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Here we are analyzing:

- K. Yamasaki, A. Gozolchiani, and S. Havlin, Climate networks around the globe are significantly effected by El Niño, June 2013.

EDIT: I removed some stuff, and renamed this thread.

## Comments

This is great so far. I have lots of criticisms of the work done so far, and I'm struggling to figure out how to do things better. But it's good to spend time understanding what's been done!

`This is great so far. I have lots of criticisms of the work done so far, and I'm struggling to figure out how to do things better. But it's good to spend time understanding what's been done!`

In particular, K. Yamasaki, A. Gozolchiani and S. Havlin seem to think El Niños break "climate links", but I'm afraid this is due to a misleading definition of those climate links.

`In particular, K. Yamasaki, A. Gozolchiani and S. Havlin seem to think El Niños break "climate links", but I'm afraid this is due to a misleading definition of those climate links.`

Namely, it seems that climate links in the Pacific Ocean are getting stronger and,

by virtue of a definition that says only the strongest links count as links, they claim links elsewhere in the world are getting broken.`Namely, it seems that climate links in the Pacific Ocean are getting stronger and, _by virtue of a definition that says only the strongest links count as links_, they claim links elsewhere in the world are getting broken.`

John wrote:

I guess you'll expand on that shortly.

I'm having difficulties with quite a few of their formulations but they are non-native english speakers (/affect/effect/)so maybe it's that rather than my ignorance.

`John wrote: > In particular, K. Yamasaki, A. Gozolchiani and S. Havlin seem to think El Niños break “climate links”, but I’m afraid this is due to a misleading definition of those climate links. I guess you'll expand on that shortly. I'm having difficulties with quite a few of their formulations but they are non-native english speakers (/affect/effect/)so maybe it's that rather than my ignorance.`

There are various things to say, Jim, but mainly:

1) It probably makes more sense to focus on this paper:

Proceedings of the National Academy of Sciences, 30 May 2013.It seems to have a simpler and more reasonable methodology than this earlier one:

2) I have the strong feeling that people in this subject - "climate network theory" - are just messing around trying to see what works. The definitions of various quantities are a bit

ad hoc, not justified by deep reasoning. This is good because it means we can mess around a bit ourselves, and try to do something that works better. But it means we can't just learn the stuff by reading papers and trusting them.`There are various things to say, Jim, but mainly: 1) It probably makes more sense to focus on this paper: * Josef Ludescher, Avi Gozolchiani, Mikhail I. Bogachev, Armin Bunde, Shlomo Havlin, and Hans Joachim Schellnhuber, [Improved El Niño forecasting by cooperativity detection](http://www.pnas.org/content/early/2013/06/26/1309353110.full.pdf+html), _Proceedings of the National Academy of Sciences_, 30 May 2013. It seems to have a simpler and more reasonable methodology than this earlier one: * K. Yamasaki, A. Gozolchiani, and S. Havlin, [Climate networks around the globe are significantly effected by El Niño](http://arxiv.org/abs/0804.1374), April 2008. 2) I have the strong feeling that people in this subject - "climate network theory" - are just messing around trying to see what works. The definitions of various quantities are a bit _ad hoc_, not justified by deep reasoning. This is good because it means we can mess around a bit ourselves, and try to do something that works better. But it means we can't just learn the stuff by reading papers and trusting them.`

John wrote:

Do you think that it is an interesting question to look at -- the effect of El Niños on the connectivity structure of the graph -- using a more appropriate definition of the climate links?

I had trouble parsing the definitions in this paper, which at times seemed ill-defined. But maybe they are basically well defined, and the trouble is that I don't have the context to supply the missing words. For instance, they didn't define their angle brackets, and why is the year a parameter of the cross-correlation function.

John, leaving aside the fact that you think the definitions were not appropriate, were you able to parse them, in detail, and did you find them to be fundamentally well-defined?

Footnote: This theoretical discussion is not idle, because it can help to guide our experimental designs.

EDIT: I was having trouble parsing the paper because my hard-copy was mangling some of the symbols!

