Comment on

_Energy Non-Conservation in Quantum Mechanics_ ,

_arXiv:2101.11052v2 [quant-ph]_

Johann Summhammer111Email<EMAIL_ADDRESS>
Technische Universität Wien

Institute of Atomic and Subatomic Physics

Stadionallee 2, 1020 Vienna, Austria

###### Abstract

In the posting arXiv:2101.11052v2 [quant-ph] an experimental protocol
involving two spins is proposed, which should show violation of energy
conservation in a quantum experiment. In the present comment an unjustified
mathematical approximation leading to that conclusion is pointed out. A
detailed analysis will restore energy conservation.

The posting cited in the title [1], and also published in [2], suggests an
experiment with two spins, with the purpose of showing non-conservation of
energy in quantum mechanics. I will refer to [1] and [2] as ENC and will argue
that there is an error in the evaluation of the Hamiltonian. When doing a
correct analysis, energy will be conserved for all possible outcomes of the
experiment. I will also mention a conceptually similar experiment, for which
the Hamiltonian and the solutions have been worked out in detail. The layout
of the experiment suggested in ENC is shown in figure 1.

Figure 1: Particle 1 is trapped in the magnetic field $\bf{B}$, while particle
2 is flying by at a distance $b$. Both particles have spin 1/2 and become
entangled by the spin-spin interaction. Particle 2 does not enter the region
of the magnetic field.

The core of the proposal of ENC is to let two spin-1/2 particles interact by
their magnetic moments. Particle 1 is trapped in a magnetic field parallel to
the z-axis, and is in a superposition of its two spin eigenstates in the
field, which are also eigenstates of energy. Particle 2 passes near particle
1, but never enters the magnetic field region. It is also in a superposition
of the two spin eigenstates relative to the z-axis. While particle 2 is flying
by, the two particles will become entangled in spin. Once particle 2 is
sufficiently far from the interaction region, a measurement of its spin is
performed. The result will be either ”up” or ”down”. Because of the
entanglement particle 1 must then be in exactly one of its two possible
eigenstates in the magnetic field. But these two states of particle 1 have
very different energy, depending on the magnitude of B. In ENC it is argued,
that the back-reaction on particle 2 during the interaction is negligible from
the energetic point of view and can be ignored. Therefore, the total energy of
the system would be different, depending on the measurement result, and this
would constitute a violation of energy conservation.

Let us look at the problem in more detail. The Hamiltonian given in ENC (eq.16
in [1]) is

$\hat{H}=-\gamma B_{z}S_{1}^{z}\otimes
11_{2}+\frac{g}{r^{3}}\left[\vec{S}_{1}\cdot\vec{S}_{2}-3(\vec{S}_{1}\cdot\vec{r})(\vec{S}_{2}\cdot\vec{r})\right].$
(1)

This Hamiltonian has no explicit time dependence, therefore it commutes with
the energy operator $-i\hbar\partial/\partial t$, and total energy will be
conserved, no matter which particular initial wavefunction for the combined
system of the two spins is assumed. This is commented on in ENC, but seems to
be understood as valid only for expectation values rather than all individual
outcomes of a measurement. On the other hand, the Hamiltonian does depend on
the position $\vec{r}$ of particle 2, and therefore it does not commute with
the momentum operator for particle 2, which involves the derivative with
respect to $\vec{r}$. As a consequence, particle 2 may be found, in general,
in different momentum states, once it is far away from the interaction region.
This is observed correctly in ENC, but there it is argued, that the difference
in kinetic energy of these states can be made negligible relative to the
energy difference which the two possible eigenstates of particle 1 will have
in the magnetic field B, because the magnetic field can be made arbitrarily
large. Hence, it is argued in ENC, if one measures the spin of particle 2, one
will be able to infer the spin state of particle 1, and since these states
will be associated with very different energies, while the different energies
of the states of particle 2 are negligible, energy must have appeared or
disappeared.

Clearly, in view of the static Hamiltonian this conclusion must be wrong.
Aside from this formal argument, one can also give a more physical one relying
on classical intuition. Imagine a classical particle 1 suspended in the
magnetic field, and for the sake of simplicity, let its magnetic moment be
anti parallel to the magnetic field, so that it has minimum potential energy.
Now along comes classical particle 2 with its magnetic moment pointing in some
other direction. When it passes near particle 1, the two particles will exert
a mutual torque on each other, and this will generally change the final
direction of their respective magnetic moments. Classically, there can be no
doubt, that the amount, by which particle 2 will be able to change the
direction of the magnetic moment of particle 1, _will depend on the magnitude
of the magnetic field_. If B is very strong, particle 1 will be so rigidly
bound in its state, that particle 2 will have a hard time to cause any change
at all. And if it does manage to influence particle 1, any change of potential
energy of particle 1 in the magnetic field will be compensated by a change of
the final linear momentum of particle 2. These considerations will also fit
together nicely with conservation of total angular momentum, because any
change of linear momentum of particle 2 implies a change of its orbital
angular momentum with respect to the position of particle 1. Based on this
classical intuition we expect something similar to happen in the quantum
mechanical case.

The mistake of ENC must consist in neglecting the back reaction of field-
trapped particle 1 on the final momentum states of particle 2. But it is not
easily recognizable how the magnetic field should become manifest in the final
states of particle 2, when this particle remains outside the field region all
the time. At least, it is not immediately obvious from the Hamiltonian of
eq.1, where $\vec{S}_{2}$ only interacts with $\vec{S}_{1}$. I did not find
papers which have solved the problem of ENC, but I did come across a paper by
P. L. Nash, which analyzes a very similar problem [3]. In particular, that
paper makes visible how the potential, to which one particle is exposed,
becomes reflected in another particle, which interacts with it, but which is
not exposed to the potential itself. The scheme is sketched in figure 2.

Figure 2: The conceptual scheme considered in [3]. A particle with mass $m$ is
scattered at another particle with mass $M$, which resides in a harmonic
oscillator potential.

A particle with mass $M$ is trapped in a harmonic oscillator potential
characterized by the angular frequency $\Omega$. Another particle with mass
$m$ flies toward the trapped particle and is scattered. The problem is
analyzed in one dimension. Here, the free particle has index 1, and the
particle oscillating in the trap has index 2. The Hamiltonian given by Nash
reads (eq.1 in [3])

$H=\frac{1}{2M}p_{2}^{2}+\frac{1}{2}M\Omega^{2}x_{2}^{2}+\frac{1}{2m}p_{1}^{2}+V(x_{1}-x_{2}).$
(2)

The interaction potential between the particles is assumed symmetric,
$V(x_{1}-x_{2})=V(x_{2}-x_{1})$. It is noteworthy, that it only depends on the
distance between the two particles, but does not depend on the parameter
characterizing the harmonic trap, $\Omega$. This is exactly as in ENC. A
further similarity with ENC is the time independence of the Hamiltonian. In
the analysis Nash makes use of the fact that the solutions of the quantum
mechanical harmonic oscillator are essentially the Hermite polynomials times
an exponential prefactor [4]. Therefore it may be convenient to introduce a
dimensionless position coordinate $X$ for particle 2 defined by

$X=x_{2}\sqrt{\frac{M\Omega}{\hbar}},$ (3)

and expand the potential between the two particles in Hermite polynomials

$V\left(x_{1}-X\sqrt{\frac{\hbar}{M\Omega}}\right)=\sum_{n}H_{n}(X)V_{n}(x_{1}).$
(4)

The $V_{n}(x_{1}$) are expansion coefficients depending on the coordinate of
particle 1 and have to be calculated. Here one can already see how the angular
frequency $\Omega$ of the harmonic oscillator enters into the potential to
which particle 1 will actually be subjected when interacting with particle 2.
This is quite surprising, because the potential between the two particles is
only assumed as some function of the distance between them. And at the face of
it, the angular frequency $\Omega$ of the harmonic oscillator, which only
concerns particle 2, should not play a role for particle 1.

Nash then proceeds to calculate the expansion coefficients $V_{n}(x_{1})$ and
from them derives the scattering matrix. From the scattering matrix the
probabilties can be determined for the exchange of an integer number of
vibrational energy quanta $\hbar\Omega$ between the oscillating mass $M$ and
the scattered mass $m$. For a given number $n$ of quanta given off by $M$ the
same number of quanta is absorbed by $m$, or vice versa. The question of
violation of conservation of energy does not even arise, because the correct
solution of the time independent Hamiltonian ensures it for each conceivable
final state of the system.

For the special case of a hard sphere potential Nash gives explicit
expressions for the elements of the scattering matrix. These are cumbersome
terms. But all of them contain a scaling factor of the form
$\sqrt{2M\Omega/\hbar}$. This shows the importance of the frequency of the
harmonic oscillator, and thus of the stiffness of the spring constant, for the
various transition probabilities. If a similarly rigorous derivation is done
for the case of ENC (Fig.1), a similar dependence of the final states of
particle 2 on the magnitude of the magnetic field B will be discovered.

I would like to thank Kenneth Augustyn for bringing the problem of ENC to my
attention.

## References

  * [1] Sean M. Carroll and Jackie Lodman, _Energy Non-conservation in Quantum Mechanics_ , arXiv:2101.11052v2 [quant-ph] 22 Jul 2021
  * [2] Sean M. Carroll and Jackie Lodman, _Energy Non-conservation in Quantum Mechanics_ , Foundations of Physics 51 (4):1-15 (2021). https://link.springer.com/article/10.1007/s10701-021-00490-5
  * [3] P. L. Nash, _Interaction potential matrix elements for single-particle scattering off a simple harmonic oscillator_ , Il Nuovo Cimento 111 B, 549 - 560 (1996). https://link.springer.com/article/10.1007/BF02726647
  * [4] https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator


$p^{l}+1$, satisfying optimal spectral properties, namely with a spectral gap
the interval $[2\sqrt{q},q+1]$, for $q=p^{l}$.

