An adjacent implication is that the radial functions are complete (i.e.
isomorphic to canonical confluent Heun polynomials). If the radial functions
are both orthogonal and complete for given value of the frequency parameter
$\tilde{\omega}$, then this would strongly imply the existence of a
biorthogonal dual for the radial solutions determined by the QNM frequencies.
Again, this would be directly analogous to the QNM’s spheroidal harmonics,
where each spheroidal harmonic requires input from a different QNM frequency
London (2023). A consequence would be that the QNMs are “quasi-complete”, i.e.
complete in the radial and angular domains, but not complete (specifically
under- or over-complete) in the time domain. Investigation of these
implications is also a topic of ongoing work that is near conclusion London
and Gurevich (2023b).

Lastly, these ideas imply a potentially useful direction of research for e.g.
BBH post-merger signals: rather than attempting to fit various QNMs’ within
numerical data, it may be possible to directly project out individual QNMs’
time series using orthogonality. In the context of ongoing efforts in
Numerical Relativity and signal modeling for future gravitational wave
detectors, this may be a fruitful direction for future research.

##  X Acknowledgements

The author is thankful to Katy Clough, Michelle Gurevich, Scott Hughes,
Bernard Whiting and Melize Ferrus for useful discussions and feedback. Lionel
London was supported at the University of Amsterdam by the GRAPPA Prize
Postdoctoral Fellowship, and then at King’s College London by the Royal
Society (URF\R1\211451).

##  Appendix A Notation

This section lists explicit expressions for the main text’s schematic
notation. The details here are provided for convenience and completeness.

In Boyer-Lindquist coordinates, Teukolsky’s master differential operator,
$\mathcal{L}_{tr\theta\phi}$, is

$\displaystyle\mathcal{L}_{tr\theta\phi}\;\;=\;\left[\frac{(r^{2}+a^{2})^{2}}{\Delta(r)}-a^{2}\sin^{2}\theta\right]\frac{\partial^{2}}{\partial{t}^{2}}$
(116)
$\displaystyle+\frac{4Mar}{\Delta(r)}\frac{\partial^{2}}{\partial{t}\partial\phi}+\left[\frac{a^{2}}{\Delta(r)}-\frac{1}{\sin^{2}\theta}\right]\frac{\partial^{2}}{\partial\phi^{2}}$
$\displaystyle-\Delta(r)^{-s}\frac{\partial}{\partial{r}}\left(\Delta(r)^{s+1}\frac{\partial}{\partial{r}}\right)-\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)$
$\displaystyle-2s\left[\frac{a(r-M)}{\Delta(r)}+\frac{i\cos\theta}{\sin^{2}\theta}\right]\frac{\partial}{\partial\phi}$
$\displaystyle-2s\left[\frac{M(r^{2}-a^{2})}{\Delta(r)}-r-ia\cos\theta\right]\frac{\partial}{\partial{t}}$
$\displaystyle+(s^{2}\cot^{2}\theta-s)\;,$

where

$\displaystyle\Delta(r)\;$ $\displaystyle=\;(r-r_{+})(r-r_{-})\;,$ (117a) and
$\displaystyle r_{\pm}\;$ $\displaystyle=M\,\pm\,\sqrt{M^{2}-a^{2}}\;.$ (117b)

Upon separation via a separable ansatz or integral transform (see Sec. III),
Teukolsky’s angular differential operator is encapsulated by the following
expression,

$\displaystyle u\;=\;\cos(\theta)\;,$ (118a)
$\displaystyle\mathcal{L}_{u}=V_{S}(u)+\partial_{u}(1-u^{2})\partial_{u}\;,$
(118b) $\displaystyle
V_{S}(u)\;=\;s+ua\tilde{\omega}(ua\tilde{\omega}-2s)-\frac{(m+su)^{2}}{1-u^{2}}\;.$
(118c)

Similarly, Teukolsky’s radial differential operator is commonly written as

$\displaystyle\mathcal{L}{{}_{r}}\;=\;$ $\displaystyle
V_{R}(r)\;+\;\Delta(r)^{-s}\partial_{r}\left[\Delta(r)^{s+1}\partial_{r}\right]\;,$
(119)

where

$\displaystyle V_{R}(r)\;$
$\displaystyle=\;\frac{K(r)^{2}-2is(r-M)K(r)}{\Delta(r)}+\;4is\tilde{\omega}{}{r}$
(120a) $\displaystyle\hskip
96.73918pt-a^{2}\tilde{\omega}{}^{2}+2am\tilde{\omega}{}$ $\displaystyle
K(r)\;$ $\displaystyle=\;(r^{2}+a^{2})\,\tilde{\omega}{}-am\;.$ (120b)

The expressions above may also be found in Refs. Teukolsky and Press (1974);
Leaver (1985); O’Sullivan and Hughes (2014).

##  Appendix B Solving Teukolsky’s angular equation with Jacobi polynomials

Teukolsky’s angular equation is

$\displaystyle\mathcal{L}_{u}\,S(u)\;=\;-\,AS(u)\;.$ (121)

In Eq. (121),

$\displaystyle u\;$ $\displaystyle=\;\cos(\theta)\;,$ (122a)
$\displaystyle\mathcal{L}_{u}$
$\displaystyle=V_{S}(u)+\partial_{u}(1-u^{2})\partial_{u}\;,$ (122b) where the
equation’s potential, $V_{S}$, is $\displaystyle V_{S}(u)\;$
$\displaystyle=\;s+u\gamma(u\gamma-2s)-\frac{(m+su)^{2}}{1-u^{2}}\;$ (122c)
with $\displaystyle\gamma\;$ $\displaystyle=\;a\tilde{\omega}\;.$ (122d)

Like Teukolsky’s radial equation, Eq. (121) is a confluent Heun equation;
however, Eq. (121) differs from the radial case in two ways of current
relevance:

  * (i)

Equation (121) has two regular singular points at $u\in\\{-1,1\\}$, and one
irregular singular point at $u\rightarrow\infty$. The irregular singular point
at $u\rightarrow\infty$ is not within the physical domain (i.e. there is no
boundary condition defined there). The relevant asymptotic boundary conditions
are essentially that $S(u)$ are regular when $u\in\\{-1,1\\}$.

  * (ii)

The degeneration of Eq. (121) to a hypergeometric equation occurs at a
physical limit ($a\rightarrow 0$), whereas the degeneration of the radial
equation to a hypergeometric one occurs at a non-physical limit for QNMs
($\tilde{\omega}\rightarrow 0$).

Respectively, these differences mean that (i) global171717By global, one means
a solution that applies to the entire domain. This is in contrast to local
solutions, such as those use in the MST method, that are only valid in the
vicinity of a singular point Mano  _et al._ (1996); Ronveaux and Arscott
(1995). solutions to the angular equation can be solved with high accuracy
using non-confluent basis functions, and that (ii) the optimal basis functions
may be constructed from physical solutions at the $a\rightarrow 0$ limit.

