\section{\label{sec:intro}Introduction}
A number of popular extensions of the Standard
Model (SM) of particle physics predict the existence of doubly charged scalar
particles $X^{\pm\pm}$. These include the Type-II seesaw \cite{Schechter:1980gr,
Magg:1980ut,Cheng:1980qt,Lazarides:1980nt,Mohapatra:1980yp,Lindner:2016bgg} and
the Zee-Babu \cite{Zee:1980ai,Babu:1988ki} models of neutrino masses, the
Left–Right model \cite{Pati:1974yy,Mohapatra:1974hk,Senjanovic:1975rk},
the Georgi–Machacek model \cite{Georgi:1985nv,Chanowitz:1985ug,Gunion:1989ci,
Gunion:1990dt,Ismail:2020zoz}, the 3-3-1 model \cite{CiezaMontalvo:2006zt,
Alves:2011kc} and the little Higgs model \cite{ArkaniHamed:2002qx}).
Doubly charged scalars appear also in simplified models, in which one merely
adds such scalars in a gauge invariant way in various representations of the
SM gauge group $SU(2)_L$ to the particle content of the SM.
The Lagrangian of the model is then complemented by
gauge-invariant interaction terms involving these new fields
\cite{Delgado:2011iz,Alloul:2013raa}.

Doubly charged scalars may be long-lived or even stable
\cite{Alloul:2013raa,Alimena:2019zri,Acharya:2020uwc,Hirsch:2021wge}.
As the simplest example, one can add to the SM an uncolored
$SU(2)_L$-singlet scalar field $X$ with hypercharge
$Y=2$ \cite{Alloul:2013raa}. The corresponding doubly charged particles will
couple to the neutral gauge bosons $\gamma$ and $Z^0$ and may also
interact with the SM Higgs boson $H$ through the  $(H^\dag H)(X^\dag X)$
term in the Higgs potential. Gauge invariance allows, in addition, the Yukawa
coupling of $X$ to right-handed charged leptons, $h_X l_R l_R X+h.c.$
This is the only coupling that makes the $X$-particles unstable
in this model;  they will be long-lived if the Yukawa coupling constants
$h_X$ are small. The Yukawa coupling of $X$  may be forbidden by e.g.\ $Z_2$
symmetry $X\to -X$, in which case the $X$-scalars will be stable.

Doubly charged scalar particles are being actively searched for
experimentally, but up to now have not been discovered.
For discussions of current experimental constraints on the doubly charged
particles and of the sensitivities to them of future experiments
see \cite{Alloul:2013raa,Alimena:2019zri,Fuks:2019clu,Padhan:2019jlc,
Acharya:2020uwc,Hirsch:2021wge,Dev:2021axj} and references therein.

In addition to interesting particle-physics phenomenology, doubly charged
scalars may have important implications for cosmology. In this paper we
will, however, consider another aspect of their possible existence.
As we shall demonstrate, doubly charged particles can catalyze fusion of
light nuclei, with potentially important applications for energy production.
The negatively charged $X^{--}$
(which we will hereafter simply refer to as
$X$) can form atomic bound systems with the nuclei of light elements, such as
deuterium, tritium or helium.
One example is the antihelium-like $(ddX)$ atom
with the $X$-particle as the ``nucleus'' and two deuterons in the 1$s$ atomic
state instead of two positrons. (Here and below we use the brackets to denote
states bound by the Coulomb force). As $X$ is expected to be very heavy, the
size of such an atomic system will in fact be determined by the deuteron
mass $m_d$ and will be of the order of the Bohr radius of the $(dX)$
ion, $a_d\simeq 7.2$ fm. Similar small-size atomic systems $(N\!N'X)$ can
exist for other light nuclei $N$ and $N'$ with charges $Z\le 2$.

Atomic binding of two nuclei to an $X$-particle brings them so close together
that this essentially eliminates the necessity for them to overcome the
Coulomb barrier in order to undergo fusion. The exothermic fusion reactions
can then occur unhindered and do not require high temperature or pressure.
The $X$-particle is not consumed in this process and can then facilitate
further nuclear fusion reactions, acting thus as a catalyst.

This $X$-catalyzed fusion mechanism is to some extent similar to muon
catalyzed fusion ($\mu$CF) (\cite{Frank,Sakharov,Zeldovich1,Alvarez:1957un,
Jackson:1957zza,Zeldovich2,Zeldovich3,bogd}, see \cite{Zeldovich4,GerPom,
bracci,Breunlich:1989vg,Ponomarev:1990pn,Zeldovich:1991pbl,bogd2} for
reviews), in which the role of the catalyst is played by singly negatively
charged muons. $\mu$CF of hydrogen isotopes was once considered a prospective
candidate for cold fusion. However, already rather early in its studies it
became clear that $\mu$CF suffers from a serious shortcoming that
may prevent it from being a viable mechanism of energy production. In the
fusion processes, isotopes of helium are produced,
and there is a chance that they will capture on their atomic orbits
the negative muons present in the final state of the fusion reactions.
Once this happens, muonic ions ($^3$He$\mu)$ or ($^4$He$\mu)$ are formed,
which, being positively charged, cannot catalyze further fusion reactions.
This effect is cumulative; the sticking to helium nuclei thus eventually
knocks the muons out from the catalytic process, i.e.\ the catalytic
poisoning occurs.

Out of all $\mu$CF reactions, the $d-t$ fusion has the smallest muon sticking
probability, $\omega_s\simeq 10^{-2}$. This means that a single muon will
catalyze $\sim$100 fusion reactions before it gets removed from the catalytic
process. The corresponding total produced energy is $\sim$1.7 GeV, which is
at least a factor of five smaller than the energy
needed to produce and handle one muon \cite{Jackson:1957zza}.
In addition, muon's short lifetime makes it impractical to try to dissolve
the produced ($^3$He$\mu)$ or ($^4$He$\mu)$ bound states by irradiating them
with particle beams in order to reuse the released muons.
These considerations have essentially killed the idea of using $\mu$CF
for energy production.

There were discussions in the literature of
the possibility of energy generation through the catalysis of
nuclear fusion by hypothetic heavy long-lived or stable singly
charged \cite{Zeldovich3,Rafelski:1989pz,
Ioffe:1979tv,Hamaguchi:2006vp} or fractionally charged
\cite{Zweig:1978sb} particles. However, it has been shown in
\cite{Zeldovich3,Ioffe:1979tv,Hamaguchi:2006vp} that these processes
suffer from the same problem of catalytic poisoning
as $\mu$CF, and therefore they cannot be useful sources of energy.
In particular, in ref.~\cite{Ioffe:1979tv} it was demonstrated that
reactivation of the catalyst particles by irradiating their atomic bound
states with helium nuclei by neutron beams, as suggested in
\cite{Zweig:1978sb}, would require beams that are about nine orders of
magnitude higher than those currently produced by most powerful nuclear
reactors.

In this paper we consider the fusion of light nuclei catalyzed by doubly
negatively charged $X$-particles and demonstrate that, unlike $\mu$CF, this
process may be a viable source of energy. We analyze $X$-catalyzed fusion
($X$CF) in deuterium environments and show that the  catalytic poisoning may
only occur in this case due to the sticking of $X$-particles
to $^6$Li nuclei, which are produced in the fusion reactions of the third
stage. The corresponding sticking probability is shown to be very low,
and, before getting bound to $^6$Li, each $X$-particle can catalyze
$\sim 3.5\cdot 10^{9}$ fusion cycles, producing $\sim 7\cdot 10^{4}$ TeV
of energy.
To the best of the present author's knowledge, nuclear fusion
catalyzed by doubly charged particles has never been considered before.

\section{\label{sec:xcg}$X$-catalyzed fusion
in deuterium}

We will be assuming that $X$-particles interact only electromagnetically,
which in any case should be a very good approximation at low energies relevant
to  nuclear fusion. Let $X$-particles be injected in pressurized D$_2$ gas or
liquid deuterium. Being very heavy and negatively charged,
the $X$-particles can easily penetrate D$_2$ molecules and D atoms,
dissociating the former and ionizing the latter and losing energy
on the way.  Once the velocity of an $X$-particle becomes comparable to
atomic velocities ($v\simeq 2e^2/\hbar\sim 10^{-2}c$), it captures a deuteron
on a highly excited atomic level of the ($dX$) system, which then very
quickly  de-excites to its ground state, mostly through electric dipole
radiation and inelastic scattering on the neighboring deuterium atoms.
As the ($dX$) ion is negatively charged, it swiftly picks up another
deuteron to form the ($ddX$) atom. The characteristic time of this
atomic phase of the $X$CF process is dominated by the $X$ moderation time
and is $\sim 10^{-10}$\,s at liquid hydrogen density
$N_0=4.25\times 10^{22}$ nuclei/cm$^3$ and $T\simeq 20$K and
about $10^{-7}$\,s in deuterium gas at $0^\circ$C and pressure of one bar
(see Appendix~\ref{sec:Xatom}).

