month explains just 1.75% of the variance of the Niño 3.4 index. That’s quite low!
Here is a scatter plot showing the Niño 3.4 index as a function of the average link strength six months earlier:
Now is 0.088. So, the link strength explains 8.8% of the variance in the Niño 3.4 index 6 months later. This is still not much—but interestingly, it’s much more than when we try to relate them at the same moment in time! And the -value is less than so the effect is statistically significant.
Of course, we could also try to use Niño 3.4 to predict itself. Here is the Niño 3.4 index plotted against the Niño 3.4 index six months earlier:
Now So, this is better than using the average link strength!
That doesn’t sound good for average link strength. But now let’s could try to predict Niño 3.4 using both itself and the average link strength 6 months earlier. Here is a scatter plot showing that:
Here the axis is an optimally chosen linear combination of average and link strength and Niño 3.4: one that maximizes .
In this case we get
What can we conclude from this?
Using a linear model, the average link strength on a given month accounts for only 8% of the variance of Niño 3.4 index 6 months in the future. That sounds bad, and indeed it is.
However, there are more interesting things to say than this!
Both the Niño 3.4 index and the average link strength can be computed from the surface air temperature of the Pacific during some window in time. The Niño 3.4 index explains 16% of its own variance 6 months into the future; the average link strength explains 8%, and taken together they explain 22%. So, these two variables contain a fair amount of independent information about the Niño 3.4 index 6 months in the future.
Furthermore, they explain a surprisingly large amount of its variance for just 2 variables.
For comparison, Mahler used a random forest variant called ExtraTreesRegressor to predict the Niño 3.4 index 6 months into the future from much larger collections of data. Out of the 778 months available he trained the algorithm on the first 400 and tested it on the remaining 378.
The result: using a full world-wide grid of surface air temperature values at a given moment in time explains only 23% of the Niño 3.4 index 6 months into the future. A full grid of surface air pressure values does considerably better, but still explains only 34% of the variance. Using twelve months of the full grid of pressure values only gets around 37%.
From this viewpoint, explaining 22% of the variance with just two variables doesn’t look so bad!
Moreover, while the Niño 3.4 index is maximally correlated with itself at the same moment in time, for obvious reasons, the average link strength is maximally correlated with the Niño 3.4 index 10 months into the future:
(The lines here occur at monthly intervals.)
However, we have not tried to determine if the average link strength as Ludescher et al define it is optimal in this respect. Graham Jones has shown that simplifying their definition of this quantity doesn’t change it much. Maybe modifying their definition could improve it. There seems to be a real phenomenon at work here, but I don’t think we know exactly what it is!
My talk has avoided discussing physical models of the ENSO, because I wanted to focus on very simple, general ideas from complex network theory. However, it seems obvious that really understanding the ENSO requires a lot of ideas from meteorology, oceanography, physics, and the like. I am not advocating a ‘purely network-based approach’.
What comes to mind when you hear the term ‘random process’? Do you think of Brownian motion? Do you think of particles hopping around? Do you think of a drunkard staggering home?
Today I’m going to tell you about a version of the drunkard’s walk with a few modifications. Firstly, we don’t have just one drunkard: we can have any positive real number of drunkards. Secondly, our drunkards have no memory; where they go next doesn’t depend on where they’ve been. Thirdly, there are special places, such as entrances to bars, where drunkards magically appear and disappear.
The second condition says that our drunkards satisfy the Markov property, making their random walk into a Markov process. The third condition is really what I want to tell you about, because it makes our Markov process into a more general ‘open Markov process’.
There are a collection of places the drunkards can be, for example:
We call this set the set of states. There are certain probabilities associated with traveling between these places. We call these transition rates. For example it is more likely for a drunkard to go from the bar to the taco truck than to go from the bar to home so the transition rate between the bar and the taco truck should be greater than the transition rate from the bar to home. Sometimes you can’t get from one place to another without passing through intermediate places. In reality the drunkard can’t go directly from the bar to the taco truck: he or she has to go from the bar to sidewalk to the taco truck.
This information can all be summarized by drawing a directed graph where the positive numbers labelling the edges are the transition rates:
For simplicity we draw only three states: home, bar, taco truck. Drunkards go from home to the bar and back, but they never go straight from home to the taco truck.
We can keep track of where all of our drunkards are using a vector with 3 entries:
We call this our population distribution. The first entry is the number of drunkards that are at home, the second is how many are at the bar, and the third is how many are at the taco truck.
There is a set of coupled, linear, first-order differential equations we can write down using the information in our graph that tells us how the number of drunkards in each place change with time. This is called the master equation:
where is a 3×3 matrix which we call the Hamiltonian. The off-diagonal entries are nonnegative:
and the columns sum to zero:
We call a matrix satisfying these conditions infinitesimal stochastic. Stochastic matrices have columns that sum to one. If we take the exponential of an infinitesimal stochastic matrix we get one whose columns sum to one, hence the label ‘infinitesimal’.
The Hamiltonian for the graph above is
John has written a lot about Markov processes and infinitesimal stochastic Hamiltonians in previous posts.
