# Dirac neutrino mass generation from Majorana messenger

Julian<EMAIL_ADDRESS>Diego
<EMAIL_ADDRESS>and Óscar
<EMAIL_ADDRESS>
1Instituto de Física, Universidad de Antioquia,
Calle 70 # 52-21, Apartado Aéreo 1226, Medellín, Colombia.
2 International Institute of Physics, Universidade Federal do Rio Grande do
Norte,
Campus Universitario, Lagoa Nova, Natal-RN 59078-970, Brazil.
3 Abdus Salam International Centre for Theoretical Physics,
Strada Costiera 11, 34151, Trieste, Italy.

###### Abstract

The radiative type-I seesaw has been already implemented to explain the
lightness of Majorana neutrinos with both Majorana and Dirac heavy fermions,
and the lightness of Dirac neutrinos with Dirac heavy fermions. In this work
we present a minimal implementation of the radiative type-I seesaw with light
Dirac neutrinos and heavy Majorana fermions. An inert doublet and a complex
singlet scalar complete the dark sector which is protected by an Abelian
fermiophobic gauge symmetry that also forbids tree level mass contributions
for the full set of light neutrinos. A fermion vector-like extension of the
model is also proposed where the light right-handed neutrinos can thermalize
in the primordial plasma and the extra gauge boson can be directly produced at
colliders. In particular, the current upper bound on $\Delta N_{\text{eff}}$
reported by PLANCK points to large ratios $M_{Z^{\prime}}/g^{\prime}\gtrsim
40\ \text{TeV}$ which can be competitive with collider constraint for
$g^{\prime}$ sufficiently large in the ballpark of the Standard Model values,
while future cosmic microwave background experiments may probe all the no
minimal models presented here.

IIPDM-2019

##  1 Introduction

The interpretation of neutrino experimental data in terms of neutrino
oscillations is compatible with both Majorana and Dirac neutrino masses [1].
The former possibility has received the most attention but, given the lack of
signals in neutrinoless double beta decay experiments [2, 3, 4, 5, 6, 7], the
latter cannot be dismissed. If neutrinos are Dirac particles, the Standard
Model (SM) particle content must be extended with right-handed neutrinos,
which can increase the effective number of light neutrinos, $N_{\text{eff}}$,
until 6. Therefore, to be compatible with the current cosmological
restrictions on $N_{\text{eff}}$, the interactions of the extra right-handed
neutrinos with the primordial plasma must be highly suppressed.

On the other hand, to give small masses to at least two Majorana or Dirac
neutrinos, as required to explain the neutrino oscillation experiments [8, 9],
the seesaw mechanism with heavy fermions is usually invoked. For the tree-
level type-I seesaw we can have either light Majorana neutrinos with heavy
Majorana mediators [10, 11, 12, 13] or light Dirac neutrinos with heavy Dirac
mediators [14, 15, 16, 17]. The radiative type-I seesaw includes both [18]
possibilities [19], but now it is also possible to have light Majorana
neutrinos with heavy Dirac mediators [20]. In this work we want to explore the
possibility to build a simple Dirac radiative type-I seesaw model with heavy
Majorana mediators. It is worth noticing that this idea have been already
illustrated in an extension of the minimal supersymmetric standard model [21]
but without show any explicit solution.

In general, solutions for light Dirac neutrino masses require a continuous
symmetry to guarantee their Diracness. This symmetry is usually identified as
the local $\operatorname{U}(1)_{B-L}$. Additionally, ad-hoc discrete
symmetries are invoked to forbid tree level Dirac or Majorana mass terms for
the light right-handed neutrinos [14, 22, 23]. However, tree-level Dirac
type-I seesaw with proper choices for the $\operatorname{U}(1)_{B-L}$ charges
have been shown to be consistent without require any extra ad-hoc discrete
symmetries [17]. In recent works, it has been shown that even for one-loop
Dirac neutrino masses, it is possible to have $\operatorname{U}(1)_{B-L}$ as
the only extra symmetry beyond the SM [24, 25, 26]444 For extensions with only
extra scalars, minimal solutions have been found with two and three loops
[26]..

As a bonus in this case, the new scalars and fermions circulating the loop can
be dark matter candidates with the stability of the lightest of them
guaranteed by the very same continuous symmetry. We focus here in solutions
for the radiative Dirac type-I seesaw with Majorana mediators, which have only
an extra local symmetry responsible for the Diracness of the light neutrinos,
the absence of any tree-level mass, and the existence of a dark sector
constituted by the particles circulating the loop.

In fact, another evidence that the SM is not a complete theory is the missing
matter content of the Universe, which is known as dark matter (DM). The main
proposals that explain DM as a particle are given in Ref. [27]. However, there
has been only gravitational evidence for the existence of dark matter so far.
Without evidence of DM as a particle, there is not a clear path to pin out the
DM properties nor the possible heavier companions of some extended dark
sector. Linking the dark sector to other specific phenomenology allows to
reduce the arbitrariness in the model building. In our construction, the dark
sector is related to the heavy sector responsible of the lightness of the
neutrinos and the same symmetry that guarantees the lightness of the Dirac
neutrinos is the responsible of the stability of the lightest dark particle
(LDP). Therefore, the number of specific models is quite restricted.

The rest of the paper is organized as follows. In the next Section we present
the model and study the scalar mass spectrum after spontaneous symmetry
breaking. In Sec. 3 we present the radiative mechanism that generates Dirac
neutrino masses and establish the lepton flavor violation constraints. The
different resulting DM scenarios are discussed in Sec. 4, and in Sec. 5 we
show the cosmological restrictions ($N_{\text{eff}}$) in a non-minimum model
for different extra Abelian symmetries.

##  2 The model

Fields | $\operatorname{SU}(2)_{L}$ | $\operatorname{U}(1)_{Y}$ | $\operatorname{U}(1)_{\mathcal{D}}$
---|---|---|---
$\eta$ | $\boldsymbol{2}$ | $1$ | $1$
$S$ | $\boldsymbol{1}$ | $0$ | $2$
$\sigma$ | $\boldsymbol{1}$ | $0$ | $3$
$\nu_{Ri}$ | $\boldsymbol{1}$ | $0$ | $-4$
$\nu_{R3}$ | $\boldsymbol{1}$ | $0$ | $5$
$\psi_{R\alpha}$ | $\boldsymbol{1}$ | 0 | $1$
Table 1: The new scalars and fermions with their respective charges. All the
SM fields are neutral under the dark $\operatorname{U}(1)_{\cal{D}}$ gauge
symmetry.

We extend the SM with a spontaneously broken Abelian gauge symmetry which
guarantees the diracness of the massive neutrinos. Only the new particles,
including the right-handed partners of the SM neutrinos, are charged under
this new $\operatorname{U}(1)_{\mathcal{D}}$ dark gauge symmetry [28, 29, 30]
to obtain an anomaly free theory. We choose the new particle set such that the
following dimension six operator is realized at one-loop level

$\displaystyle\mathcal{O}_{6D}=\frac{1}{\Lambda^{2}}\overline{L}\tilde{H}\nu_{R}S^{2}\,,$
(1)

where $S$ is the singlet scalar field which spontaneously breaks the
$\operatorname{U}(1)_{\cal{D}}$ symmetry needed to forbid the Dirac and
Majorana neutrino mass terms at tree level.

With the aim to illustrate the one-loop Dirac neutrino mass generation we
consider the particle content shown in Table 1 as a possible realization of
the effective operator $\mathcal{O}_{6D}$. Specifically, we introduce three
scalar fields $\eta,\sigma$ and $S$, where only $S$ develops a nonzero vacuum
expectation value (VEV), a set of three singlet fermions, $\nu_{Rj}$ ($j=1,2$)
and $\nu_{R3}$, and another set of three heavy Majorana fermions,
$\psi_{R\alpha}$ ($\alpha=1,2,3$). The $\operatorname{U}(1)_{\mathcal{D}}$
charges for the new particles are defined by the anomaly cancellation
conditions and the gauge invariance in Yukawa and scalar interactions.

The most general Lagrangian for some gauge $\operatorname{U}(1)_{X}$ symmetry,
which includes the trivial case $X=\mathcal{D}$, must contains the following
gauge, Yukawa and scalar interactions in order to realize $\mathcal{O}_{6D}$
at one-loop:

$\displaystyle\mathcal{L}\supset$
$\displaystyle-\,g^{\prime}\,Z_{\mu}^{\prime}\sum_{F}q_{F}\overline{F}\gamma^{\mu}F+\sum_{\phi}\left|\left(\partial_{\mu}+i\,g^{\prime}\,q_{\phi}\,Z^{\prime}_{\mu}\right)\phi\right|^{2}$
$\displaystyle-[h_{i\alpha}\overline{L_{i}}\tilde{\eta}\psi_{R\alpha}+y_{j\alpha}\overline{\nu_{R_{j}}}\sigma^{*}\psi^{c}_{R\alpha}+\kappa_{\alpha\beta}\overline{\psi^{c}_{R\alpha}}\psi_{R\beta}S^{*}+\text{h.c.}]-\mathcal{V}(H,S,\eta,\sigma)\,.$
(2)

In the first row $g^{\prime}$ is the gauge coupling associated to the
$\operatorname{U}(1)_{X}$ group and $Z_{\mu}^{\prime}$ is its corresponding
gauge boson, $F$ ($\phi$) denote the new fermions (scalars), and
$q_{F,\,\phi}$ their $X$ charges. In the second row $L_{i}$ ($i=1,2,3$) and
$H$ are the SM lepton and Higgs doublets, respectively,
$\widetilde{\eta}=i\sigma_{2}\eta^{*}$, and $h$, $y$ and $\kappa$ are matrices
in the flavor space. The scalar potential can be cast as