The construction of analytic solutions begins by finding a similarity
transformation that results in the transformed potential being regular when
$u\in[-1,1]$. The similarity transformation is equivalent to taking the
solution ansatz,

$\displaystyle S(u)\;=\;\eta(u)\,g(u)\;,$ (123)

where,

$\displaystyle\eta(u)\;=\;e^{uk_{2}}\;(1-u)^{k_{0}}\;(1+u)^{k_{1}}\;,$ (124)

and then applying it to the angular equation thusly,

$\displaystyle\left[\eta(u)^{-1}\,\mathcal{L}{{}_{u}}\,\eta(u)\right]g(u)\;$
$\displaystyle=\;-\left[\eta(u)^{-1}A\,\eta(u)\right]g(u)\;$ (125a)
$\displaystyle\mathcal{L}{{}_{u}^{\prime}}\;$
$\displaystyle=\;\left[\eta(u)^{-1}\,\mathcal{L}{{}_{u}}\,\eta(u)\right]$
(125b) $\displaystyle\mathcal{L}{{}_{u}^{\prime}}\,g(u)\;$
$\displaystyle=\;-\,A\,g(u)\;.$ (125c)

In Eq. (125a), the similarity transformation of $\mathcal{L}{{}_{u}}$ is
explicitly performed. In Eq. (125b), the transformed operator,
$\mathcal{L}{{}_{u}^{\prime}}$, is defined, and in Eq. (125c) its eigenvalue
problem is stated.

The values of $k_{0}$, $k_{1}$ and $k_{2}$ that result in the potential of
$\mathcal{L}{{}_{u}^{\prime}}$ being regular and linear on $u\in\\{-1,1\\}$
are,

$\displaystyle k_{0}\;$ $\displaystyle=\;\frac{1}{2}|m+s|\;,$ (126a)
$\displaystyle k_{1}\;$ $\displaystyle=\;\frac{1}{2}|m-s|\;,$ (126b)
$\displaystyle k_{2}\;$ $\displaystyle=\;\gamma\;.$ (126c)

Equivalently, $k_{0}$, $k_{1}$ and $k_{2}$ are the singular exponents that may
be found via the method of Frobenius Courant and Hilbert. (1954); Arfken  _et
al._ (2013); Pinchover and Rubinstein (2005). In effect, $k_{0}$ and $k_{1}$
remove divergences from the potential at the regular singular points, and
$k_{2}$ removes the potential’s quadratic dependence on $u$. The appearance of
the absolute values, e.g. $|m+s|$, ensures that $\eta$ is always finite when
$u\in\\{-1,1\\}$, regardless of the values of $m$ and $s$. This is equivalent
to enforcing asymptotic boundary conditions on $S(u)$. In Eq. (126c),
$-\gamma$ is also a viable solution for $k_{2}$; however, $k_{2}=+\gamma$ is
chosen by convention Leaver (1985). The result for
$\mathcal{L}{{}_{u}^{\prime}}$ has two schematic forms of relevance,

$\displaystyle\mathcal{L}_{u}^{\prime}\;=\;$
$\displaystyle(\text{a}_{0}+\text{a}_{1}\,u)$ (127a)
$\displaystyle+\;\left(\text{a}_{2}+\text{a}_{3}\,u+\text{a}_{4}\,[1-u^{2}]\right)\,\partial_{u}$
(127b) $\displaystyle+\;\left(1-u^{2}\right)\partial^{2}_{u}\;,$ (127c)

and

$\displaystyle\mathcal{L}_{u}^{\prime}\;=\;$
$\displaystyle(\text{a}_{0}+\text{a}_{1}\,u)\,+\,\mathcal{D}^{\prime}_{u}\;.$
(128)

In Eq. (127), $\text{a}_{0}$ through $\text{a}_{4}$ are constant coefficients
defined in Eqs. (129a-129e). They should not to be confused with the similarly
labeled coefficients defined in the main text.

$\displaystyle\text{a}_{0}\;$
$\displaystyle=\;-k_{0}(2k_{1}+1)-k_{1}-\frac{1}{2}[m^{2}+s(s+2)]$ (129a)
$\displaystyle\;\;\;\;\;\,+\gamma^{2}+2\gamma(k_{1}-k_{0})$
$\displaystyle\text{a}_{1}\;$ $\displaystyle=\;-2\gamma(k_{0}+k_{1}+s+1)$
(129b) $\displaystyle\text{a}_{2}\;$ $\displaystyle=\;-2(k_{0}-k_{1})$ (129c)
$\displaystyle\text{a}_{3}\;$ $\displaystyle=\;-2(k_{0}+k_{1}+1)$ (129d)
$\displaystyle\text{a}_{4}\;$ $\displaystyle=\;2\gamma$ (129e)

Equation (128) defines $\mathcal{D}^{\prime}_{u}$ to be the differential part
of Eq. (127) (i.e. Eqs. 127b and 127c). The operator,
$\mathcal{D}^{\prime}_{u}$, is useful for understanding the eigenproblem (Eq.
125c) in the non-spinning limit (i.e. $\gamma=a\tilde{\omega}=0$), and for
general spins.

In the non-spinning limit, $\mathcal{D}^{\prime}_{u}$ is simply
$\mathcal{D}^{\text{Jac.}}_{u}$, the differential operator for which the
Jacobi polynomials are eigenfunctions:

$\displaystyle\mathcal{D}^{\prime}_{u}|_{\gamma=0}\;$
$\displaystyle=\;\mathcal{D}^{\text{Jac.}}_{u}\;.$ (130)

The Jacobi operator is

$\displaystyle\mathcal{D}^{\text{Jac.}}_{u}\;=\;$
$\displaystyle-(\,\alpha-\beta+(\alpha+\beta+2)\,u\,)\partial_{u}$ (131)
$\displaystyle+(1-u^{2})\partial^{2}_{u}\;,$

where

$\displaystyle\alpha\;$ $\displaystyle=\;2\,k_{0}\,,$ (132)
$\displaystyle\beta\;$ $\displaystyle=\;2\,k_{1}\,.$ (133)

The simplest functions generated by $\mathcal{D}^{\text{Jac.}}_{u}$ are the
Jacobi polynomials, $p_{n}(u)$. Their eigenvalue relationship is

$\displaystyle\mathcal{D}^{\text{Jac.}}_{u}\,p_{n}(u)\;$
$\displaystyle=\;-n(\alpha+\beta+n+1)\,p_{n}(u)\;.$ (134)

In Eq. (134), the eigenvalue is determined by the requirement that the series
solution for $p_{n}(u)$ terminates after $n$ terms.

The zero-spin eigenvalue, $A|_{a=0}=A|_{\gamma=0}$, can be determined by
requiring equivalence between the physical eigenvalue problem (Eq. 125c), and
the Jacobi one (Eq. 134). This amounts to holding the non-derivative terms in
each eigenvalue relation to be equivalent in the zero spin limit,

$\displaystyle(a_{0}+A)|_{\gamma\rightarrow 0}\;=\;n(\alpha+\beta+n+1)$ (135)

Equation (135) can be solved for the eigenvalue,

$\displaystyle A_{0}\;$ $\displaystyle=\;A|_{\gamma=0}$ (136a)
$\displaystyle=\;-\left(\ell-s\right)\left(\ell+s+1\right)\;.$ (136b)

In Eq. (136b), $A_{0}$ is the usual spin weighted spherical harmonic
eigenvalue Newman and Penrose (1966); Leaver (1985); Fackerell and Crossman
(2008), and index $\ell$ is related to the polynomial order, $n$, via

$\displaystyle\ell\;$ $\displaystyle=\;\ell_{\min}+n\,,$ (137a)
$\displaystyle\ell_{\min}\;$ $\displaystyle=\;k_{0}+k_{1}\;.$ (137b)

For non-zero spin, term by term comparison of $\mathcal{D}^{\prime}_{u}$ and
$\mathcal{D}^{\text{Jac.}}_{u}$ yields

$\displaystyle\mathcal{D}^{\prime}_{u}\;$
$\displaystyle=\;\mathcal{D}^{\text{Jac.}}_{u}+\text{a}_{4}(1-u^{2})\partial_{u}\;.$
(138)

Equation (138), in effect, casts $\mathcal{D}^{\prime}_{u}$ as a deformation
of the Jacobi operator. This implies that it is natural to consider the
spheroidal functions $g(u)$ to be deformations of the Jacobi polynomials. In
particular, we will take $g(u)$ to be exactly represented by Jacobi
polynomials,

$\displaystyle g(u)\;=\;\sum_{n=0}^{\infty}\,a_{n}\,p_{n}(u)\;.$ (139)

Equation (139) is justified not only because the Jacobi polynomials are known
to be complete, but also because $g(u)$ and $p_{n}(u)$ satisfy the same
asymptotic boundary conditions on the same domain.