Given two vectors describing the populations of drunkards which obey the same master equation, we can calculate the relative entropy of relative to :
This is an example of a ‘divergence’. In statistics, a divergence a way of measuring the distance between probability distributions, which may not be symmetrical and may even not obey the triangle inequality.
The relative entropy is important because it decreases monotonically with time, making it a Lyapunov function for Markov processes. Indeed, it is a well known fact that
This is true for any two population distributions which evolve according to the same master equation, though you have to allow infinity as a possible value for the relative entropy and negative infinity for its time derivative.
Why is entropy decreasing? Doesn’t the Second Law of Thermodynamics say entropy increases?
Don’t worry: the reason is that I have not put a minus sign in my definition of relative entropy. Put one in if you like, and then it will increase. Sometimes without the minus sign it’s called the Kullback–Leibler divergence. This decreases with the passage of time, saying that any two population distributions and get ‘closer together’ as they get randomized with the passage of time.
That itself is a nice result, but I want to tell you what happens when you allow drunkards to appear and disappear at certain states. Drunkards appear at the bar once they’ve had enough to drink and once they are home for long enough they can disappear. The set of places where drunkards can appear or disappear is called the set of boundary states. So for the above process
is the set of boundary states. This changes the way in which the population of drunkards changes with time!
The drunkards at the taco truck obey the master equation. For them,
still holds. But because the populations can appear or disappear at the boundary states the master equation no longer holds at those states! Instead it is useful to define the flow of drunkards into the state by
This quantity describes by how much the rate of change of the populations at the boundary states differ from that given by the master equation.
The reason why we are interested in open Markov processes is because you can take two open Markov processes and glue them together along some subset of their boundary states to get a new open Markov process! This allows us to build up or break down complicated Markov processes using open Markov processes as the building blocks.
For example we can draw the graph corresponding to the drunkards’ walk again, only now we will distinguish boundary states from internal states by coloring internal states blue and having boundary states be white:
Consider another open Markov process with states
where
are the boundary states, leaving
as an internal state:
Since the boundary states of this process overlap with the boundary states of the first process we can compose the two to form a new Markov process:
Notice the boundary states are now internal states. I hope any Markov process that could approximately model your behavior has more interesting nodes! There is a nice way to figure out the Hamiltonian of the composite from the Hamiltonians of the pieces, but we will leave that for another time.
We can ask ourselves, how does relative entropy change with time in open Markov processes? You can read my paper for the details, but here is the punchline:
This is a version of the Second Law of Thermodynamics for open Markov processes.
It is important to notice that the sum is only over the boundary states! This inequality tells us that relative entropy still decreases inside our process, but depending on the flow of populations through the boundary states the relative entropy of the whole process could either increase or decrease! This inequality will be important when we study how the relative entropy changes in different parts of a bigger more complicated process.
That is all for now, but I leave it as an exercise for you to imagine a Markov process that describes your life. How many states does it have? What are the relative transition rates? Are there states you would like to spend more or less time in? Are there states somewhere you would like to visit?
Here is my paper, which proves the above inequality:
• Blake Pollard, A Second Law for open Markov processes.
If you have comments or corrections, let me know!
Since I was in Banff, my student Franciscus Rebro took over this week and explained more about cospan categories. These are a tool for constructing categories where the morphisms are networks such as electrical circuit diagrams, signal flow diagrams, Markov processes and the like. For some more details see:
• John Baez and Brendan Fong, A compositional framework for passive linear networks.
Cospan categories are really best thought of as bicategories, and Franciscus gets into this aspect too.
and sends it to its behavior as viewed from outside. From outside, all you can see is the relation between currents and potentials at the ‘terminals’—the little bits of wire that poke out of the black box:
I came close to defining the black box functor, but I didn’t quite make it! This time let’s finish the job.
The black box functor
goes from the category where morphisms are circuits made of resistors, to the category where morphisms are linear relations. Let me remind you how these categories work, and introduce a bit of new notation.
Here is the category
• an object is a finite set;
• a morphism from to is an isomorphism class of cospans
in the category of graphs with edges labelled by resistances: numbers in Here we think of the finite sets and as graphs with no edges. We call the set of inputs and the set of outputs.
• we compose morphisms in by composing isomorphism classes of cospans.
And here is the category
• an object is a finite-dimensional real vector space;
• a morphism from to is a linear relation meaning a linear subspace
• we compose a linear relation and a linear relation in the usual way we compose relations, getting:
In case you’re wondering: I’ve just introduced the wiggly arrow notation
for a linear relation from to because it suggests that a relation is a bit like a function but more general. Indeed, a function is a special case of a relation, and composing functions is a special case of composing relations.
Now, how do we define the black box functor?
Defining it on objects is easy. An object of is a finite set and we define
The idea is that could be a set of inputs or outputs, and then
is a list of numbers: the potentials and currents at those inputs or outputs.
So, the interesting part is defining the black box functor on morphisms!
For this we start with a morphism in :
The labelled graph consists of:
• a set of nodes,
• a set of edges,
• maps sending each edge to its source and target,
• a map sending each edge to its resistance.