`John wrote: > In particular, K. Yamasaki, A. Gozolchiani and S. Havlin seem to think El Niños break “climate links”, but I’m afraid this is due to a misleading definition of those climate links. Do you think that it is an interesting question to look at -- the effect of El Niños on the connectivity structure of the graph -- using a more appropriate definition of the climate links? I had trouble parsing the definitions in this paper, which at times seemed ill-defined. But maybe they are basically well defined, and the trouble is that I don't have the context to supply the missing words. For instance, they didn't define their angle brackets, and why is the year a parameter of the cross-correlation function. John, leaving aside the fact that you think the definitions were not appropriate, were you able to parse them, in detail, and did you find them to be fundamentally well-defined? Footnote: This theoretical discussion is not idle, because it can help to guide our experimental designs. EDIT: I was having trouble parsing the paper because my hard-copy was mangling some of the symbols!`

They seemed well-defined to me.

They are using abbreviations and shorthands the way physicists and many other scientists do. For example, when they have a function of several variables, they don't bother to write the variables that they are uninterested in at that moment. The idea is that you're supposed to figure out what the notation means by thinking about what would be the only reasonable thing for it to mean. The idea is that writing things out in a completely unambiguous way makes for big ugly expressions which they'd rather avoid.

Since this seems to drive computer programmers crazy, I will modify their notation a bit.

Angle brackets are a standard symbol for an arithmetic mean, or average. So is an overline or a capital letter $E$.

So, here is what they are doing. They start with a bunch of temperatures $\tilde{T}(\ell,y,d)$: one for each grid point $\ell$, year $y$ ($1 \le y \le N$) and day of the year $d$ ($1 \le d \le 365$). Then they consider

$$ T(\ell,r,y,d) := \tilde{T}(\ell,y,d) - \frac{1}{N} \sum_{y = 1}^N \tilde{T}(\ell,y,d) $$ This tells you how much "hotter it is today than it usually is here at this time of year".

Then, for two grid points $\ell$ and $r$ and a year $y$ they work out a kind of correlation between $\tilde{T}(\ell,y,d)$ and $\tilde{T}(\ell,y,d+\tau)$ where $\tau = 1,2,3,\dots$) :

$$ X(\ell,r,y,\tau) = \frac{1}{365} \sum_{d = 1}^{365} T(\ell,y,d) T(r,y,d+\tau) $$ I assume that when $d+\tau$ exceeds 365 you go over to the next year, since that's the reasonable thing to do.

The point is that $X(\ell,y, \tau)$ tells you how much the temperature at grid point $\ell$ is correlated to the temperature at grid point $r$, $\tau$ days later, during year $y$.

Does this make sense so far?

So far this is what anyone would be tempted to do when studying how a randomly fluctuating quantity

here and nowis correlated to the same quantitysomewhere else and a bit later. The fun part starts later.`> John, leaving aside the fact that you think the definitions were not appropriate, were you able to parse them, in detail, and did you find them to be fundamentally well-defined? They seemed well-defined to me. They are using abbreviations and shorthands the way physicists and many other scientists do. For example, when they have a function of several variables, they don't bother to write the variables that they are uninterested in at that moment. The idea is that you're supposed to figure out what the notation means by thinking about what would be the only reasonable thing for it to mean. The idea is that writing things out in a completely unambiguous way makes for big ugly expressions which they'd rather avoid. Since this seems to drive computer programmers crazy, I will modify their notation a bit. > For instance, they didn’t define their angle brackets... Angle brackets are a standard symbol for an arithmetic mean, or average. So is an overline or a capital letter $E$. So, here is what they are doing. They start with a bunch of temperatures $\tilde{T}(\ell,y,d)$: one for each grid point $\ell$, year $y$ ($1 \le y \le N$) and day of the year $d$ ($1 \le d \le 365$). Then they consider $$ T(\ell,r,y,d) := \tilde{T}(\ell,y,d) - \frac{1}{N} \sum_{y = 1}^N \tilde{T}(\ell,y,d) $$ This tells you how much "hotter it is today than it usually is here at this time of year". Then, for two grid points $\ell$ and $r$ and a year $y$ they work out a kind of correlation between $\tilde{T}(\ell,y,d)$ and $\tilde{T}(\ell,y,d+\tau)$ where $\tau = 1,2,3,\dots$) : $$ X(\ell,r,y,\tau) = \frac{1}{365} \sum_{d = 1}^{365} T(\ell,y,d) T(r,y,d+\tau) $$ I assume that when $d+\tau$ exceeds 365 you go over to the next year, since that's the reasonable thing to do. The point is that $X(\ell,y, \tau)$ tells you how much the temperature at grid point $\ell$ is correlated to the temperature at grid point $r$, $\tau$ days later, during year $y$. Does this make sense so far? So far this is what anyone would be tempted to do when studying how a randomly fluctuating quantity _here and now_ is correlated to the same quantity _somewhere else and a bit later_. The fun part starts later.`