###### Proof.

Such $k$-cube complexes were constructed in [26, Section 6]. They correspond
to congruence quotients of arithmetic groups. The number of vertices of such
complexes is given by the order of the group $\operatorname{PGL}(2,p^{l})$. ∎

######  Remark 5.6.

There are also non-residually finite complexes which have interesting
$k$-graphs, although they do not necessarily exhibit the optimal spectral gap.
Such complexes with one vertex were constructed in [26, Section 5]. Applying
Lemma 4.4, we get such complexes with $2$ vertices for all values $k\geq 1$.

Now we extend the notion of the Ramanujan cube complexes to higher-rank
graphs.

######  Definition 5.7.

We say that a coordinate matrix of a $k$-graph is $L$-regular for
$L\in{\mathbb{N}}$, if the sum of all row entries is equal to $L$.

######  Definition 5.8.

Let $\Lambda$ be a $k$-graph with $L_{i}$-regular coordinate matrices $M_{i}$
having positive second eigenvalue $\lambda_{i}$, for $i=1,\dots,k$. We say
that the $k$-graph $\Lambda$ is Ramanujan if

$\lambda_{i}\leq 2\sqrt{L_{i}-1}\text{ for all }i=1,\dots,k.$

######  Theorem 5.9.

For each $k\geq 2$, there is an infinite family of Ramanujan k-graphs with
$N\geq 2$ vertices. More precisely, $N$ is determined as the index of
congruence subgroups of the RSV-groups $\Gamma_{M,\delta}$ from subsection
2.8.

###### Proof.

This is a direct application of Theorem 5.5 in conjunction with Theorem 4.7.
For a complex $\mathcal{X}$ with $N$ vertices, there is a $k$-graph
$\Lambda(\mathcal{X})$ with $N$ vertices by an application of Theorem 4.7. ∎

We note that both one-vertex cube complexes covered by product of $k$ trees
and one-vertex higher rank graphs are trivially Ramanujan, so we will require
in addition the number of vertices to be greater than, for example, the
maximum of $(L_{i}-1)^{2},i=1,...,k$.

######  Example 5.10.

We now describe an explicit Ramanujan $3$-graph with $25$ vertices in the
above infinite family. Let $p=5$ and consider the group $\Gamma_{1}$ from
section 2.8 acting on a product of three trees with valencies $(6,6,6)$. Let
$P$ denote the one-vertex $3$-complex associated to $G$. The existence of the
claimed $3$-graph is assured by Proposition 4.5 and Theorem 4.7 because
$\Gamma_{1}$ has a quotient $L$ of order $25$ (indeed, it has quotients of
order $5^{l}$ for all $l\geq 1$). Let $\mathcal{X}$ denote the resulting
complex with $25$ vertices, and let $\Lambda$ be its associated $3$-graph.

Certain subsets of generators of $\Gamma_{1}$ already generate a group of
order $25$ in the cover, as may be verified using MAGMA. For example, the
image $Q(a_{1})$ in $S_{25}$ is the product of disjoint cycles

$Q(a_{1})=(1,15,24,8,17)(2,11,25,9,18)(3,12,21,10,19)(4,13,22,6,20)(5,14,23,7,16).$

With the notation of Proposition 4.5, we have isomorphisms of groups

$\displaystyle L$ $\displaystyle\cong\langle
Q(a_{1}),Q(a_{5}),Q(a_{9})\rangle$ $\displaystyle L$
$\displaystyle\cong\langle Q(b_{2}),Q(b_{6}),Q(b_{10})\rangle$ $\displaystyle
L$ $\displaystyle\cong\langle Q(c_{3}),Q(c_{7}),Q(c_{11})\rangle,$

thus all three groups in the right hand side are abstractly isomorphic to a
finite group of order $25$. Let $K_{1}$, $K_{2}$ and $K_{3}$, respectively,
denote the Cayley graphs of the finite group of order $25$ coming from the
three preceding isomorphisms.

This means that while the presentation of the infinite group $\Gamma_{1}$
requires generators of all three colours $a_{i},b_{j},c_{k}$ (as $\Gamma_{1}$
is irreducible), in the presentation of the finite group of order $25$,
generators of only one colour suffice. The finite cover is the complex
$\mathcal{X}$, and fixing each colour yields the Cayley graph of a finite
group of order $25$. In other words, each of the generating sets $A_{1}$,
$A_{2}$ and $A_{3}$ give Cayley graphs in three different sets of generators
(colours) of the _same_ finite group.

The adjacency matrices $M_{i}$ of the Cayley graphs $K_{i}$, $1=1,2,3$, may be
computed using MAGMA, using that $Q(G)$ acts as permutations in $S_{25}$. It
turns out that $M_{1},M_{2},M_{3}$ are equal. As noted in the proof of
Proposition 5.1, the adjacency matrices $M_{1},M_{2},M_{3}$ of the complex are
also the adjacency matrices of the $3$-graph $\Lambda$. Each $M_{i}$ is
$6$-regular in the sense of Definition 5.7, for $i=1,2,3$, as may be seen from
the concrete description of the matrices obtained with MAGMA. An application
of Theorem 5.9 gives that $\Lambda$ is a Ramanujan $3$-graph, so the second
largest eigenvalue $\lambda_{i}$ of $M_{i}$ is dominated by $2\sqrt{5}$, for
$i=1,2,3$.

In this example, the spectral gap is strictly in the optimal bound, namely,
the second eigenvalue of $M_{1}$ is dominated by $3.24$, according to MAGMA
computations. This bound is lower than the theoretically predicted
$2\sqrt{5}$.

Also using MAGMA shows that the product $M_{1}M_{2}M_{3}$ is not a
$(0,1)$-matrix, which distinguishes this example from [28] and all papers
inspired by it. For example, the diagonal entries in $M_{1}M_{2}M_{3}$ are all
equal to $12$. The remaining entries are $6,7$ or $15$.

##  6\. Matrices of the Cayley graph of an order $25$-group

$\setcounter{MaxMatrixCols}{25}\begin{bmatrix}0&1&1&1&1&1&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
1&0&1&0&0&0&1&1&1&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
1&1&0&1&0&0&0&0&1&0&1&1&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
1&0&1&0&1&0&0&0&0&0&0&1&1&1&0&0&0&0&0&0&0&0&0&0&0\\\
1&0&0&1&0&1&0&0&0&0&0&0&0&1&1&1&0&0&0&0&0&0&0&0&0\\\
1&0&0&0&1&0&1&0&0&0&0&0&0&0&0&1&1&1&0&0&0&0&0&0&0\\\
1&1&0&0&0&1&0&0&0&1&0&0&0&0&0&0&0&1&1&0&0&0&0&0&0\\\
0&1&0&0&0&0&0&0&1&1&0&0&0&0&1&0&0&0&0&1&1&0&0&0&0\\\
0&1&1&0&0&0&0&1&0&0&1&0&0&0&0&0&0&0&0&1&0&1&0&0&0\\\
0&1&0&0&0&0&1&1&0&0&0&0&0&0&0&0&0&0&1&0&1&0&1&0&0\\\
0&0&1&0&0&0&0&0&1&0&0&1&0&0&0&0&1&0&0&0&0&1&0&1&0\\\
0&0&1&1&0&0&0&0&0&0&1&0&1&0&0&0&0&0&0&0&0&0&1&1&0\\\
0&0&0&1&0&0&0&0&0&0&0&1&0&1&0&0&0&0&1&0&0&0&1&0&1\\\
0&0&0&1&1&0&0&0&0&0&0&0&1&0&1&0&0&0&0&1&0&0&0&0&1\\\
0&0&0&0&1&0&0&1&0&0&0&0&0&1&0&1&0&0&0&1&1&0&0&0&0\\\
0&0&0&0&1&1&0&0&0&0&0&0&0&0&1&0&1&0&0&0&1&0&0&1&0\\\
0&0&0&0&0&1&0&0&0&0&1&0&0&0&0&1&0&1&0&0&0&1&0&1&0\\\
0&0&0&0&0&1&1&0&0&0&0&0&0&0&0&0&1&0&1&0&0&1&0&0&1\\\
0&0&0&0&0&0&1&0&0&1&0&0&1&0&0&0&0&1&0&0&0&0&1&0&1\\\
0&0&0&0&0&0&0&1&1&0&0&0&0&1&1&0&0&0&0&0&0&1&0&0&1\\\
0&0&0&0&0&0&0&1&0&1&0&0&0&0&1&1&0&0&0&0&0&0&1&1&0\\\
0&0&0&0&0&0&0&0&1&0&1&0&0&0&0&0&1&1&0&1&0&0&0&0&1\\\
0&0&0&0&0&0&0&0&0&1&0&1&1&0&0&0&0&0&1&0&1&0&0&1&0\\\
0&0&0&0&0&0&0&0&0&0&1&1&0&0&0&1&1&0&0&0&1&0&1&0&]\\\
0&0&0&0&0&0&0&0&0&0&0&0&1&1&0&0&0&1&1&1&0&1&0&0&0\\\ \end{bmatrix}$