The cospan gives maps
These say how the inputs and outputs are interpreted as nodes in the circuit. We’ll call the nodes that come from inputs or outputs ‘terminals’. So, mathematically,
is the set of terminals: the union of the images of and
In the simplest case, the maps and are one-to-one, with disjoint ranges. Then each terminal either comes from a single input, or a single output, but not both! This is a good picture to keep in mind. But some subtleties arise when we leave this simplest case and consider other cases.
Now, the black box functor is supposed to send our circuit to a linear relation. I’ll call the circuit for short, though it’s really the whole cospan
So, our black box functor is supposed to send this circuit to a linear relation
This is a relation between the potentials and currents at the input terminals and the potentials and currents at the output terminals! How is it defined?
I’ll start by outlining how this works.
First, our circuit picks out a subspace
This is the subspace of allowed potentials and currents on the terminals. I’ll explain this and why it’s called a bit later. Briefly, it comes from the principle of minimum power, described last time.
Then, the map
gives a linear relation
This says how the potentials and currents at the inputs are related to those at the terminals. Similarly, the map
gives a linear relation
This says how the potentials and currents at the outputs are related to those at the terminals.
Next, we can ‘turn around’ any linear relation
to get a relation
defined by
Here we are just switching the input and output potentials, but when we switch the currents we also throw in a minus sign. The reason is that we care about the current flowing in to an input, but out of an output.
Finally, one more trick: given a linear subspace
of a vector space we get a linear relation
called the identity restricted to , defined like this:
If is all of this relation is actually the identity function on Otherwise it’s a partially defined function that’s defined only on and is the identity there. (A partially defined function is an example of a relation.) My notation is probably bad, but I don’t know a better one, so bear with me.
Let’s use all these ideas to define
To do this, we compose three linear relations:
1) We start with
2) We compose this with
3) Then we compose this with
Note that:
1) says how the potentials and currents at the inputs are related to those at the terminals,
2) picks out which potentials and currents at the terminals are actually allowed, and
3) says how the potentials and currents at the terminals are related to those at the outputs.
So, I hope all makes sense, at least in some rough way. In brief, here’s the formula:
Now I just need to fill in some details. First, how do we define and They work exactly the same way, by ‘copying potentials and adding currents’, so I’ll just talk about one. Second, how do we define the subspace This uses the principle of minimum power.
Any function between finite sets
gives a linear map
Mathematicians call this linear map the pullback along and for any it’s defined by
In our application, we think of as a list of potentials at terminals. The function could map a bunch of inputs to the same terminal, and the above formula says the potential at this terminal gives the potential at all those inputs. So, we are copying potentials.
We also get a linear map going the other way:
Mathematicians call this the pushforward along and for any it’s defined by
In our application, we think of as a list of currents entering at some inputs. The function could map a bunch of inputs to the same terminal, and the above formula says the current at this terminal is the sum of the currents at all those inputs. So, we are adding currents.
Putting these together, our map
gives a linear relation
where the pair is related to the pair iff
and
So, here’s the rule of thumb when attaching the points of to the input terminals of our circuit: copy potentials, but add up currents. More formally:
Finally, how does our circuit define a subspace
of allowed potential-current pairs at the terminals? The trick is to use the ideas we discussed last time. If we know the potential at all nodes of our circuit, say , we know the power used by the circuit:
We saw last time that if we fix the potentials at the terminals, the circuit will choose potentials at the other nodes to minimize this power. We can describe the potential at the terminals by
So, the power for a given potential at the terminals is
Actually this is half the power: I stuck in a factor of 1/2 for some reason we’ll soon see. This is a quadratic function
so its derivative is linear. And, our work last time showed something interesting: to compute the current flowing into a terminal we just differentiate with respect to the potential at that terminal:
This is the reason for the 1/2: when we take the derivative of we bring down a 2 from differentiating all those squares, and to make that go away we need a 1/2.
The space of allowed potential-current pairs at the terminals is thus the linear subspace
And this completes our precise description of the black box functor!
The hard part is this:
Theorem. is a functor.
In other words, we have to prove that it preserves composition:
For that, read our paper:
• John Baez and Brendan Fong, A compositional framework for passive linear networks.
• Naftali Tishby, Sensing and acting under information constraints—a principled approach to biology and intelligence, 28 October 2014.
Tishby’s talk wasn’t easy for me to follow—he assumed you already knew rate-distortion theory and the Bellman equation, and I didn’t—but it was great!
It was about the ‘action-perception loop':
This is the feedback loop in which living organisms—like us—take actions depending on our goals and what we perceive, and perceive things depending on the actions we take and the state of the world.
How do we do this so well? Among other things, we need to balance the cost of storing information about the past against the payoff of achieving our desired goals in the future.
Tishby presented a detailed yet highly general mathematical model of this! And he ended by testing the model on experiments with cats listening to music and rats swimming to land.
It’s beautiful stuff. I want to learn it. I hope to blog about it as I understand more. But for now, let me just dive in and say some basic stuff. I’ll start with the two buzzwords I dropped on you. I hate it when people use terminology without ever explaining it.