After the $(ddX)$ atom has been formed, the deuterons
undergo nuclear fusion through several channels, see below.
Simple estimates show that the fusion rates are many orders of magnitudes
faster than the rates of the atomic formation processes.
That is, once ($ddX$) [or similar ($N\!N'X$)] atoms are formed, the
fusion occurs practically instantaneously. The time scale of $X$CF is
therefore determined by the atomic formation times.
The rates of the fusion reactions, however,  determine the branching ratios of
various fusion channels, which are important for the kinetics of the
catalytic cycle.

At the first stage of $X$CF in deuterium two deuterons fuse to produce
$^3$He, $^3$H or $^4$He. In each case there is at least one channel
in which the final-state $X$ forms an atomic bound state with one of the
produced nuclei. Stage I fusion reactions are
\begin{align}
&(ddX)\to {^3\rm He}+n+X &(Q=2.98~{\rm MeV},~29.1\%)
\tag{1a} \label{eq:r1a}\\
&(ddX)\to ({\rm ^3He}X)+n &(Q=3.89~{\rm MeV},
~19.4\%) \tag{1b} \label{eq:r1b}
\end{align}
\vglue-8mm
\begin{align}
&(ddX)\to {\rm ^3H}+p+X &&(Q=3.74~{\rm MeV},~34.4\%)
\tag{2a} \label{eq:r2a}\\
&(ddX)\to ({\rm ^3H}X)+p &&(Q=4.01~{\rm MeV}, ~6.2\%)
\tag{2b} \label{eq:r2b}\\
&(ddX)\to {\rm ^3H}+(pX) &&(Q=3.84~{\rm MeV},
~0.5\%)
\tag{2c} \label{eq:r2c}
\end{align}
\vglue-8mm
\begin{align}
&(ddX)\to {\rm ^4He}+\gamma+X &&(Q=23.6~{\rm MeV}, ~4\!\cdot\!10^{-9})
\tag{3a} \label{eq:r3a}\\
&(ddX)\to ({\rm ^4He}X)+\gamma &&(Q=24.7~{\rm MeV}, ~3\!\cdot\! 10^{-8})
\tag{3b} \label{eq:r3b}\\
&(ddX)\to {\rm ^4He}+X &&(Q=23.6~{\rm MeV}, ~10.4\%)
\tag{3c} \label{eq:r3c}
\end{align}
Here in the parentheses the $Q$-values and the branching ratios of the
reactions are shown. In evaluating the $Q$-values we have taken into account
that the atomic binding of the two deuterons to $X$ in the initial state
reduces $Q$, whereas the binding to $X$ of one of the
final-state nuclei increases it. As the Bohr radii of most of the $X$-atomic
states we consider are either comparable to or smaller than the nuclear
radii, in calculating the Coulomb binding energies one has to allow for the
finite nuclear sizes.  We do that by making use of a variational approach,
as described in Appendix~\ref{sec:Bind}.

The rates of reactions (\ref{eq:r1b}), (\ref{eq:r2b}), (\ref{eq:r2c}) and
(\ref{eq:r3b}) with bound $X$-particles in the final states
are proportional to the corresponding $X$-particle sticking probabilities,
$\omega_{s}$. The existence of such channels obviously affects the branching
ratios of the analogous reactions with free $X$ in the final states.

Radiative reactions (\ref{eq:r3a}) and (\ref{eq:r3b}) have tiny branching
ratios, which is related to their electromagnetic nature and to the fact that
for their $X$-less version, $d+d\to{\rm ^4He}+\gamma$, transitions
of E1 type are strictly forbidden. This comes about because the two fusing
nuclei are identical, which, in particular, means that they have the same
charge-to-mass ratio. This reaction therefore proceeds mainly
through E2 transitions \cite{bogd}. When the deuterons are bound to $X$,
the strict prohibition of E1 transitions is lifted due to possible
transitions through intermediate excited atomic states.%
\footnote{\label{fn:1}The author is grateful to M.~Pospelov for raising this
issue and suggesting an example of a route through which E1 transitions could
proceed in reaction (\ref{eq:r3b}).}
However, as shown in Appendix~\ref{sec:lift},
the resulting E1 transitions are in this case heavily hindered and their
rates actually fall below the rates of the E2 transitions.

Reaction (\ref{eq:r3c}) is an internal conversion process.
Note that, unlike for reactions (\ref{eq:r1a}) - (\ref{eq:r3b}),
the $X$-less version of (\ref{eq:r3c}) does not exist: the process
$d+d\to ^4$He is forbidden by kinematics. For the details of the calculation
of the rate of reaction (\ref{eq:r3c}) as well as of the rates of the other
discussed in this paper reactions, see Appendix~\ref{sec:Sfactor}.
The relevant $Q$-values of the reactions and sticking probabilities are
evaluated in Appendices~\ref{sec:Bind} and \ref{sec:sticking}, respectively.

The final states of reactions (\ref{eq:r1a}), (\ref{eq:r2a}), (\ref{eq:r3a})
and (\ref{eq:r3c}) contain free $X$-particles which are practically at rest
and can immediately capture deuterons of the medium, forming again the
($ddX$) atoms. Thus, they can again catalyze $d-d$ fusion through stage I
reactions (\ref{eq:r1a})-(\ref{eq:r3c}).
The same is also true for the $X$-particles in the final state of reaction
(\ref{eq:r2c}) which  emerge being bound to protons. Collisions of ($pX$)
with deuterons of the medium lead to fast replacement of the protons
by deuterons through the exothermic charge exchange reaction
$(pX)+d\to (dX)+p$ with the energy release $\sim$90~keV
(see Appendix~\ref{sec:charge}).
The produced $(dX)$ ion then picks up a deuteron to form the $(ddX)$ atom,
which can again participate in stage I reactions
(\ref{eq:r1a})-(\ref{eq:r3c}).

The situation is different for the $X$-particles
in the final states of reactions (\ref{eq:r1b}) and (\ref{eq:r2b})
forming the bound states with $^3$He and $^3$H, respectively.
They can no longer directly participate in stage I $d-d$ fusion reactions.
However, they are not lost for the fusion process: the produced
(${\rm ^3He}X$) and (${\rm ^3H}X$)  can still pick up deuterons of the medium
to form the atomic bound states (${\rm ^3He}dX$) and
(${\rm ^3H}dX$), which can give rise to stage II fusion reactions, which we
will consider next.

Before we proceed, a comment is in order. While (${\rm ^3H}X$) is a singly
negatively charged ion which can obviously pick up a positively charged
deuteron to form an (${\rm ^3H}dX$) atom, (${\rm ^3He}X$) is a neutral
$X$-atom. It is not immediately obvious whether it can form a stable bound
state with $d$, which, if exists, would be a positive ion. In the case of
the usual atomic systems, analogous (though negatively charged) states
do exist -- a well-known example is the negative ion of hydrogen H$^-$.
However, the stability of (${\rm ^3He}\,dX$) cannot be directly deduced
from the stability of H$^-$: in the latter case the two particles
orbiting the nucleus are identical electrons, whereas for
(${\rm ^3He}dX$) these are different entities -- nuclei with differing masses
and charges. Nevertheless, from the results of a general analysis of
three-body Coulomb systems carried out in \cite{Martin:1998zc,krikeb,armour}
it follows that the state (${\rm ^3He}dX$) (as well as the bound state
(${\rm ^4He}dX$) which we will discuss later on) should exist and be stable.
For additional information see Appendix~\ref{sec:posIons}.