Equation (139) is an analytic solution for $g(u)$ that is motivated by the
structure of Teukolsky’s angular equation. In effect, Eq. (139) is a spectral
decomposition of $g(u)$ into Jacobi moments, $a_{n}$. Thus, finding the values
of each $a_{n}$ is equivalent to finding $g(u)$.

To do so, the Jacobi polynomials’ recursive and orthogonality properties must
be used. The relevant recursion rules for Jacobi polynomials are

$\displaystyle(1-u^{2})\,\partial_{u}p_{n}\;$
$\displaystyle=\;\text{b}_{0}\,p_{n-1}+\text{b}_{1}\,p_{n}+\text{b}_{2}\,p_{n+1}\;,$
(140a) $\displaystyle u\,p_{n}\;$
$\displaystyle=\;\text{b}_{3}\,p_{n-1}+\text{b}_{4}\,p_{n}+\text{b}_{5}\,p_{n+1}\;.$
(140b)

Normalization of the Jacobi polynomials is defined with respect to the
weighted inner product,

$\displaystyle{\langle{p_{n}}\,|\,{p_{n^{\prime}}}\rangle}\;$
$\displaystyle=\;\int_{-1}^{1}p_{n}(u)\,p_{n^{\prime}}(u)\;\text{W}^{\text{Jac.}}(u)\,\text{d}u$
(141a) $\displaystyle=\;\text{b}_{6}\,\delta_{n^{\prime}n}\;,$ (141b)

where the Jacobi weight function, $\text{W}^{\text{Jac.}}(u)$, is

$\displaystyle\text{W}^{\text{Jac.}}\;=\;(1-u)^{\alpha}(1+u)^{\beta}\;.$ (142)

In Eqs. (140) and (141), $\text{b}_{0}$ through $\text{b}_{6}$ are $n$
dependent constants defined in Eqs. (143a-143g).

$\displaystyle\text{b}_{0}(n)\;$
$\displaystyle=\;-\frac{2(\alpha+n)(\beta+n)(\alpha+\beta+n+1)}{(\alpha+\beta+2n)(\alpha+\beta+2n+1)}$
(143a) $\displaystyle\text{b}_{1}(n)\;$
$\displaystyle=\;-\frac{2n(\beta-\alpha)(\alpha+\beta+n+1)}{(\alpha+\beta+2n)(\alpha+\beta+2n+2)}$
(143b) $\displaystyle\text{b}_{2}(n)\;$
$\displaystyle=\;\frac{2n(n+1)(\alpha+\beta+n+1)}{(\alpha+\beta+2n+1)(\alpha+\beta+2n+2)}$
(143c) $\displaystyle\text{b}_{3}(n)\;$
$\displaystyle=\;-\frac{2(\alpha+n)(\beta+n)}{(\alpha+\beta+2n)(\alpha+\beta+2n+1)}$
(143d) $\displaystyle\text{b}_{4}(n)\;$
$\displaystyle=\;-\frac{\alpha^{2}-\beta^{2}}{(\alpha+\beta+2n)(\alpha+\beta+2n+2)}$
(143e) $\displaystyle\text{b}_{5}(n)\;$
$\displaystyle=\;-\frac{2(n+1)(\alpha+\beta+n+1)}{(\alpha+\beta+2n+1)(\alpha+\beta+2n+2)}$
(143f) $\displaystyle\text{b}_{6}(n)\;$
$\displaystyle=\;\frac{\Gamma(n+\alpha+1)\Gamma(n+\beta+1)}{(\alpha+\beta+2n+1)\Gamma(n+1)\Gamma(n+\alpha+\beta+1)}$
(143g)

Observe that $\text{W}^{\text{Jac.}}(u)$ is very similar to the scaling
function $\eta(u)$ introduced in Eq. (124). This can be understood as
resulting from the fact that the transformation present in Eq. (125a) is
conformal, meaning that

$\displaystyle\int_{-1}^{1}p_{n}(u)\,p_{n^{\prime}}$
$\displaystyle(u)\,\text{W}^{\text{Jac.}}(u)\,\text{d}u$ (144a)
$\displaystyle=\;$ $\displaystyle\int_{-1}^{1}([p_{n}(u)\eta(u)$
$\displaystyle]\,[p_{n^{\prime}}(u)\eta(u)])|_{\gamma=0}\,\text{d}u$ (144b)

There is no weight function in Eq. (144b) because Eq. (122b) is explicitly in
Sturm-Liouville form (with weight function $1$) Arfken  _et al._ (2013). The
integrands of Eqs. (144a) and (144b) must be equivalent, meaning that
$\text{W}^{\text{Jac.}}(u)$ and $\eta(u)|_{\gamma=0}$ are related,

$\displaystyle\text{W}^{\text{Jac.}}(u)\;=\;\eta(u)^{2}|_{\gamma=0}\;.$ (145)

In general, it can be shown that if $f_{1}$ satisfies the second order linear
differential equation $\mathcal{L}_{1}f_{1}=0$, where $\mathcal{L}$ is self-
adjoint with weight function $\lambda_{1}$, then $f_{2}=\eta^{-1}f_{1}$ will
satisfy $\mathcal{L}_{2}g=0$, where
$\mathcal{L}_{2}=\eta^{-1}\mathcal{L}_{1}\eta$ is self-adjoint with new weight
function $\lambda_{2}=\eta^{2}\lambda_{1}$.

Equations (140-143) can be used to determine the values of $a_{n}$ as follows.
Given the scalar product, Eq. (141a), we will use bra-ket notation to simplify
to representation of various steps. In that notation, the completeness of the
Jacobi polynomials is expressed as

$\displaystyle\sum_{n=0}^{\infty}\,|{p_{n}}\rangle\langle{p_{n}}|\;=\;{\mathbb{I}}\;.$
(146)

In Eq. (146), ${\mathbb{I}}$ is the identity operator for all functions that
have the same asymptotic behavior as the Jacobi polynomials on $u\in[-1,1]$,
and the polynomials $|{p_{n}}\rangle$ are implicitly normalized so that
${\langle{p_{n}}\,|\,{p_{n}}\rangle}=1$. The bra-ket notation form of the
physical eigenvalue relation is,

$\displaystyle\mathcal{L}{{}_{u}^{\prime}}\,|{g}\rangle\;$
$\displaystyle=\;-\,A\,|{g}\rangle\;.$ (147)

From this perspective, the Jacobi moments, $a_{n}$, are simply the components
of $g(u)$ in the basis of Jacobi polynomials. This is made explicit by using
Eq. (146) to express both sides of Eq. (147) in the Jacobi polynomial basis,

$\displaystyle\sum_{n,n^{\prime}}|{p_{n^{\prime}}}\rangle\langle{p_{n^{\prime}}}|\mathcal{L}{{}_{u}^{\prime}}|{p_{n}}\rangle{\langle{p_{n}}\,|\,{g}\rangle}\;$
$\displaystyle=\;-\,A\,\sum_{n}|{p_{n}}\rangle{\langle{p_{n}}\,|\,{g}\rangle}\;.$
(148)

Note that the right hand side of Eq. (148) is equivalent to that of Eq. (138)
(i.e. $a_{n}={\langle{p_{n}}\,|\,{g}\rangle}$). Equation (148) may be recast
as a simple matrix equation

$\displaystyle\hat{L}\,\vec{g}\;$ $\displaystyle=\;-A\,\vec{g}\;,$ (149)

where $\hat{L}$ and $\vec{g}$ are defined from Eq. (148) as

$\displaystyle\hat{L}\;$
$\displaystyle=\;\\{\langle{p_{n^{\prime}}}|\mathcal{L}{{}_{u}^{\prime}}|{p_{n}}\rangle\\}_{n,n^{\prime}}\;,$
(150a) $\displaystyle\vec{g}\;$
$\displaystyle=\;\\{{\langle{p_{n}}\,|\,{g}\rangle}\\}_{n}\;.$ (150b)

Thus, taking a finite dimensional truncation of $\hat{L}$ allows $\vec{g}$
(i.e. $a_{n}$) to be determined using standard linear algebraic methods Axler
(2015); Virtanen  _et al._ (2020).