Rate-distortion theory is a branch of information theory which seeks to find the minimum rate at which bits must be communicated over a noisy channel so that the signal can be approximately reconstructed at the other end without exceeding a given distortion. Shannon’s first big result in this theory, the ‘rate-distortion theorem’, gives a formula for this minimum rate. Needless to say, it still requires a lot of extra work to determine and achieve this minimum rate in practice.
For the basic definitions and a statement of the theorem, try this:
• Natasha Devroye, Rate-distortion theory, course notes, University of Chicago, Illinois, Fall 2009.
One of the people organizing this conference is a big expert on rate-distortion theory, and he wrote a book about it.
• Toby Berger, Rate Distortion Theory: A Mathematical Basis for Data Compression, Prentice–Hall, 1971.
Unfortunately it’s out of print and selling for $259 used on Amazon! An easier option might be this:
• Thomas M. Cover and Joy A. Thomas, Elements of Information Theory, Chapter 10: Rate Distortion Theory, Wiley, New York, 2006.
The Bellman equation reduces the task of finding an optimal course of action to choosing what to do at each step. So, you’re trying to maximize the ‘total reward’—the sum of rewards at each time step—and Bellman’s equation says what to do at each time step.
If you’ve studied physics, this should remind you of how starting from the principle of least action, we can get a differential equation describing the motion of a particle: the Euler–Lagrange equation.
And in fact they’re deeply related. The relation is obscured by two little things. First, Bellman’s equation is usually formulated in a context where time passes in discrete steps, while the Euler–Lagrange equation is usually formulated in continuous time. Second, Bellman’s equation is really the discrete-time version not of the Euler–Lagrange equation but a more or less equivalent thing: the ‘Hamilton–Jacobi equation’.
Ah, another buzzword to demystify! I was scared of the Hamilton–Jacobi equation for years, until I taught a course on classical mechanics that covered it. Now I think it’s the greatest thing in the world!
Briefly: the Hamilton–Jacobi equation concerns the least possible action to get from a fixed starting point to a point in space at time If we call this least possible action the Hamilton–Jacobi equation says
where is the particle’s momentum at the endpoint of its path, and is its energy there.
If we replace derivatives by differences, and talk about maximizing total reward instead of minimizing action, we get Bellman’s equation:
• Bellman equation, Wikipedia.
Bellman’s equation can be useful whenever you’re trying to figure out an optimal course of action. An important example is a ‘Markov decision process’. To prepare you for Tishby’s talk, I should say what this is.
In a Markov process, something randomly hops around from state to state with fixed probabilities. In the simplest case there’s a finite set of states, and time proceeds in discrete steps. At each time step, the probability to hop from state to state is some fixed number
This sort of thing is called a Markov chain, or if you feel the need to be more insistent, a discrete-time Markov chain.
A Markov decision process is a generalization where an outside agent gets to change these probabilities! The agent gets to choose actions from some set If at a given time he chooses the action the probability of the system hopping from state to state is Needless to say, these probabilities have to sum to one for any fixed
That would already be interesting, but the real fun is that there’s also a reward This is a real number saying how much joy or misery the agent experiences if he does action and the system hops from to
The problem is to choose a policy—a function from states to actions—that maximizes the total expected reward over some period of time. This is precisely the kind of thing Bellman’s equation is good for!
If you’re an economist you might also want to ‘discount’ future rewards, saying that a reward time steps in the future gets multiplied by where is some discount factor. This extra tweak is easily handled, and you can see it all here:
• Markov decision process, Wikipedia.
There’s a further generalization where the agent can’t see all the details of the system! Instead, when he takes an action and the system hops from state to state he sees something: a point in some set of observations. He makes the observation with probability
(I don’t know why this probability depends on but not Maybe it ultimately doesn’t matter much.)
Again, the goal is to choose a policy that maximizes the expected total reward. But a policy is a bit different now. The action at any time can only depend on all the observations made thus far.
Partially observable Markov decision processes are also called POMPDs. If you want to learn about them, try these:
• Partially observable Markov decision process, Wikipedia.
• Tony Cassandra, Partially observable Markov decision processes.
The latter includes an introduction without any formulas to POMDPs and how to choose optimal policies. I’m not sure who would study this subject and not want to see formulas, but it’s certainly a good exercise to explain things using just words—and it makes certain things easier to understand (though not others, in a way that depends on who is trying to learn the stuff).
I already explained the action-perception loop, with the help of this picture from the University of Bielefeld’s Department of Cognitive Neuroscience:
Nafthali Tishby has a nice picture of it that’s more abstract:
We’re assuming time comes in discrete steps, just to keep things simple.
At each time
• the world has some state and
• the agent has some state
Why the letter ? This stands for memory: it can be the state of the agent’s memory, but I prefer to think of it as the state of the agent.
At each time
• the agent takes an action which affects the world’s next state, and
• the world provides a sensation to the agent, which affect’s the agent’s next state.