Once (${\rm ^3He}X$) and (${\rm ^3H}X$), produced in reactions (\ref{eq:r1b})
and (\ref{eq:r2b}), have picked up deuterons from the medium and formed the
atomic bound states (${\rm ^3He}dX$) and (${\rm ^3H}dX$),
the following stage II fusion reactions occur:
\begin{align}
\!\!\!\!\!&(^3{\rm He}dX)\to {\rm ^4He}+p+X\!\!\!
&&(Q=17.4~{\rm MeV},\,94\%)
\!\!\!\tag{4a} \label{eq:r4a}\\
\!\!\!\!\!&({\rm ^3He}dX)\to ({\rm ^4He}X)+p\!\!\!
&&(Q=18.6~{\rm MeV},\,6\%) \nonumber
\!\!\!\tag{4b} \label{eq:r4b}\\
\!\!\!\!\!&({\rm ^3He}dX)\to {\rm ^4He}+(pX)\!\!\!
&&(Q=17.5~{\rm MeV},\,3\!\cdot\!10^{-4})
\!\!\!
\tag{4c} \label{eq:r4c}
\end{align}
\vglue-8.0mm
\begin{align}
\!\!\!\!\!\!&(^3{\rm H} d X)
\to {\rm ^4He}+n+X
\!\!\!&&(Q=17.3~{\rm MeV},~96\%)
\tag{5a} \label{eq:r5a}\\
\!\!\!\!\!\!&(^3{\rm H} d X)
\to ({\rm ^4He}X)+n \!\!\!
&&(Q=18.4~{\rm MeV}, ~4\%)
\tag{5b} \label{eq:r5b}
\end{align}
In these reactions vast majority of $X$ bound to $^3$He and $^3$H are
liberated; the freed $X$-particles can again form $(ddX)$ states and catalyze
stage I fusion reactions (\ref{eq:r1a})-(\ref{eq:r3c}). The same applies to
the final-state $X$-particles bound to protons, as was discussed above.
The remaining relatively small fraction of $X$-particles come out of
stage II reactions in the form of $({\rm ^4He}X)$ atoms.
Together with a very small amount of $({\rm ^4He}X)$ produced in
reaction (\ref{eq:r3b}), they pick up deuterons from the medium and form
$({\rm ^4He}dX)$ states, which undergo stage III $X$CF reactions:
\begin{align}
&(^4{\rm He} d X)\to {\rm ^6Li}+\gamma+X
\!\!\!
\!\!\!
&&(Q=0.32~{\rm MeV},
~10^{-13})
\tag{6a} \label{eq:r6a}\\
&({\rm ^4He} d X)\to ({\rm ^6Li}X)+\gamma
\!\!\!
\!\!\!
&&(Q=2.4~{\rm MeV}, ~\,2\!\cdot\! 10^{-8})
\tag{6b} \label{eq:r6b}\\
&({\rm ^4He} d X)\to {\rm ^6Li}+X \!\!\!
\!\!\!\!\!&&(Q=0.32~{\rm MeV},
\,\simeq100\%)
\tag{6c} \label{eq:r6c}
\end{align}
In these reactions, almost all previously bound $X$-particles are liberated
and are free to catalyze again nuclear fusion through $X$CF reactions of
stages I and II. The remaining tiny fraction of $X$-particles end up being
bound to the produced ${\rm ^6Li}$ nuclei through reaction (\ref{eq:r6b}).
However, as small as it is, this fraction is very important for the
kinetics of $X$CF. The bound states $({\rm ^6Li}X)$ are ions of
charge +1; they cannot form bound state with positively charged nuclei and
participate in further $X$CF reaction. That is, with their formation
catalytic poisoning occurs and the catalytic process stops.

{}From the branching ratios of stage I, II, and III $X$CF reactions one
finds that the fraction of the initially injected $X$-particles which
end up in the $({\rm ^6Li}X)$ bound state is $\sim 2.8\times 10^{-10}$.
This means that each initial $X$-particle, before getting stuck to a $^6$Li
nucleus, can catalyze $\sim 3.5\times 10^{9}$ fusion cycles.

Direct inspection shows that, independently of which sub-channels were
involved, the net effect of stage I, II and III
$X$CF reactions is  the conversion of four deuterons to
a $^6$Li nucleus, a proton and a neutron:
\begin{equation}
4d\to {\rm ^6Li}+p+n+23.1\,{\rm MeV}\,.
\tag{7} \label{eq:7}
\end{equation}
Therefore, each initial $X$-particle will produce about $7\times 10^4$ TeV of
energy before it gets knocked out of the catalytic process.  It should be
stressed that this assumes that the $X$-particles are sufficiently
long-lived to survive during $3.5\times 10^9$ fusion cycles. From our
analysis it follows that the slowest processes in the $X$CF cycle are the
formation of positive ions $({\rm ^3He}dX)$ and $({\rm ^4He}dX)$.
The corresponding formation times are estimated to be of the order
of $10^{-8}$\,s.
(see sec.~\ref{sec:posIons} of the Supplemental material).
Therefore, for the $X$-particles to survive during $3.5\times 10^9$ fusion
cycles and produce $\sim 7\times 10^4$ TeV of energy, their lifetime $\tau_X$
should exceed $\sim 10^2$\,s. For shorter lifetimes the energy produced by a
single $X$-particle before it gets stuck to a $^6$Li nucleus
is reduced accordingly.

\section{\label{sec:acquis} Acquisition and reactivation of $X$-particles}

The amount of energy produced by a single $X$-particle  has to be compared
with energy expenditures related to its production.
$X$-particles can be produced in pairs in accelerator experiments, either in
$l^+l^-$ annihilation at lepton colliders or through the Drell-Yan processes
at hadronic machines. Although the energy $E\sim 7\times 10^4$ TeV
produced by one $X$-particle before it gets knocked out of the catalytic
process is quite large on microscopic scale, it is only about
10\,mJ. This means that
$\gtrsim 10^{8}$ $X$-particles are needed to generate 1\,MJ of energy.
While colliders are better suited for discovery of new particles,
for production of large numbers of $X$-particles fixed-target accelerator
experiments are more appropriate. For such experiments the beam energy must
exceed the mass of the $X$-particle significantly. Currently, plans
for building such machines are being discussed \cite{Benedikt:2020ejr}.

The problem is, however, that the $X$-particle production cross section is
very small. This comes about because of their expected large mass
($m_X\gtrsim 1$\,TeV/$c^2$) and the fact
that for their efficient moderation needed to make the formation of $(dX)$
atoms possible, $X$-particles should be produced with relatively low velocities.
The cross section $\sigma_p$ of production of $X$-particles
with mass $m_X\simeq 1$\,TeV/$c^2$ and
$\beta=v/c\simeq 0.3$ is only $\sim 1$ fb (note that for scalar $X$-particles
$\sigma_p\propto \beta^3$). As a result, the energy spent on production of an
$X^{++}X^{--}$ pair will be  by far larger than the energy that can be
generated by one $X^{--}$ before it gets bound to a $^6$Li nucleus. This
means that reactivating and reusing the bound $X$-particles multiple times
would be mandatory in this case. This, in turn, implies that only very
long lived $X$-particles with $\tau_X\gtrsim 3\times 10^{4}$\,yr
will be suitable for energy production.

Reactivation of $X$-particles bound to $^6$Li requires dissociation
of $({\rm ^6Li}X)$ ions. This could be achieved by irradiating them
with particle beams, similarly to what was suggested for reactivation of
lower-charge catalyst particles in ref.~\cite{Zweig:1978sb}.
However, it would be much more efficient to use instead $({\rm ^6Li}X)$ ions
as projectiles and irradiate a target with their beam.%
\footnote{We thank M.~Pospelov for this suggestion.}
The Coulomb binding energy of $X$ to ${\rm ^6Li}$ is about 2 MeV; to strip
them off by scattering on target nuclei with the average atomic number
$A\simeq 40$ one would have to accelerate $({\rm ^6Li}X)$ ions to velocities
$\beta\simeq 0.01$ which, for $m_X\simeq 1$\,TeV/$c^2$,
corresponds to beam energy $\sim 0.05$ GeV. At these
energies the cross section of the stripping reaction is $\gtrsim 0.1$\,b,
and $X$-particles can be liberated with high efficiency in relatively small
targets. The energy spent on the reactivation of one $X$-particle will then
only be about $10^{-9}$ of the energy it can produce before sticking to a
$^6$Li nucleus.

If $X$-particles are stable or practically stable, i.e.\ their lifetime
$\tau_X$ is comparable to the age of the Universe, there may exist a
terrestrial population of relic $X$-particles bound to nuclei or (in the
case of $X^{++}$) to electrons and thus forming exotic nuclei or atoms. The
possibility of the existence of exotic bound states containing charged
massive particles was suggested in ref.~\cite{Cahn:1980ss} (see also
\cite{DeRujula:1989fe}) and has been studied by many authors.
The concentration of such exotic atoms on the Earth may be very low if
reheating after inflation occurs at sufficiently low temperatures.
Note that reheating temperatures as low as a few MeV are consistent
with observations \cite{Hannestad:2004px}.
A number of searches for such superheavy exotic isotopes has been
carried out using a variety of experimental techniques,
and upper limits on their concentrations were established, see
\cite{Burdin:2014xma} for a review.

Exotic helium atoms $(X^{++}ee)$ were searched for in the Earth's
atmosphere using laser spectroscopy technique, and the limit of their
concentration $10^{-12}-10^{-17}$ per atom over the mass range 20 $-$
$10^4$\,GeV/$c^2$ was established~\cite{Mueller:2003ji}.
In the case of doubly negatively charged $X$, their Coulomb binding to nuclei
of charge $Z$ would produce superheavy exotic isotopes with nuclear
properties of the original nuclei but chemical properties of atoms with
nuclear charge $Z-2$. Such isotopes could have accumulated in
continental crust and marine sediments.
Singly positively charged ions
($^6$Li$X$) and ($^7$Li$X$) chemically behave as superheavy protons; they can
capture electrons and form anomalously heavy hydrogen atoms.
Experimental searches for anomalous hydrogen in normal water have put upper
limits on its concentration at the level of $\sim 10^{-28} - 10^{-29}$ for the
mass range 12 to 1200 GeV/$c^2$ \cite{smith1} and $\sim 6\times 10^{-15}$ for
the masses between 10 and $10^5$ TeV/$c^2$ \cite{verkerk}.