Yes, very clear. Can we keep proceeding with this discussion, at this level of clarity -- which is more explicit and less contextual. By the end of the discussion we will have a clear recipe for writing programs.

`Yes, very clear. Can we keep proceeding with this discussion, at this level of clarity -- which is more explicit and less contextual. By the end of the discussion we will have a clear recipe for writing programs.`

Then they give a formula for negative $\tau$:

$$ X(\ell,r,y,-\tau) = X(r,\ell,y,\tau) $$

`Then they give a formula for negative $\tau$: $$ X(\ell,r,y,-\tau) = X(r,\ell,y,\tau) $$`

Proceeding with their development...

Next, define:

$$ Max(\ell,r,y) = Max { |X(r,\ell,y,\tau)| : : \tau_{min} \leq \tau \leq \tau_{max} } $$ $$ Std(\ell,r,y) = Std { |X(r,\ell,y,\tau)| : : \tau_{min} \leq \tau \leq \tau_{max} } $$ $$ W(\ell,r,y) = Max(\ell,r,y) \, / \, Std(\ell,r,y) $$ $ W(\ell,r,y) $ is called the

correlation strengthof the link between $\ell$ and $r$ during year $y$.The value of $\tau$ at which $ Max(\ell,r,y) $ is achieved is the

time delaybetween points $\ell$ and $r$ during year $y$.`Proceeding with their development... Next, define: $$ Max(\ell,r,y) = Max \{ |X(r,\ell,y,\tau)| : \: \tau_{min} \leq \tau \leq \tau_{max} \} $$ $$ Std(\ell,r,y) = Std \{ |X(r,\ell,y,\tau)| : \: \tau_{min} \leq \tau \leq \tau_{max} \} $$ $$ W(\ell,r,y) = Max(\ell,r,y) \, / \, Std(\ell,r,y) $$ $ W(\ell,r,y) $ is called the _correlation strength_ of the link between $\ell$ and $r$ during year $y$. The value of $\tau$ at which $ Max(\ell,r,y) $ is achieved is the _time delay_ between points $\ell$ and $r$ during year $y$.`

EDIT: moved the analysis of Ludescher et. al to another thread.

`EDIT: moved the analysis of Ludescher et. al to another thread.`

EDIT: moved the analysis of Ludescher et. al to another thread.

`EDIT: moved the analysis of Ludescher et. al to another thread.`

Thanks for catching those typos, David. Almost everything you say looks fine to me!

But this formula

still has a serious typo in it, due to me - there should not be tildes on those $T$s. Here's the right formula:

$$ X(\ell,r,y,\tau) = \frac{1}{365} \sum_{d = 1}^{365} T(\ell,y,d) T(r,y,d+\tau) $$ The point is that it would be silly to define a correlation between temperatures $\tilde{T}$ using the top formula; we need to subtract off their means and use $T$s instead: we don't want the correlation to be big just because it's hot all the time!

I have taken the liberty of rewriting history and fixing my original comment, taking into account your typo fixes and also this.