$\displaystyle\mathcal{V}(H,S,\eta,\sigma)=$ $\displaystyle
V(H)+V(S)+V(\eta)+V(\sigma)$
$\displaystyle+\lambda_{1}(H^{\dagger}H)(S^{*}S)+\lambda_{2}(H^{\dagger}H)(\sigma^{*}\sigma)+\lambda_{3}(H^{\dagger}H)(\eta^{\dagger}\eta)$
$\displaystyle+\lambda_{4}(S^{*}S)(\sigma^{*}\sigma)+\lambda_{5}(S^{*}S)(\eta^{\dagger}\eta)+\lambda_{6}(\eta^{\dagger}\eta)(\sigma^{*}\sigma)+\lambda_{7}(\eta^{\dagger}H)(H^{\dagger}\eta)$
$\displaystyle+\lambda_{8}(\eta^{\dagger}HS^{*}\sigma+\text{h.c.})\,,$ (3)

with
$V(\omega)=\mu^{2}_{\omega}\omega^{\dagger}\omega+\lambda_{\omega}(\omega^{\dagger}\omega)^{2}$.
It is worth to emphasize that $\operatorname{U}(1)_{\mathcal{D}}$
automatically allows for all the terms in Eqs. (2) and (2) but other
realizations will be checked later on. Note that after the spontaneous
symmetry breaking of $\operatorname{U}(1)_{X}$ the $\lambda_{8}$ term gives
rise to the mixing between the neutral parts of $\eta$ and $\sigma$, which is
mandatory to generate non-zero radiative masses. We assume $\lambda_{8}$ and
$\langle S\rangle$ reals to preserve CP symmetry in the scalar sector, and
$\mu^{2}_{\eta},\mu^{2}_{\sigma}>0$ to avoid tree-level mixing terms among the
fermions. Moreover, we also assume $\lambda_{1}\ll 1$ such that the scalar $S$
and $H$ do not mix allowing us to identify the CP even scalar particle in $H$
as the SM Higgs boson. To establish the scalar spectrum we expand the scalar
fields as

$\displaystyle H=\begin{pmatrix}G^{+}\\\
\frac{1}{\sqrt{2}}(h+v_{H}+iG)\end{pmatrix}\,,$ $\displaystyle\hskip
28.45274pt\eta=\begin{pmatrix}\eta^{+}\\\
\frac{1}{\sqrt{2}}(\eta_{R}+i\eta_{I})\end{pmatrix}\,,$ $\displaystyle
S=\frac{1}{\sqrt{2}}(S_{R}+v_{S}+iS_{I})\,,$ $\displaystyle\hskip
28.45274pt\sigma=\frac{1}{\sqrt{2}}(\sigma_{R}+i\sigma_{I}),$

with $v_{H}=246.22$ GeV. Of the original twelve scalar degrees of freedom in
the model, the gauge bosons $W^{\pm}$, $Z^{0}$ and $Z^{\prime}$ absorb four of
them (the Goldstone bosons $G^{\pm}$,$G$ and $S_{I}$). Thus, the scalar
spectrum contains two sets of two neutral CP-even states ($h$ and $S_{R}$, and
$\sigma_{R}$ and $\eta_{R}$), two CP-odd scalar states ($\sigma_{I}$ and
$\eta_{I}$) and one charged scalar ($\eta^{\pm}$). The mass spectrum for the
unmixed scalars reads

$\displaystyle
m_{\eta^{\pm}}^{2}=\mu_{\eta}^{2}+\frac{1}{2}(\lambda_{3}\upsilon^{2}_{H}+\lambda_{5}\upsilon^{2}_{S})\,,\,\,\,\,m_{H}^{2}=\lambda_{H}\upsilon_{H}^{2}\,,\,\,\,\,m_{S}^{2}=\lambda_{S}\upsilon_{S}^{2}\,.$

The other mass eigenstates are defined as

$\displaystyle\begin{pmatrix}\chi_{(R,I)_{1}}\\\
\chi_{(R,I)_{2}}\end{pmatrix}=$
$\displaystyle\begin{pmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{pmatrix}\begin{pmatrix}\sigma_{(R,I)}\\\
\eta_{(R,I)}\end{pmatrix}\,,$

where $\tan\theta=2c/(b-a)$, with
$a=m^{2}_{\eta^{\pm}}+\frac{1}{2}\lambda_{7}\upsilon^{2}_{H}$,
$b=\mu^{2}_{\sigma}+\frac{1}{2}(\lambda_{2}\upsilon^{2}_{H}+\lambda_{4}\upsilon^{2}_{S})$
and $c=\frac{1}{2}\lambda_{8}v_{H}v_{S}$ . Note that the CP-even states
$\chi_{R(1,2)}$ are mass degenerate with CP-odd ones $\chi_{I(1,2)}$, with
masses $m_{\chi_{(1,2)}}^{2}=[a+b\mp\sqrt{(a-b)^{2}+4c^{2}}]/2$.

On the other hand, we assume that the heavy Majorana fermions are already in
the diagonal basis in such a way their masses are
$M_{\psi_{\alpha}}=\kappa_{\alpha\alpha}v_{S}/\sqrt{2}$, with
$M_{\psi_{1}}<M_{\psi_{2}}<M_{\psi_{3}}$. Finally, the mass of the new gauge
boson is given by

$\displaystyle M_{Z^{\prime}}=q_{S}g^{\prime}v_{S}.$ (4)

##  3 Neutrino masses and charged lepton flavor violation

Figure 1: One-loop realization of the dimension-6 operator
$\overline{L}\tilde{H}\nu_{R}S^{2}$ leading to Dirac neutrino masses with
Majorana mediators.

Neutrino masses are generated at one-loop level according to the diagram in
Fig. 1. The expression for the effective neutrino mass matrix
$\mathcal{M}_{\nu}$ can be cast as

$\displaystyle(\mathcal{M}_{\nu})_{ij}=\frac{1}{32\pi^{2}}\frac{\lambda_{8}v_{S}v_{H}}{m_{\chi_{R_{2}}}^{2}-m_{\chi_{R_{1}}}^{2}}\sum_{\alpha=1}^{3}h_{i\alpha}M_{\psi_{\alpha}}y^{*}_{j\alpha}\left[F\left(\frac{m_{\chi_{R_{2}}}^{2}}{M_{\psi_{\alpha}}^{2}}\right)-F\left(\frac{m_{\chi_{R_{1}}}^{2}}{M_{\psi_{\alpha}}^{2}}\right)\right]+(R\to
I)\,,$ (5)

where $F(x)=x\log x/(x-1)$. Note that the structure of the effective neutrino
mass matrix, given by the product $(M_{\nu})_{ij}\propto
h_{i\alpha}y_{j\alpha}$, is similar to the structure of the neutrino mass
matrix for the tree-level seesaw mechanism for Dirac neutrinos [31]. It is
also worth mentioning that if only one fermion $\psi_{R}$ is added, then there
will be two massless neutrinos, which would be ruled out by the current
neutrino oscillation data [32]. In our case, we assume the existence of three
of such fermions, generating Dirac scotogenic masses for the two left-handed
neutrinos ($\nu_{3}=\nu_{L3}+\nu_{R3}$ is massless due to the charge
assignment).

In order to estimate the possible values for the parameters involved in the
neutrino masses we consider the case where $\lambda_{2}$, $\lambda_{4}$,
$\lambda_{7}\ll 1$ and $m_{\chi}^{2}\equiv
m_{\eta^{\pm}}^{2}=\mu^{2}_{\sigma}\gg\frac{1}{2}\lambda_{8}vv_{S}$, which
leads to $a\approx b\gg c$. Taking into account that for the mentioned case
$m^{2}_{\chi_{R_{2}}}-m^{2}_{\chi_{R_{1}}}=\lambda_{8}vv_{S}$ and
$m^{2}_{\chi_{R_{2}}}+m^{2}_{\chi_{R_{1}}}=2m^{2}_{\chi}$, we have that

$\displaystyle(\mathcal{M}_{\nu})_{ij}=\frac{\lambda_{8}v_{S}v}{16\pi^{2}}\sum_{\alpha=1}^{3}\frac{h_{i\alpha}M_{\psi_{\alpha}}y^{*}_{j\alpha}}{m_{\chi}^{2}-M_{\psi_{\alpha}}^{2}}\left[1-\frac{M_{\psi_{\alpha}}^{2}}{m_{\chi}^{2}-M_{\psi_{\alpha}}^{2}}\log\left(\frac{m_{\chi}^{2}}{M_{\psi_{\alpha}}^{2}}\right)\right]\,,$
(6)

and by further assuming $m_{\chi}^{2}\gg M_{\psi_{\alpha}}^{2}$ one finds

$\displaystyle(\mathcal{M}_{\nu})_{ij}$
$\displaystyle=\frac{\lambda_{8}v_{S}v}{16\pi^{2}m_{\chi}^{2}}\sum_{\alpha=1}^{3}h_{i\alpha}M_{\psi_{\alpha}}y^{*}_{j\alpha}\,,$
(7) $\displaystyle\sim
0.04~{}\text{eV}\left(\frac{\lambda_{8}}{10^{-4}}\right)\left(\frac{v_{S}}{200\,\text{GeV}}\right)\left(\frac{M_{\psi_{\alpha}}}{50\,\text{GeV}}\right)\left(\frac{2\,\text{TeV}}{m_{\chi}}\right)^{2}\left(\frac{h_{i\alpha}y_{j\alpha}}{10^{-4}}\right)\,.$
(8)

In this way, in addition to the loop suppression it is possible to have
further suppression in the neutrino mass matrix for small values of either
$v_{S}$, $\lambda_{8}$ or $h_{i\alpha}y_{j\alpha}$.