## References

  * [1] W. Ballmann, M. Brin, Orbihedra of nonpositive curvature, Inst. Hautes Études Sci. Publ. Math. No. 82 (1995), 169-209.
  * [2] E. Brieskorn and K. Saito, _Artin-Gruppen und Coxeter-Gruppen_ , Invent. Math., 17(1972), 245–271.
  * [3] M. Burger, Sh. Mozes, Lattices in product of trees, Inst. Hautes Études Sci. Publ. Math. 92 (2000), 151–194.
  * [4] J. Cuntz and W. Krieger, _A class of $C^{*}$-algebras and topological Markov chains_, Invent. Math. 56 (1980), 251–268.
  * [5] P. Dehornoy, F. Digne, E. Godelle, D. Krammer and J. Michel, Foundations of Garside Theory, Number 22 in EMS Tracts in Mathematics, European Math. Soc., 2015.
  * [6] D.G. Evans, _On the K-theory of higher rank graph $C^{*}$-algebras_, New York J. Math. 14 (2008), 1–31.
  * [7] N. Fowler and A. Sims, _Product systems over right-angled Artin semigroups_ , Trans. Amer. Math. Soc. 354 (2002), 1487–1509.
  * [8] M. Gromov, Hyperbolic groups, Essays in Group Theory (S.M. Gersten, ed.), 8, Math. Sci. Res. Inst. Publ., (1988) , 75–263.
  * [9] A. Hatcher, Algebraic Topology, electronic version available from author’s webpage.
  * [10] R. Hazlewood, I. Raeburn, A. Sims and S.B.G. Webster _Remarks on some fundamental results about higher-rank graphs and their $C^{*}$-algebras_, Proc. Edinb. Math. Soc. (2) 56 (2013), no. 2, 575–597.
  * [11] A. an Huef, M. Laca, I. Raeburn and A. Sims, _KMS states on the C ∗-algebra of a higher-rank graph and periodicity in the path space_, J. Funct. Anal. 268 (2015), no. 7, 1840–-1875.
  * [12] J.S. Kimberley and G. Robertson, _Groups acting on products of trees, tiling systems and analytic K-theory_ , New York J. Math. 8 (2002), 111–131.
  * [13] A. Kumjian and D. Pask, _Higher rank graph $C^{*}$-algebras_ New York J. Math. 6 (2000), 1–20.
  * [14] M. Laca, N.S. Larsen, S. Neshveyev, A. Sims, S.B.G. Webster, Von Neumann algebras of strongly connected higher-rank graphs, Math. Ann. 363 (2015), no. 1-2, 657–678.
  * [15] M. V. Lawson, A. Vdovina, Higher dimensional generalizations of the Thompson groups, Adv. in Math. 369 (2020), 107191.
  * [16] M. V. Lawson, A. Sims and A. Vdovina, Higher dimensional generalizations of the Thompson groups via higher rank graphs, preprint, arXiv:2010.08960v1[math.GR].
  * [17] S. Mac Lane, Categories for the working mathematician, Second edition, 1978 Springer-Verlag New York.
  * [18] S.A. Mutter, _K-theory of 2-rank graphs associated to complete bipartite graphs_ , preprint, arXiv:2004:11602v1[math.CO].
  * [19] S.A. Mutter, A.C. Radu and A. Vdovina, _C*-algebras of higher-rank graphs from groups acting in buildings, and explicit computation of their K-theory_ , preprint, arXiv:2012.05561v1[math.OA].
  * [20] R. Okayasu, _Type III factors arising from Cuntz-Krieger algebras_ , Proc. Amer. Math. Soc. 131 (2003), no. 7, 2145–2153.
  * [21] O. Ore, Theory of graphs, Amer. Math. Soc. Colloquium Publications, vol XXXVIII, R. I. 1962.
  * [22] N.C. Phillips, _A classification theorem for nuclear purely infinite simple $C^{*}$-algebras_, Doc. Math. 5 (2000), 49–114.
  * [23] S.C. Power,_Classifying higher rank analytic Toeplitz algebras_ , New York J. Math. 13 (2007), 271–298.
  * [24] I. Raeburn, A. Sims and T. Yeend, Higher-rank graphs and their $C^{*}$-algebras, Proc. Edinburgh Math. Soc. 46 (2003), 99–115.
  * [25] D.Rattaggi, Computations in groups acting on a product of trees: Normal subgroup structures and quaternion lattices. Thesis (Dr.sc.math.) ETH Zürich (Switzerland), 2004, 305 pages, ProQuestLLC.
  * [26] N. Rungtanapirom, J. Stix, A. Vdovina, Infinite series of quaternionic 1-vertex cube complexes, the doubling construction, and explicit cubical Ramanujan complexes. International Journal of Algebra Computation 29 (2019), no. 6, 951–1007.
  * [27] G. Robertson, _Boundary $C^{*}$-algebras of triangle geometries_, J. Funct. Anal. 266 (2014), no. 11, 6319–6334.
  * [28] G. Robertson and T. Steger, _Affine buildings, tiling systems and higher rank Cuntz-Krieger algebras_ , J. Reine Angew. Math. 513 (1999), 115–144.
  * [29] G. Robertson and T. Steger, _Asymptotic K-theory for groups acting on $\tilde{A}_{2}$ buildings_, Canad. J. Math. 53 (2001), no. 4, 809–833.
  * [30] M. Sageev, CAT(0) cube complexes and groups, in: Geometric Group Theory, editors: M. Bestvina, M. Sageev, K. Vogtmann, IAS/Park City Mathematics Series Volume 21, 2014, 399pp.
  * [31] K. Saito, Growth functions for Artin monoids, Proc. Japan Acad., 85, Ser. A (2009), 84–88.
  * [32] A. Sims, _Gauge-invariant ideals in the $C^{*}$-algebras of finitely aligned higher rank graphs_, Canad. J. Math. 58 (2006), no. 6, 1268-1290.
  * [33] J. Spielberg, _Free product groups, Cuntz-Krieger algebras, and covariant maps_ , Internat. J. Math. 2(1991), 457–476.
  * [34] A. Vdovina, Drinfeld-Manin solutions of the Yang-Baxter equation coming from cube complexes, Internat. J. Algebra Comput. 31 (2021), no. 4, 775–788.
  * [35] A. Vdovina, _Combinatorial structure of some hyperbolic buildings_ , Math. Z. 241 (2002), 471–478.
  * [36] D. Wise, Non-positively curved squared complexes: Aperiodic tilings and non-residually finite groups. Thesis (Ph.D.)Princeton University. 1996
  * [37] D. Yang, The interplay between $k$-graphs and the Yang-Baxter equation, J. Algebra 451 (2016), 494-525.
  * [38] D. Yang, _Factoriality and type classification of $k$-graph von Neumann algebras_, Proc. Edinb. Math. Soc. 60 (2017), no. 2, 499-–518.


# Teleconnection patterns of different El Niño types revealed by climate
network curvature

Felix M. Strnad
Cluster of Excellence Machine Learning: New Perspectives for Science
Universität Tübingen
72072 Tübingen, Germany Jakob Schlör 11footnotemark: 1
Cluster of Excellence Machine Learning: New Perspectives for Science
Universität Tübingen
72072 Tübingen, Germany Christian Fröhlich
Cluster of Excellence Machine Learning: New Perspectives for Science
Universität Tübingen
72072 Tübingen, Germany Bedartha Goswami
Cluster of Excellence Machine Learning: New Perspectives for Science
Universität Tübingen
72072 Tübingen, Germany These two authors contributed equally to the paper.

###### Abstract

The diversity of El Niño events is commonly described by two distinct flavors,
the Eastern Pacific (EP) and Central Pacific (CP) types. While the remote
impacts, i.e. teleconnections, of EP and CP events have been studied for
different regions individually, a global picture of their teleconnection
patterns is still lacking. Here, we use Forman-Ricci curvature applied on
climate networks constructed from 2-meter air temperature data to distinguish
regional links from teleconnections. Our results confirm that teleconnection
patterns are strongly influenced by the El Niño type. EP events have primarily
tropical teleconnections whereas CP events involve tropical-extratropical
connections, particularly in the Pacific. Moreover, the central Pacific region
does not have many teleconnections, even during CP events. It is mainly the
eastern Pacific that mediates the remote influences for both El Niño types.

_Keywords_ Climate networks $\cdot$ El Niño diversity $\cdot$ Ricci-curvature

##  1 Introduction

The El Niño Southern Oscillation (ENSO) is the largest interannual variation
in the global climate system. It is a dynamical atmospheric and oceanic
phenomenon characterized by anomalously warm (El Niño) or cold (La Niña)
phases of sea surface temperatures (SST) in the equatorial Pacific. Both
phases are known to impact earth’s climate significantly on long spatial
scales, typically referred to as teleconnections (Trenberth, 1997).

The interrelation between the ‘type’ of El Niño and its impact has been
investigated in many studies over the past two decades (Capotondi et al.,
2015, Timmermann et al., 2018, Capotondi et al., 2020). The diversity of El
Niño events is currently thought to be characterized by two modes: The
“canonical” or Eastern Pacific (EP) El Niño with peak SST anomalies in the
eastern equatorial Pacific, and the “El Niño Modoki” or Central Pacific (CP)
El Niño with peak SST anomalies in the central equatorial Pacific. Although
the impact of EP and CP El Niños on different parts of the climate system —
such as the Indian Ocean (IO), maritime continent, tropical Atlantic, and
Northern America — have been studied thoroughly (see Okumura and Taschetto et
al. for an overview), previous work has mainly focused on single
teleconnections of the El Niño types. In comparison, little is known about
differences in global teleconnection patterns between EP and CP. In this
study, we close this gap by presenting a machine learning approach based on
the Ricci-curvature of climate networks which detects global teleconnection
patterns of the El Niño types and highlights their differences.