In general people write $\langle X \rangle $ for the mean of any random variable $X$, so the standard deviation of $X$, say $Std(X)$ in your notation, is

$$ Std(X)^2 = \langle (X - \langle X \rangle)(X - \langle X \rangle) \rangle $$ Given two random variables $X$ and $Y$ we compute their correlation by first working out

$$ \langle (X - \langle X \rangle)(Y - \langle Y \rangle) \rangle $$ and then normalizing it by dividing by $ Std(X) Std(Y)$: this makes the result lie between $-1$ and $1$.

I hope you see that

$$ X(\ell,r,y,\tau) = \frac{1}{365} \sum_{d = 1}^{365} T(\ell,y,d) T(r,y,d+\tau) $$ is almost an example of this

$$ \langle (X - \langle X \rangle)(Y - \langle Y \rangle) \rangle $$ concept: we've taken $ \tilde{T}(\ell,y,d)$ and $\tilde{T}(r,y,d+\tau) $, subtracted off their mean (average over all years), multiplied them, and then taken the mean of that (but now averaging over days of a given year). In the paper it looks like they use $\langle \rangle_d$ and $\langle \rangle_y$ to mean averaging over days of a given year and averaging over years.

More to say, but it's my bedtime!

`Thanks for catching those typos, David. Almost everything you say looks fine to me! But this formula > $$ X(\ell,r,y,\tau) = \frac{1}{365} \sum_{d = 1}^{365} \tilde{T}(\ell,y,d) \tilde{T}(r,y,d+\tau) $$ still has a serious typo in it, due to me - there should not be tildes on those $T$s. Here's the right formula: $$ X(\ell,r,y,\tau) = \frac{1}{365} \sum_{d = 1}^{365} T(\ell,y,d) T(r,y,d+\tau) $$ The point is that it would be silly to define a correlation between temperatures $\tilde{T}$ using the top formula; we need to subtract off their means and use $T$s instead: we don't want the correlation to be big just because it's hot all the time! I have taken the liberty of rewriting history and fixing my [original comment](http://forum.azimuthproject.org/discussion/1357/climate-networks/?Focus=10714#Comment_10714), taking into account your typo fixes and also this. In general people write $\langle X \rangle $ for the mean of any random variable $X$, so the standard deviation of $X$, say $Std(X)$ in your notation, is $$ Std(X)^2 = \langle (X - \langle X \rangle)(X - \langle X \rangle) \rangle $$ Given two random variables $X$ and $Y$ we compute their correlation by first working out $$ \langle (X - \langle X \rangle)(Y - \langle Y \rangle) \rangle $$ and then normalizing it by dividing by $ Std(X) Std(Y)$: this makes the result lie between $-1$ and $1$. I hope you see that $$ X(\ell,r,y,\tau) = \frac{1}{365} \sum_{d = 1}^{365} T(\ell,y,d) T(r,y,d+\tau) $$ is almost an example of this $$ \langle (X - \langle X \rangle)(Y - \langle Y \rangle) \rangle $$ concept: we've taken $ \tilde{T}(\ell,y,d)$ and $\tilde{T}(r,y,d+\tau) $, subtracted off their mean (average over all years), multiplied them, and then taken the mean of that (but now averaging over days of a given year). In the paper it looks like they use $\langle \rangle_d$ and $\langle \rangle_y$ to mean averaging over days of a given year and averaging over years. More to say, but it's my bedtime!`

John, there is one consistent set of typos in your formulas for $ X(\ell,r,y,\tau) $ -- you've left out the second parameter $r$. Can you edit them, so we can make this thread look nice.

`John, there is one consistent set of typos in your formulas for $ X(\ell,r,y,\tau) $ -- you've left out the second parameter $r$. Can you edit them, so we can make this thread look nice.`

Next, they state the idea of a physical threshold $Q$ so that only pairs of nodes with a link strength greater than $Q$ will be regarded as significantly linked.

To this end, they define a new matrix, that takes on the value 1 whenever two nodes are significantly linked, otherwise it is 0:

$ \rho(\ell,r,y) = \Theta(W(\ell,r,y) - Q) $

where $\Theta$ is the Heaviside function, mapping negative values to 0, positives values to 1 (and 0 to 0.5).