On the other hand, the $h_{i\alpha}$ Yukawa interaction in Eq. (2) leads to
charged lepton flavor violation (CLFV) processes induced at one-loop level and
mediated by the charged scalars $\eta^{\pm}$ as the ones shown in Fig. 2 for
the $\ell_{i}\to\ell_{j}\gamma$ type. By using the current experimental
constraint on $\operatorname{Br}(\mu\to e\gamma)<5.7\times 10^{-13}$ [33] and
for the case $m_{\chi}^{2}=m_{\eta^{\pm}}^{2}\gg M_{\psi_{a}}^{2}$ we can
obtain an upper bound for the product of Yukawa couplings

$\displaystyle\left|\sum_{\alpha}h_{2\alpha}h_{1\alpha}^{*}\right|\lesssim
0.02\left(\frac{m_{\chi}}{2\,\text{TeV}}\right)^{2}.$ (9)

It is worth noticing that $y_{j\alpha}$ is not constrained by the non
observation of CLFV processes.

Figure 2: Feynman diagram for the processes $\ell_{i}\to\ell_{j}\gamma$

##  4 Dark matter

From the model charge assignment in Table 1 we have that a residual $Z_{2}$
symmetry is left over after the $\operatorname{U}(1)_{\mathcal{D}}$ symmetry
breaking, with the particles circulating the one-loop neutrino mass diagram
(see Fig. 1) and $\nu_{R3}$ being odd under it whereas $\nu_{R1},\nu_{R2}$,
$S$ and all the SM particles being even. Thus the lightest electrically-
neutral $Z_{2}$-odd particle becomes a dark matter candidate. In other words,
this model also provides a solution to the DM puzzle via either fermion
($\psi_{1}$) or scalar ($\chi_{1}$) dark matter555Note that $Z^{\prime}$
cannot constitute a DM candidate due to the instability associated to the
$Z^{\prime}\to\bar{\nu}_{R3}\nu_{R3}$ decay channel, which cannot be
kinematically closed, and that neither $\psi_{1}$ and $\chi_{1}$ can decay
into $\nu_{R3}$ since they do not share Yukawa interactions (see Eq. (2))..

Since $\psi_{1}$ is a singlet under the SM gauge group its thermal relic
density is controlled by the Yukawa couplings and
$\operatorname{U}(1)_{\mathcal{D}}$ gauge interactions. The scenario where
$\psi_{1}$ self-annihilates dominantly through $h_{i\alpha}$-mediated
interactions resembles the very well known scotogenic model [18], where
sizable $h_{i\alpha}$ are required to reproduce the correct DM abundance,
which in turn leads to a mild tension with experimental upper bounds on the
rates for rare charged lepton decays [34, 35, 36]. On the other hand, when the
$Z^{\prime}$ portal [37, 38] is the main gate to visible sector, $\psi_{1}$
largely annihilates into neutrinos in such a way the observed DM abundance can
be reproduced without entering in conflict with the DM searches, which follows
from the fact that the $Z^{\prime}$ does not couple to quarks and charged
leptons (see [38, 39, 40, 41, 42] for phenomenological studies on
$Z^{\prime}$-mediated Majorana DM).

The elastic scattering of $\psi_{1}$ particles off nuclei may occur via
different mediators: inert scalars ($\sigma,\eta$), SM Higgs (through the
mixing with $S$) and $Z^{\prime}$. When the inert scalars mediate DM nucleon
scatterings the corresponding spin independent cross section is loop
suppressed due to their quark-phobic nature [36], whereas the scatterings via
the $Z^{\prime}$ require necessarily a kinetic mixing (since no SM fermion
couples directly to $Z^{\prime}$) and lead to a spin-dependent cross section.
Lastly, the scatterings via the Higgs exchange are also suppressed by the
mixing parameter $\lambda_{1}$, which is required to be small to correctly
reproduce the SM Higgs phenomenology. All in all, it is feasible to expect
that the rates for the elastic scattering of $\psi$ particles off nuclei lie
well below the sensitivity of present direct detection experiments.

In contrast to fermion DM, the scalar DM candidate has additional interaction
terms to the Yukawa and gauge interactions. This entails that the later ones
can be used to alter the relic density predictions in scenarios with mixed
scalar DM. Since in the present model the CP-even and CP-odd neutral
$Z_{2}$-odd particles are mass degenerate, we have the scenario of singlet-
doublet complex DM [43, 44] where the DM candidate is a mixture of a complex
singlet [45] and a $SU(2)_{L}$ doublet [46, 47]. It follows that for
negligible Yukawa and gauge interactions there are two DM mass regions that
allow us to properly reproduce the relic abundance, one corresponds the Higgs
funnel region and the second one demands masses above $100$ GeV [48].
Regarding direct detection signals, we expect similar scattering rates as
those in the complex scalar singlet model [49, 50].

##  5 Beyond the minimal model: cosmological and collider constraints

It turns that the $Z^{\prime}$ portal also allows to probe the model through
modifications on the cosmological history of the Universe, namely, via
additional contributions to the effective number of relativistic degrees of
freedom $N_{\text{eff}}$ [51]. In this model, these contributions arise from
the presence of the right-handed neutrinos and may be expected to be sizeable
precisely due to the large $\operatorname{U}(1)_{\mathcal{D}}$ charges of the
$\nu_{R}$’s. Nevertheless, since the right-handed neutrinos do not couple
directly to the rest of the SM particles (see Eq. (2)) they decouple early
enough (when the DM does or even before) from the thermal bath and, therefore,
modify the SM prediction for $N_{\text{eff}}$ with an extra contribution at
most of $\sim 0.2$ [52].

With the aim of extending the thermalization period of the right-handed
neutrinos with the primordial plasma in the early Universe, we consider an
modification of the previous setup by using a general anomaly free Abelian
gauge symmetry with generation-independent charge assignments for the SM
fermions, $\operatorname{U}(1)_{X}$. In the Appendix we show that the
solutions to the anomaly cancellation conditions allow us to write the
$X$-charges of the SM fields in terms of two parameters [53, 28, 54, 55], that
we choose as

$\displaystyle X(r,l)=rR-lY\,,$ (10)

where $R$ is the generator of $\operatorname{U}(1)_{R}$ (the gauge Abelian
symmetry where only the right-handed SM fermions have non-vanishing
$X$-charges), $Y$ is the hypercharge, $l$ is the $X$-charge of the lepton
doublets, and $r$ parametrizes the contribution to the linear and mixed gauge-
gravitational anomalies of any extra set of chiral fermions, as given in Eq.
(19) of the Appendix. In this way, if only extra vector-like fermions are
allowed beyond the SM ($r=0$) the solution must be proportional to
hypercharge. Without lost of generality, we can write the solutions in terms
of just one parameter [56, 57, 58], that we choose to be $l$ after fix $r=1$.
Then, as shown in the Appendix, the full family of solutions for a fixed $l$
can be obtained after rescaling all the $X$-charges by a factor $r$. In
particular, the fermiophobic [28, 29, 30] solution used in the previous
sections, $\operatorname{U}(1)_{\mathcal{D}}$, corresponds to the rescaling
$r=0$ of the $\operatorname{U}(1)_{R}$ solution: $\mathcal{D}=X(0,0)$.

The one parameter solution is shown in column $\operatorname{U}(1)_{X}$ of
Table 2. In order to analyse the phenomenology, we fix $l$ to recover some
already studied Abelian gauge groups $X=B-L,R,D,G,$ as defined in Ref.
[28]666We change $\operatorname{U}(1)_{B}$ for the more suitable name of
$\operatorname{U}(1)_{R}$ [59].. The last column corresponds to the rescaling
with $r=0$ of $\operatorname{U}(1)_{R}$.

Since we are interested here in keeping the exotic set of $X$-charges charges
$\\{\mp 4$, $\mp 4$, $\pm 5\\}$ in such a way that when assigned to the right-
handed neutrinos the tree level Dirac and Majorana masses can be forbidden
[24], the set of Majorana mediators $\psi_{R\alpha}$ which realize the
dimension-6 operator at one-loop would spoil the anomaly cancellation
condition. In view of that $\psi_{R\alpha}$ have necessarily nonzero
$X$-charges (see the Appendix), we further add an extra set of chiral fermions
in such a way the full set heavy fermions do not affect the anomaly
cancellation (their charges cancel each other in a vector like way). The
resulting charge assignment is displayed in Table 2, where the fields
$\xi_{L\alpha}$ constitute the new set of chiral fermion that guarantee the
anomaly cancellation. In this way we end up with a model with four Majorana
fermions, two more than in the minimal solution, since third generation of
chiral fermions is not needed to cancel the $\operatorname{U(1)}_{X}$
anomalies (alternatively we may consider another simple setup by adding a
single set of $\psi_{R}$ and $\xi_{L}$ and two set of scalars $\eta_{\alpha}$,
$\sigma_{\alpha}$ [60])777 Seeing that the two additional left-handed fields
$\xi_{L\alpha}$, with a $X$-charge $r=3/4$, have both Dirac and Majorana mass
terms $\overline{\xi_{L\alpha}}\psi_{R\beta}$ and
$\overline{\xi_{L\alpha}^{c}}\xi_{L\beta}\langle S\rangle$, the full set of
Majorana fields are massive and heavy. Moreover, because the
$\operatorname{U}(1)_{X}$ left out a remnant $Z_{2}$ discrete symmetry which
guarantees the stability of the lightest $Z_{2}$-odd particle, we have that
the lightest of these Majorana fields may play the role of DM candidate..