Climate networks (Dijkstra et al., 2019) have gained increasing interest for
the analysis of spatial dependencies of climatic variables through their
ability to reduce data to relevant climatic patterns. They have been widely
used in the analysis of ENSO, starting with Tsonis and Swanson, who
investigated the topology of El Niño and La Niña networks of surface air
temperature. Yamasaki et al. studied the global impact of El Niño on various
geographical zones while Donges et al. and Zhou et al. examined geographical
long-range teleconnections of ENSO.

Impacts of ENSO diversity have been studied by evolving climate network
analyses (Radebach et al., 2013, Kittel et al., 2021). Wiedermann et al. use
climate networks to find a robust way to distinguish different types of El
Niños and La Niñas. Similarly, Lu et al. use an analysis of climate networks
of the Pacific Ocean to distinguish EP and CP events and estimate their
expected impacts.

Ricci-curvature measures the deviation of a continuous space from being
locally flat and has been recently generalized to discrete spaces like complex
networks (Ollivier, 2009, Sreejith et al., 2016). In networks, the Ricci-
curvature of a link describes the deviation of its surrounding from a regular
grid (each node is connected to its four neighboring nodes). Ricci-curvature
highlights whether an edge connects nodes within a community or bridges
communities and thereby helps to understand visualize the network structure
intuitively, applied for example in the analysis of financial markets (Sandhu
et al., 2016), gene expressions (Sandhu et al., 2015, Pouryahya et al., 2018),
brain connectivity (Farooq et al., 2019), urban transportation (Gao et al.,
2019), power grids (Jonckheere and Grippo, 2019), and epidemiology (de Souza
et al., 2021). We use it to reveal global teleconnection patterns and are
thereby able to uncover structurally different El Niño impacts.

##  2 Data and Methods

###  2.1 Data

We use daily 2-meter air temperature (or surface air temperature, SAT) data
for the years 1950–2020 from the ERA5 Global Reanalysis database (Hersbach et
al., 2020). We first detrend each time series, then subtract the daily
climatology over the whole time period. We use next-neighbor interpolation to
map the data to a grid of spatially approximately uniformly distributed points
using the Fekete algorithm (Bendito et al., 2007) to avoid spurious
correlation patterns close to the poles (Ebert-Uphoff and Deng, 2012). The
distance between grid points in the Fekete grid corresponds to 2.5° for points
at the equator of a Gaussian grid, resulting in a total of $\approx 6000$ grid
points.

###  2.2 Classification of EP and CP El Niño conditions

We classify each day as EP (CP) based on whether the average Dec–Feb SST
anomaly in the Niño-3 region is greater (less) than that of the Niño-4 region
and Niño-3 (Niño-4) larger than 0.5 (Capotondi et al., 2020). Days that are
neither EP, CP, nor La Niña are labeled as ‘normal.’ The SST anomalies are
calculated with the Oceanic Niño index baselines, i.e., using multiple
centered 30-year base periods, successively updated in 5-year steps. For the
end (beginning) of the time period, the last (first) available 30-year base
period is used.

###  2.3 2-meter air temperature Dec–Feb climate networks

Following earlier work on climate networks (Donges et al., 2009a, Wiedermann
et al., 2016, Ciemer et al., 2020), we estimate the weighted adjacency matrix
$\mathbf{W}$ of the climate network by placing links between pairs of
locations which have a correlation value among the 2% strongest absolute
correlations. Thus, for the correlation threshold
$\rho_{0.98}=Q_{|\rho|}(0.98)$, where $Q_{X}(\cdot)$ denotes the quantile
function for $X$, we define

$\displaystyle\mathbf{W}_{ij}=\begin{cases}|\rho_{ij}|,&|\rho_{ij}|>\rho_{0.98}\\\
0,&\textrm{otherwise}\end{cases},$ (1)

where $i,j\in\\{1,\dots,N\\}$ index spatial locations and $\rho_{ij}$ is the
Dec–Feb Spearman’s rank-order correlation between locations $i$ and $j$.
$\mathbf{W}_{ij}$ thus defines a network $G:=(V,E)$ with a set of edges (or
links) $e_{ij}\in E$ that connect pairs of nodes $(v_{i},v_{j})\in V$ with
link weight $w_{ij}=|\rho_{ij}|$.

Accounting for autocorrelation in the data, a two-sided test for non-random
correlations at a confidence level of 1% yields a threshold
$|\hat{\rho}|=0.35$. While our threshold is far higher, due to the high number
of hypotheses tests ($3.6\times 10^{7}$), we nevertheless expect a non-
negligible number of network links to be false positives. We thus additionally
use a spatial null model which assumes that correlations caused by physical
mechanisms are likely part of spatially extended patterns (Boers et al.,
2019). For each spatial location, we randomly rewire its network links $2000$
times and use a Gaussian kernel density estimator (KDE) to get the likelihood
of a link to the chosen location. An observed network link to the chosen
location is considered statistically significant if the spatial likelihood of
the link (also obtained via Gaussian KDE) is above the $99.9$-th percentile of
the local null model link distribution.

###  2.4 Curvature of complex networks

Figure 1: Curvature of complex networks. Forman-curvature of an edge with
weight $w_{ij}$ connecting nodes $v_{i}$ and $v_{j}$ is obtained by Eq. 2 (a).
In the synthetic network constructed with a Stochastic Block Model, Forman-
curvature (b) provides values to distinguish within-community links (red) from
between-community links (blue) in contrast to betweenness centrality (c) with
only a few links with high scores. The network with four communities are
visualized using the spring layout from NetworkX (Hagberg et al., 2008).

The Ricci-curvature of a network link describes how the connectivity of its
network neighborhood differs from the connectivity of a regular grid. Out of
two numerical approximations of Ricci curvature on networks — Forman-Ricci
curvature (Forman, 2003, Sreejith et al., 2016) and Ollivier-Ricci curvature
(Ollivier, 2010) — we use Forman-Ricci curvature (henceforth simply Forman
curvature) as it is computationally cheaper and as both definitions are highly
correlated, barring slight differences in extreme values (Samal et al., 2018).
Forman curvature of edge $e_{ij}$ in an undirected network with weight
$w_{ij}\in\mathbf{W}$ is estimated as,

$\displaystyle F_{ij}=w_{ij}\left(|\mathcal{T}_{ij}|\cdot
w_{ij}+\frac{2}{w_{ij}}-\sum_{\begin{subarray}{c}k=1\\\
w_{ik}w_{kj}=0\end{subarray}}^{N}\sum_{\begin{subarray}{c}l\in\\{i,j\\}\\\
w_{lk}>0\end{subarray}}\frac{1}{\sqrt{w_{ij}\cdot w_{lk}}}\right),$ (2)

where $\mathcal{T}_{ij}:=\\{v_{k}:w_{ik}w_{jk}>0\\}$ denotes the set of nodes
in the neighborhood of $v_{i}$ and $v_{j}$ which form triangles containing
edge $e_{ij}$ (green edges in Fig. 1a) and $|\cdot|$ denotes set cardinality.
The last term in Eq. 2 counts the number of edges adjacent to node $v_{i}$ and
$v_{j}$ which do not form triangles with edge $e_{ij}$ (black edges in Fig. 1
a). Equation 2 approximates the “augmented” Forman curvature (cf. Samal et
al., Eq. 9) by considering only triangles, no node weights, and no cycle
weights.

Since curvature is in general not symmetrically distributed around zero, we
use the 10% most positive curvatures,

$\displaystyle F_{ij}^{+}$ $\displaystyle:=\\{F_{ij}:F_{ij}>Q_{F}(0.9)\\},$
(3)

and the 10% most negative curvatures $F_{ij}^{-}$ (defined similarly as values
less than $Q_{F}(0.1)$. We also define the Forman-curvature of a network node,

$\displaystyle f_{i}:=\frac{\sum_{j}F_{ij}}{\sum_{j}\mathbf{W}_{ij}}$ (4)

to identify geographical locations, or ‘hotspots,’ connected to strongly
negatively or positively curved links. As the value ranges of curvature
differs for different networks, we use the min-max transform to normalize it
to $(-1,1)$, denoted by $\tilde{F}_{ij}$. Corresponding to
$\tilde{F}_{ij}^{+}$, we use Eq. 4 to define $\tilde{f}_{i}^{+}$ as positive
node-curvature hotspots and similarly, $\tilde{f}_{i}^{-}$ for negative node-
curvature hotspots.

Consider a random network with four communities (Fig. 1 b, c) generated using
a stochastic block model from NetworkX (Hagberg et al., 2008). Forman-
curvature clearly separates the between-community links from the within-
community links. Within-community links are typically part of triangles,
indicating local convergence of shortest paths, i.e. positive curvature.
Conversely, links connecting nodes with a high degree that are not part of
triangles indicate local divergence of shortest paths and negative curvature.
Forman-curvature provides a continuous measure over network links that
indicates if an edge is inside a community or if it straddles two communities.
By comparison, the edge betweenness centrality (see SI sec A)— typically used
to identify edges that connect communities — fails to identify many between-
community links (Fig. 1c). This is likely due to the binary notion of a
shortest path - either a path is the shortest or it is not, implying that
“almost-shortest” paths are not considered (SI Eq. 8).

##  3 Results and Discussion

###  3.1 Spatial organization of teleconnections depends on El Niño type

Figure 2: Forman curvature of normal, EP, and CP El Niño climate network
links. The networks are computed from 2m air temperature anomalies for normal
(left column), EP(middle column) and CP (right column) conditions. The first
row (a, b, c) depicts the spatial link length distribution for all (black),
most positively $F_{ij}^{+}$ (red) and most negatively curved $F_{ij}^{-}$
(blue) edges. The second row (d, e, f) shows $F_{ij}^{+}$, the third row (g,
h, i) $F_{ij}^{-}$. Colorbars indicate the number of edges to the node. For
visual reasons only every 20th edge is plotted in d-i.