This is simplified but very nice. Note that there’s a symmetry interchanging the world and the agent!
We could make it fancier by having lots of agents who all interact, but there are a lot of questions already. The big question Tishby focuses on is optimizing how much the agent should remember about the past if they
• get a reward depending on the action taken and the resulting state of the world
but
• pay a price for the information stored from sensations.
Tishby formulates this optimization question as something like a partially observed Markov decision process, uses rate-distortion theory to analyze how much information needs to be stored to achieve a given reward, and uses Bellman’s equation to solve the optimization problem!
So, everything I sketched fits together somehow!
I hope what I’m saying now is roughly right: it will take me more time to get the details straight. If you’re having trouble absorbing all the information I just threw at you, don’t feel bad: so am I. But the math feels really natural and good to me. It involves a lot of my favorite ideas (like generalizations of the principle of least action, and relative entropy), and it seems ripe to be combined with network theory ideas.
For details, I highly recommend this paper:
• Naftali Tishby and Daniel Polani, Information theory of decisions and actions, in Perception-Reason-Action Cycle: Models, Algorithms and System. Vassilis, Hussain and Taylor, Springer, Berlin, 2010.
I’m going to print this out, put it by my bed, and read it every night until I’ve absorbed it.
I’m giving a short 30-minute talk at a workshop on Biological and Bio-Inspired Information Theory at the Banff International Research Institute.
I’ll say more about the workshop later, but here’s my talk, in PDF and video form:
• Biodiversity, entropy and thermodynamics.
Most of the people at this workshop study neurobiology and cell signalling, not evolutionary game theory or biodiversity. So, the talk is just a quick intro to some things we’ve seen before here. Starting from scratch, I derive the Lotka–Volterra equation describing how the distribution of organisms of different species changes with time. Then I use it to prove a version of the Second Law of Thermodynamics.
This law says that if there is a ‘dominant distribution’—a distribution of species whose mean fitness is at least as great as that of any population it finds itself amidst—then as time passes, the information any population has ‘left to learn’ always decreases!
Of course reality is more complicated, but this result is a good start.
This was proved by Siavash Shahshahani in 1979. For more, see:
• Lou Jost, Entropy and diversity.
• Marc Harper, The replicator equation as an inference dynamic.
• Marc Harper, Information geometry and evolutionary game theory.
and more recent papers by Harper.
This time we use the principle of minimum power to determine what a circuit made of resistors actually does. Its ‘behavior’ is described by a functor sending circuits to linear relations between the potentials and currents at the input and output terminals. We call this the ‘black box’ functor, since it takes a circuit:
and puts a metaphorical ‘black box’ around it:
hiding the circuit’s internal details and letting us see only how it acts as viewed ‘from outside’.
For more, see the lecture notes here:
and puts a ‘black box’ around it:
forgetting the internal details of the circuit and remembering only how the it behaves as viewed from outside. As viewed from outside, all the circuit does is define a relation between the potentials and currents at the inputs and outputs. We call this relation the circuit’s behavior. Lots of different choices of the resistances would give the same behavior. In fact, we could even replace the whole fancy circuit by a single edge with a single resistor on it, and get a circuit with the same behavior!
The idea is that when we use a circuit to do something, all we care about is its behavior: what it does as viewed from outside, not what it’s made of.
Furthermore, we’d like the behavior of a system made of parts to depend in a simple way on the external behaviors of its parts. We don’t want to have to ‘peek inside’ the parts to figure out what the whole will do! Of course, in some situations we do need to peek inside the parts to see what the whole will do. But in this particular case we don’t—at least in the idealization we are considering. And this fact is described mathematically by saying that black boxing is a functor.
So, how do circuits made of resistors behave? To answer this we first need to remember what they are!
Remember that for us, a circuit made of resistors is a mathematical structure like this:
It’s a cospan where:
• is a graph labelled by resistances. So, it consists of a finite set of nodes, a finite set of edges, two functions
sending each edge to its source and target nodes, and a function
that labels each edge with its resistance.
• is a map of graphs labelled by resistances, where has no edges. A labelled graph with no edges has nothing but nodes! So, the map is just a trick for specifying a finite set of nodes called inputs and mapping them to Thus picks out some nodes of and declares them to be inputs. (However, may not be one-to-one! We’ll take advantage of that subtlety later.)
• is another map of graphs labelled by resistances, where again has no edges, and we call its nodes outputs.
So what does a circuit made of resistors do? This is described by the principle of minimum power.
Recall from Part 27 that when we put it to work, our circuit has a current flowing along each edge This is described by a function
It also has a voltage across each edge. The word ‘across’ is standard here, but don’t worry about it too much; what matters is that we have another function
describing the voltage across each edge
Resistors heat up when current flows through them, so they eat up electrical power and turn this power into heat. How much? The power is given by
So far, so good. But what does it mean to minimize power?