Fields | $\operatorname{SU}(2)_{L}$ | $\operatorname{U}(1)_{Y}$ | $\operatorname{U}(1)_{X}$ | $\operatorname{U}(1)_{B-L}$ | $\operatorname{U}(1)_{R}$ | $\operatorname{U}(1)_{D}$ | $\operatorname{U}(1)_{G}$ | $\operatorname{U}(1)_{\mathcal{D}}$
---|---|---|---|---|---|---|---|---
$L$ | $\boldsymbol{2}$ | $-1$ | $l$ | $-1$ | $0$ | $-3/2$ | $-1/2$ | $0$
$d_{R}$ | $\boldsymbol{1}$ | $-2/3$ | $1+2l/3$ | $1/3$ | $1$ | $0$ | $2/3$ | $0$
$u_{R}$ | $\boldsymbol{1}$ | $+4/3$ | $-1-4l/3$ | $1/3$ | $-1$ | $1$ | $-1/3$ | $0$
$Q$ | $\boldsymbol{2}$ | $1/3$ | $-l/3$ | $1/3$ | $0$ | $1/2$ | $1/6$ | $0$
$e_{R}$ | $\boldsymbol{1}$ | $-2$ | $1+2l$ | $-1$ | $1$ | $-2$ | $0$ | $0$
$H$ | $\boldsymbol{2}$ | $1$ | $-1-l$ | $0$ | $-1$ | $1/2$ | $-1/2$ | $0$
$\eta$ | $\boldsymbol{2}$ | $1$ | $3/4-l$ | $7/4$ | $3/4$ | $9/4$ | $5/4$ | $1$
$S$ | $\boldsymbol{1}$ | $0$ | $3/2$ | $3/2$ | $3/2$ | $3/2$ | $3/2$ | $2$
$\sigma$ | $\boldsymbol{1}$ | $0$ | $13/4$ | $13/4$ | $13/4$ | $13/4$ | $13/4$ | $3$
$\nu_{Ri}$ | $\boldsymbol{1}$ | $0$ | $-4$ | $-4$ | $-4$ | $-4$ | $-4$ | $-4$
$\nu_{R3}$ | $\boldsymbol{1}$ | $0$ | $+5$ | $+5$ | $5$ | $5$ | $5$ | $5$
$\psi_{R\alpha}$ | $\boldsymbol{1}$ | $0$ | $3/4$ | $3/4$ | $3/4$ | $3/4$ | $3/4$ | $1$
$\xi_{L\alpha}$ | $\boldsymbol{1}$ | $0$ | $3/4$ | $3/4$ | $3/4$ | $3/4$ | $3/4$ | $-$
Table 2: General one-parameter solution with some examples of rational
solutions ($X=B-L,R,D,G$ and $\mathcal{D}$) for the radiative type-I seesaw
realization of the effective operator $\mathcal{O}_{6D}$ for Dirac neutrino
masses. The last column corresponds to the solution in Table 1.

Accordingly, we may expect that the three right-handed neutrinos within the
not-so minimal model contribute to the radiation energy density of the
Universe, since now the interaction between the $Z^{\prime}$ with the SM
fermions opens up the possibility to thermalize them with the primordial
plasma. In other words, this leads to a modification in the relativistic
degrees of freedom as [61, 62]

$\displaystyle\Delta
N_{\text{eff}}=N_{\text{eff}}-N^{\text{SM}}_{\text{eff}}=N_{\nu_{R}}\left(\frac{T_{\nu_{R}}}{T_{\nu_{L}}}\right)^{4}=N_{\nu_{R}}\left(\frac{g(T^{\nu_{L}}_{\text{dec}})}{g(T^{\nu_{R}}_{\text{dec}})}\right)^{4/3},$
(11)

where $N_{\nu_{R}}$ is the number of right-handed neutrinos with the same $X$
charge and $g(T)$ is the number of relativistic degrees of freedom at a
temperature $T$ in the SM [63]. The decoupling temperature of the SM neutrinos
is $T^{\nu_{L}}_{\text{dec}}\approx 2.3$ MeV, when
$g(T^{\nu_{L}}_{\text{dec}})=43/4$ corresponding to the three $\nu_{L}$,
$e^{\pm}$ and the photon [64, 65]. Since the interaction of the right-handed
neutrinos with the SM is only mediated by the gauge boson $Z^{\prime}$, the
corresponding rate can be cast as [66]

$\displaystyle\Gamma_{\nu_{R}}(T)$
$\displaystyle=n_{\nu_{R}}(T)\langle\sigma(\overline{\nu_{R}}\nu_{R}\to\overline{f}f)\upsilon\rangle\,$
$\displaystyle=\frac{g^{2}_{\nu_{R}}}{n_{\nu_{R}}(T)}\int\frac{d^{3}p}{(2\pi)^{3}}f_{\nu_{R}}(p)\int\frac{d^{3}k}{(2\pi)^{3}}f_{\nu_{R}}(k)\sigma_{f}(s)\upsilon\,,$
(12)

where $f_{\nu_{R}}(k)=1/(e^{k/T}+1)$ is the Fermi-Dirac distribution,
$g_{\nu_{R}}=2$, $\upsilon=1-\cos{\varphi}$ is the Moller velocity,
$s=2pk(1-\cos{\varphi})$, $p$ and $k$ are the momenta of the particle with
$\varphi$ the angle between them, and the number density of right-handed
neutrinos is given by

$\displaystyle
n_{\nu_{R}}(T)=g_{\nu_{R}}\int\frac{d^{3}k}{(2\pi)^{3}}f_{\nu_{R}}(k)\,.$

The cross section for the case of a heavy mediator
($T^{\nu_{R}}_{\text{dec}}\ll M_{Z^{\prime}}$), with $s\ll M_{Z^{\prime}}$ and
neglecting the fermion masses in the final state reads [67]

$\displaystyle\sigma_{f}(s)\approx
N^{C}_{f}\frac{s}{12\pi}\left(\frac{g^{\prime}}{M_{Z^{\prime}}}\right)^{4}q^{2}_{\nu_{R}}(q^{2}_{f_{L}}+q^{2}_{f_{R}})\,,$
(13)

where $N^{C}_{f}=1(3)$ for leptons (quarks), and $q_{f}$ is the $X$-charge of
the SM fermion. Accordingly, the interaction rate takes the form

$\displaystyle\Gamma_{\nu_{R}}(T)=\frac{49\pi^{5}T^{5}}{97200\zeta(3)}\left(\frac{g^{\prime}}{M_{Z^{\prime}}}\right)^{4}\sum_{f}N^{C}_{f}q^{2}_{f}\,.$
(14)

In this expression the sum is performed over all SM fermions that are in
thermal equilibrium with the plasma at temperature $T$. To estimate the
contribution to the relativistic degrees of freedom from right-handed
neutrinos, it is necessary to calculate the decoupling temperature of the
right-handed neutrinos ($T^{\nu_{R}}_{\text{dec}}$). The latter occurs when
the interaction rate $\Gamma_{\nu_{R}}(T)$ drops below the rate of expansion
of the Universe,
$\Gamma(T^{\nu_{R}}_{\text{dec}})=H(T^{\nu_{R}}_{\text{dec}})\,,$ with
$H(T)=[4\pi^{3}G_{N}(g(T)+21/4)/45]^{1/2}T^{2}\,.$ Note that the factor $21/4$
corresponds to the contribution of the right-handed neutrinos to the
relativistic degrees of freedom.

Figure 3: Contribution to the number of extra relativistic degrees of freedom
($\Delta N_{\text{eff}}$) in function of $M_{Z^{\prime}}/g^{\prime}$. The
region above the solid gray line is excluded by the measurements at $2\sigma$
reported by the PLANCK Collaboration [68]. For comparison purposes the upper
bound at $2\sigma$ (black line) obtained from the BBN analysis [69] is also
shown. The horizontal dashed lines show the projected sensitivity of the
future experiments SPT-3G/SO [70, 71] and CMB-S4 [72]. $\Delta
N_{\text{eff}}=0.4$ represents the extra contribution required to relieve the
tension on the inferred $H_{0}$ values from high- and low-redshift
observations [73, 74].

The results for $\Delta N_{\text{eff}}$ as a function of
$M_{Z^{\prime}}/g^{\prime}$ are displayed in Fig. 3, where it can be observed
that for small ratios $M_{Z^{\prime}}/g^{\prime}$ the variation in the number
of relativistic degrees of freedom of the new species is large ($\Delta
N_{\text{eff}}\gtrsim 0.3$ corresponds to decoupling temperatures
$T^{\nu_{R}}_{\text{dec}}\lesssim 2$ GeV) and vice versa. It follows that for
the $\operatorname{U}(1)_{G}$ model the Planck+BAO upper bound at $2\sigma$
(solid gray line) demands that $M_{Z^{\prime}}/g^{\prime}\gtrsim 36$ TeV while
for the $\operatorname{U}(1)_{D}$ model a more stringent lower bound applies
$M_{Z^{\prime}}/g^{\prime}\gtrsim 55$ TeV (these bounds become slightly weaker
once the Planck+BAO+$H_{0}$ combination [68] is considered, but in such a case
the BBN bound from the primordial abundances of light elements would rule
[69]). In other words, bearing in mind that $M_{Z^{\prime}}/g^{\prime}\approx
3v_{S}/2$888Because of the LEP constraint (see below) the mixing between the
$Z^{\prime}$ and the SM $Z$ boson is negligible [75]. we have that the energy
scale of the $\operatorname{U}(1)_{X}$ symmetry breaking must be at least
$\sim 24$ TeV (note that the $\operatorname{U}(1)_{X}$ having the lowest
$M_{Z^{\prime}}/g^{\prime}$ ratio features a X charge $l=-6/11\approx-1/2$).
Similar constraints would apply to all gauged and anomaly free
$\operatorname{U}(1)_{X}$ extensions of the SM with Dirac neutrino masses,
where light right handed neutrinos are associated to the solution $(-4,-4,+5)$
[17, 76, 77, 25, 24, 78]. On the other hand, it is remarkable the fact the
next generation of CMB experiments [70, 71, 72] has the potential to entirely
probe all the not-so minimal models [79, 80].