The essential feature of this approach is that, starting with Eq. (149), the
recursion and orthogonality relationships, Eqs. (140-143), can be used to show
that $\hat{L}$ is at most tridiagonal. In particular, when
$\mathcal{L}{{}_{u}^{\prime}}$ acts on $|{p_{n}}\rangle$, only lower to upper
diagonal terms contribute,

$\displaystyle\mathcal{L}{{}_{u}^{\prime}}|{p_{n}}\rangle\;=\;\sigma_{n}\,|{p_{n-1}}\rangle+\mu_{n}\,|{p_{n}}\rangle+\nu_{n}\,|{p_{n+1}}\rangle\;.$
(151)

In Eq. (151), the $n$ dependent constants, $\sigma_{n}$, $\mu_{n}$, and
$\nu_{n}$ are

$\displaystyle\sigma_{n}\;$
$\displaystyle=\;\frac{\text{b}_{6}(n-1)}{\text{b}_{6}(n)}[\text{a}_{1}\text{b}_{3}(n)+\text{a}_{4}\text{b}_{0}(n)]$
(152a) $\displaystyle\mu_{n}\;$
$\displaystyle=\;\text{a}_{0}+\text{a}_{1}\text{b}_{4}(n)+\text{a}_{4}\text{b}_{1}(n)$
(152b) $\displaystyle\quad\quad\quad\quad\quad\quad-n(\alpha+\beta+n+1)$
$\displaystyle\nu_{n}\;$
$\displaystyle=\;\frac{\text{b}_{6}(n+1)}{\text{b}_{6}(n)}[\text{a}_{1}\text{b}_{5}(n)+\text{a}_{4}\text{b}_{2}(n)]$
(152c)

For any $N$ dimensional truncation of $\hat{L}$, say $\hat{L}_{N}$, Eq. (152)
provides the matrix elements,

$\displaystyle\hat{L}_{N}\;$
$\displaystyle=\;\left(\begin{array}[]{ccccc}\mu_{0}&\nu_{0}&0&\cdots&0\\\
\sigma_{1}&\mu_{1}&\nu_{1}&\ddots&0\\\ 0&\sigma_{2}&\mu_{2}&\ddots&\vdots\\\
\vdots&\ddots&\ddots&\ddots&\nu_{N-1}\\\
0&0&\cdots&\sigma_{N}&\mu_{N}\end{array}\right)\;.$ (158)

The use of Eq. (158) to determine angular functions $S(u)=\eta(u)g(u)$, and
their eigenvalues $A(u)$, amounts to determining the eigenvectors and
eigenvalues of $\hat{L}_{N}$. Since this method explicitly estimates the point
spectra of $\hat{L}$, it is a purely spectral method (e.g. not a generalized
spectral method) for the construction of analytic solutions to Teukolsky’s
angular equation.

This construction facilitates the following conclusions: Since the transformed
angular equation degenerates to the Jacobi equation when $\gamma=0$, the
spheroidal functions $g(u)$ may be exactly identified with Jacobi polynomials
in that limit. For general values of $\gamma$, the fact that $\hat{L}$ is
tridiagonal means that for every $g(u)$ associated with index $\ell$, there
always exists a unique Jacobi polynomial with that same index London (2023).
In this way, the QNM index $\ell$ is inherently linked to a Jacobi polynomial
of order $n=\ell-\ell_{\text{min}}$.

##  Appendix C The confluent Heun equation in natural general form

The confluent Heun equation has many forms that differ with respect to
transformations of the independent and dependent variables Ronveaux and
Arscott (1995). Here, we briefly outline the connection between Teukolsky’s
radial equation and the confluent Heun equation in natural general form.

Perhaps the most referenced (e.g. Refs. Hortacsu (2012); Leaver (1986); Cook
(2020)) form of confluent Heun equation is,

$\displaystyle\mathcal{L}^{(1)}_{\text{cH}}\,y(z)\;=\;0\;,$ (159a) where
$\displaystyle\mathcal{L}^{(1)}_{\text{cH}}\;=\;(\alpha$ $\displaystyle\nu
z-\sigma)$ (159b) $\displaystyle\;+\,$
$\displaystyle\left[\gamma(z-1)+z(\delta+\nu(z-1))\right]\,\partial_{z}$
$\displaystyle\;+\,$ $\displaystyle(z-1)z\,\partial^{2}_{z}\;.$

Equation (159) may be derived from the Heun equation (which has $4$ regular
singular points) by confluence of two singular points (see e.g. Arfken  _et
al._ (2013)) into a single irregular one. Equation (159a) is simply a
definition of the confluent Heun equation in terms of its associated linear
differential operator, $\mathcal{L}^{(1)}_{z}$. In Eq. (159b), $\alpha$,
$\nu$, $\sigma$, $\gamma$ and $\delta$ are complex valued parameters, and $z$
is nominally real valued.

Equation (159), is the confluent Heun equation in what’s called nonsymmetrical
canonical form Ronveaux and Arscott (1995); Cook and Zalutskiy (2014); Fiziev
(2009b). It happens that this is actually not the most general form of the
confluent Heun equation. As a result, a naive comparison between Eq. (159) and
Teukolsky’s radial equation may generate unnecessary confusion. To facilitate
the upcoming discussion, we will henceforth use $r$ rather than $z$, and we
will discuss forms of the confluent Heun equation by referring to the related
differential operators, $\mathcal{L}^{(i)}_{\text{cH}}$, such that

$\displaystyle\mathcal{L}^{(i)}_{\text{cH}}\,y\;=\;0\;.$ (160)

The most general form of the confluent Heun differential operator with an
irregular singular point at infinity is (see e.g. Ref. Ronveaux and Arscott
(1995))

$\displaystyle\mathcal{L}^{(2)}_{\text{cH}}\;=\;$
$\displaystyle\left({\alpha_{0}}+{\alpha_{1}}r+{\alpha_{2}}r^{2}+\frac{{\alpha_{4}}}{r-{r_{-}}}+\frac{{\alpha_{3}}}{r-{r_{+}}}\right)$
(161)
$\displaystyle+\;\left({\alpha_{5}}+{\alpha_{6}}r+{\alpha_{7}}r^{2}\right)\,\partial_{r}$
$\displaystyle+\;(r-{r_{-}})(r-{r_{+}})\,\partial^{2}_{r}\;.$

In Eq. (161), $\mathcal{L}^{(2)}_{\text{cH}}$ corresponds to the confluent
Heun equation in what is called natural general form Ronveaux and Arscott
(1995).