`Next, they state the idea of a physical threshold $Q$ so that only pairs of nodes with a link strength greater than $Q$ will be regarded as significantly linked. To this end, they define a new matrix, that takes on the value 1 whenever two nodes are significantly linked, otherwise it is 0: $ \rho(\ell,r,y) = \Theta(W(\ell,r,y) - Q) $ where $\Theta$ is the Heaviside function, mapping negative values to 0, positives values to 1 (and 0 to 0.5).`

Then they observe that some of the elements of $\rho$ may "blink" as a function of $y$, i.e., and disappear.

They state that:

and they state the goal of distinguishing them from the more robust links.

`Then they observe that some of the elements of $\rho$ may "blink" as a function of $y$, i.e., and disappear. They state that: > Blinking links seem to be a signature of structural changes and they state the goal of distinguishing them from the more robust links.`

Then they introduce a new matrix $M$ which counts the number of times a link appeared before continuously (without a blink):

$$ M(\ell,r,y) = \sum_{n = 0}^{y-1} \prod_{m = y-n}^{y} \rho(\ell,r,m) $$ I.e. it counts the length of the maximal run of 1's, going backwards from year $y$.

`Then they introduce a new matrix $M$ which counts the number of times a link appeared before continuously (without a blink): $$ M(\ell,r,y) = \sum_{n = 0}^{y-1} \prod_{m = y-n}^{y} \rho(\ell,r,m) $$ I.e. it counts the length of the maximal run of 1's, going backwards from year $y$.`

Let $k$ be a parameter.

Define a link to be

k-robustif $M(\ell,r,y) \ge k$.`Let $k$ be a parameter. Define a link to be _k-robust_ if $M(\ell,r,y) \ge k$.`

Let $n_k(y)$ be the count of the number of k-robust links in year $y$:

$$ n_k(y) = \sum_{\ell = 0}^{N} \sum_{r=l+1}^{N} \Theta(M(\ell,r,y) - k + 1) $$

`Let $n_k(y)$ be the count of the number of k-robust links in year $y$: $$ n_k(y) = \sum_{\ell = 0}^{N} \sum_{r=l+1}^{N} \Theta(M(\ell,r,y) - k + 1) $$`

Here are the parameters chosen by the authors:

$k = 5$

$Q = 2$

`Here are the parameters chosen by the authors: * $k = 5$ * $Q = 2$`

Then they say that:

Does this mean that they're using an arbitrary notion of what a year means, and that their "years" are not synched up with the calendar? I find this confusing, because their graphs refer to real, empirical years.

`Then they say that: > ...we choose the $y$ resolution (the jumps between two subsequent dates represented by $y$) to be 50 days Does this mean that they're using an arbitrary notion of what a year means, and that their "years" are not synched up with the calendar? I find this confusing, because their graphs refer to real, empirical years.`

Then they choose four representative zones around the globe.

For each zone they examine the evolution of $n_k(y)$ over time, and compare it with standard indices for El Niño -- the difference in sea level pressure between Tahiti and Darwin, and mean sea surface temperature in the standard basin.

`Then they choose four representative zones around the globe. For each zone they examine the evolution of $n_k(y)$ over time, and compare it with standard indices for El Niño -- the difference in sea level pressure between Tahiti and Darwin, and mean sea surface temperature in the standard basin.`

In general, their graphs show that $n_k(y)$ drops during times of El Niño, which leads them to their main conclusion that:

`In general, their graphs show that $n_k(y)$ drops during times of El Niño, which leads them to their main conclusion that: > ...the number of surviving links comprises a specific and sensitive measure for El Niño events.`

They also claim that these results:

`They also claim that these results: > ...are not sensitive to the choice of $k$. However, choosing too large $k$ values reduces the number of surviving links significantly, and therefore eliminates much of the effet. Choosing too small values of $k$, on the other hand, does not enable the elimination of blinking links, and therefore causes $n_k(y)$ to be more noisy, but the significant effect of breaking links is still evident.`