If superheavy isotopes containing relic $X$-particles of cosmological origin
exist, they can be extracted
from minerals  e.g.\ by making use of mass spectrometry techniques, and
their $X$-particles can then be stripped off. To estimate the required
energy, we conservatively assume that it is twice the energy needed to
vaporize the matter sample. As an example, it takes about 10 kJ to vaporize
1\,g of granite \cite{Woskov}; denoting the concentration of $X$-particles in
granite (number of $X$ per molecule) by $c_X$, we find that the energy
necessary to extract one $X$-particle is
$\sim 2.3\times 10^{-18}\,{\rm J}/c_X$.
Requiring that it does not exceed the energy one $X$-particle can produce
before getting stuck to a $^6$Li nucleus
leads to the constraint $c_X\gtrsim 2.3\times 10^{-16}$.
If it is satisfied, extracting
$X$-particles from granite would allow $X$CF to produce more energy than it
consumes, even without reactivation and recycling of the $X$-particles.
Another advantage of the extraction of relic $X$-particles from minerals
compared with their production at accelerators is that it could work even for
$X$-particles with mass $m_X\gg 1$\,TeV/$c^2$.

In assessing the viability of $X$CF as a mechanism of energy generation,
in addition to pure energy considerations one should
obviously address  many technical issues
related to its practical implementation,
such as collection and moderation of the produced
$X$-particles and prevention of their binding to the surrounding nuclei
(or their liberation if such binding occurs), etc. However, the corresponding
technical difficulties seem to be surmountable~\cite{Goity:1993ih}.

\section{\label{sec:disc} Discussion}

There are several obvious ways in which our analysis of $X$CF can be
generalized. Although we only considered nuclear fusion catalyzed by
scalar $X$-particles, doubly charged particles of non-zero spin can do
the job as well. While we studied $X$CF in deuterium, fusion processes
with participation of other hydrogen isotopes can also be catalyzed by
$X$-particles.

We considered $X$CF taking place in $X$-atomic states.
The catalyzed fusion can also proceed through in-flight reactions occurring
e.g.\ in $d+(dX)$ collisions. However, because even at the highest attainable
densities the average distance $\bar{r}$ between deuterons is
much larger than it is in $(ddX)$ atoms, the rates of in-flight reactions
are suppressed by a factor of the order of $(\bar{r}/a_d)^3\gtrsim 10^{9}$
compared with those of reactions  occurring in $X$-atoms.

Our results depend sensitively on the properties of positive ions
$({\rm ^3He} d X)$ and $({\rm ^4He} d X)$, for which we obtained only crude
estimates. More accurate calculations of these properties and of the formation
times of these positive ions would be highly desirable.

The existence of long-lived doubly charged particles may have important
cosmological consequences. In particular, they may form exotic
atoms, which have been discussed in connection with the dark matter
problem \cite{Fargion:2005ep,Belotsky:2006pp,Cudell:2015xiw}. They may also
affect primordial nucleosynthesis in an important way.
In ref.~\cite{Pospelov:2006sc} it was suggested
that singly negatively charged heavy metastable particles may catalyze
nuclear fusion reactions at the nucleosynthesis era, possibly solving the
cosmological lithium problem. The issue has been subsequently studied by many
authors, see refs.~\cite{Pospelov:2010hj,Kusakabe:2017brd} for reviews.
Doubly charged scalars $X$ may also catalyze nuclear fusion reactions in the
early Universe and thus may have significant impact on primordial
nucleosynthesis. On the other hand, cosmology may provide important
constraints on the $X$CF mechanism discussed here. Therefore,
a comprehensive study of cosmological implications of the existence of
$X^{\pm\pm}$ particles would be of great interest.

To conclude, we have demonstrated that long-lived or stable doubly
negatively charged scalar particles $X$, if exist, can catalyze nuclear fusion
and provide a viable source of energy. Our study gives a strong  additional
motivation for continuing and extending the experimental searches for such
particles.

{\it Note added.} Recently, the ATLAS Collaboration has reported a
3.6$\sigma$ (3.3$\sigma$) local (global) excess of events with large
specific ionization energy loss $|dE/dx|$ in their search for long-lived
charged particles at LHC \cite{ATLAS:2022pib}. In the complete LHC Run 2
dataset, seven events were found for which the values of $|dE/dx|$ were
in tension with the time-of-flight velocity measurements, assuming that
the corresponding particles were of unit charge. It has been shown in
\cite{Giudice:2022bpq} that this excess could be explained as being due
to relatively long-lived doubly charged particles. It would be very
interesting to see if the reported excess will survive with increasing
statistics of the forthcoming LHC Run 3.

\section{Acknowledgments}
The author is grateful to Manfred Lindner, Alexei Smirnov and Andreas Trautner
for useful discussions. Special thanks are due to Maxim Pospelov for numerous
helpful discussions of various aspects of $X$-catalyzed fusion and
constructive criticism.


\section{Introduction} \label{sec:introduction}
Hierarchical clustering for graphs plays an important role in the structural analysis of a given data set. Understanding hierarchical structures on the levels of multiple granularities is fundamental in various disciplines including artificial intelligence, physics, biology, sociology, etc \cite{brown1992class,eisen1998cluster,gorban2008principal,culotta2007author}. Hierarchical clustering requires a cluster tree that represents a recursive partitioning of a graph into smaller clusters as the tree nodes get deeper. In this cluster tree, each leaf represents a graph node while each non-leaf node represents a cluster containing its descendant leaves. The root is the largest one containing all leaves.

During the last two decades, flat clustering has attracted great attentions, which breeds plenty of algorithms, such as $k$-means \cite{hartingan1979kmeans}, DBSCAN \cite{ester1996density}, spectral clustering \cite{alpert1995spectral}, and so on. From the combinatorial perspective, we have cost functions, i.e., modularity and centrality measures, to evaluate the quality of partition-based clustering. Therefore, community detection is usually formulated as an optimization problem to optimize these objectives. By contrast, no comparative cost function with a clear and reasonable combinatorial explanation was developed until Dasgupta \cite{dasgupta2016cost} introduced a cost function for cluster trees. In this definition, similarity or dissimilarity between data points is represented by weighted edges. Taking similarity scenario as an example, a cluster is a set of nodes with relatively denser intra-links compared with its inter-links, and in a good cluster tree, heavier edges tend to connect leaves whose lowest common ancestor is as deep as possible. This intuition leads to Dasgupta's cost function that is a weighted linear combination of the sizes of lowest common ancestors over all edges.

Motivated by Dasgupta's cost function, Cohen-Addad et al. \cite{cohen2019hierarchical} proposed the concept of admissible cost function. In their definition, the size of each lowest common ancestor in Dasgupta's cost function is generalized to be a function of the sizes of its left and right children. For all similarity graphs generated from a minimal ultrametric, a cluster tree achieves the minimum cost if and only if it is a generating tree that is a ``natural'' ground truth tree in an axiomatic sense therein. A necessary condition of admissibility of an objective function is that it achieves the same value for every cluster tree for a uniformly weighted clique that has no structure in common sense. However, any slight deviation of edge weights would generally separate the two end-points of a light edge on a high level of its optimal (similarity-based) cluster tree. Thus, it seems that admissible objective functions, which take Dasgupta's cost function as a specific form, ought to be an unchallenged criterion in evaluating cluster trees since they are formulated by an axiomatic approach.

However, an admissible cost function seems imperfect in practice. The arbitrariness of optima of cluster trees for cliques indicates that the division of each internal nodes on an optimal cluster tree totally neglects the \emph{balance} of its two children. Edge weight is the unique factor that decides the structure of optimal trees. But a balanced tree is commonly considered as an ideal candidate in hierarchical clustering compared to an unbalanced one. So even clustering for cliques, a balanced partition should be preferable for each internal node. At least, the height of an optimal cluster tree which is logarithm of graph size $n$ is intuitively more reasonable than that of a caterpillar shaped cluster tree whose height is $n-1$. Moreover, a simple proof would imply that the optimal cluster tree for any connected graphs is binary. This property is not always useful in practical use since a real system usually has its inherent number of hierarchies and a natural partition for each internal clusters. For instance, the natural levels of administrative division in a country is usually intrinsic, and it is not suitable to differentiate hierarchies for parallel cities in the same state. This structure cannot be obtained by simply minimizing admissible cost functions.

In this paper, we investigate the hierarchical clustering from the perspective of information theory. Our study is based on Li and Pan's structural information theory \cite{li2016structural} whose core concept named structural entropy measures the complexity of hierarchical networks. We formulate a new objective function from this point of view, which builds the bridge for combinatorial and information-theoretic perspectives for hierarchical clustering. For this cost function, the balance of cluster trees will be involved naturally as a factor just like we design optimal codes, for which the balance of probability over objects is fundamental in constructing an efficient coding tree. We also define cluster trees with a specific height, which is coincident with our cognition of natural clustering. For practical use, we develop a novel algorithm for natural hierarchical clustering for which the number of hierarchies can be determined automatically. The idea of our algorithm is essentially different from the popular recursive division or agglomeration framework. We formulate two basic operations called \emph{stretch} and \emph{compress} respectively on cluster trees to search for the sparsest level iteratively. Our algorithm HCSE terminates when a specific criterion that intuitively coincides with the natural hierarchies is met. Our extensive experiments on both synthetic and real datasets demonstrate that HCSE outperforms the present popular heuristic algorithms LOUVAIN \cite{blondel2008fast} and HLP \cite{rossi2020fast}. The latter two algorithms proceed simply by recursively invoking flat clustering algorithms based on modularity and label propagation, respectively. For both of them, the hierarchy number is solely determined by the round numbers when the algorithm terminates, for which the interpretability is quite poor. Our experimental results on synthetic datasets show that HCSE has a great advantage in finding the intrinsic number of hierarchies, and the results on real datasets show that HCSE achieves competitive costs over LOUVAIN and HLP.