Regarding collider searches, the recasting of the latest ATLAS results for the
search of dilepton resonances using $139\ \text{fb}^{-1}$ [81] was done in
Ref. [82] for the $\operatorname{U}(1)_{B-L}$ model. The green (upper) region
in Fig. 4 shows the excluded region at $95\%\ \text{C.L.}$ To ease the
comparison with other results, we show the exclusion as function of
$M_{Z^{\prime}}/g^{\prime}$. In particular, the limit from LEP for
$\operatorname{U}(1)_{B-L}$ model is [75, 83, 84]999The constraint as function
of the $X$-charge $h=-1-l$ is given in Ref. [85].

$\displaystyle M_{Z^{\prime}}/g^{\prime}>6.7\ \text{TeV}\,,$
$\displaystyle\text{at $95\%$ C.L }\,,$ (15)

which is obtained from the search for effective four-lepton operators and is
valid for $M_{Z^{\prime}}\gg 200\ \text{GeV}$. This constraint corresponds to
the excluded magenta (lower) region of Fig. 4 and start to be relevant for
$M_{Z^{\prime}}>5.8\ \text{TeV}$.

Figure 4: Collider and cosmological constraints for the
$\operatorname{U}(1)_{B-L}$ model, and contours of constant $g^{\prime}$ in
the plane of $q_{S}v_{S}$ as a function of $M_{Z^{\prime}}$ with $q_{S}=3/2$.

The constraint of $\Delta N_{\text{eff}}$ for $\operatorname{U}(1)_{B-L}$ is
shown in the blue (middle) region and start to be better than current ATLAS
limit for $M_{Z^{\prime}}\gtrsim 4.8\ \text{TeV}$. In the figure we also show
contours of constant $g^{\prime}$ at $0.01$, $0.1$ and $0.5$ with the solid,
dashed and dot-dashed lines respectively. We can see that $\Delta
N_{\text{eff}}$ start to constraint $g^{\prime}$ for values larger than $0.1$
in $\operatorname{U}(1)_{B-L}$. Similar restrictions can be obtained for the
other models quoted in Fig. 3.

##  6 Conclusions

The mechanism behind the neutrino mass generation and the Dirac/Majorana
character of massive neutrinos still remain a conundrum despite the clear
understanding of the neutrino oscillation pattern and the great experimental
efforts behind the neutrinoless double-beta decay. Alongside this is the fact
that there are no clues on the nature of the DM particle and the properties of
the dark sector it belongs.

In view of this we have proposed a new mechanism for Dirac neutrino masses
which making use of heavy Majorana mediators generate nonzero masses at one-
loop level. The model presented in this paper is a one-loop realization of the
dimension 6 operator $\overline{L}\tilde{H}\nu_{R}S^{2}\,$ and enters into the
list of Dirac radiative type-I seesaw models with the novel feature that it
involves Majorana mediators rather than Dirac mediators. The Diracness of the
massive neutrinos is protected by only one extra $\operatorname{U}(1)_{X}$
gauge symmetry, which in turn ensures the stability of the lightest particle
mediating the one-loop neutrino mass diagram. In this way the model offers in
a non trivial way a the solution to both neutrino and DM puzzles. Moreover, we
have shown that going beyond the minimal model other interesting
phenomenological aspects arise such that possible signals at colliders and new
contributions to the number of extra relativistic species. In particular, the
current upper bound on $\Delta N_{\text{eff}}$ reported by PLANCK points to
large ratios $M_{Z^{\prime}}/g^{\prime}\gtrsim 40\ \text{TeV}$. Future cosmic
microwave background experiments may probe all the no minimal models presented
here.

## Acknowledgments

Work supported by Sostenibilidad-UdeA and the UdeA/CODI Grant 2017-16286, and
by COLCIENCIAS through the Grant 111577657253. D.R would also like to thank
the IIP-UFRN, for its support, hospitality and the nice and stimulating
atmosphere during the Visiting Professors program where part of this work was
completed. O.Z. acknowledges the ICTP Simons associates program.

##  Appendix A $\operatorname{U}(1)_{X}$ anomaly cancellation

We use $f$ ($f$) to denote the general $\operatorname{U}(1)_{X}$ generation-
independent charge assignments of the field $f_{R}$ ($F_{L}$). The three
linear anomalies in $\operatorname{U(1)_{X}}$ [28]

$\displaystyle\left[\mathrm{SU}(3)_{C}\right]^{2}\mathrm{U}(1)_{X}:$
$\displaystyle[3u+3d]-[3\cdot 2q]=$ $\displaystyle 0,$
$\displaystyle\left[\mathrm{SU}(2)_{L}\right]^{2}\mathrm{U}(1)_{X}:$
$\displaystyle-[2l+3\cdot 2q]=$ $\displaystyle 0,$
$\displaystyle\left[\mathrm{U}(1)_{Y}\right]^{2}\mathrm{U}(1)_{X}:$
$\displaystyle{\left[(-2)^{2}e+3\left(\tfrac{4}{3}\right)^{2}u+3\left(-\tfrac{2}{3}\right)^{2}d\right]-\left[2(-1)^{2}l+3\cdot
2\left(\tfrac{1}{3}\right)^{2}q\right]}=$ $\displaystyle 0,$ (16)

allows to express three $X$-charges in terms of the other two

$\displaystyle u=$ $\displaystyle-e+\frac{2l}{3}\,,$ $\displaystyle d=$
$\displaystyle e-\frac{4l}{3}\,,$ $\displaystyle q=$
$\displaystyle-\frac{l}{3}\,.$ (17)

The quadratic anomaly condition is automatically satisfied, while the mixed
gauge-gravitational and cubic anomalies depend of any extra singlet quiral
fermions of zero hypercharge, like the right-handed counterpart of the Dirac
neutrinos. For $N$ extra quiral fields with $X$-charge $n_{\alpha}$, these
conditions read

$\displaystyle\left[\text{Grav}\right]^{2}\mathrm{U}(1)_{X}:\
\sum_{\alpha=1}^{N}n_{\alpha}+3(e-2l)=$ $\displaystyle 0\,,$
$\displaystyle\left[\mathrm{U}(1)_{X}\right]^{3}:\
\sum_{\alpha=1}^{N}n_{\alpha}^{3}+3(e-2l)^{3}=$ $\displaystyle 0\,.$ (18)

We choose the solutions with $r\equiv e-2l$, such that

$\displaystyle\sum_{\alpha=1}^{N}n_{\alpha}=$ $\displaystyle-3r\,,$
$\displaystyle\sum_{\alpha=1}^{N}n_{\alpha}^{3}=$ $\displaystyle-3r^{3}\,.$
(19)

The full set of anomaly free SM $X$-charges in terms of two parameters [53,
28, 54] that we choose as $l$ and $r$, is just

$\displaystyle u=$ $\displaystyle-r-\frac{4l}{3}\,,$ $\displaystyle d=$
$\displaystyle r+\frac{2l}{3}\,,$ $\displaystyle q=$
$\displaystyle-\frac{l}{3}\,,$ $\displaystyle e=$ $\displaystyle r+2l\,,$
$\displaystyle h=$ $\displaystyle-r-l\,.$ (20)

where the condition in the charged lepton Yukawa couplings have been used to
fix $h$, and is automatically consistent with the conditions in the quark
Yukawa couplings. By setting $l=0$ in the previous equations, we can define
the Abelian symmetry in which only the right-handed charged fermions have non-
vanishing $X$-charges as $\operatorname{U}(1)_{R}$. Then the general anomaly
free two-parameter solution can be written as

$\displaystyle X(r,l)=rR-lY\,.$ (21)

If we now change $f\to f^{\prime}=f/r$ for all the charged fermion $X$-charges
[54], the first set of anomaly cancellation conditions Eq. (A) remains
invariant, and without lost of generality it is always possible to normalize
the solutions such that the last set Eq. (19) is just

$\displaystyle\sum_{\alpha=1}^{N}n_{\alpha}^{\prime}=$ $\displaystyle-3\,,$
$\displaystyle\sum_{\alpha=1}^{N}n_{\alpha}^{\prime\,3}=$ $\displaystyle-3\,.$
(22)

For example, the solution with $r=3$: $n_{\alpha}=\left(-2,-2,-4,-1\right)$
[53] can be easily normalized to the form in Eq. (22) with $f\to f/3$ to
$n_{\alpha}^{\prime}=\left(-2/3,-2/3,-4/3,-1/3\right)$ as used in Ref. [86].
In this way, without lost of generality, we will work with the normalized
solution in terms of a single parameter [56, 57, 58] that we choose to be $l$,
by setting $r=1$ as summarized in column $\operatorname{U}(1)_{X}$ of Table 2,
which is just

$\displaystyle X(l)=R-l\,Y\,.$ (23)

In particular, this includes the solution $n_{\alpha}=(-4,-4,+5)$ [53]. In
general, we have that for $\nu=n_{1}=n_{2}$, the extra fermion inside the one-
loop neutrino mass diagram in Fig. 1, must have charges

$\displaystyle\psi=$ $\displaystyle-\frac{\nu+1}{4}\,,$ $\displaystyle\eta=$
$\displaystyle-\frac{\nu+1}{4}-l\,,$ $\displaystyle\sigma=$
$\displaystyle\frac{1-3\nu}{4}\,.$ (24)

The case for $\nu=-4$ is also displayed in Table 2.

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# Training Graph Neural Networks on Growing Stochastic Graphs

###### Abstract

Graph Neural Networks (GNNs) rely on graph convolutions to exploit meaningful
patterns in networked data. Based on matrix multiplications, convolutions
incur in high computational costs leading to scalability limitations in
practice. To overcome these limitations, proposed methods rely on training
GNNs in smaller number of nodes, and then transferring the GNN to larger
graphs. Even though these methods are able to bound the difference between the
output of the GNN with different number of nodes, they do not provide
guarantees against the optimal GNN on the very large graph. In this paper, we
propose to learn GNNs on very large graphs by leveraging the limit object of a
sequence of growing graphs, the graphon. We propose to grow the size of the
graph as we train, and we show that our proposed methodology – learning by
transference – converges to a neighborhood of a first order stationary point
on the graphon data. A numerical experiment validates our proposed approach.