Fig. 2 a–c shows the distribution of great-circle lengths of the most
positively curved $F_{ij}^{+}$ and most negatively curved $F_{ij}^{-}$ network
links for normal, EP, and CP conditions respectively. While $F_{ij}^{-}$ links
occur at all spatial scales, $F_{ij}^{+}$ links occur only at regional scales
less than $5000$ km. Long-range teleconnections are thus modulated almost
always via negatively curved links; and as the curvature estimation of climate
network links does not include any geographical information, likely, the
correlation between negative climate network link curvature and long spatial
scales is an intrinsic topological property of the SAT dynamics. Furthermore,
the spatial distribution of $F_{ij}^{+}$ (Fig. 2 d–f) and $F_{ij}^{-}$ (Fig. 2
g–i) shows that while positive curvature occurs at only regional scales,
negative curvature results in link bundles that are related to well-known
teleconnection patterns, e.g., the connection between the eastern equatorial
Pacific ENSO tongue pattern, and the tropical Atlantic.

The number of most negatively curved links for El Niño conditions does not
describe all spatially long-range links, in contrast to normal conditions
(Fig. 2 a–c). However, even with the lower number of links, $F_{ij}^{-}$ links
undergo a drastic spatial reorganization for El Niño conditions, especially
from normal conditions (Fig. 2 g) to CP conditions (Fig. 2 i). During CP El
Niños, e.g., the tropical Pacific shows an enhanced connection to the
extratropical Pacific as well as to the mid-latitude North Atlantic region
(see Sec. 3.3). Even for EP conditions (Fig. 2 h), we observe that the
connection between the tropical Pacific and the southern Atlantic is
strengthened in comparison to normal months. Although we find subtle changes
in the spatial organization of the most positively curved links between normal
(Fig. 2 d) and El Niño conditions (Fig. 2 e, f), such as weakening of regional
correlation structures in the tropical Atlantic and strengthening of
correlation in the West African monsoon belt, these changes are not as drastic
as the ones that occur in the negatively curved links.

Note, we mainly discover teleconnections within the global oceans because
correlations are generally higher over oceans than over land due to slower
oceanic SAT variability (Lambert et al., 2011).

###  3.2 EP El Niño teleconnections are tropical, while CP El Niño
teleconnections are at all latitudes

Figure 3: Node curvature of normal, EP El Niño, and CP El Niño climate
networks. Node curvature for most positively curved, $\tilde{f}_{i}^{+}$ (a-c)
and most negatively curved, $\tilde{f}_{i}^{-}$ (d-f) links are shown. EP
hotspots (middle column) are more constrained to the tropics than for CP
(right column). $\tilde{f}_{i}^{+/-}$ for normal conditions (left column)
shows more similarity to CP than to EP. Zonal medians of $\tilde{f}_{i}^{+}$
(red) and $\tilde{f}_{i}^{+}$ (blue) further confirm this similarity. (g-i).

Positive curvature hotspots, $\tilde{f}_{i}^{+}$, for EP conditions reveal the
well-known ENSO tongue (Fig. 3b) typically observed in empirical orthogonal
function analysis of sea surface temperature data (Johnson, 2013). We also
find pronounced regions of $\tilde{f}_{i}^{+}$ in the IO and tropical
Atlantic, which are known to be affected by strong EP El Niño events (Klein et
al., 1999, Zhang et al., 2015, Rodrigues et al., 2015). Under CP conditions,
$\tilde{f}_{i}^{+}$ is spread over all latitudes and over different regions of
the globe (Fig. 3c). For instance, we observe a hotspot in the tropical
Pacific similar to the El Niño tongue, which is however shifted towards the
dateline and also extended southwards. The spatial pattern under CP conditions
is more similar to normal months (Fig. 3a) than under EP conditions.
$\tilde{f}_{i}^{+}$ shows strong similarity to node degree (SI 6).

Negative node-curvature hotspots, $\tilde{f}_{i}^{-}$, for EP conditions (Fig.
3e) show enhanced teleconnections arising from South China Sea, tropical IO,
eastern tropical Pacific, and the tropical Atlantic. This coincides with a
decrease in teleconnections arising from the extratropical Pacific, southern
IO, North Atlantic (near Greenland), and the Southern Ocean. As with
$\tilde{f}_{i}^{+}$, here too the spatial pattern of CP conditions (Fig. 3f)
is spread all over the globe and looks more similar to normal conditions (Fig.
3d), barring the northern tropical Pacific and the North Atlantic, both of
which show a decrease in teleconnections.

Overall, EP conditions result in confinement of hotspots around the tropics
for both regional links and teleconnections, which is further confirmed by
comparing the zonal medians shown in Fig. 3 g–i. CP conditions, on the other
hand, show only minor differences from normal months, although the magnitude
of curvature values increase (SI 9). This partly corroborates previous work
(Wiedermann et al., 2016, Lu et al., 2020) which also reports a strong
localization of climate network links during EP conditions. These studies,
however, did not focus on the impact of ENSO diversity on climate network
teleconnections.

###  3.3 EP and CP El Niño teleconnection patterns of eastern and central
Pacific Ocean, IO, and Labrador Sea

Figure 4: Teleconnections arising from eastern and central Pacific Ocean, IO,
and Labrador Sea. $F_{ij}^{-}$, i.e. teleconnections, for EP (purple) and CP
(green) El Niño events connected to the four selected regions (black
rectangles): Niño 3 (a), Niño 4 (b), IO (c), and Labrador Sea (d). For visual
clarity, only a third of all links are shown.

Teleconnections link the Niño 3 region (Fig. 4a) to the tropical Atlantic
during both EP and CP conditions, supporting earlier work showing that strong
El Niños can lead to warming in the tropical Atlantic mediated by the
tropospheric temperature mechanism (Chang et al., 2006) and the atmospheric
bridge via the Pacific North Atlantic (PNA) pattern (Rodrigues et al., 2011).
There are also links from the Niño 3 region that appear only under EP
conditions, such as those that connect to the IO, likely arising because of
the influence of eastern tropical Pacific SSTs on the IO during and after El
Niño events (Klein et al., 1999). Some links from the Niño 3 box arise only
under CP conditions, primarily connecting the extratropical Pacific. These
links are likely due to the North and South Pacific Meridional Mode (NPMM and
SPMM), as atmospheric and oceanic anomalies in the extratropics associated
with SPMM and NPMM affect the intensity and flavor of El Niño (You and
Furtado, 2018). The relationship of NPMM and SPMM with ENSO diversity is,
however, still under debate (Amaya, 2019).

The teleconnections of the Niño 4 region (Fig. 4b) are far fewer compared to
the Niño 3 region (Fig. 4a). It is rather surprising that the Ninõ 4 region is
not well connected even during CP conditions, although a CP El Niño is
primarily characterised by higher SST anomalies in the Niño 4 region than in
the Niño 3 region. A likely explanation is that although the temperature
fields are more anomalous in the central Pacific during CP conditions, the
impacts of these anomalies are nevertheless mediated via the eastern Pacific.
We find the same result for climate networks created from networks with
shorter time ranges as well.

The IO has a large number of most negatively curved links during EP conditions
but not for CP (Fig. 3 c, d). The EP event teleconnections link the IO to the
tropical Atlantic and Pacific basin (Fig. 4 c). The links to the tropical
Atlantic could be either attributed to indirect links mitigated by the impact
of the Ninõ 3 region to the tropical Atlantic or might resemble direct impact
between the oceans as recently described by Zhang and Han. CP conditions do
not result in teleconnections in the IO as CP events are generally weaker
(Zhang et al., 2015).

The Labrador Sea (Fig. 4d) is another pronounced area of most negative node-
curvature in the CP network not present in the EP network (compare Fig. 3
c,d). Edges with the most negative curvature adjacent to the Labrador Sea
connect to the eastern tropical Pacific and the extra topical Atlantic. This
pattern may be attributed to the North Atlantic Oscillation (NAO), which
refers to sea level pressure changes in the Arctic and subtropical Atlantic
(Jiménez-Esteve and Domeisen, 2018). El Niño is known to cause a negative NAO
pattern driven by the PNA pattern where a negative NAO phase is attributable
to CP El Niño events via the subtropical bridge (Graf and Zanchettin, 2012,
Domeisen et al., 2019).

Regressing the SAT fields against the data in the four boxes corroborates the
findings from the curvature-based analysis (Fig. 5, SI Fig. 11, 12, 13). The
Niño 3 region is highly teleconnected during both EP and CP conditions, but
the spatial organization is different between the two (Fig. 5 a, b). The
tropical Indian and Atlantic oceans are well modeled during EP and the
tropical-extratropical links in the Pacific and the Labrador Sea are well
modeled during CP. By contrast, the Niño 4 box explains very little of the
temperature fields irrespective of the El Niño type (Fig. 5 c, d). Similarly,
we find that the IO is highly active during EP but not during CP conditions
(Fig. 5 e, f); and that the Labrador Sea is correlated to the tropics only
during CP (Fig. 5 g, i). Note that we can to identify the teleconnection
patterns only because we have the results from the curvature-based climate
network analysis to guide our interpretations of the correlation maps. Without
the curvature analysis, i.e. using simple correlation maps (SI Fig. 10) or
classical complex network measures (SI Fig. 6, 7, 8), it is not trivial to
figure out the most important regions for each El Niño flavor.