We organize this paper as follows. The structural information theory and its relationship with combinatorial cost functions will be introduced in Section \ref{sec:cost_functions}, and the algorithm will be given in Section \ref{sec:algorithm}. The experiments and their results are presented in Section \ref{sec:experiments}. We conclude the paper in Section \ref{sec:conclusions}.

\section{A cost function from information-theoretic perspective}
\label{sec:cost_functions}

In this section, we introduce Li and Pan's structural information theory \cite{li2016structural} and the combinatorial cost functions of Dasgupta \cite{dasgupta2016cost} and Cohen-Addad et al. \cite{cohen2019hierarchical}. Then we propose a new cost function that is developed from structural information theory and establish the relationship between the information-theoretic and combinatorial perspectives.

\subsection{Notations}
\label{subsec:notations}

Let $G=(V,E,w)$ be an undirected weighted graph with a set of vertices $V$, a set of edges $E$ and a weight function $w:E\rightarrow\mathbb{R}^+$, where $\mathbb{R}^+$ denotes the set of all positive real numbers. An unweighted graph can be viewed as a weighted one whose weights are unit. For each vertex $u\in V$, denote by $d_u=\sum_{(u,v)\in E} w(u,v)$ the weighted degree of $u$.\footnote{From now on, whenever we say the degree of a vertex, we always refer to the weighted degree.} For a subset of vertices $S\subseteq V$, define the volume of $S$ to be the sum of degrees of vertices. We denote it by $\textrm{vol}(S)=\sum_{u\in S} d_u$.

A cluster tree $T$ for graph $G$ is a rooted tree with $|V|$ leaves, each of which is labeled by a distinct vertex $v\in V$. Each non-leaf node on $T$ is labeled by a subset $S$ of $V$ that consists of all the leaves treating $S$ as ancestor. For each node $\alpha$ on $T$, denote by $\alpha^-$ the parent of $\alpha$. For each pair of leaves $u$ and $v$, denote by $u\vee v$ the least common ancestor (LCA) of them on $T$.

\subsection{Structural information and structural entropy}
\label{subsec:structural_information}

The idea of structural information is to encode a random walk with a certain rule by using a high-dimensional encoding system for a graph $G$. It is well known that a random walk, for which a neighbor is randomly chosen with probability proportional to edge weights, has a stationary distribution on vertices that is proportional to vertex degree.\footnote{For connected graphs, this stationary distribution is unique, but not for disconnected ones. Here, we consider this one for all graphs.} So to position a random walk under its stationary distribution, the amount of information needed is typically the Shannon's entropy, denoted by $$\mathcal{H}^{(1)}(G)=-\sum_{v\in V} \frac{d_v}{\textrm{vol}(V)} \log \frac{d_v}{\textrm{vol}(V)} \footnote{In this paper, the omitted base of logarithm is always $2$.}.$$ By Shannon's noiseless coding theorem, $\mathcal{H}^{(1)}(G)$ is the limit of average code length generated from the \emph{memoryless} source for one step of the random walk. However, dependence of locations may shorten the code length. For each level on cluster trees, the uncertainty of locations is measured by the entropy of the stationary distribution on the clusters of this level. Consider an encoding for every cluster, including the leaves. Each non-root node $\alpha$ is labeled by its order among the children of its parent $\alpha^-$. So the self-information of $\alpha$ within this local parent-children substructure is $-\log(\textrm{vol}(\alpha)/\textrm{vol}(\alpha^-))$, which is also roughly the length of Shannon code for $\alpha$ and its siblings. The codeword of $\alpha$ consists of the sequential labels of nodes along the unique path from the root (excluded) to itself (included). The key idea is as follows. For one step of the random walk from $u$ to $v$ in $G$, to indicate $v$, we omit from $v$'s codeword the longest common prefix of $u$ and $v$ that is exactly the codeword of $u\vee v$. This means that the random walk takes this step in the cluster $u\vee v$ (and also in $u\vee v$'s ancestors) and the uncertainty at this level may not be involved. Therefore, intuitively, a quality similarity-based cluster tree would trap the random walk with high frequency in the deep clusters that are far from the root, and long codeword of $u\vee v$ would be omitted. This shortens the average code length of the random walk. Note that we ignore the uniqueness of decoding since a practical design of codewords is not our purpose. We utilize this scheme to evaluate and differentiate hierarchical structures.

Then we formulate the above scheme and measure the average code length as follows. Given a weighted graph $G=(V,E,w)$ and a cluster tree $T$ for $G$, note that under the stationary distribution, the random walk takes one step out of a cluster $\alpha$ on $T$ with probability $g_\alpha/\textrm{vol}(V)$, where $g_\alpha$ is the sum of weights of edges with exactly one end-point in $\alpha$. Therefore, the aforementioned uncertainty measured by the average code length is
\[\mathcal{H}^T (G)=-\sum_{\alpha\in T} \frac{g_\alpha}{\textrm{vol}(V)} \log \frac{\textrm{vol}(\alpha)}{\textrm{vol}(\alpha^-)}.\footnote{For notational convenience, for the root $\lambda$ of $T$, set $\lambda^-=\lambda$. So the term for $\lambda$ in the summation is $0$.}\]
We call $\mathcal{H}^T(G)$ the \emph{structural entropy of $G$ on $T$}. We define the \emph{structural entropy of $G$} to be the minimum one among all cluster trees, denoted by
$\mathcal{H}(G)=\min_{T}\{\mathcal{H}^T (G)\}.$
Note that the structural entropy of $G$ on the trivial $1$-level cluster tree is consistent with the previously defined $\mathcal{H}^{(1)}(G)$. It doesn't have any non-trivial cluster.

\subsection{Combinatorial explanation of structural entropy}

The cost function of a cluster tree $T$ for graph $G=(V,E)$ introduced by Dasgupta \cite{dasgupta2016cost} is defined to be $c^T(G)=\sum_{(u,v)\in E} w(u,v) |u\vee v|$, where $|u\vee v|$ denotes the size of cluster $u\vee v$. The admissible cost function introduced by Cohen-Addad et al. \cite{cohen2019hierarchical} generalizes the term $|u\vee v|$ in the definition of $c^T(G)$ to be a general function $g(|u|,|v|)$, for which Dasgupta defined $g(x,y)=x+y$. For both definitions, the optimal hierarchical clustering of $G$ is in correspondence with a cluster tree of minimum cost in the combinatorial sense that heavy edges are cut as far down the tree as possible.
The following theorem establishes the relationship between structural entropy and this kind of combinatorial form of cost functions.

\begin{theorem} \label{thm:equvi_cost_func}
	For a weighted graph $G=(V,E,w)$, to minimize $H^T(G)$ (over $T$) is equivalent to minimize the cost function
	\begin{equation} \label{eqn:SE_cost_form}
		\textrm{cost}^T(G)=\sum_{(u,v)\in E} w(u,v) \log\textrm{vol}(u\vee v).
	\end{equation}
\end{theorem}

\begin{proof}
	Note that
	\begin{eqnarray*}
		H^T(G) &=& -\sum_{\alpha\in T} \frac{g_\alpha}{\textrm{vol}(V)}\log\frac{\textrm{vol}(\alpha)}{\textrm{vol}(\alpha^-)}\\
		&=& -\sum_{\alpha\in T} \sum_{(u,v)\in g_\alpha} \frac{w(u,v)}{\textrm{vol}(V)}\log\frac{\textrm{vol}(\alpha)}{\textrm{vol}(\alpha^-)}\\
		&=& -\sum_{(u,v)\in E} \left(\frac{w(u,v)}{\textrm{vol}(V)} \sum_{\alpha:(u,v)\in g_\alpha} \log\frac{\textrm{vol}(\alpha)}{\textrm{vol}(\alpha^-)}\right).
	\end{eqnarray*}
	For a single edge $(u,v)\in E$, all the terms $\log(\textrm{vol}(\alpha)/\textrm{vol}(\alpha^-))$ for leaf $u$ satisfying $(u,v)\in g_\alpha$ sum (over $\alpha$) up to $\log (d_u/\textrm{vol}(u\vee v))$ along the unique path from $u$ to $u\vee v$. It is symmetric for $v$. Therefore, considering ordered pair $(u,v)\in E$,
	\begin{eqnarray*}
		H^T(G) &=& -\sum_{\text{ordered }(u,v)\in E} \frac{w(u,v)}{\textrm{vol}(V)}\log\frac{d_u}{\textrm{vol}(u\vee v)}\\
		&=& \frac{1}{\textrm{vol}(V)} \left( -\sum_{u\in V} d_u \log d_u + \sum_{\text{ordered }(u,v)\in E} w(u,v) \log \textrm{vol}(u\vee v) \right)\\
		&=& \frac{1}{\textrm{vol}(V)} \left( -\sum_{u\in V} d_u \log d_u + 2\cdot\sum_{(u,v)\in E} w(u,v) \log \textrm{vol}(u\vee v) \right).
	\end{eqnarray*}
	The second equality follows from the fact $\sum_{u\in V}d_u=\sum_{\text{ordered }(u,v)\in E} w(u,v)=\textrm{vol}(V)$ and the last equality from the symmetry of $(u,v)$. Since the first summation is independent of $T$, to minimize $H^T(G)$ is equivalent to minimize $\sum_{\{u,v\}\in E} w(u,v)\log \textrm{vol}(u\vee v)$.
\end{proof}