Index Terms—  Graph Neural Networks, Graphons

##  1 Introduction

Graph Neural Networks (GNNs) are deep convolutional architectures for non-
Euclidean data supported on graphs [1, 2]. GNNs are formed by a succession of
layers, each of which is composed of a graph convolutional filter and a point-
wise non-linearity. In practice, GNNs have shown state-of-the-art performance
in several learning tasks [3, 4], and have found applications in biology [5,
6, 7], recommendation systems [8, 9, 10] and robotics [11, 12]. In theory,
part of their success is credited to their stability to graph perturbations
[2], the fact that they are invariant to relabelings [13, 14], their
expressive power [15, 16], and their generalization capabilities [17].

The ability of GNNs to exploit patterns in network data is largely due to
their graph convolutional layers. Graph convolutional filters leverage the
graph diffusion sequence to extract features that are shared across the graph
[2]. However, these filters rely on computing matrix-vector multiplications,
which become computationally costly when the number of nodes is large. On the
other hand, the number of parameters in a GNN is independent of the number of
nodes due to its local parametrization, which motivates training GNNs on small
graphs and then deploying them on larger graphs.

Several works have been proposed which exploit this transferability property
of the GNN [18, 19, 20]. Generally speaking, these works upper bound the
output difference between two GNNs with the same parameters supported on
graphs with different number of nodes. They show that if the graphs belong to
a family of graphs modeled by a graphon [21], as the number of nodes increases
the so-called transferability error decreases. But while this result is
useful, it does not guarantee than the optimal parameters achieved by training
the GNN on the small graph are optimal, or sufficiently close to optimal, on
the large graph. This subtle albeit crucial observation is what motivates our
work.

More specifically, we leverage the transferability properties of GNNs to
introduce a method that trains GNNs by successively increasing the size of the
graph during the training process. We call this procedure learning by
transference. We prove that the learning directions (gradients) on the graphon
and on the graph are aligned as long as the number of nodes in the GNN is
sufficiently large. Under a minimum graph size at each epoch, we show that our
method converges to a neighborhood of the first order stationary point on
graphon by taking gradient steps on the graph. We benchmark our procedure in a
multi-agent system problem, where a GNN is trained to learn a decentralized
control policy for agents communicating via a proximity graph.

Related work. This work extends upon previous works that consider the nodes to
be sampled from a regular partition of the graphon [22]. In this work, we
consider the nodes to be uniformly sampled, which not only provides a more
general modeling of GNNs, but also more closely correlates with graph signals
seen in practice. This more general sampling strategy induces further sampling
errors (in addition to the edge sampling error) [19], which worsen the
approximation of graphons and graphon data by graphs and graph data thus
requiring additional theoretical considerations.

##  2 Graph and Graphon Neural Networks

A graph is represented by the triplet
${\mathbf{G}}_{n}=({\mathcal{V}},{\mathcal{E}},W)$, where
${\mathcal{V}},|{\mathcal{V}}|=n$ is the set of nodes,
${\mathcal{E}}\subseteq{\mathcal{V}}\times{\mathcal{V}}$ is the set of edges,
and $W:{\mathcal{E}}\to{\mathbb{R}}$ is a map assigning weights to each edge.
Alternatively, we can represent the graph ${\mathbf{G}}_{n}$ by its graph
shift operator (GSO) ${\mathbf{S}}_{n}\in{\mathbb{R}}^{n\times n}$. Examples
of GSO are the adjecency matrix ${\mathbf{A}}$, the graph laplacian
${\mathbf{L}}=\mbox{diag}({\mathbf{A}}{\mathbf{1}})-{\mathbf{A}}$, to name a
few.

Data on graphs is defined as a graph signal
${\mathbf{x}}=[x_{1},\dots,x_{n}]$, where the $i$th component of
${\mathbf{x}}$ corresponds to the value of the signal on node $i$. A graph
signal ${\mathbf{x}}$ can be aggregated through the graph, by applying the GSO
${\mathbf{S}}_{n}$ as follows,

$\displaystyle{\mathbf{z}}={\mathbf{S}}_{n}{\mathbf{x}}_{n}.$ (1)

Intuitively, the value of signal ${\mathbf{z}}$ at coordinate $i$ is the
weighted average of the information present in its one hop neighborhood, i.e.
$z_{i}=\sum_{j\in{\mathcal{N}}(i)}[{\mathbf{S}}_{n}]_{ij}x_{j}$. We can
construct $k$-hop diffusion by applying $k$ power of ${\mathbf{S}}_{n}$. We
define the graph convolutions, as a weighted average over the powers of
${\mathbf{S}}_{n}$. Explicitly, by letting the coefficients be
${\mathbf{h}}=[h_{0},\dots,h_{K-1}]$, we define the graph convolution as,

$\displaystyle{\mathbf{y}}_{n}={\mathbf{h}}_{*{\mathbf{S}}_{n}}{\mathbf{x}}_{n}=\sum_{k=0}^{K-1}h_{k}{\mathbf{S}}_{n}^{k}{\mathbf{x}}_{n}$
(2)

where ${\mathbf{x}}_{n},{\mathbf{y}}_{n}$ are graph signals, and
${\mathbf{h}}_{*{\mathbf{S}}_{n}}$ denotes the convolution operator with GSO
${\mathbf{S}}_{n}$.

In the case of undirected graphs, ${\mathbf{S}}_{n}/n$ becomes symmetric,
admitting a spectral decomposition
${\mathbf{S}}_{n}={\mathbf{V}}\boldsymbol{\Lambda}{\mathbf{V}}^{T}$, where the
columns of ${\mathbf{V}}$ are the eigenvectors of ${\mathbf{S}}_{n}$, and
$\boldsymbol{\Lambda}$ is a diagonal matrix with $-1\leq\dots\leq\dots 1$.
Since the eigenvector of ${\mathbf{S}}_{n}$ for a basis of ${\mathbb{R}}^{n}$,
we can project filter ${\mathbf{y}}_{n}$ into this basis to obtain,

$\displaystyle h(\lambda)=\sum_{k=0}^{K-1}h_{k}\lambda^{k}$ (3)

Note that $h(\lambda)$ only depends on $h_{k}$, and on the eigenvalues of the
GSO. By the Cayley-Hamilton theorem, convolutional filters may be used to
represent any graph filter with spectral representation
$h(\lambda)=f(\lambda)$, where $f$ is analytic.

Graph Neural Networks are layered architectures, composed of graph
convolutions followed by point-wise non-linearities. Formally, introducing a
point-wise non-linearity $\rho$, and by stacking all the graph signals at
layer $l$ in a matrix
${\mathbf{X}}_{l}=[{\mathbf{x}}_{nl}^{1},\dots,{\mathbf{x}}_{nl}^{F_{l}}]\in{\mathbb{R}}^{n\times
F_{l}}$, where $F_{l}$ indicates the $F_{l}$ features at layer $l$. Notice
that for $l=0$, the input matrix is a concatenation of the input signal, i.e.
${\mathbf{X}}_{0}=[{\mathbf{x}}_{n},\dots,{\mathbf{x}}_{n}]$. We can define
the $l$ layer of a GNN as,

$\displaystyle{\mathbf{X}}_{l}=\rho\bigg{(}\sum_{k=1}^{K}{\mathbf{X}}_{n}^{k}{\mathbf{X}}_{l_{1}}{\mathbf{H}}_{lk}\bigg{)},$
(4)

where matrix ${\mathbf{H}}_{lk}\in{\mathbb{R}}^{F_{l-1}\times F_{l}}$
represents the $k$ coefficient of the graph convolution. By grouping all the
learnable parameters we can obtain a more succint representation of the GNN as
$\phi({\mathbf{x}};{\mathcal{H}},{\mathbf{S}})$, with
${\mathcal{H}}=\\{{\mathbf{H}}_{lk}\\}_{lk},1\leq l\leq L$, and $0\leq k\leq
K-1$.

###  2.1 Graphon Neural Networks

Graphons are the limit object of a converging sequence of dense undirected
graphs. Formally, a graphon is a symmetric, bounded, and measurable function
${\mathbf{W}}:[0,1]^{2}\to[0,1]$. Sequences of dense graphs converge to a
graphon in the sense that the densities of adjacency preserving graph motifs
converge to the densities of these same motifs on the graphon [18, 21].

Analogously to graph signals, we can define a graphon signal as a function
$X\in L^{2}([0,1])$. A graphon signal can be diffused over the graphon by
applying the linear integrator operator given by,

$\displaystyle T_{\mathbf{W}}X(v)=\int_{0}^{1}{\mathbf{W}}(u,v)X(v)du$ (5)

which we denote graphon shift operator (WSO). Since graphon ${\mathbf{W}}$ is
bounded and symmetric, $T_{\mathbf{W}}$ is a Hilbert-Schmidt and self adjoint
operator [23]. Therefore, we can express ${\mathbf{W}}$ by its eigen
decomposition, ${\mathbf{W}}(u,v)=\sum_{i\in{\mathbb{Z}}\smallsetminus
0}\lambda_{i}\phi_{i}(u)\phi_{i}(v)$, where $\lambda_{i}$ is the eigenvalue
associated with eigenfunction $\phi_{i}$. The absolute value of the
eigenvalues is bounded by $1$, and the eigenfunctions form an orthogonal basis
of $L^{2}([0,1])$. Utilizing the graphon shift, we define the graphon
convolution with parameters ${\mathbf{h}}=[h_{0},\dots,h_{K-1}]$, as

$\displaystyle
Y=T_{\mathbf{h}}X={\mathbf{h}}_{*{\mathbf{W}}}X=\sum_{k=0}^{K-1}h_{k}(T_{\mathbf{W}}^{(k)}X)(v)\text{
with }$ (6)
$\displaystyle(T_{\mathbf{W}}^{(k)}X)(v)=\int_{0}^{1}{\mathbf{W}}(u,v)(T_{\mathbf{W}}^{(k-1)}X)(u)du$
(7)

where $X,Y$ are graphon signals, $*{\mathbf{W}}$ defines the convolution
operator, and $T^{(0)}={\mathbf{I}}$ is the identity[24]. Projecting the
filter (6) onto the eigenbasis $\\{\phi_{i}\\}_{i\in{\mathbb{Z}}\smallsetminus
0}$, the graphon convolution admits a spectral representation given by,

$\displaystyle h(\lambda)=\sum_{k=0}^{K-1}h_{k}\lambda^{k}.$ (8)

Like its graphon counterpart (cf. (3)) the graphon convolution admits a
spectral representation that only depends on the coefficients of the filter
${\mathbf{h}}$.