To understand this, we need to manipulate the formula for power using the laws of electrical circuits described in Part 27. First, Ohm’s law says that for linear resistors, the current is proportional to the voltage. More precisely, for each edge
where is the resistance of that edge. So, the bigger the resistance, the less current flows: that makes sense. Using Ohm’s law we get
Now we see that power is always nonnegative! Now it makes more sense to minimize it. Of course we could minimize it simply by setting all the voltages equal to zero. That would work, but that would be boring: it gives a circuit with no current flowing through it. The fun starts when we minimize power subject to some constraints.
For this we need to remember another law of electrical circuits: a spinoff of Kirchhoff’s voltage law. This says that we can find a function called the potential
such that
for each In other words, the voltage across each edge is the difference of potentials at the two ends of this edge.
Using this, we can rewrite the power as
Now we’re really ready to minimize power! Our circuit made of resistors has certain nodes called terminals:
These are the nodes that are either inputs or outputs. More precisely, they’re the nodes in the image of
or
The principle of minimum power says that:
If we fix the potential on all terminals, the potential at other nodes will minimize the power
subject to this constraint.
This should remind you of all the other minimum or maximum principles you know, like the principle of least action, or the way a system in thermodynamic equilibrium maximizes its entropy. All these principles—or at least, most of them—are connected. I could talk about this endlessly. But not now!
Now let’s just use the principle of minimum power. Let’s see what it tells us about the behavior of an electrical circuit.
Let’s imagine changing the potential by adding some multiple of a function
If this other function vanishes at the terminals:
then doesn’t change at the terminals as we change the number
Now suppose obeys the principle of minimum power. In other words, supposes it minimizes power subject to the constraint of taking the values it does at the terminals. Then we must have
whenever
This is just the first derivative test for a minimum. But the converse is true, too! The reason is that our power function is a sum of nonnegative quadratic terms. Its graph will look like a paraboloid. So, the power has no points where its derivative vanishes except minima, even when we constrain by making it lie on a linear subspace.
We can go ahead and start working out the derivative:
To work out the derivative of these quadratic terms at we only need to keep the part that’s proportional to The rest gives zero. So:
The principle of minimum power says this is zero whenever is a function that vanishes at terminals. By linearity, it’s enough to consider functions that are zero at every node except one node that is not a terminal. By linearity we can also assume
Given this, the only nonzero terms in the sum
will be those involving edges whose source or target is We get
So, the principle of minimum power says precisely
for all nodes that aren’t terminals.
What does this mean? You could just say it’s a set of linear equations that must be obeyed by the potential So, the principle of minimum power says that fixing the potential at terminals, the potential at other nodes must be chosen in a way that obeys a set of linear equations.
But what do these equations mean? They have a nice meaning. Remember, Kirchhoff’s voltage law says
and Ohm’s law says
Putting these together,
so the principle of minimum power merely says that
for any node that is not a terminal.
This is Kirchhoff’s current law: for any node except a terminal, the total current flowing into that node must equal the total current flowing out! That makes a lot of sense. We allow current to flow in or out of our circuit at terminals, but ‘inside’ the circuit charge is conserved, so if current flows into some other node, an equal amount has to flow out.
In short: the principle of minimum power implies Kirchoff’s current law! Conversely, we can run the whole argument backward and derive the principle of minimum power from Kirchhoff’s current law. (In both the forwards and backwards versions of this argument, we use Kirchhoff’s voltage law and Ohm’s law.)
When the node is a terminal, the quantity
need not be zero. But it has an important meaning: it’s the amount of current flowing into that terminal!
We’ll call this the current at the terminal This is something we can measure even when our circuit has a black box around it:
So is the potential at the terminal It’s these currents and potentials at terminals that matter when we try to describe the behavior of a circuit while ignoring its inner workings.
Now let me quickly sketch how black boxing becomes a functor.
A circuit made of resistors gives a linear relation between the potentials and currents at terminals. A relation is something that can hold or fail to hold. A ‘linear’ relation is one defined using linear equations.
A bit more precisely, suppose we choose potentials and currents at the terminals:
Then we seek potentials and currents at all the nodes and edges of our circuit:
that are compatible with our choice of and Here compatible means that
and
whenever but also
for every and
whenever (The last two equations combine Kirchoff’s laws and Ohm’s law.)
There either exist and making all these equations true, in which case we say our potentials and currents at the terminals obey the relation… or they don’t exist, in which case we say the potentials and currents at the terminals don’t obey the relation.
The relation is clearly linear, since it’s defined by a bunch of linear equations. With a little work, we can make it into a linear relation between potentials and currents in
and potentials and currents in
Remember, is our set of inputs and is our set of outputs.
In fact, this process of getting a linear relation from a circuit made of resistors defines a functor:
Here is the category where morphisms are circuits made of resistors, while is the category where morphisms are linear relations.
More precisely, here is the category
• an object of is a finite set;
• a morphism from to is an isomorphism class of circuits made of resistors:
having as its set of inputs and as its set of outputs;
• we compose morphisms in by composing isomorphism classes of cospans.
(Remember, circuits made of resistors are cospans. This lets us talk about isomorphisms between them. If you forget the how isomorphism between cospans work, you can review it in Part 31.)