Theorem \ref{thm:equvi_cost_func} indicates that when we view $g$ as a function of vertices rather than of numbers and define $g(u,v)=\log \textrm{vol}(u\vee v)$, the ``admissible'' function becomes equivalent to structural entropy in evaluating cluster trees, although it is not admissible any more.

So what is the difference between these two cost functions? As stated by Cohen-Addad et al. \cite{cohen2019hierarchical}, an important axiomatic hypothesis for admissible function, thus also for Dasgupta's cost function, is that the cost for every binary cluster tree of an unweighted clique is identical. This means that any binary tree for clustering on cliques is reasonable, which coincides with the common sense that structureless datasets can be organized hierarchically free. However, for structural entropy, the following theorem indicates that balanced organization is of importance even though for structureless dataset.

\begin{theorem} \label{thm:SE_for_cliques}
	For any positive integer $n$, let $K_n$ be the clique of $n$ vertices with identical weight on every edge. Then a cluster tree $T$ of $K_n$ achieves minimum structural entropy if and only if $T$ is a balanced binary tree, that is, the two children clusters of each sub-tree of $T$ have difference in size at most $1$.
\end{theorem}

The proof of Theorem \ref{thm:SE_for_cliques} is a bit technical, and we defer it to Appendix \ref{sec:proof_thm_2.2}. The intuition behind Theorem \ref{thm:SE_for_cliques} is that balanced codes are the most efficient encoding scheme for unrelated data. So the codewords of the random walk that jumps freely among clusters on each level of a cluster tree have the minimum average length if all the clusters on this level are in balance. This is able to be guaranteed exactly by a balanced cluster tree.

In the cost function (\ref{eqn:SE_cost_form}), which we call cost(SE) from now on, $\log\textrm{vol}(u\vee v)$ is a concave function of the volume of $u\vee v$. In the context of regular graphs (e.g. cliques), replacing $\textrm{vol}(u\vee v)$ by $|u\vee v|$ is equivalent for optimization. Dasgupta \cite{dasgupta2016cost} claimed that for the clique $K_4$ of four vertices, a balanced tree is preferable when replace $|u\vee v|$ by $g(|u\vee v|)$ for any strictly increasing concave function $g$ with $g(0)=0$. However, it is interesting to note that this generalization does not hold for all those concave functions. For example, it is easy to check that for $g(x)=1-e^{-x}$, the cluster tree of $K_6$ that achieves minimum cost partitions $K_6$ into $K_2$ and $K_4$ on the first level, rather than $K_3$ and $K_3$. Theorem \ref{thm:SE_for_cliques} shows that for all cliques, balanced trees are preferable when $g$ is a logarithmic function.

It is worth noting the admissible function introduced by Cohen-Addad et al. \cite{cohen2019hierarchical} is defined from the viewpoint that a generating tree $T$ of a similarity-based graph $G$ that is generated from a minimal ultrametric achieves the minimum cost. In this definition, the monotonicity of edge weights between clusters on each level from bottom to top on $T$, which is given by Cohen-Addad et al. \cite{cohen2019hierarchical} as a property of a ``natural'' ground-truth hierarchical clustering, is the unique factor when evaluating $T$. However, Theorem \ref{thm:SE_for_cliques} implies that for cost(SE), besides cluster weights, the balance of cluster trees is implicitly involved as another factor. Moreover, for cliques, the minimum cost should be achieved on every subtree, which makes an optimal cluster tree balanced everywhere. This optimal clustering for cliques is also robust in the sense that a slight perturbation to the minimal ultrametric, which can be considered as slight variations to the weights of a batch of edges, will not change the optimal cluster tree structure wildly due to the holdback force of balance.

\section{Our hierarchical clustering algorithm} \label{sec:algorithm}

In this section, we develop an algorithm to optimize cost(SE) (Eq. (\ref{eqn:SE_cost_form}), equivalent to optimizing structure entropy) and yield the associated cluster tree. At present, all existing algorithms for hierarchical clustering can be categorized into two frameworks: top-down division and bottom-up agglomeration \cite{cohen2019hierarchical}. The top-down division approach usually yields a binary tree by recursively dividing a cluster into two parts with a cut-related criterion. But a binary clustering tree is far from a practical one as we introduced in Section \ref{sec:introduction}. For practical use, bottom-up agglomeration that is also known as hierarchical agglomerative clustering (HAC) is commonly preferable. It constructs a cluster tree from leaves to the root recursively, during each round of which the newly generated clusters shrink into single vertices.

Our algorithm jumps out of these two frameworks. We establish a new one that stratifies the \emph{sparsest} level of a cluster tree recursively rather than in a sequential order. In general, in guide with cost(SE), we construct a $k+1$-level cluster tree from the previous $k$-level one, during which the level whose stratification makes the average local cost that is incorporated in a local reduced subgraph decrease most is differentiated into two levels. The process of stratification consists of two basic operations: \emph{stretch} and \emph{compression}. In stretch steps, given an internal node of a cluster tree, a local binary subtree is constructed by an agglomerative approach, while in compression steps, the paths that are overlength from the root to leaves on the binary tree is compressed by shrinking tree edges that make the cost reduce most. This framework can be collocated with any cost function.

Then we define the operations ``stretch'' and ``compression'' formally. Given a cluster tree $T$ for graph $G=(V,E)$, let $u$ be an internal node on $T$ and $v_1,v_2,\ldots,v_\ell$ be its children. We call this local parent-children structure to be a \emph{$u$-triangle} of $T$, denoted by $T_u$. These two operations are defined on $u$-triangles. Note that each child $v_i$ of $u$ is a cluster in $G$. We reduce $G$ by shrinking each $v_i$ to be a single vertex $v_i'$ while maintaining each intra-link and ignoring each internal edge of $v_i$. This reduction captures the connections of clusters at this level in the parent cluster $u$. The stretch operation proceeds in HAC approach for $u$-triangle. That is, initially, view each $v_i'$ as a cluster and recursively combine two clusters into a new one for which cost(SE) drops most. The sequence of combinations yields a binary subtree $T_u'$ rooted at $u$ which has $v_1,v_2,\ldots,v_\ell$ as leaves. Then the compression operation is proposed to reduce the height of $T_u'$ to be $2$. Let $\hat{E}(T')$ be the set of edges on $T'$ each of which appears on a path of length more than $2$ from the root of $T'$ to some leaf. Denote by $\Delta(e)$ for edge $e$ be the amount of structural entropy enhanced by the shrink of $e$. We pick from $\hat{E}(T_u')$ the edge $e$ with least $\Delta(e)$. Note that the compression of a tree edge makes the grandchildren of some internal node to be children, it must amplify the cost. The compression operation picks the least amplification. The process of stretch and compression is illustrated in Figure \ref{fig:stretch_compress} and stated in Algorithms \ref{alg:stretch} and \ref{alg:compress}, respectively.

\begin{figure}[ht]
	\centering
	\includegraphics[scale=0.5]{stretch_compress.jpg}
	\caption{Illustration of stretch and compression for a $u$-triangle. A binary tree is constructed first by stretch, and then edge $e$ is compressed.}
	\label{fig:stretch_compress}
\end{figure}

\begin{algorithm} \label{alg:stretch}
	\caption{Stretch}
	\KwIn{a $u$-triangle $T_u$}
	\KwOut{a binary tree rooted at $u$}
	Let $\{v_1,v_2,\ldots,v_\ell\}$ be the set of leaves of $T_u$\;
	Compute $\eta(a,b)$ which is structural entropy reduced by merging siblings $a,b$ into a single cluster\;
	\For{$t\in [\ell-1]$}{
		$(\alpha,\beta) \gets \arg\max_{(a,b) \text{ are siblings}} \{\eta(a,b)\}$\;
		Add a new node $\gamma$\;
		$\gamma.parent \gets \alpha.parent$\;
		$\alpha.parent = \gamma$\;
		$\beta.parent = \gamma$\;
	}
	return $T_u$
\end{algorithm}

\begin{algorithm} \label{alg:compress}
	\caption{Compress}
	\KwIn{a binary tree $T$}