Graphon Neural Networks (WNNs) are the extension of GNNs to graphon data. Each
layer of a WNN is composed by a graphon convolution (6), and a non-linearity
$\rho$. Denoting $F_{l}$ the features at layer $l$, we can group the
parameters of the $F_{l-1}\times F_{l}$ convolutions into $K$ matrices
$\\{{\mathbf{H}}_{lk}\\}\in{\mathbb{R}}^{F_{l-1}\times F_{l}}$, to write the
$f$ th feature of the $l$th layer as follows,

$\displaystyle
X_{l}^{f}=\rho\bigg{(}\sum_{g=1}^{F_{l-1}}\sum_{k=1}^{K-1}(T_{\mathbf{W}}^{(k)}X_{l-1}^{g})[{\mathbf{H}}_{lk}]_{gf}\bigg{)}$
(9)

for $1\leq g\leq F_{0}$. For an $L$ layered WNN, $X_{0}^{g}$ is given by the
input data $X^{g}$, and (9) is repeated for $1\leq l\leq L$, and the output of
the WNN is given by $Y=X_{L}$. Analogoues to GNNS, a more succint
representation of the WNN can be obtained be grouping all the coefficients,
i.e. $\phi(X;{\mathcal{H}},{\mathbf{W}})$, with
${\mathcal{H}}=\\{{\mathbf{H}}_{lk}\\}_{lk},1\leq l\leq L$, and $0\leq k\leq
K-1$.

###  2.2 From Graphons to Graphs, and Back

In this paper we focus on graphons, and graphon signals as generative models
of graphs, and graph signals. Let $\\{u_{i}\\}_{i=1}^{N}$ be $n$ points
sampled independently at uniformly from $[0,1]$, $u_{i}\sim\text{unif}(0,1)$.
The $n$-node stochastic GSO of graph ${\mathbf{G}}_{n}$ is obtained from
graphon ${\mathbf{W}}$ as follows,

$\displaystyle[{\mathbf{S}}_{n}]_{ij}=[{\mathbf{S}}_{n}]_{ji}\sim\text{Bernoulli}({\mathbf{W}}(u_{i},u_{j})))$
(10)

A graph signal ${\mathbf{x}}$ can be obtained by evaluating the graphon
signal,

$\displaystyle[{\mathbf{x}}]_{i}=X(u_{i})\text{ for all }i\in[n].$ (11)

Therefore, we can obtain stochastic graphs ${\mathbf{S}}_{n}$, and graph
signals ${\mathbf{x}}$ from graphon data ${\mathbf{W}},X$.

A graphon can be induced by a graph even if the node labels
$\\{u_{i}\\}_{i=1}^{N}$ are unknown. Let
${\mathbf{S}}_{n}\in{\mathbb{R}}^{n\times n}$ be the GSO of a graph, and
define $u_{i}=(i-1)/n$ for $1\leq i\leq n$. We construct the intervals
$I_{i}=[u_{i},u_{i+1}]$ for $1\leq i\leq n$. Letting ${\mathbf{1}}$ be the
indicator function, the induced graphon ${\mathbf{W}}_{\mathbf{S}}$, and graph
signal ${\mathbf{x}}$ can be obtained as,

$\displaystyle{\mathbf{W}}_{\mathbf{S}}(u,v)$
$\displaystyle=\sum_{i=1}^{n}\sum_{j=1}^{n}[{\mathbf{S}}_{n}]_{ij}{\mathbf{1}}(u\in
I_{i}){\mathbf{1}}(v\in I_{j}),\text{ and }$ (12) $\displaystyle X_{n}(u)$
$\displaystyle=\sum_{i=1}^{n}[{\mathbf{x}}_{n}]_{i}{\mathbf{1}}(u\in I_{i}).$
(13)

Algorithm 1 Learning by transference

1:Initialize ${\mathcal{H}}_{0},n_{0}$ and sample graph ${\mathbf{G}}_{n_{0}}$
from graphon ${\mathbf{W}}$

2:repeat for epochs $0,1,\ldots$

3: for $k$ =1,…, $|{\mathcal{D}}|$ do

4: Obtain sample $(Y,X)\sim{\mathcal{D}}$

5: Construct graph signals ${\mathbf{y}}_{n},{\mathbf{x}}_{n}$ [cf. (11)]

6: Take learning step:

7:
${\mathcal{H}}_{k+1}={\mathcal{H}}_{k}-\eta_{k}\nabla\ell({\mathbf{y}}_{n},\boldsymbol{\Phi}({\mathbf{x}}_{n};{\mathcal{H}}_{k},{\mathbf{S}}_{n}))$

8: end for

9: Increase number of nodes $n$

10: Sample points uniformly $u_{i}\sim unif(0,1)$, $i=[1,\dots,n]$

11: Sample ${\mathbf{S}}_{n}$ Bernoulli from graphon ${\mathbf{W}}$ [cf. (10)]

12:until convergence

##  3 Learning By Transference

We are interested in solving a statistical learning problem where the data is
supported in very large graphs. In the limit, this corresponds to a graphon,
so we consider the problem of learning a WNN.

Let $\ell:{\mathbb{R}}\times{\mathbb{R}}\to{\mathbb{R}}$ be a non-negative
loss function such that $\ell(x,y)=0$ if and only if $x=y$, and let $p(X,Y)$
be an unknown probability distribution over the space of the graphon signal.
The graphon statistical learning problem is defined as

$\displaystyle\min_{{\mathcal{H}}}{\mathbb{E}}_{p(X,Y)}[\ell(Y,\phi(X;{\mathcal{H}},{\mathbf{W}}))].$
(14)

Given that the joint probability distribution $p(X,Y)$ is unknown, a solution
of 14 cannot be derived in close form but, under the learning paradigm, we
assume that we have access to samples of the distribution
${\mathcal{D}}=\\{(X^{j},Y^{j})\sim p(X,Y),j=1,\dots,|{\mathcal{D}}|\\}$.
Provided that the samples in ${\mathcal{D}}$ are obtained independently, and
that $|{\mathcal{D}}|$ is sufficiently large, 14 can be approximated by its
empirical version

$\displaystyle\min_{{\mathcal{H}}}\frac{1}{|{\mathcal{D}}|}\sum_{j=1}^{|{\mathcal{D}}|}\ell(Y^{j},\phi(X^{j};{\mathcal{H}},{\mathbf{W}})).$
(15)

Like most other empirical risk minimization problems, the graphon empirical
learning problem (15) is solved in an iterative fashion by gradient descent.
The $k$th gradient descent iteration is given by

$\displaystyle{\mathcal{H}}_{k+1}={\mathcal{H}}_{k}-\eta_{k}\frac{1}{|{\mathcal{D}}|}\sum_{j=1}^{|{\mathcal{D}}|}\nabla_{\mathcal{H}}\ell(Y^{j},\phi(X^{j};{\mathcal{H}}_{k},{\mathbf{W}}))$
(16)

where $\eta_{k}\in(0,1)$ is the step size at iteration $k$.

In practice, however, the limit graphon ${\mathbf{W}}$ is either unknown or
hard to measure, and step (16) cannot be computed. But this can be addressed
by noting that this gradient can be approximated by sampling graphs
${\mathbf{S}}_{n}$ (cf. (16)) and computing the gradient descent iteration on
the graph. Explicitly,

$\displaystyle{\mathcal{H}}_{k+1}={\mathcal{H}}_{k}-\eta_{k}\frac{1}{|{\mathcal{D}}|}\sum_{j=1}^{|{\mathcal{D}}|}\nabla_{\mathcal{H}}\ell({\mathbf{y}}^{j},\phi({\mathbf{x}}^{j};{\mathcal{H}}_{k},{\mathbf{S}}_{n})).$
(17)

To show that the graphon (16) and the graph gradient descent iterations (17)
are close, we need the following assumptions.

######  Definition 1 (Lipschitz Functions)

A function $f$ is $A$-Lipschitz on the variables $u_{1},\ldots,u_{d}$ if it
satisfies $|f(v_{1},v_{2},\ldots,v_{d})-f(u_{1},u_{2},\ldots,u_{d})|\leq
A\sum_{i=1}^{d}|v_{i}-u_{i}|$. If $A=1$, we say that this function is
normalized Lipschitz.

######  Definition 2 (Node Stochasticity)

For a fixed probability ${\mathcal{X}}_{1}\in[0,1]$, the node stochasticity
constant on $n$ nodes, denoted $\alpha({\mathcal{X}}_{1},n)$ is defined as
$\alpha({\mathcal{X}}_{1},n)=\log((n+1)^{2}/\log(1-{\mathcal{X}}_{1})^{-1})$.