Here is their justification for choosing the threshold $Q = 2$:

`Here is their justification for choosing the threshold $Q = 2$: > When observing the probability density function of $W$ it is clear that for non-El Niño time regimes, W = 2 is actually the minimal value that exists. It therefore appears that choosing this threshold makes the network very sensitive to El Niño events while remaining insensitive to other changes in climate. The reason is that the distribution of $W(\ell,r,y)$ tends to typical lower values of $W(\ell,r,y)$ during El Niño, as can clearly be seen in Fig. 3 (A-E). A remarkable property of this softening is that the lower limit of the distribution drops from being close to 2 to some significantly lower value. Changes in climate around the world due to El Niño events thus share a unified property of the correlation pattern, which can be tracked in a reliable way by the number of surviving links $n_k(y)$ in the climate network.`

This article is rife with software-experimentally testable statements!

Go, ye R progammers, go.

`This article is rife with software-experimentally testable statements! Go, ye R progammers, go.`

Now let's explore John's criticism:

They are claiming that links are getting broken during El Niños, so do you mean to say that the links in the Pacific Ocean get stronger during El Niños?

`Now let's explore John's criticism: > Namely, it seems that climate links in the Pacific Ocean are getting stronger and, _by virtue of a definition that says only the strongest links count as links_, they claim links elsewhere in the world are getting broken. They are claiming that links are getting broken during El Niños, so do you mean to say that the links in the Pacific Ocean get stronger during El Niños?`

Ye R programmers, can you verify that climate links in the Pacific Ocean are getting stronger?

`Ye R programmers, can you verify that climate links in the Pacific Ocean are getting stronger?`

John, now that we have a terminology, can you explain your statement that with their definitions only the strongest links count as links.

I don't see any normalization of the link strengths taking place.

`John, now that we have a terminology, can you explain your statement that with their definitions only the strongest links count as links. I don't see any normalization of the link strengths taking place.`

The correlation strengths are always compared to a fixed number $Q$. These strengths are defined as a ratio of a Max value divided by a Std value, where the Max and Std is taken over a set of numbers that is per $\ell,r$ pair.

`The correlation strengths are always compared to a fixed number $Q$. These strengths are defined as a ratio of a Max value divided by a Std value, where the Max and Std is taken over a set of numbers that is per $\ell,r$ pair.`

In any case, all of these claims provide great exercises for us to start working with the data and developing software to analyze it.

`In any case, all of these claims provide great exercises for us to start working with the data and developing software to analyze it.`

I can't see any indication of this in Fig 2 of

Proceedings of the National Academy of Sciences, 30 May 2013.If anything, since about 2000, they've been weaker than typical.

`> Ye R programmers, can you verify that climate links in the Pacific Ocean are getting stronger? I can't see any indication of this in Fig 2 of * Josef Ludescher, Avi Gozolchiani, Mikhail I. Bogachev, Armin Bunde, Shlomo Havlin, and Hans Joachim Schellnhuber, [Improved El Niño forecasting by cooperativity detection](http://www.pnas.org/content/early/2013/06/26/1309353110.full.pdf+html), _Proceedings of the National Academy of Sciences_, 30 May 2013. If anything, since about 2000, they've been weaker than typical. <img src = "http://math.ucr.edu/home/baez/ecological/ludescher_el_nino_cooperativity_2.jpg" alt = ""/>`

In my revised version of message 28, I asked John:

Graham, do you see any any indication of this in the data?

`In my revised version of message 28, I asked John: > They are claiming that links are getting broken during El Niños, so do you mean to say that the links in the Pacific Ocean get stronger during El Niños? Graham, do you see any any indication of this in the data?`

Yes, again from Fig 2. (It will be some time before I could answer using my own analysis!) The red line (= signal strength ~=count of of links) do seem to go down during most El Niños.

`Yes, again from Fig 2. (It will be some time before I could answer using my own analysis!) The red line (= signal strength ~=count of of links) do seem to go down during most El Niños. <img src = "http://math.ucr.edu/home/baez/ecological/ludescher_el_nino_cooperativity_2.jpg" alt = ""/>`

Hi Graham, yes I see that, which is the evidence that they are showing for their claim.