Figure 5: SAT fields regressed on temperatures from the four selected regions.
Rank-normalized SAT time series at each location is regressed against all the
rank-normalized SAT time series inside the four regions (black rectangles,
same as in Fig. 4) for EP events (left column) and CP events (right column)
and the average linear regression coefficient is shown.

##  4 Conclusion

We presented a new approach to estimate global teleconnection patterns of
2-meter air temperature and used it to investigate the teleconnections of
Eastern Pacific and Central Pacific El Niño events. Our approach involves
correlation-based climate networks and a recently established network measure
based on Ricci-curvature. In particular, we used Forman-Ricci curvature to
distinguish links related to small-scale regional structures (positively
curved links) from long-range teleconnections connecting regions from
different parts of the globe (negatively curved links).

We showed that El Niños diversity drastically impacts the spatial organization
of teleconnections. Using node-based curvature, we identified teleconnection
hotspots for both EP and CP conditions. Our results showed that
teleconnections in the EP climate network were mainly confined to the tropics,
whereas CP network teleconnections were found at northern and southern mid-
latitudes as well. We further investigated the impact of ENSO diversity on the
teleconnection patterns of four specific regions: the Niño 3 region, the Niño
4 region, the northern IO, and the Labrador Sea. We found that the Niño 3
region in the eastern Pacific has a large number of teleconnections
irrespective of whether we consider EP or CP conditions, whereas the Niño 4
region in the central Pacific has, by comparison, very few teleconnections
under both EP and CP conditions. We thus conclude that the eastern Pacific is
the primary mediator of El Niño impacts irrespective of the El Niño type, and
acknowledge that further work on the role of the eastern Pacific during CP El
Niños is needed. We found that the northern IO and the Labrador Sea show
teleconnections almost only under EP and CP conditions respectively. While the
teleconnections of the IO region to the Niño 3 region during EP conditions are
well-known, the links between the Labrador Sea to the eastern tropical Pacific
and northern tropical Atlantic are not fully understood.

## Open Research

Datasets for this research are available from Copernicus Climate Change
Service. The data from 1979 till date was taken from Hersbach et al. and the
data from 1950 to 1979 from Bell et al.. The code for generating and analyzing
the networks is made publicly available under Strnad and Schlör. The code for
reproducing the analysis of the network curvature described in this paper is
publicly available under Schlör and Strnad.

## Acknowledgment

Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under Germany’s Excellence Strategy – EXC number 2064/1 – Project
number 390727645. The authors thank the International Max Planck Research
School for Intelligent Systems (IMPRS-IS) for supporting Jakob Schlör and
Felix Strnad.