In Eq. (161), $\mathcal{L}^{(2)}_{\text{cH}}$ has been formatted to ease
comparison with Teukolsky’s radial differential operator. In particular, the
two regular singular points have been placed at $r_{+}$ and $r_{-}$, just as
they are in the case of Teukolsky’s radial differential operator (Eq. 119),
which can be formatted as

$\displaystyle\mathcal{L}{{}_{r}}\;=\;$
$\displaystyle\left({\text{A}_{0}}+{\text{A}_{1}}r+\text{A}_{2}\,r^{2}+\frac{{\text{A}_{3}}}{r-{r_{-}}}+\frac{{\text{A}_{4}}}{r-{r_{+}}}\right)$
(162) $\displaystyle+\;({\text{A}_{5}}+{\text{A}_{6}}r)\,\partial_{r}$
$\displaystyle+\;(r-{r_{-}})(r-{r_{+}})\,\partial_{r}^{2}\;.$

For completeness, parameters $\text{A}_{1}$ through $\text{A}_{6}$ are defined
in terms of $r_{+}$, $r_{-}$ and $s$ as

$\displaystyle\text{A}_{0}\;=$
$\displaystyle\;\tilde{\omega}{}({r_{-}}+{r_{+}})(\tilde{\omega}{}({r_{-}}+{r_{+}})-is)$
(163a) $\displaystyle\text{A}_{1}\;=$
$\displaystyle\;\tilde{\omega}{}(\tilde{\omega}{}({r_{-}}+{r_{+}})+2is)$
(163b) $\displaystyle\text{A}_{2}\;=$ $\displaystyle\;\tilde{\omega}{}^{2}$
(163c) $\displaystyle\text{A}_{3}\;=$
$\displaystyle\;\frac{{r_{-}}^{2}\left(m^{2}+4{r_{-}}^{2}\tilde{\omega}{}^{2}\right)}{{r_{-}}-{r_{+}}}$
(163d)
$\displaystyle-\;{r_{-}}\left(m^{2}+{r_{-}}\tilde{\omega}{}(3{r_{-}}\tilde{\omega}{}+is)\right)$
$\displaystyle+\;\frac{4m{r_{-}}^{2}\tilde{\omega}{}\sqrt{{r_{-}}{r_{+}}}}{{r_{+}}-{r_{-}}}+ims\sqrt{{r_{-}}{r_{+}}}$
$\displaystyle+\;2m{r_{-}}\tilde{\omega}{}\sqrt{{r_{-}}{r_{+}}}-{r_{-}}{r_{+}}\tilde{\omega}{}({r_{-}}\tilde{\omega}{}+is)$
$\displaystyle\text{A}_{4}\;=$
$\displaystyle\;-\frac{{r_{+}}^{2}\left(m^{2}+4{r_{+}}^{2}\tilde{\omega}{}^{2}\right)}{{r_{-}}-{r_{+}}}$
(163e)
$\displaystyle-\;{r_{+}}\left(m^{2}+{r_{+}}\tilde{\omega}{}(3{r_{+}}\tilde{\omega}{}+is)\right)$
$\displaystyle+\;\frac{4m{r_{+}}^{2}\tilde{\omega}{}\sqrt{{r_{-}}{r_{+}}}}{{r_{-}}-{r_{+}}}+ims\sqrt{{r_{-}}{r_{+}}}$
$\displaystyle+\;2m{r_{+}}\tilde{\omega}{}\sqrt{{r_{-}}{r_{+}}}-{r_{-}}{r_{+}}\tilde{\omega}{}({r_{+}}\tilde{\omega}{}+is)$
$\displaystyle\text{A}_{5}\;=$ $\displaystyle\;(s+1)(-({r_{-}}+{r_{+}}))$
(163f) $\displaystyle\text{A}_{6}\;=$ $\displaystyle\;2(s+1)\;.$ (163g)

The fact that Teukolsky’s radial operator is a special case of Heun’s
confluent operator is clear by comparing Eq. (161) to Eq. (162). The confluent
Heun nature of the operator is not changed under transformations of the
dependent or independent variables, but such transformations can impact the
physical nature of solutions. For example, application of the QNM boundary
conditions (see e.g. Eqs. 11-13) has the effect of transforming Eq. (162) into
a different version of Eq. (161). However, if the solution of the transformed
equation (e.g. Eq. (15)) is to be consistent with QNM boundary conditions,
then it cannot have asymptotic behavior that affects a transformation into yet
another version of the confluent Heun equation. Application of a compactified
coordinate (see Eq. 10) has the effect of changing the coordinate position of
singular points, but not their regular or irregular nature.

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# Identifying edges that facilitate the generation of extreme events in
networked dynamical systems

Timo Bröhl<EMAIL_ADDRESS>Department of Epileptology, University of
Bonn Medical Centre, Venusberg Campus 1, 53127 Bonn, Germany  Helmholtz
Institute for Radiation and Nuclear Physics, University of Bonn, Nussallee
14–16, 53115 Bonn, Germany  Klaus Lehnertz<EMAIL_ADDRESS>Department of Epileptology, University of Bonn Medical Centre, Venusberg
Campus 1, 53127 Bonn, Germany  Helmholtz Institute for Radiation and Nuclear
Physics, University of Bonn, Nussallee 14–16, 53115 Bonn, Germany
Interdisciplinary Center for Complex Systems, University of Bonn, Brühler
Straße 7, 53175 Bonn, Germany

###### Abstract

The collective dynamics of complex networks of FitzHugh-Nagumo units exhibits
rare and recurrent events of high amplitude (extreme events) that are preceded
by so-called proto-events during which a certain fraction of the units become
excited. Although it is well known that a sufficiently large fraction of
excited units is required to turn a proto-event into an extreme event, it is
not yet clear how the other units are being recruited into the final
generation of an extreme event. Addressing this question and mimicking typical
experimental situations, we investigate the centrality of edges in time-
dependent interaction networks. We derived these networks from time series of
the units’ dynamics employing a widely used bivariate analysis technique.
Using our recently proposed edge centrality concepts together with an edge-
based network decomposition technique, we observe that the recruitment is
primarily facilitated by sets of certain edges that have no equivalent in the
underlying topology. Our finding might aid to improve the understanding of
generation of extreme events in natural networked dynamical systems.

> Many natural, technological, or social systems are capable of recurrently
> generating large events that can lead to disasters when interacting with
> exposed or vulnerable human or natural systems. The understanding of the
> dynamical underpinnings of the generation of such extreme events has
> recently attracted much attention. While certain dynamical mechanisms have
> already been identified, only little is know about potential pathways in
> networked dynamical systems, that may play a vital role in facilitating the
> build-up of precursor structures that eventually lead to an extreme event.
> We here use the concept of centrality – originally proposed in the social
> sciences for network vertices and recently extended for network edges – to
> identify such pathways in networks of coupled, weakly interacting nonlinear
> oscillators. These networks are prototypical for excitable systems and are
> capable of self-generating and self-terminating extreme events. We
> demonstrate that particularly interactions and only rarely edges in the
> coupling topology facilitate the build-up of precursor structures of extreme
> events.