And here is the category
• an object of is a finite-dimensional real vector space;
• a morphism from to is a linear relation meaning a linear subspace of the vector space
• we compose a linear relation and a linear relation in the usual way we compose relations, getting:
So far I’ve set up most of the necessary background but not precisely defined the black boxing functor
There are some nuances I’ve glossed over, like the difference between inputs and outputs as elements of and and their images in If you want to see the precise definition and the proof that it’s a functor, read our paper:
• John Baez and Brendan Fong, A compositional framework for passive linear networks.
The proof is fairly long: there may be a much quicker one, but at least this one has the virtue of introducing a lot of nice ideas that will be useful elsewhere.
Next time I’ll define the black box functor more carefully.
This time I explain more about how ‘cospans’ represent gadgets with two ends, an input end and an output end:
I describe how to glue such gadgets together by composing cospans. We compose cospans using a category-theoretic construction called a ‘pushout’, so I also explain pushouts. At the end, I explain how this gives us a category where the morphisms are electrical circuits made of resistors, and sketch what we’ll do next: study the behavior of these circuits.
These lecture notes provide extra details:
• Y. Berezin, Avi Gozolchiani, O. Guez and Shlomo Havlin, Stability of climate networks with time, Scientific Reports 2 (2012).
Some of the authors are the same, and the way they define climate networks is very similar. But their goal here is different: they want to see see how stable climate networks are over time. This is important, since the other paper wants to predict El Niños by changes in climate networks.
They divide the world into 9 zones:
For each zone they construct several climate networks. Each one is an array of numbers , one for each year and each pair of grid points in that zone. They call a link strength: it’s a measure of how how correlated the weather is at those two grid points during that year.
I’ll say more later about how they compute these link strengths. In Part 3 we explained one method for doing it. This paper uses a similar but subtly different method.
The paper’s first big claim is that doesn’t change much from year to year, “in complete contrast” to the pattern of local daily air temperature and pressure fluctuations. In simple terms: the strength of the correlation between weather at two different points tends to be quite stable.
Moreover, the definition of link strength involves an adjustable time delay, . We can measure the correlation between the weather at point at any given time and point at a time days later. The link strength is computed by taking a maximum over time delays . Naively speaking, the value of that gives the maximum correlation is “how long it typically takes for weather at point to affect weather at point ”. Or the other way around, if is negative.
This is a naive way of explaining the idea, because I’m mixing up correlation with causation. But you get the idea, I hope.
Their second big claim is that when the link strength between two points and is big, the value of that gives the maximum correlation doesn’t change much from year to year. In simple terms: if the weather at two locations is strongly correlated, the amount of time it takes for weather at one point to reach the other point doesn’t change very much.
How do Berezin et al define their climate network?
They use data obtained from here:
This is not exactly the same data set that Ludescher et al use, namely:
“Reanalysis 2″ is a newer attempt to reanalyze and fix up the same pile of data. That’s a very interesting issue, but never mind that now!
Berezin et al use data for:
• the geopotential height for six different pressures
and
• the air temperature at those different heights
The geopotential height for some pressure says roughly how high you have to go for air to have that pressure. Click the link if you want a more precise definition! Here’s the geopotential height field for the pressure of 500 millibars on some particular day of some particular year:
The height is in meters.
Berezin et al use daily values for this data for:
• locations world-wide on a grid with a resolution of 5° × 5°,
during:
• the years from 1948 to 2006.
They divide the globe into 9 zones, and separately study each zone:
So, they’ve got twelve different functions of space and time, where space is a rectangle discretized using a 5° × 5° grid, and time is discretized in days. From each such function they build a ‘climate network’.
How do they do it?
Berezin’s method of defining a climate network is similar to Ludescher et al‘s, but different. Compare Part 3 if you want to think about this.
Let be any one of their functions, evaluated at the grid point on day of year .
Let be minus its climatological average. For example, if is June 1st and is 1970, we average the temperature at location over all June 1sts from 1948 to 2006, and subtract that from to get . In other words:
where is the number of years considered.
For any function of time , let be the average of the function over all days in year . This is different than the ‘running average’ used by Ludescher et al, and I can’t even be 100% sure that Berezin mean what I just said: they use the notation .
Let and be two grid points, and any number of days in the interval . Define the cross-covariance function at time by:
I believe Berezin mean to consider this quantity, because they mention two grid points and . Their notation omits the subscripts and so it is impossible to be completely sure what they mean! But what I wrote is the reasonable quantity to consider here, so I’ll assume this is what they meant.
They normalize this quantity and take its absolute value, forming:
They then take the maximum value of over delays , subtract its mean over delays in this range, and divide by the standard deviation. They write something like this:
and say that the maximum, mean and standard deviation are taken over the (not written) variable .
Each number is called a link strength. For each year, the matrix of numbers where and range over all grid points in our zone is called a climate network.
We can think of a climate network as a weighted complete graph with the grid points as nodes. Remember, an undirected graph is one without arrows on the edges. A complete graph is an undirected graph with one edge between any pair of nodes:
A weighted graph is an undirected graph where each edge is labelled by a number called its weight. But right now we’re also calling the weight the ‘link strength’.