## References

  * Amaya (2019) D. J. Amaya.  The Pacific Meridional Mode and ENSO: a Review.  _Curr. Clim. Change Rep._ , 5(4):296–307, Dec 2019.  ISSN 2198-6061.  doi:10.1007/s40641-019-00142-x.
  * Bell et al. (2020) B. Bell, H. Hersbach, P. Berrisford, P. Dahlgren, A. Horányi, J. Muñoz Sabater, J. Nicolas, R. Radu, D. Schepers, A. Simmons, C. Soci, and J.-N. Thépaut.  Era5 hourly data on single levels from 1950 to 1978 (preliminary version), 2020.  URL https://cds.climate.copernicus-climate.eu/cdsapp#!/dataset/reanalysis-era5-single-levels-preliminary-back-extension.  Accessed on 02-03-2022.
  * Bendito et al. (2007) E. Bendito, A. Carmona, A. M. Encinas, and J. M. Gesto.  Estimation of Fekete points.  _J. Comput. Phys._ , 225(2):2354–2376, Aug 2007\.  ISSN 0021-9991.  doi:10.1016/j.jcp.2007.03.017.
  * Boers et al. (2013) N. Boers, B. Bookhagen, N. Marwan, J. Kurths, and J. Marengo.  Complex networks identify spatial patterns of extreme rainfall events of the South American Monsoon System.  _Geophys. Res. Lett._ , 40(16):4386–4392, Aug 2013.  ISSN 0094-8276.  doi:10.1002/grl.50681.
  * Boers et al. (2019) N. Boers, B. Goswami, A. Rheinwalt, B. Bookhagen, B. Hoskins, and J. Kurths.  Complex networks reveal global pattern of extreme-rainfall teleconnections.  _Nature_ , 566(7744):373–377, 2019.  ISSN 14764687.  doi:10.1038/s41586-018-0872-x.  URL http://dx.doi.org/10.1038/s41586-018-0872-x.
  * Capotondi et al. (2015) A. Capotondi, A. T. Wittenberg, M. Newman, E. Di Lorenzo, J.-Y. Yu, P. Braconnot, J. Cole, B. Dewitte, B. Giese, E. Guilyardi, F.-F. Jin, K. Karnauskas, B. Kirtman, T. Lee, N. Schneider, Y. Xue, and S.-W. Yeh.  Understanding ENSO Diversity.  _Bull. Am. Meteorol. Soc._ , 96(6):921–938, Jun 2015.  ISSN 0003-0007.  doi:10.1175/BAMS-D-13-00117.1.
  * Capotondi et al. (2020) A. Capotondi, A. T. Wittenberg, J.-S. Kug, K. Takahashi, and M. J. McPhaden.  ENSO Diversity.  In _El Niño Southern Oscillation in a Changing Climate_ , pages 65–86. American Geophysical Union (AGU), Nov 2020\.  ISBN 978-1-11954816-4.  doi:10.1002/9781119548164.ch4.
  * Chang et al. (2006) P. Chang, Y. Fang, R. Saravanan, L. Ji, and H. Seidel.  The cause of the fragile relationship between the Pacific El Niño and the Atlantic Niño.  _Nature_ , 443:324–328, Sep 2006.  ISSN 1476-4687.  doi:10.1038/nature05053.
  * Ciemer et al. (2018) C. Ciemer, N. Boers, H. M. J. Barbosa, J. Kurths, and A. Rammig.  Temporal evolution of the spatial covariability of rainfall in South America.  _Clim. Dyn._ , 51(1):371–382, Jul 2018.  ISSN 1432-0894.  doi:10.1007/s00382-017-3929-x.
  * Ciemer et al. (2020) C. Ciemer, L. Rehm, J. Kurths, R. V. Donner, R. Winkelmann, and N. Boers.  An early-warning indicator for Amazon droughts exclusively based on tropical Atlantic sea surface temperatures.  _Environ. Res. Lett._ , 15(9):094087, Sep 2020\.  ISSN 1748-9326.  doi:10.1088/1748-9326/ab9cff.
  * de Souza et al. (2021) D. B. de Souza, J. T. S. da Cunha, E. F. dos Santos, J. B. Correia, H. P. da Silva, J. L. de Lima Filho, J. Albuquerque, and F. A. N. Santos.  Using discrete Ricci curvatures to infer COVID-19 epidemic network fragility and systemic risk.  _J. Stat. Mech.: Theory Exp._ , 2021(5):053501, May 2021.  ISSN 1742-5468.  doi:10.1088/1742-5468/abed4e.
  * Dijkstra et al. (2019) H. A. Dijkstra, E. Hernández-García, C. Masoller, and M. Barreiro.  _Networks in Climate_.  Cambridge University Press, Cambridge, England, UK, Feb 2019.  ISBN 978-1-31627575-7.  doi:10.1017/9781316275757.
  * Domeisen et al. (2019) D. I. V. Domeisen, C. I. Garfinkel, and A. H. Butler.  The Teleconnection of El Niño Southern Oscillation to the Stratosphere.  _Rev. Geophys._ , 57(1):5–47, Mar 2019.  ISSN 8755-1209.  doi:10.1029/2018RG000596.
  * Donges et al. (2009a) J. F. Donges, Y. Zou, N. Marwan, and J. Kurths.  The backbone of the climate network.  _EPL (Europhysics Letters)_ , 87(4):48007, sep 2009a.  ISSN 0295-5075.  doi:10.1209/0295-5075/87/48007.
  * Donges et al. (2009b) J. F. Donges, Y. Zou, N. Marwan, and J. Kurths.  Complex networks in climate dynamics.  _The European Physical Journal Special Topics_ , 174(1):157–179, 2009b.
  * Donges et al. (2015) J. F. Donges, I. Petrova, A. Loew, N. Marwan, and J. Kurths.  How complex climate networks complement eigen techniques for the statistical analysis of climatological data.  _Clim. Dyn._ , 45(9):2407–2424, Nov 2015.  ISSN 1432-0894.  doi:10.1007/s00382-015-2479-3.
  * Ebert-Uphoff and Deng (2012) I. Ebert-Uphoff and Y. Deng.  A new type of climate network based on probabilistic graphical models: Results of boreal winter versus summer.  _Geophysical Research Letters_ , 39(18):1–7, 2012\.  ISSN 00948276.  doi:10.1029/2012GL053269.
  * Farooq et al. (2019) H. Farooq, Y. Chen, T. T. Georgiou, A. Tannenbaum, and C. Lenglet.  Network curvature as a hallmark of brain structural connectivity - Nature Communications.  _Nat. Commun._ , 10(4937):1–11, Oct 2019.  ISSN 2041-1723.  doi:10.1038/s41467-019-12915-x.
  * Forman (2003) R. Forman.  Bochner’s method for cell complexes and combinatorial Ricci curvature.  _Discrete and Computational Geometry_ , 29(3):323–374, 2003.
  * Freeman (1977) L. C. Freeman.  A Set of Measures of Centrality Based on Betweenness.  _Sociometry_ , 40(1):35–41, 1977.  URL https://www.jstor.org/stable/3033543.
  * Gao et al. (2019) L. Gao, X. Liu, Y. Liu, P. Wang, M. Deng, Q. Zhu, and H. Li.  Measuring road network topology vulnerability by Ricci curvature.  _Physica A_ , 527:121071, Aug 2019.  ISSN 0378-4371.  doi:10.1016/j.physa.2019.121071.
  * Graf and Zanchettin (2012) H.-F. Graf and D. Zanchettin.  Central Pacific El Niño, the “subtropical bridge,” and Eurasian climate.  _J. Geophys. Res. Atmos._ , 117(D1), Jan 2012.  ISSN 0148-0227.  doi:10.1029/2011JD016493.
  * Hagberg et al. (2008) A. A. Hagberg, D. A. Schult, and P. J. Swart.  Exploring network structure, dynamics, and function using networkx.  In G. Varoquaux, T. Vaught, and J. Millman, editors, _Proceedings of the 7th Python in Science Conference_ , pages 11 – 15, Pasadena, CA USA, 2008.  URL http://conference.scipy.org/proceedings/SciPy2008/paper_2.
  * Hersbach et al. (2018) H. Hersbach, B. Bell, P. Berrisford, G. Biavati, A. Horányi, J. Muñoz Sabater, J. Nicolas, C. Peubey, R. Radu, I. Rozum, D. Schepers, A. Simmons, C. Soci, D. Dee, and J.-N. Thépaut.  Era5 hourly data on single levels from 1979 to present, 2018.  Accessed on 02-03-2022.
  * Hersbach et al. (2020) H. Hersbach, B. Bell, P. Berrisford, S. Hirahara, A. Horányi, J. Muñoz-Sabater, J. Nicolas, C. Peubey, R. Radu, D. Schepers, A. Simmons, C. Soci, S. Abdalla, X. Abellan, G. Balsamo, P. Bechtold, G. Biavati, J. Bidlot, M. Bonavita, G. De Chiara, P. Dahlgren, D. Dee, M. Diamantakis, R. Dragani, J. Flemming, R. Forbes, M. Fuentes, A. Geer, L. Haimberger, S. Healy, R. J. Hogan, E. Hólm, M. Janisková, S. Keeley, P. Laloyaux, P. Lopez, C. Lupu, G. Radnoti, P. de Rosnay, I. Rozum, F. Vamborg, S. Villaume, and J. N. Thépaut.  The ERA5 global reanalysis.  _Quarterly Journal of the Royal Meteorological Society_ , 146(730):1999–2049, 2020.  ISSN 1477870X.  doi:10.1002/qj.3803.
  * Jiménez-Esteve and Domeisen (2018) B. Jiménez-Esteve and D. I. V. Domeisen.  The Tropospheric Pathway of the ENSO–North Atlantic Teleconnection.  _J. Clim._ , 31(11):4563–4584, Jun 2018.  ISSN 0894-8755.  doi:10.1175/JCLI-D-17-0716.1.
  * Johnson (2013) N. C. Johnson.  How Many ENSO Flavors Can We Distinguish?  _J. Clim._ , 26(13):4816–4827, Jul 2013.  ISSN 0894-8755.  doi:10.1175/JCLI-D-12-00649.1.
  * Jonckheere and Grippo (2019) E. Jonckheere and E. Grippo.  Ollivier-Ricci Curvature Approach to Cost-Effective Power Grid Congestion Management.  In _2019 Chinese Control And Decision Conference (CCDC)_ , pages 2118–2123. IEEE, Jun 2019.  doi:10.1109/CCDC.2019.8832819.
  * Kittel et al. (2021) T. Kittel, C. Ciemer, N. Lotfi, T. Peron, F. Rodrigues, J. Kurths, and R. V. Donner.  Evolving climate network perspectives on global surface air temperature effects of ENSO and strong volcanic eruptions.  _Eur. Phys. J. Spec. Top._ , 230(14):3075–3100, Oct 2021.  ISSN 1951-6401.  doi:10.1140/epjs/s11734-021-00269-9.
  * Klein et al. (1999) S. A. Klein, B. J. Soden, and N.-C. Lau.  Remote Sea Surface Temperature Variations during ENSO: Evidence for a Tropical Atmospheric Bridge.  _J. Clim._ , 12(4):917–932, Apr 1999.  ISSN 0894-8755.  doi:10.1175/1520-0442(1999)012<0917:RSSTVD>2.0.CO;2.
  * Lambert et al. (2011) F. H. Lambert, M. J. Webb, and M. M. Joshi.  The Relationship between Land–Ocean Surface Temperature Contrast and Radiative Forcing.  _J. Clim._ , 24(13):3239–3256, Jul 2011.  ISSN 0894-8755.  doi:10.1175/2011JCLI3893.1.
  * Lu et al. (2020) Z. Lu, N. Yuan, L. Chen, and Z. Gong.  On the Impacts of El Niño Events: A New Monitoring Approach Using Complex Network Analysis.  _Geophys. Res. Lett._ , 47(6):e2019GL086533, Mar 2020.  ISSN 0094-8276.  doi:10.1029/2019GL086533.
  * Okumura (2019) Y. M. Okumura.  ENSO Diversity from an Atmospheric Perspective.  _Curr. Clim. Change Rep._ , 5(3):245–257, Sep 2019.  ISSN 2198-6061.  doi:10.1007/s40641-019-00138-7.
  * Ollivier (2009) Y. Ollivier.  Ricci curvature of markov chains on metric spaces.  _Journal of Functional Analysis_ , 256(3):810–864, 2009.
  * Ollivier (2010) Y. Ollivier.  A survey of Ricci curvature for metric spaces and markov chains.  In _Probabilistic approach to geometry_ , pages 343–381. Mathematical Society of Japan, 2010.
  * Pouryahya et al. (2018) M. Pouryahya, J. H. Oh, J. C. Mathews, J. O. Deasy, and A. R. Tannenbaum.  Characterizing Cancer Drug Response and Biological Correlates: A Geometric Network Approach - Scientific Reports.  _Sci. Rep._ , 8(6402):1–12, Apr 2018.  ISSN 2045-2322.  doi:10.1038/s41598-018-24679-3.
  * Radebach et al. (2013) A. Radebach, R. V. Donner, J. Runge, J. F. Donges, and J. Kurths.  Disentangling different types of El Ni\~no episodes by evolving climate network analysis.  _Phys. Rev. E_ , 88(5):052807, Nov 2013.  ISSN 2470-0053.  doi:10.1103/PhysRevE.88.052807.
  * Rodrigues et al. (2011) R. R. Rodrigues, R. J. Haarsma, E. J. D. Campos, and T. Ambrizzi.  The Impacts of Inter–El Niño Variability on the Tropical Atlantic and Northeast Brazil Climate.  _J. Clim._ , 24(13):3402–3422, Jul 2011.  ISSN 0894-8755.  doi:10.1175/2011JCLI3983.1.
  * Rodrigues et al. (2015) R. R. Rodrigues, E. J. D. Campos, and R. Haarsma.  The Impact of ENSO on the South Atlantic Subtropical Dipole Mode.  _J. Clim._ , 28(7):2691–2705, Apr 2015.  ISSN 0894-8755.  doi:10.1175/JCLI-D-14-00483.1.
  * Samal et al. (2018) A. Samal, R. P. Sreejith, J. Gu, S. Liu, E. Saucan, and J. Jost.  Comparative analysis of two discretizations of Ricci curvature for complex networks - Scientific Reports.  _Sci. Rep._ , 8(8650):1–16, Jun 2018.  ISSN 2045-2322.  doi:10.1038/s41598-018-27001-3.
  * Sandhu et al. (2015) R. Sandhu, T. Georgiou, E. Reznik, L. Zhu, I. Kolesov, Y. Senbabaoglu, and A. Tannenbaum.  Graph Curvature for Differentiating Cancer Networks - Scientific Reports.  _Sci. Rep._ , 5(12323):1–13, Jul 2015.  ISSN 2045-2322.  doi:10.1038/srep12323.
  * Sandhu et al. (2016) R. S. Sandhu, T. T. Georgiou, and A. R. Tannenbaum.  Ricci curvature: An economic indicator for market fragility and systemic risk.  _Sci. Adv._ , May 2016.  URL https://www.science.org/doi/10.1126/sciadv.1501495#F2.
  * Schlör and Strnad (2022) J. Schlör and F. Strnad.  netcurvature.  _Zenodo_ , v1.0.0, 2022.  doi:https://doi.org/10.5281/zenodo.6325299.
  * Sreejith et al. (2016) R. Sreejith, K. Mohanraj, J. Jost, E. Saucan, and A. Samal.  Forman curvature for complex networks.  _Journal of Statistical Mechanics: Theory and Experiment_ , 2016(6):063206, 2016.
  * Strnad and Schlör (2022) F. Strnad and J. Schlör.  climnet.  _Zenodo_ , v1.0.0, 2022.  doi:https://doi.org/10.5281/zenodo.6325661.
  * Taschetto et al. (2020) A. S. Taschetto, C. C. Ummenhofer, M. F. Stuecker, D. Dommenget, K. Ashok, R. R. Rodrigues, and S.-W. Yeh.  ENSO Atmospheric Teleconnections.  In _El Niño Southern Oscillation in a Changing Climate_ , pages 309–335. American Geophysical Union (AGU), Nov 2020.  ISBN 978-1-11954816-4.  doi:10.1002/9781119548164.ch14.
  * Timmermann et al. (2018) A. Timmermann, S.-I. An, J.-S. Kug, F.-F. Jin, W. Cai, A. Capotondi, K. M. Cobb, M. Lengaigne, M. J. McPhaden, M. F. Stuecker, K. Stein, A. T. Wittenberg, K.-S. Yun, T. Bayr, H.-C. Chen, Y. Chikamoto, B. Dewitte, D. Dommenget, P. Grothe, E. Guilyardi, Y.-G. Ham, M. Hayashi, S. Ineson, D. Kang, S. Kim, W. Kim, J.-Y. Lee, T. Li, J.-J. Luo, S. McGregor, Y. Planton, S. Power, H. Rashid, H.-L. Ren, A. Santoso, K. Takahashi, A. Todd, G. Wang, G. Wang, R. Xie, W.-H. Yang, S.-W. Yeh, J. Yoon, E. Zeller, and X. Zhang.  El Niño–Southern Oscillation complexity - Nature.  _Nature_ , 559:535–545, Jul 2018.  ISSN 1476-4687.  doi:10.1038/s41586-018-0252-6.
  * Trenberth (1997) K. E. Trenberth.  The Definition of El Niño.  _Bull. Am. Meteorol. Soc._ , 78(12):2771–2778, 1997.  ISSN 0003-0007.  doi:10.1175/1520-0477(1997)078<2771:TDOENO>2.0.CO;2.
  * Tsonis and Swanson (2008) A. A. Tsonis and K. L. Swanson.  Topology and Predictability of El Ni\~no and La Ni\~na Networks.  _Phys. Rev. Lett._ , 100(22):228502, Jun 2008\.  ISSN 1079-7114.  doi:10.1103/PhysRevLett.100.228502.
  * Wiedermann et al. (2016) M. Wiedermann, A. Radebach, J. F. Donges, J. Kurths, and R. V. Donner.  A climate network-based index to discriminate different types of El Niño and La Niña.  _Geophys. Res. Lett._ , 43(13):7176–7185, Jul 2016.  ISSN 0094-8276.  doi:10.1002/2016GL069119.
  * Yamasaki et al. (2008) K. Yamasaki, A. Gozolchiani, and S. Havlin.  Climate Networks around the Globe are Significantly Affected by El Ni\~no.  _Phys. Rev. Lett._ , 100(22):228501, Jun 2008\.  ISSN 1079-7114.  doi:10.1103/PhysRevLett.100.228501.
  * You and Furtado (2018) Y. You and J. C. Furtado.  The South Pacific Meridional Mode and Its Role in Tropical Pacific Climate Variability.  _J. Clim._ , 31(24):10141–10163, Dec 2018.  ISSN 0894-8755.  doi:10.1175/JCLI-D-17-0860.1.
  * Zhang and Han (2021) L. Zhang and W. Han.  Indian Ocean Dipole leads to Atlantic Niño - Nature Communications.  _Nat. Commun._ , 12(5952):1–9, Oct 2021.  ISSN 2041-1723.  doi:10.1038/s41467-021-26223-w.
  * Zhang et al. (2015) W. Zhang, Y. Wang, F.-F. Jin, M. F. Stuecker, and A. G. Turner.  Impact of different El Niño types on the El Niño/IOD relationship.  _Geophys. Res. Lett._ , 42(20):8570–8576, Oct 2015.  ISSN 0094-8276.  doi:10.1002/2015GL065703.
  * Zhou et al. (2015) D. Zhou, A. Gozolchiani, Y. Ashkenazy, and S. Havlin.  Teleconnection Paths via Climate Network Direct Link Detection.  _Phys. Rev. Lett._ , 115(26):268501, Dec 2015\.  ISSN 1079-7114.  doi:10.1103/PhysRevLett.115.268501.