##  I Introduction

Extreme weather events and other natural hazards, large-scale blackouts in
power supply networks, market crashes, mass panics, wars, harmful algal blooms
in marine ecosystems, or epileptic seizures in the human brain are recurrent,
large-impact events that occur spontaneously in many natural, technological or
social dynamical systems Hobsbawm (1994); Bunde, Kropp, and Schellnhuber
(2002); Sornette (2003); Albeverio, Jentsch, and Kantz (2006); Ghil  _et al._
(2011); McPhillips  _et al._ (2018); Farazmand and Sapsis (2019). For systems
that can be described by a time-dependent (or evolving) interaction network,
novel methods have been developed over the last years that allow one to
identify precursors of extreme events Havlin  _et al._ (2012). This holds true
particularly for climate extremes Malik  _et al._ (2012); Boers  _et al._
(2013); Scarsoglio, Laio, and Ridolfi (2013); Boers  _et al._ (2014);
Ludescher  _et al._ (2014); Marwan and Kurths (2015); Weimer  _et al._ (2015);
Gelbrecht, Boers, and Kurths (2017); Ozturk  _et al._ (2018); Boers  _et al._
(2019); Dijkstra  _et al._ (2019); Ozturk  _et al._ (2019), seismic extremes
Dueñas-Osorio, Craig, and Goodno (2007), hydrological extremes Konapala and
Mishra (2017), economic extremes Constantin, Peltonen, and Sarlin (2018);
Bosma, Koetter, and Wedow (2019); Faggini, Bruno, and Parziale (2019), and
epileptic seizures Lehnertz  _et al._ (2016); Rings, von Wrede, and Lehnertz
(2019). Methods employed so far either aim at assessing global networks
properties (e.g., clustering-coefficient-related or path-related measures) or
local network properties – mostly vertex centralities Lü  _et al._ (2016). For
interaction networks – in which an edge represents attributes of an
interaction (strength, direction, coupling function) between two vertices – an
improved characterization of edge properties could add to advance
understanding, prediction, and control of such networks Slotine and Liu
(2012). To this end, and in order to find which edges in a network are
important between other pairs of vertices, we recently modified various,
widely used centrality concepts for vertices to those for edges Bröhl and
Lehnertz (2019). We also proposed a network decomposition technique that is
based on edge centrality and allows one to identify a hierarchy of sets of
edges, with each set being associated with a different level of importance
Bröhl and Lehnertz (2019).

We here apply these novel concepts to investigate precursor structures of
extreme events in the dynamics of complex networks of excitable units of
FitzHugh–Nagumo type. Previous studies Ansmann  _et al._ (2013); Karnatak  _et
al._ (2014); Bialonski, Ansmann, and Kantz (2015); Ansmann, Lehnertz, and
Feudel (2016); Rings, Ansmann, and Lehnertz (2017) have shown these systems to
be capable of self-generating and self-terminating strong, rare, short-
lasting, and recurrent deviations from their regular dynamics without the
influence of noise or parameter change. These extreme events are preceded by
local excitations (so-called proto-events Ansmann  _et al._ (2013); Rings,
Ansmann, and Lehnertz (2017)) in a certain fraction of units that play a
decisive role in their generation. Similar phenomena were also observed in
other excitable systems Alvarez, Borkar, and Masoller (2017); Bonatto and
Endler (2017); Jin, Siyu, and Masoller (2017); Karnatak, Kantz, and Bialonski
(2017); Kingston  _et al._ (2017); Saha and Feudel (2017); Mishra  _et al._
(2018); Saha and Feudel (2018); Moitra and Sinha (2019); Ray  _et al._ (2019).
It is, however, not yet clear how the other units in a network are being
recruited into the final generation of an extreme event, and we here
conjecture that the recruitment is facilitated by certain edges. We
demonstrate the suitability of our novel concepts for the analysis of
empirical data by mimicking typical experimental situations.

##  II Methods

###  II.1 Networks of excitable units

We consider networks of $V$ diffusively coupled, excitable FitzHugh-Nagumo
units $(n\in\left\\{1,\ldots,V\right\\})$, where the equations of motion of
unit $n$ read

$\displaystyle\dot{x}_{n}$ $\displaystyle=$ $\displaystyle
x_{n}(a_{n}-x_{n})(x_{n}-1)-y_{n}$ $\displaystyle+$
$\displaystyle\frac{K}{V-1}\sum_{m=1}^{V}A_{nm}(x_{m}-x_{n})$
$\displaystyle\dot{y}_{n}$ $\displaystyle=$ $\displaystyle
b_{n}x_{n}-c_{n}y_{n}.$

The unit’s internal control parameters are $a_{n}$, $b_{n}$, and $c_{n}$, and
the coupling strength is denoted by $K$. The symmetric adjacency matrix
$\bm{A}\in\left\\{0,1\right\\}^{V\times V}$ has entries $A_{nm}=A_{nm}=1$, iff
units $n$ and $m$ are coupled. We employ parameter settings that had been
identified in previous studies Ansmann  _et al._ (2013); Karnatak  _et al._
(2014); Rings, Ansmann, and Lehnertz (2017) to allow robust generation of
extreme events in complex networks. In particular, we set parameters $a$ and
$c$ identical for all units: $a_{n}=a=0.0274\,\forall n$ and
$c_{n}=c=0.018\,\forall n$; the parameter $b$ is mismatched with
$b_{n}=0.006+\frac{n}{V-1}0.008,\forall n$, and the coupling strength $K$ is
chosen individually for each network. We regard three coupling topologies each
of which connects $V=20$ vertices but with different number of edges $E$:

  * •

$\mathrm{T}_{\mathrm{A}}$: a binary network with a small-world topology Watts
and Strogatz (1998) with $E=100$ (number of nearest neighbors: $5$; rewiring
probability: $0.25$) and $K=0.128$;

  * •

$\mathrm{T}_{\mathrm{B}}$: a binary network with a small-world topology Watts
and Strogatz (1998) with $E=40$ (number of nearest neighbors: $2$; rewiring
probability: $0.25$) and $K=0.33$;

  * •

$\mathrm{T}_{\mathrm{C}}$: a binary network with a scale-free topology
Barabási and Albert (1999) with $E=36$ and $K=0.1128$. The degree ($\kappa$)
distribution $\mathcal{F}$ of the network follows a power law of the form
$\mathcal{F}(\kappa)\propto\kappa^{-3}$.

Each networks’ dynamics was integrated using an adaptive, explicit Runge-Kutta
method of 5th order Ansmann (2018) with a step size of $11$. We discarded at
least $10^{4}$ initial time units, and time series (here: $x$-components) used
for further analyses consisted of $10^{6}$ data points. The choice of the
initial conditions (near the attractor) had no influence on our observations.

In Fig. 1 we show, for each coupling topology, excerpts of the time series of
the average of the first dynamical variable
$\overline{x}(t)=\frac{1}{V}\sum_{n}^{V}x_{n}(t)$. Generally, we observe
$\overline{x}(t)$ to exhibit irregular, low-amplitude oscillations Ansmann
_et al._ (2013); Karnatak  _et al._ (2014); Rings, Ansmann, and Lehnertz
(2017) with $-0.15<\overline{x}(t)<0.15$. Occasionally, we observe stereotyped
events at which all units become excited and thus $\overline{x}(t)$ clearly
exceeds – by at least a factor of six – the amplitude of the collective low-
amplitude oscillations. We consider these rare but recurring high-amplitude
events as extreme events (time interval beginning with $\overline{x}(t)$
exceeding a threshold $\theta=0.5$). We find 195 such events for
$\mathrm{T}_{\mathrm{A}}$, 138 events for $\mathrm{T}_{\mathrm{B}}$, and 830
events for $\mathrm{T}_{\mathrm{C}}$. For $\mathrm{T}_{\mathrm{A}}$ and
$\mathrm{T}_{\mathrm{C}}$, for which we often observe double extreme events
Ansmann  _et al._ (2013); Rings, Ansmann, and Lehnertz (2017), only the
leading one is considered.

Figure 1: (left) Exemplary temporal evolutions of the average of the first
dynamical variable $\overline{x}$ for topologies $\mathrm{T}_{\mathrm{A}}$ –
$\mathrm{T}_{\mathrm{C}}$ (top to bottom). (right) Excerpt around an extreme
event (colored black) of the respective time series.