A lot of what’s usually called ‘network theory’ is the study of weighted graphs. You can learn about it here:
• Ernesto Estrada, The Structure of Complex Networks: Theory and Applications, Oxford U. Press, Oxford, 2011.
Suffice it to say that given a weighted graph, there are lot of quantities you can compute from it, which are believed to tell us interesting things!
I will not delve into the real meat of the paper, namely what they actually do with their climate networks! The paper is free online, so you can read this yourself.
I will just quote their conclusions and show you a couple of graphs.
The conclusions touch on an issue that’s important for the network-based approach to El Niño prediction. If climate networks are ‘stable’, not changing much in time, why would we use them to predict a time-dependent phenomenon like the El Niño Southern Oscillation?
We have established the stability of the network of connections between the dynamics of climate variables (e.g. temperatures and geopotential heights) in different geographical regions. This stability stands in fierce contrast to the observed instability of the original climatological field pattern. Thus the coupling between different regions is, to a large extent, constant and predictable. The links in the climate network seem to encapsulate information that is missed in analysis of the original field.
The strength of the physical connection, , that each link in this network represents, changes only between 5% to 30% over time. A clear boundary between links that represent real physical dependence and links that emerge due to noise is shown to exist. The distinction is based on both the high link average strength and on the low variability of time delays .
Recent studies indicate that the strength of the links in the climate network changes during the El Niño Southern Oscillation and the North Atlantic Oscillation cycles. These changes are within the standard deviation of the strength of the links found here. Indeed in Fig. 3 it is clearly seen that the coefficient of variation of links in the El Niño basin (zone 9) is larger than other regions such as zone 1. Note that even in the El Niño basin the coefficient of variation is relatively small (less than 30%).
Beside the stability of single links, also the hierarchy of the link strengths in the climate network is preserved to a large extent. We have shown that this hierarchy is partially due to the two dimensional space in which the network is embedded, and partially due to pure physical coupling processes. Moreover the contribution of each of these effects, and the level of noise was explicitly estimated. The spatial effect is typically around 50% of the observed stability, and the noise reduces the stability value by typically 5%–10%.
The network structure was further shown to be consistent across different altitudes, and a monotonic relation between the altitude distance and the correspondence between the network structures is shown to exist. This yields another indication that the observed network structure represents effects of physical coupling.
The stability of the network and the contributions of different effects were summarized in specific relation to different geographical areas, and a clear distinction between equatorial and off–equatorial areas was observed. Generally, the network structure of equatorial regions is less stable and more fluctuative.
The stability and consistence of the network structure during time and across different altitudes stands in contrast to the known unstable variability of the daily anomalies of climate variables. This contrast indicates an analogy between the behavior of nodes in the climate network and the behavior of coupled chaotic oscillators. While the fluctuations of each coupled oscillators are highly erratic and unpredictable, the interactions between the oscillators is stable and can be predicted. The possible outreach of such an analogy lies in the search for known behavior patterns of coupled chaotic oscillators in the climate system. For example, existence of phase slips in coupled chaotic oscillators is one of the fingerprints for their cooperated behavior, which is evident in each of the individual oscillators. Some abrupt changes in climate variables, for example, might be related to phase slips, and can be understood better in this context.
On the basis of our measured coefficient of variation of single links (around 15%), and the significant overall network stability of 20–40%, one may speculatively assess the extent of climate change. However, for this assessment our current available data is too short and does not include enough time from periods before the temperature trends. An assessment of the relation between the network stability and climate change might be possible mainly through launching of global climate model “experiments” realizing other climate conditions, which we indeed intend to perform.
A further future outreach of our work can be a mapping between network features (such as network motifs) and known physical processes. Such a mapping was previously shown to exist between an autonomous cluster in the climate network and El Niño. Further structures without such a climate interpretation might point towards physical coupling processes which were not observed earlier.
(I have expanded some acronyms and deleted some reference numbers.)
Finally, here two nice graphs showing the average link strength as a function of distance. The first is based on four climate networks for Zone 1, the southern half of South America:
The second is based on four climate networks for Zone 9, a big patch of the Pacific north of the Equator which roughly corresponds to the ‘El Niño basin':
As we expect, temperatures and geopotential heights get less correlated at points further away. But the rate at which the correlation drops off conveys interesting information! Graham Jones has made some interesting charts of this for the rectangle of the Pacific that Ludescher et al use for El Niño prediction, and I’ll show you those next time.
• El Niño project (part 1): basic introduction to El Niño and our project here.
• El Niño project (part 2): introduction to the physics of El Niño.
• El Niño project (part 3): summary of the work of Ludescher et al.
• El Niño project (part 4): how Graham Jones replicated the work by Ludescher et al, using software written in R.
• El Niño project (part 5): how to download R and use it to get files of climate data.
• El Niño project (part 6): Steve Wenner’s statistical analysis of the work of Ludescher et al.
• El Niño project (part 7): the definition of El Niño.
• El Niño project (part 8): Berezin et al on the stability of climate networks.