## Supplementary Information

##  Appendix A Complex Network Measures

Assume complex network graph $G$, defined by its set of nodes $V$, connected
by its set of edges $E$, where $e_{ij}$ denotes an edge connecting node
$v_{i}$ to node $v_{j}$. This type of a graph is described by its adjacency
matrix $\mathbf{A}$:

$\displaystyle\mathbf{A}_{ij}=\begin{cases}1,&e_{ij}\in E\\\
0,&\textrm{otherwise}\end{cases},$

If the edges $e_{ij}$ are weighted by weights $w_{ij}$, the weighted graph is
described by

$\displaystyle\mathbf{W}_{ij}=\begin{cases}w_{ij},&e_{ij}\in E\\\
0,&\textrm{otherwise}\end{cases},$

#### Node degree

The node degree $\tilde{k}_{i}$ and the weighted node degree $k_{i}$ is a
frequently applied measure to study complex networks (Donges et al., 2009b,
Boers et al., 2013, Donges et al., 2015). It counts the (weighted) number of
neighboring nodes to node $v_{i}$, defined by:

$\displaystyle k_{i}$ $\displaystyle=\sum_{j}^{N}\mathbf{W}_{ij}\;,$ (5)
$\displaystyle\tilde{k}_{i}$ $\displaystyle=\sum_{j}^{N}\mathbf{A}_{ij}\;,$
(6)

where $N$ denotes the number of nodes $v_{i}\in V$.

In Figure 6 the node degree is visualized for normal (6 a), EP (6 b) and CP (6
c) networks.

Figure 6: Node degree for normal, EP and CP conditions. The weighted node
degree is computed following Eq. 5. The climate networks are obtained for
normal (a), on Eastern Pacific El Niño (b) and on Central Pacific El Niño (c)
conditions.

#### Betweenness Centrality

Betweenness Centrality is defined for nodes $v$ and edges $e$ as,

$\displaystyle BC_{v}(v_{i})$
$\displaystyle=\sum_{s,t}^{N}\frac{\sigma(v_{s},v_{t}|v_{i})}{\sigma(v_{s},v_{t})}\;,$
(7) $\displaystyle BC_{e}(e_{ij})$
$\displaystyle=\sum_{s,t}^{N}\frac{\sigma(v_{s},v_{t}|e_{ij})}{\sigma(v_{s},v_{t})}\;,$
(8)

where $\sigma(v_{s},v_{t})$ denotes the number of shortest paths between nodes
$v_{s}$ and $v_{t}$ and $\sigma(v_{s},v_{t}|v_{i})\leq\sigma(v_{s},v_{t})$ the
number of all shortest paths that include node $v_{i}$. Similarly for edge
betweenness centrality $\sigma(v_{s},v_{t}|e_{ij})\leq\sigma(v_{s},v_{t})$
yields the number of all shortest paths that include edge $e_{ij}$. For
instance, betweenness centrality measures were used in (Freeman, 1977, Donges
et al., 2009a, Boers et al., 2013, Ciemer et al., 2018). $BC_{n}$ is often
referred to as pathway of a variable through the network and therefore taken
as an indicator for the flow of the variable of interest. In Figure 7 the node
degree is visualized for normal year (7 a), EP (7 b) and CP (7 c) conditions.

Figure 7: Node Betweenness Centrality for normal, EP and CP conditions. The
betweenness centrality is computed using Eq. 7. The climate networks obtained
for normal (a), on Eastern Pacific El Niño (b) and on Central Pacific El Niño
(c) conditions.

#### Clustering Coefficient

The clustering coefficient of a node $v_{i}$ is defined by the fraction of
possible triangles through the actual number of nodes in the network
normalized by the unweighted node degree $\tilde{k}_{i}$:

$\displaystyle
c_{i}=\frac{2\mathcal{T}(v_{i})}{\tilde{k}_{i}(\tilde{k}_{i}-1)}\;.$ (9)

Here, $\mathcal{T}(v_{i})$ describes the number of triangles including node
$v_{i}$.

Figure 8: Clustering Coefficient for normal, EP and CP conditions. The
clustering coefficient is computed using Eq. 9. Networks are obtained for
normal (a), EP (b) and CP (c) conditions. Figure 9: Same as Fig. 3 but with
equal scales. The network curvature scales are fixed between the networks
(a-i). The differences between the distributions of node curvature
$\tilde{f}_{i}$ of normal (j), EP (k) and CP (l) networks could be attributed
to their network topologies. The quantiles $Q_{F}(0.1)$ and $Q_{F}(0.1)$ are
indicated by black lines (j-l). In order to plot the quantiles of node
curvature we have to adapt the scales, as done in the main text Fig. 3. Figure
10: Average Spearmann’s correlation coefficient. SAT time series at one
location is Spearman-correlated against all other time series of the global
dataset. The average Spearman’s correlation coefficient is shown for the
climate networks constructed based on normal (a), Eastern Pacific El Niño (b)
and Central Pacific El Niño (c) conditions. Figure 11: SAT fields regressed on
temperatures from the four selected regions. SAT time series (not normalized)
at each location is regressed against all the rank-normalized SAT time series
inside the four regions (black rectangles). Displayed are the Niño 3 box (1.
row), Niño 4 box (2. row), Indian Ocean (3. row), and Labrador Sea (4. row),
with first column denoting EP and second column CP conditions. Figure 12: SAT
fields (Z-score normalized) regressed on temperatures from the four selected
regions. SAT time series at each location is regressed against all the rank-
normalized SAT time series inside the four regions (black rectangles).
Displayed are the Niño 3 box (1. row), Niño 4 box (2. row), Indian Ocean (3.
row), and Labrador Sea (4. row), with first column denoting EP and second
column CP conditions. Figure 13: SAT fields (min-max normalized) regressed on
temperatures from the four selected regions. SAT time series at each location
is regressed against all the rank-normalized SAT time series inside the four
regions (black rectangles). Displayed are the Niño 3 box (1. row), Niño 4 box
(2. row), Indian Ocean (3. row), and Labrador Sea (4. row), with first column
denoting EP and second column CP conditions.