###  II.2 Data-driven construction of time-dependent interaction networks

Mimicking typical experimental situations Bialonski, Horstmann, and Lehnertz
(2010); Lehnertz  _et al._ (2014); Gao, Small, and Kurths (2017); Brugere,
Gallagher, and Berger-Wolf (2018), we derive time-dependent interaction
networks by estimating – using a sliding-window approach – the strength of
interaction between pairs $\left\\{n,m\right\\}\in\left\\{1,\ldots,V\right\\}$
of time series of the first dynamical variables $x$. To do so, we employed an
established method for investigating time-variant changes in phase
synchronization. The mean phase coherence Mormann  _et al._ (2000) is defined
as

$R_{nm}=\left|\frac{1}{T}\sum_{t=0}^{T-1}{e^{{i\mkern
1.0mu}\left(\Phi_{n}(t)-\Phi_{m}(t)\right)}}\right|,$ (2)

where $\Phi_{n}$ are the instantaneous phases of time series from unit $n$ (we
here use the Hilbert transform Mormann  _et al._ (2000); Boashash (1992)), and
$T$ denotes the number of data points. By definition, $R_{nm}$ is confined to
the interval [0,1], where $R_{nm}=1$ indicates fully phase-synchronized units.
Note that the window size $T$ is a critical parameter since it affects the
sensitivity of the mean phase coherence. We here chose $T$ such that a window
captured at least one full cycle of either an extreme event or a low-amplitude
oscillation.

Having calculated $R_{nm}$ for all pairs $(n,m)$ of units, we derived – for
each window – a synchronization matrix whose non-diagonal elements were
associated with an adjacency matrix $\tilde{\bm{A}}$. This matrix represents
an undirected, weighted snapshot network. To simplify notation, in the
following we define $\tilde{A}_{nn}\mathrel{\mathop{:}}=0\,\forall\,n$.
Depending on the underlying coupling topology we refer to these time-dependent
interaction networks as networks A, B, and C.

We position a reference window (window number $0$) around the extreme event,
such that the window center coincides with the first time point for which the
amplitude of $\overline{x}(t)$ exceeds the threshold $\theta$. The window
number increases while going back in time, with time windows of number $6$ or
larger are assumed to represent typical inter-event dynamics.

###  II.3 Estimating edge importance in time-dependent interaction networks

For our investigations, we employ two opposing Bröhl and Lehnertz (2019)
concepts to estimate the centrality of edges in each snapshot network, namely
edge betweenness centrality $\mathcal{C}^{\mathrm{B}}_{\mathrm{e}}$ and edge
eigenvector centrality $\mathcal{C}^{\mathrm{E}}_{\mathrm{e}}$.

Edge betweenness centrality (of edge $k$) can be defined as Freeman (1977);
Girvan and Newman (2002)

$\mathcal{C}^{\rm B}_{\rm e}(k)=\frac{2}{V(V-1)}\sum_{i\neq
j}\frac{q_{ij}(k)}{G_{ij}},$ (3)

where $k\in\left\\{1,\ldots,E\right\\}$,
$\left\\{i,j\right\\}\in\left\\{1,\ldots,V\right\\}$, $q_{ij}(k)$ is the
number of shortest paths between vertices $i$ and $j$ running through edge
$k$, and $G_{ij}$ is the total number of shortest paths between vertices $i$
and $j$. A shortest path is defined as the path between two edges for which
the sum of the inverse weights of edges along this path is minimal Bröhl and
Lehnertz (2019).

Edge eigenvector centrality (of edge $k$) is defined Bröhl and Lehnertz (2019)
as the $k$th entry of the eigenvector $\vec{v}$ corresponding to the dominant
eigenvalue $\lambda_{\max}$ of matrix ${\bf M}$, which we derive from the
eigenvector equation ${\bf M}\vec{v}=\lambda\vec{v}$ using the power iteration
method:

$\mathcal{C}^{\rm E}_{\rm
e}(k)=\frac{1}{\lambda_{\max}}\sum_{l}M_{kl}\,\mathcal{C}^{\rm E}_{\rm e}(l),$
(4)

with $\left\\{k,l\right\\}\in\left\\{1,\ldots,E\right\\}$. Here ${\bf M}$
denotes the edge adjacency matrix
$\tilde{\bm{A}}^{\rm(e)}\in\mathbb{R}_{+}^{E\times E}$ whose entries
$\tilde{\bm{A}}^{\rm(e)}_{ij}$ are assigned the average weight of edges $i$
and $j$ if these edges are connected to a same vertex, and 0 otherwise.

With the aforementioned definitions, we regard an edge with the highest
centrality value as most important (rank 1) and the one with the lowest
centrality value as least important (rank $E$). In case of equal centrality
values we rank in order of appearance.

###  II.4 Identifying important sets of edges in time-dependent interaction
networks

With the aforementioned edge centrality concepts, we employ our previously
proposed edge-centrality-based network decomposition technique Bröhl and
Lehnertz (2019) that allows us to identify a bottom-up hierarchy of sets of
edges (or “webs”), where each set is associated with a different level of
importance. The decomposition technique consists of the following steps:

  1. 0.

initialize algorithm: set $E^{\prime}=E$ and set iteration $q=1$;

  2. 1.

estimate centrality $\mathcal{C}_{\rm e}(k)$ for all edges
$k\in\left\\{1,\ldots,E^{\prime}\right\\}$ in the current network;

  3. 2.

choose the lowest centrality value as threshold value
$\Theta=\min_{k}\mathcal{C}_{\rm e}(k)$, in order to eliminate less central
edges;

  4. 3.

every edge $k^{\prime}$ with $\mathcal{C}_{\rm e}(k^{\prime})\leq\Theta$ is
assigned to the web of rank $q$ and is removed from the current network (which
decreases $E^{\prime}$; note that the $<$ sign holds for repetitions of step 3
within the $q$th iteration);

  5. 4.

repeat step 1 and step 3 until no further edge is assigned to the web of rank
$q$;

  6. 5.

continue with next iteration (increase $q$ by 1) at step 1, as long as there
are remaining edges to be assigned to webs;

  7. 6.

reverse ranking of webs; the most important web has rank 1.

We note that this network decomposition can lead to two divisions of a network
that are not helpful in identifying sets of edges associated with different
levels of importance. These cases are either an assignment of all edges to
only one web (number of webs $N_{W}=1$) or an assignment of each edge to a web
($N_{W}=E$). We also note that edges in a web do not have to be connected with
each other.

##  III Results

As shown earlier Ansmann  _et al._ (2013); Rings, Ansmann, and Lehnertz
(2017), extreme events in the dynamics of coupled FitzHugh-Nagumo oscillators
are preceded by proto-events during which a fraction of the units (those with
low values of the control parameter $b$) become excited and which turn into
extreme events, if and only if this fraction is sufficiently large (note that
not all proto-events are followed by an extreme event). It is, however, not
yet clear how the other units are being recruited into the final generation of
an extreme event. We conjecture that the recruitment is facilitated by certain
edges (or sets thereof), and in the following, we will identify and
characterize these edges employing the edge-centrality concepts and the edge-
based network decomposition technique. Given that most of the complexity of a
interaction network is encoded into the topology of interactions among its
vertices (i.e., edges) and into the layout of the interactions’ weights Barrat
_et al._ (2004); Park, Lai, and Ye (2004); Van Mieghem and Magdalena (2005);
Yang  _et al._ (2009), we first investigate how the edge weight distributions
change when our time-dependent interaction networks transit into an extreme
event. Since edge weights represent the strengths of interaction between units
(estimated with $R_{nm}$; see Eq. 2), we expect a narrow range of large edge
weights despite the constant and rather low coupling strengths $K$. Indeed,
the edge weight distribution peaks close to the maximum value of $R_{nm}=1$
with a rather narrow spread, by construction (see Fig. 2).