	\While{$T$'s height is more than $2$}{
		$e \gets \arg\min_{e'\in\hat{E}(T)} \{\Delta(e')\}$\;
		Denote $e=(u,v)$ where $u$ is the parent of $v$\;
		\For{$w\in v.children$}{
			$w.parent \gets u$\;
		}
	}

\end{algorithm}

Then we define the sparsest level of a cluster tree $T$. Let $U_j$ be the set of $j$-level nodes on $T$, that is, $U_j$ is the set of nodes each of which has distance $j$ from $T$'s root. Suppose that the height of $T$ is $k$, then $U_0,U_1,\ldots,U_{k-1}$ is a partition for all internal nodes of $T$. For each internal node $u$, define $\mathcal{H}(u)=-\sum_{v:v^-=u} \frac{g_u}{\textrm{vol}(V)} \log \frac{\textrm{vol}(v)}{\textrm{vol}(u)}$. Note that $\mathcal{H}(u)$ is the partial sum contributed by $u$ in $\mathcal{H}^T(G)$. After a ``stretch-and-compress'' round on $u$-triangle, denote by $\Delta\mathcal{H}(u)$ the structural entropy by which the new cluster tree reduces. Since the reconstruction of $u$-triangle stratifies cluster $u$, $\Delta\mathcal{H}(u)$ is always non-negative. Define the sparsity of $u$ to be
$\text{Spar}(u)=\frac{\Delta\mathcal{H}(u)}{\mathcal{H}(u)}$, which is the relative variation of structural entropy in cluster $u$. From the information-theoretic perspective, this means that the uncertainty of random walk can be measured locally in any internal cluster, which reflects the quality of clustering of this local area. At last, we define the \emph{sparsest level} of $T$ to be the $j$-th level such that the average sparsity of triangles rooted at nodes in $U_j$ is maximum, that is $\arg\max_j \{\overline{\text{Spar}}_j(T)\}$, where $\overline{\text{Spar}}_j(T)=\sum_{u\in U_j}\text{Spar(u)}/|U_j|$. Then the operation of stratification stretches and compresses on the sparsest level of $T$. This is illustrated in Figure \ref{fig:stratify}.

\begin{figure}[htbp]
\begin{center}
	\subfigure[]{
		\begin{minipage}[t]{0.5\linewidth}
			\centering
			\includegraphics[width=6cm]{up.jpg}
			\label{fig:up}

		\end{minipage}%
	}%
	\subfigure[]{
		\begin{minipage}[t]{0.5\linewidth}
			\centering
			\includegraphics[width=6cm]{below.jpg}
			\label{fig:below}

		\end{minipage}%

	\caption{Illustration of stratification for a $2$-level cluster tree. The preference of (a) and (b) depends on the average sparsity of triangles at each level.}
	\label{fig:stratify}
\end{center}
\end{figure}

For a given positive integer $k$, to construct a cluster tree of height $k$ for graph $G$, we start from the trivial $1$-level cluster tree that involves all vertices of $G$ as leaves. Then we do not stop stratifying at the sparsest level recursively until a $k$-level cluster tree is obtained. This process is described in Algorithm \ref{alg:k-HCSE}.

\begin{algorithm} \label{alg:k-HCSE}
	\caption{$k$-Hierarchical clustering based on structural entropy ($k$-HCSE)}
	\KwIn{a graph $G = (V,E)$, $k\in\mathbb{Z}^+$}
	\KwOut{a $k$-level cluster tree $T$}
	Initialize $T$ to be the $1$-level cluster tree\;
	$h=\text{height(T)}$\;
	\While{$h<k$}{
		$j' \gets \arg\max_{j} \{\overline{\text{Spar}}_{j}(T)\}$; \quad // Find the sparsest level of $T$ (breaking ties arbitraily)\;
		\If{$\overline{\text{Spar}}_{j'}(T)=0$}{
			break;	\quad // No cost will be saved by any further clustering\;
		}
		\For{$u\in U_{j'}$}{
			$T_u \gets$ Stretch($u$-triangle $T_u$)\;
			Compress($T_u$)\;
			$h \gets h+1$\;
		}
		\For{$j\in [j'+1,h]$}{
			Update $U_j$\;
		}
	}
	return $T$
\end{algorithm}

To determine the height of the cluster tree automatically, we derive the natural clustering from the variation of sparsity on each level. Intuitively, a natural hierarchical cluster tree $T$ should have not only sparse boundary on clusters, but also low sparsity for triangles of $T$, which means that stratification within the reduced subgraphs corresponding to the triangles on the sparsest level make little sense. For this reason, we consider the inflection points of the sequence $\{\delta_t(\mathcal{H})\}_{t=1,2,\ldots}$, where $\delta_t(\mathcal{H})$ is the structural entropy by which the $t$-th round of stratification reduces. Formally, denote $\Delta_t\mathcal{H}=\delta_t(\mathcal{H})-\delta_{t-1}(\mathcal{H})$ for each $t\geq 2$. We say that $\Delta_t\mathcal{H}$ is an inflection point if both $\Delta_t\mathcal{H} \geq \Delta_{t-1}\mathcal{H}$ and $\Delta_t\mathcal{H} \geq \Delta_{t+1}\mathcal{H}$ hold. Our algorithm finds the least $t$ such that $\Delta_t\mathcal{H}$ is an inflection point and fix the height of the cluster tree to be $t$ (Note that after $t-1$ rounds of stratification, the number of levels is $t$). This process is described as Algorithm \ref{alg:HCSE}.

\begin{algorithm} \label{alg:HCSE}
	\caption{Hierarchical clustering based on structural entropy (HCSE)}
	\KwIn{a graph $G = (V,E)$}
	\KwOut{a cluster tree $T$}
	$t \gets 2$\;
	\While{$\Delta_t\mathcal{H} < \Delta_{t-1}\mathcal{H}$ or $\Delta_t\mathcal{H} < \Delta_{t+1}\mathcal{H}$}{
		\If{$\max_{j} \{\overline{\text{Spar}}_{j}(T)\}$=0}{
			break\;
		}
		$t \gets t+1$\;
	}
	return $t$-HCSE$(T)$
\end{algorithm}

\section{Experiments} \label{sec:experiments}

Our experiments are given both on synthetic networks generated from the Hierarchical Stochastic Block Model (HSBM) and on real datasets. We compare our algorithm HSE with the popular practical algorithms LOUVAIN \cite{blondel2008fast} and HLP \cite{rossi2020fast}. Both of these two algorithms construct a non-binary cluster tree with the same framework, that is, the hierarchies are formed from bottom to top one by one. In each round, they invoke different flat clustering algorithms, Modularity and Label Propagation, respectively. To avoid over-fitting to higher levels, which possibly results in under-fitting to lower levels, LOUVAIN admits a sequential input of vertices. Usually, to avert the worst-case trap, the order in which the vertices come is random, and so the resulting cluster tree depends on this order. HLP invokes the common LP algorithm recursively, and so it cannot be guaranteed to avoid under-fitting in each round. This can be seen in our experiments on synthetic datasets, for which these two algorithms usually miss a ground-truth level.

For real datasets, we do the comparative experiments on real networks. Some of them have (possibly hierarchical) ground truth, e.g., Amazon, while most of them do not have. We evaluate the resulting cluster trees for Amazon networks by Jaccard index, and for others without ground truth by both cost(SE) and Dasgupta's cost function cost(Das).

\subsection{HSBM}

The stochastic block model (SBM) is a type of probability generation model proposed based on the idea of random equivalence. The SBM of flat graphs needs to design the number of clusters, the number of vertices in each cluster, the probability of generating an edge for each pair of vertices in clusters, and the probability of generating an edge for each pair of vertices that are from different clusters. For HSBM, a ground truth of hierarchies should be presumed. Suppose that there are $m$ clusters at the bottom level. Then probability of generating edges is determined by a symmetric $m\times m$ matrix, in which the $(i,j)$-th entry is the probability of connecting each pair of vertices from cluster $i$ and $j$, respectively. Two clusters that have higher LCA on the ground-truth tree have lower probability.

Our experiments utilize $4$-level HSBM. For simplicity, let $\vec{p}=(p_0,p_1,p_2,p_3)$ be the probability vector for which $p_i$ is the probability of generating edges for vertex pairs whose LCA on the ground-truth cluster tree has depth $i$. Note that the $0$-depth node is the root. We compare the Normalized Mutual Information (NMI) at each level of the ground-truth cluster tree to those of three algorithms. Note that the randomness in LOUVAIN, and breaking-ties rule as well as convergence of HLP make different results, we choose the most effective strategy and pick the best results in five runs for both of them. Compared to their uncertainty, our algorithm HCSE yields stable results.