Sorry my question didn't make clear what I was really asking about, which is the second part of the sentence: is there evidence that the links in the Pacific Ocean get stronger (1) over time in general, and (2) during El Niños? I believe this would take our own analysis to answer, so we may have to build towards addressing such questions.

`Hi Graham, yes I see that, which is the evidence that they are showing for their claim. Sorry my question didn't make clear what I was really asking about, which is the second part of the sentence: is there evidence that the links in the Pacific Ocean get stronger (1) over time in general, and (2) during El Niños? I believe this would take our own analysis to answer, so we may have to build towards addressing such questions.`

One more point to note about their methodology.

It is true that they talk about discarding the weakest links, but this is just a matter of discarding the "blinking links," i.e., only retaining links that have been over the threshold $Q$ for at least 5 consecutive years going back from the present. But I don't see how this set of surviving, robust links would get reduced by increasing of other link strengths in the Pacific. If anything, I would predict that an increasing of link strengths in the Pacific would just increase the number of robust links.

`One more point to note about their methodology. It is true that they talk about discarding the weakest links, but this is just a matter of discarding the "blinking links," i.e., only retaining links that have been over the threshold $Q$ for at least 5 consecutive years going back from the present. But I don't see how this set of surviving, robust links would get reduced by increasing of other link strengths in the Pacific. If anything, I would predict that an increasing of link strengths in the Pacific would just increase the number of robust links.`

David wrote:

I've attempted to fix this everywhere; I've deleted your comment about this, and I will delete this one if seems all my $ X(\ell,y,\tau) $s have been successfully corrected to $ X(\ell,r,y,\tau) $'s. Thanks!

`David wrote: > John, there is one consistent set of typos in your formulas for $ X(\ell,r,y,\tau) $ -- you've left out the second parameter $r$. Can you edit them, so we can make this thread look nice? I've attempted to fix this everywhere; I've deleted your comment about this, and I will delete this one if seems all my $ X(\ell,y,\tau) $s have been successfully corrected to $ X(\ell,r,y,\tau) $'s. Thanks!`

David wrote:

Hmm, I had thought their cutoff $Q$ varied with time, so that the presence of strong links increased $Q$ and made weaker links no longer count as links! This seems to have been a hallucination on my part. So, I withdraw this criticism. Thanks!

Part of my confusion is this: it seems that Yamasaki, Gozolchiani and Havlin seem to think El Niños "break climate links worldwide”, while Ludescher et al predict El Niños by detecting "increased cooperativity in the El Niño basin"

beforethe El Niños.These aren't contradictory, and of course one or both might be false.

But still, I want to get some better intuitive picture of what the authors of both papers think is going on, and see if there's a consistent reasonable-sounding story behind both accounts. Right now I don't have that.

`David wrote: > John, now that we have a terminology, can you explain your statement that with their definitions only the strongest links count as links? I don’t see any normalization of the link strengths taking place. Hmm, I had thought their cutoff $Q$ varied with time, so that the presence of strong links increased $Q$ and made weaker links no longer count as links! This seems to have been a hallucination on my part. So, I withdraw this criticism. Thanks! Part of my confusion is this: it seems that Yamasaki, Gozolchiani and Havlin seem to think El Niños "break climate links worldwide”, while [Ludescher et al](http://forum.azimuthproject.org/discussion/1360/paper-ludescher-et-al-improved-el-nino-forecasting-by-cooperativity-detection/) predict El Niños by detecting "increased cooperativity in the El Niño basin" _before_ the El Niños. These aren't contradictory, and of course one or both might be false. But still, I want to get some better intuitive picture of what the authors of both papers think is going on, and see if there's a consistent reasonable-sounding story behind both accounts. Right now I don't have that.`

By the way, I've added an image of Figure 2 from Ludescher's paper to Graham's comments where he referred to that.

`By the way, I've added an image of Figure 2 from Ludescher's paper to Graham's comments where he referred to that.`