Figure 2: Edge weight distributions (means and standard deviations obtained
from observations of the respective amount of extreme events) of time-
dependent interaction networks A, B, and C (from top to bottom) for each time
window. Time window $0$ is positioned around the extreme events, and the
window number increases while going back in time. Figure 3: Relative frequency
of an edge with a given rank to be connected to a vertex whose dynamics
exhibits proto-events. Edge rank estimated via ranking of a) edge eigenvector
centrality and b) edge betweenness centrality. Time window $0$ is positioned
around the extreme events, and the window number increases while going back in
time. Data from 195, 138 and 830 extreme events in the time-dependent
interaction networks A, B, and C (from top to bottom). Red dots indicate edges
in the time-depended interaction networks and black dots edges from the
underlying coupling topology.

For our time-dependent interaction networks, we find edge weights from the
time window capturing an extreme event to compare to those from most preceding
windows. Interestingly though, we observe decreased edge weights in the time
window directly preceding the extreme event (window 1), and this decrease is
most pronounced for networks A and B. With our analysis approach, proto-events
thus reflect a desynchronized state during which only few units are
simultaneously excited, while the other units are not. We note that similar
desynchronization phenomena were observed prior to epileptic seizures recorded
in humans Mormann  _et al._ (2003) and in a simple dynamical model of two
interacting networks of integrate-and-fire neurons that mimics such an event
Feldt  _et al._ (2007).

Since edge weights impact on the centrality concepts employed here (cf. Sec.
II.3) and given our previous observations, we next hypothesize that a certain
amount of edges in the interaction networks from the time windows prior to the
extreme event will rank among the ones with highest centrality and are
therefore possibly more relevant for the recruitment of further units. In
order to check this hypothesis, we estimate – for each time window – the
probability $\mathcal{P}$ for an edge to be identified as most important
(i.e., highest centrality value and thus highest rank) with the respective
edge centrality. For each network, and independently of the used centrality,
we observe (data not shown) in each time window (including time window 0) the
respective probability distributions to peak around a small amount of edges
(if we neglect edges with $\mathcal{P}<0.2$). In addition, we observe that
these distributions differ in the time window prior to the extreme event,
indicating that during this time window other edges are most important.

Given these findings, we further investigate which edges are connected to
vertices whose dynamics exhibit proto-events and whether these edges have a
high rank and can be traced back to the underlying coupling topology (direct
edge) or not (indirect edge). As shown in Fig. 3, we observe edges with low
rank to be (on average) more frequently connected to such vertices in all time
windows preceding time window 1. In time window 1, we additionally observe few
more high-ranked edges to be frequently connected to these vertices, however,
this findings holds for importance estimated using
$\mathcal{C}^{\mathrm{B}}_{\mathrm{e}}$ only. If importance was estimated
using $\mathcal{C}^{\mathrm{E}}_{\mathrm{e}}$, the low-ranked edges are even
more frequently connected to these vertices. Whereas the underlying coupling
topologies had no influence on these findings, the differences seen for the
two edge centralities can be related to the differences in their conceptual
basis. While $\mathcal{C}^{\mathrm{E}}_{\mathrm{e}}$ considers the centrality
of the neighborhood of a given edge, $\mathcal{C}^{\mathrm{B}}_{\mathrm{e}}$
is a path-based approach to identify a central edge. For either centrality
concept it is rather straightforward to understand that in time windows far
from the extreme event, high-ranked edges are not expected to be connected to
the few vertices whose dynamics exhibit proto-events. In the time window prior
to the extreme event, the opposite can be observed with
$\mathcal{C}^{\mathrm{B}}_{\mathrm{e}}$. This indicates that the recruitment
of non-excited vertices is facilitated via short paths from excited vertices,
making it more likely for edges that are directly connected to such vertices
to have a high rank. On the other hand, with
$\mathcal{C}^{\mathrm{E}}_{\mathrm{e}}$ highest-ranked edges connect non-
excited vertices as these are mostly stronger connected (larger edge weights)
to other non-excited vertices.

For direct edges and independent of their centrality ranking, we furthermore
observe a general decrease, in time window 1 compared to other time windows,
in their relative frequency to be connected to a vertex whose dynamics
exhibits proto events. One can thus deduce, that most of the edges that are
connected to a vertex whose dynamics exhibits a proto-event represent indirect
edges, with few exceptions found with betweenness centrality.

Summarizing our findings discussed so far, we conclude that the recruitment of
non-excited units into the generation of an extreme event is facilitated by
the most important and the least important indirect edges. As a last point, we
investigate whether these edges form specific sets. To this end, we employ our
edge-centrality-based network decomposition technique to identify – for each
time window – the most and the least important web and eventually detail their
characteristics. Our results presented in Figs. 4 and 5 indicate that the
least important webs for time window 1 consist of smaller sets of edges than
the ones in the least important webs identified for windows preceding window 1
or even for the window that captures the extreme event. These sets consists to
a greater amount of indirect edges than of direct edges. In general,
differences are most distinct for networks A and C and for the
$\mathcal{C}^{\mathrm{B}}_{\mathrm{e}}$-based network decomposition. We note
that we achieved similar findings when considering the most important webs
(data not shown).

Interestingly, the sets seen for time window 1 are composed of edges (either
direct or indirect ones) connected to vertices whose dynamics exhibit proto-
events. As expected the $\mathcal{C}^{\mathrm{B}}_{\mathrm{e}}$-based
decomposition mostly identifies long-range connections while the
$\mathcal{C}^{\mathrm{E}}_{\mathrm{e}}$-based decomposition mostly identifies
(nearest or next-nearest) neighboring connections within the web.

Figure 4: Adjacency matrices of least important webs projected onto the
underlying coupling topology (grey) with direct and indirect edges marked
black and red, respectively. Only edges with a relative frequency (occurrence
in data from respective amount of extreme events) higher than 75 % were
considered. $\mathcal{C}^{\mathrm{E}}_{\mathrm{e}}$-based network
decomposition. Vertices sorted in ascending order of the control parameter
$b$. Figure 5: Same as Fig. 4 but for
$\mathcal{C}^{\mathrm{B}}_{\mathrm{e}}$-based network decomposition.

It can be summarized that distinct sets of primarily indirect edges appear to
play a vital role (in the time window prior to the extreme event) for the
recruitment of non-excited units into excitation leading up to an extreme
event.

##  IV Conclusions

We investigated which edges in networks of coupled, excitable FitzHugh-Nagumo
units facilitate the recruitment of non-excited units into the final
generation of an extreme event. With an eye on typical experimental situations
that explore excitable system, we investigated the importance of edges in
time-dependent interaction networks. We derived these networks from
investigating the strength of interaction between time series of the units’
dynamics in a time-resolved manner. Importance of edges and sets thereof were
characterized with the concept of edge centrality and an edge-centrality-based
network decomposition technique respectively. Our findings indicate, that the
recruitment of non-excited units is facilitated primarily by sets of certain
most and least important edges, both of which have no equivalent in the
underlying topology. A more comprehensive understanding of the role of such
indirect edges and their relationship to the underlying coupling topology
might aid to gain further insights into the generation of extreme events in
natural networked dynamical systems.

## Acknowledgements

The authors would like to thank Lina Zabawa and Thorsten Rings for interesting
discussions and for critical comments on earlier versions of the manuscript.
This work was supported by the Deutsche Forschungsgemeinschaft (Grant No: LE
660/7-1).

## Data availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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