Table \ref{tab:HSBM_NMI} demonstrates the results in three groups of probabilities, for which the clarity of hierarchical structure gets higher one by one. Our algorithm HSE is always able to find the right number of levels, while LOUVAIN always misses the top level, and HLP misses the top level in two groups. The inflection points for choosing the intrinsic hierarchy number $t=4$ of hierarchies are demonstrated in Figure \ref{fig:inflection_points}.

\makeatletter
\newcommand\figcaption{\def\@captype{figure}\caption}
\newcommand\tabcaption{\def\@captype{table}\caption}
\makeatother

\begin{figure}[tb]
	\centering
	\begin{minipage}{0.42\textwidth}
		\centering
		\begin{tabular}{ccccc}
			\toprule

			&  $\vec{p}$  & HCSE     & HLP      & LOU  \\
			\midrule
			$p_2$ &  4.5E(-2)  & 0.89  & 0.79      & \textbf{0.92}\\
			$p_1$ &  1.5E(-3)  & \textbf{0.93} & 0.75       & 0.92 \\
			$p_0$ &  6E(-6)  & \textbf{0.62}       & 0.58       & \verb|--|\\
			\midrule
			$p_2$ &  5.5E(-2)  & 0.87  & \textbf{0.89}      & 0.89\\
			$p_1$ &  1.5E(-3)  & \textbf{0.95} & 0.87       & 0.87 \\
			$p_0$ &  4E(-6)  & \textbf{0.72}       & \verb|--|       & \verb|--| \\
			\midrule
			$p_2$ &  6.5E(-2)  & 0.96  & 0.95      & \textbf{0.99}\\
			$p_1$ &  4.5E(-3)  & 0.94 & 0.81       & \textbf{0.99} \\
			$p_0$ &  2.5E(-6)  & \textbf{0.80}       & \verb|--|       & \verb|--| \\
			\bottomrule
		\end{tabular}
		\tabcaption{\footnotesize NMI for three algorithms. Each dataset has $2,500$ vertices, and the cluster numbers at three levels are $5$, $25$ and $250$, respectively, for which the size of each cluster is accordingly
			generated at random. $p_3=0.9$ for each graph. ``$--$'' means the algorithm did not find this level.}
		\label{tab:HSBM_NMI}
	\end{minipage}
	\hspace{0.5in}
	\begin{minipage}[h]{0.42\linewidth}
		\centering
		\includegraphics[scale=0.4]{delta.png}
		\figcaption{$\delta_t(\mathcal{H})$ variations for HCSE. It can be observed easily that the inflection points for all the three datasets appear on $t=4$, which is also the ground-truth number of hierarchies.}
		\label{fig:inflection_points}
	\end{minipage}
\end{figure}

\subsection{Real datasets}

First, we do our experiments on Amazon network \footnote{http://snap.stanford.edu/data/} for which the set of ground-truth clusters has been given. For two sets $A,B$, the \emph{Jaccard Index} of them is defined as $J(A,B)=|A\cap B|/|A\cup B|$. We pick the largest cluster which is a subgraph with $58283$ vertices and $133178$ edges. We run HCSE algorithm on it. For each ground-truth cluster $c$ that appears in this subgraph, we find from the resulting cluster tree an internal node that has maximum Jaccard index with $c$. Then we calculate the average Jaccard index $\overline{J}$ over all such $c$. We also calculate cost(SE) and cost(Das). The results are demonstrated in Table \ref{tab:Amazon}. HCSE performs better for $\overline{J}$ and cost(SE), while LOUVAIN performs better for cost(Das). Because of unbalance in over-fitting and under-fitting traps, HLP outperforms none of the other two algorithms for all criteria.

\begin{table}[htbp]
	\centering
	\begin{tabular}{cccc}
		\toprule

		index     & \hspace{0.3em}HCSE     & HLP      & LOUVAIN  \\
		\midrule
		$\overline{J}$ & \textbf{0.20}  & 0.16      & 0.17\\
		cost(SE)    & \textbf{1.85E6} & 2.05E6 & 1.89E6 \\
		cost(Das)   & 5.57E8 & 3.99E8 & \textbf{3.08E8} \\
		\bottomrule
	\end{tabular}
	\smallskip
	\caption{\footnotesize Comparisons of the average Jaccard index ($\overline{J}$), cost function based on structural entropy (cost(SE)) and Dasgupta's cost function (cost(Das)).}
	\label{tab:Amazon}
\end{table}

Second, we do our experiments on a series of real networks \footnote{http://networkrepository.com/index.php} without ground truth. We compare cost(SE) and cost(Das), respectively. Since the different level numbers given by the three algorithms influence the costs seriously, that is, lower costs are obtained just due to greater heights, we only list in Table \ref{tab:real_datasets} the networks for which the three algorithms yield similar level numbers that differ by at most $1$ or $2$. It can be observed that HLP does not achieve optima for any network, while HCSE performs best w.r.t. cost(Das) for all networks, but does not outperform LOUVAIN for most networks. This is mainly due to the fact that LOUVAIN always finds no less number of hierarchies than HCSE, and the better cost probably benefits from its depth.

\begin{table}
	\centering
	\begin{tabular}{cccccccccc}
		\toprule

		Networks     & HCSE     & HLP      & LOUVAIN  \\
		\midrule
		CSphd         & 1.30E4 / \textbf{5.19E4} / 5  & 1.54E4 / 5.58E4 / 4      & \textbf{1.28E4} / 7.61E4 / 5\\
		\midrule
		fb-pages-government & 2.48E6 / \textbf{1.18E8} / 4  & 2.53E6 / 1.76E8 / 3      & \textbf{2.43E6} / 1.33E8 / 4\\
		\midrule
		email-univ & 1.16E5 / \textbf{2.20E6} / 3  & 1.46E5 / 6.14E6 / 3      & \textbf{1.14E5} / 2.20E6 / 4\\
		\midrule
		fb-messages & 1.58E5 / \textbf{4.50E6} / 4  & 1.76E5 / 8.12E6 / 3      & \textbf{1.52E5} / 4.96E6 / 4\\
		\midrule
		G22 & \textbf{5.56E5} / \textbf{2.68E7} / 4  & 6.11E5 / 4.00E7 / 3      & 5.63E5 / 2.80E7 / 5\\
		\midrule
		As20000102 & 2.64E5 / \textbf{2.36E7} / 4  & 3.62E5 / 7.63E7 / 3      & \textbf{2.42E5} / 2.42E7 / 5\\
		\midrule
		bibd-13-6 & \textbf{7.41E5} / \textbf{2.56E7} / 3  & 8.05E5 / 4.41E7 / 2      & 7.50E5 / 2.75E7 / 4\\
		\midrule
		delaunay-n10 & 4.65E4 / \textbf{3.39E5} / 4  & 4.87E4 / 3.55E5 / 4      & \textbf{4.24E4} / 4.25E5 / 5\\
		\midrule
		p2p-Gnutella05 & 9.00E5 / \textbf{1.48E8} / 3  & 1.01E6 / 2.78E8 / 3      & \textbf{8.05E5} / 1.49E8 / 5\\
		\midrule
		p2p-Gnutella08 & 5.59E5 / \textbf{5.51E7} / 4  & 6.36E5 / 1.28E8 / 4      & \textbf{4.88E5} / 6.03E7 / 5\\
		\bottomrule
	\end{tabular}
	\smallskip
	\caption{``cost(SE) / cost(Das) / $k$'' for three algorithms, where $k$ is the hierarchy number that the algorithm finds.}
	\label{tab:real_datasets}
\end{table}

\section{Conclusions and future discussions} \label{sec:conclusions}

In this paper, we investigate the hierarchical clustering problem from an information-theoretic perspective and propose a new objective function that relates to the combinatoric cost functions raised by Dasgupta \cite{dasgupta2016cost} and Cohen-Addad et al. \cite{cohen2019hierarchical}. We define the optimal $k$-level cluster tree for practical use and devise an hierarchical clustering algorithm that stratifies the sparsest level of the cluster tree recursively. This is a general framework that can be collocated with any cost function. We also propose an interpretable strategy to find the intrinsic number of levels without any hyper-parameter. The experimental results on $k$-level HSBM demonstrate that our algorithm HCSE has a great advantage in finding $k$ compared to the popular but strongly heuristic algorithms LOUVAIN and HLP. Our results on real datasets show that HCSE also achieves competitive costs compared to these two algorithms.

There are several directions that are worth further study. The first problem is about the relationship between the concavity of $g$ of the cost function and the balance of the optimal cluster tree. We have seen that for cliques, being concave is not a sufficient condition for total balance, so whether is it a necessary condition? Moreover, is there any explicit necessary and sufficient condition for total balance of the optimal cluster tree for cliques? The second problem is about approximation algorithms for both structural entropy and cost(SE). Due to the non-linear and volume-related function $g$, the proof techniques for approximation algorithms in \cite{cohen2019hierarchical} becomes unavailable. The third one is about more precise characterizations for ``natural'' hierarchical clustering whose depth is limited. Since any reasonable choice of $g$ makes the cost function achieve optimum on some binary tree, a blind pursuit of minimization of cost functions seems not to be a rational approach. More criteria in this scenario need to be studied.

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