######  Definition 3 (Edge Stochasticity)

For a fixed probability ${\mathcal{X}}_{2}\in[0,1]$, the edge stochasticity
constant on $n$ nodes, denoted $\beta({\mathcal{X}}_{2},n)$ is defined as
$\beta({\mathcal{X}}_{2},n)=\sqrt{n\log(2n/{\mathcal{X}}_{2})}$.

###### AS 1

The graphon ${\mathbf{W}}$ and graphon signals $X,Y$ are normalized Lipschitz.

###### AS 2

The convolutional filters $h$ are normalized Lipschitz and non-amplifying,
i.e., $\|h(\lambda)\|<1$.

###### AS 3

The activation functions and their gradients are normalized Lipschitz, and
$\rho(0)=0$.

###### AS 4

The loss function $\ell:{\mathbb{R}}\times{\mathbb{R}}\to{\mathbb{R}}^{+}$ and
its gradient are normalized Lipschitz, and $\ell(x,x)=0$.

###### AS 5

For a fixed value of ${\mathcal{X}}_{3}\in(0,1)$, $n$ is such that
$n-{\log(2n/{\mathcal{X}}_{3})}/{d_{\mathbf{W}}}>{2}/{d_{\mathbf{W}}}$ where
$d_{\mathbf{W}}$ denotes the maximum degree of the graphon ${\mathbf{W}}$,
i.e., $d_{\mathbf{W}}=\max_{v}\int^{1}_{0}{\mathbf{W}}(u,v)du$.

###### Theorem 1

Consider the ERM problem in (15) and let
$\boldsymbol{\Phi}(X;{\mathcal{H}},{\mathbf{W}})$ be an $L$-layer WNN with
$F_{0}=F_{L}=1$, and $F_{l}=F$ for $1\leq l\leq L-1$. Let $c\in(0,1]$ and
assume that the graphon convolutions in all layers of this WNN have $K$ filter
taps [cf. (6)]. Let
$\boldsymbol{\Phi}({\mathbf{x}}_{n};{\mathcal{H}},{\mathbf{S}}_{n})$ be a GNN
sampled from $\boldsymbol{\Phi}(X;{\mathcal{H}},{\mathbf{W}})$ as in (10), and
(11). Under assumptions AS1–AS5, it holds that

$\displaystyle{\mathbb{E}}[\|\nabla_{{\mathcal{H}}}$
$\displaystyle\ell(Y,\phi(X;{\mathcal{H}},{\mathbf{W}}))-\nabla_{{\mathcal{H}}}\ell(Y_{n},\phi(X_{n};{\mathcal{H}},{\mathbf{W}}_{n}))\|]$
$\displaystyle\leq\gamma
c+{\mathcal{O}}\bigg{(}\max(\alpha(1/\sqrt{n},n),\beta(1/\sqrt{n},n))\bigg{)}$
(18)

where $Y_{n}$ is the graphon signal induced by
$[{\mathbf{y}}_{n}]_{i}=Y(u_{i})$ [cf. (12)], and
$\gamma=12\sqrt{K^{5}F^{5L-5}}$.

As expected, the bound in (1) decreases with the number of nodes, but it also
contains a constant term called the non-transferable bound, which corresponds
to high-frequency spectral components that converge more slowly with $n$ (see
[19, Sec. IV.C]).

(a) Starting with $10$ nodes

(b) Starting with $20$ nodes

Fig. 1: Velocity variation of the flocking problem for the whole trajectory in
the testing set relative to the centralized controller.

###  3.1 Algorithm Construction

Algorithm 1 presents a simple strategy for training GNNs for large graphs:
increasing the number of nodes and resampling the graph at regular intervals
during the training process. The insight is that at the beginning of the
learning process, the gradient on the graphon is large, given that the
parameters ${\mathcal{H}}_{k}$ are distant from the optimal values. As we
train, the norm of the gradient on the graphon decreases, and we require a
larger graph, i.e., a better approximation of the graphon, to follow the right
learning direction on the graph.

The advantage of Algorithm 1 versus training on the large $N$-node graph
directly is that given that the convolutions implement matrix-vector
multiplications, this would require ${\mathcal{O}}(N^{2})$ computations. By
implementing Algorithm 1, we are able to bring this down to
${\mathcal{O}}(n^{2})$ computations, $n<N$, without compromising optimality.
What is more, Algorithm 1 might not require evaluating gradients on the very
large graph, as it could be the case that smaller graphs render the desired
optimality conditions.

###  3.2 Algorithm Convergence

To show the convergence of Algorithm 1, we need additional Lipschitz
assumptions.

###### AS 6

The graphon neural network $\boldsymbol{\Phi}(X;{\mathcal{H}},{\mathbf{W}})$
is ${A_{\boldsymbol{\Phi}}}$-Lipschitz, and its gradient
$\nabla_{\mathcal{H}}\boldsymbol{\Phi}(X;{\mathcal{H}},{\mathbf{W}})$ is
${A_{\nabla\boldsymbol{\Phi}}}$-Lipschitz, with respect to the parameters
${\mathcal{H}}$ [cf. Definition 1].

###### Theorem 2

Consider the ERM problem in (15) and let
$\boldsymbol{\Phi}(X;{\mathcal{H}},{\mathbf{W}})$ be an $L$-layer WNN with
$F_{0}=F_{L}=1$, and $F_{l}=F$ for $1\leq l\leq L-1$. Let $c\in(0,1]$,
$\epsilon>0$, step size $\eta<{{A_{\nabla\ell}}}^{-1}$, with
${A_{\nabla\ell}}={A_{\nabla\Phi}}+{A_{\boldsymbol{\Phi}}}F^{2L}\sqrt{K}$ and
assume that the graphon convolutions in all layers of this WNN have $K$ filter
taps [cf. (6)]. Let
$\boldsymbol{\Phi}({\mathbf{x}}_{n};{\mathcal{H}},{\mathbf{S}}_{n})$ be a GNN
sampled from $\boldsymbol{\Phi}(X;{\mathcal{H}},{\mathbf{W}})$ as in (12).
Consider the iterates generated by equation (17), under Assumptions AS1-AS6,
if at each step $k$ the number of nodes $n$ verifies

$\displaystyle{\mathbb{E}}[\|\nabla_{{\mathcal{H}}_{k}}$
$\displaystyle\ell(Y,\boldsymbol{\Phi}(X;{\mathcal{H}}_{k},{\mathbf{W}}))-\nabla_{{\mathcal{H}}}\ell(Y_{n},\boldsymbol{\Phi}(X_{n};{\mathcal{H}},{\mathbf{W}}_{n}))\|]$
$\displaystyle+\epsilon<\|\nabla_{{\mathcal{H}}}\ell(Y,\boldsymbol{\Phi}(X;{\mathcal{H}}_{k},{\mathbf{W}}))\|$
(19)

then Algorithm 1 converges to an $\epsilon$-neighborhood of the solution of
the Graphon Learning problem (15) in at most
$k^{*}={\mathcal{O}}(1/\epsilon^{2})$ iterations, with
$\gamma=12\sqrt{K^{5}F^{5L-5}}$].

Theorem 2 provides the conditions under which Algorithm 1 converges to a
neighborhood of the first order stationary point of the empirical graphon
learning problem (15). Other than the aforementioned smoothness assumptions,
we only need to satisfy condition (2) at every epoch, i.e., the norm of the
gradient on the graphon has to be larger than the difference between the
gradients on the graph and graphon.

##  4 Experiments

We consider the problem of coordinating the velocity of a set of $n$ agents
while avoiding collisions. At each time $t$ each agents knows its own position
$r(t)_{i}\in{\mathbb{R}}^{2}$, and speed $v(t)_{i}\in{\mathbb{R}}^{2}$, and
reciprocally exchanges it with its neighbors. Communication links
$[{\mathbf{S}}]_{ij}$ exists if the distance between two agents $i,j$ is
smaller that $2$ meters. At each time $t$ the controller sets an acceleration
that remains constant over an interval of $T_{s}=20ms$. The system dynamics is
governed by,

$\displaystyle r_{i}(t+1)$
$\displaystyle=u_{i}(t)T_{s}^{2}/2+v_{i}(t)T_{s}+r_{i}(t),$ (20)
$\displaystyle v_{i}(t+1)$ $\displaystyle=u_{i}(t)T_{s}+v_{i}(t).$ (21)

We define the velocity variation of the team as
$\sigma_{{\mathbf{v}}(t)}=\sum_{i=1}^{n}\|v_{i}(t)-\bar{\mathbf{v}}(t)\|^{2}$,
and the collision avoidance potential

$\displaystyle\begin{split}CA_{ij}=\begin{cases}\frac{1}{\|r_{i}-r_{j}\|^{2}}-\log(\|r_{i}-r_{j}\|^{2})&\text{if
$\|r_{i}-r_{j}\|\leq R_{CA}$}\\\
\frac{1}{R_{CA}^{2}}-\log(R_{CA}^{2})&\text{otherwise,}\end{cases}\end{split}$

with $R_{CA}=1m$. We consider a centralized controller whose action is given
by
$u_{i}(t)^{*}=-n(v_{i}-\bar{\mathbf{v}})+\sum_{j=1}^{n}\nabla_{r_{i}}CA(r_{i},r_{j})$
[25].

The learning setting is created by training a GNN that mimics the output of
the centralized controller. To do so, we define the empirical risk
minimization problem over the dataset
${\mathcal{D}}=\\{{\mathbf{u}}^{*}_{m},{\mathbf{x}}_{m}\\}_{m}$ for
$m\in[0,400]$,

$\displaystyle\min_{\mathcal{H}}\sum_{m=1}^{|{\mathcal{D}}|}\|{\mathbf{u}}^{*}_{m}-\boldsymbol{\Phi}({\mathbf{x}}_{m};{\mathcal{H}},{\mathbf{S}})\|^{2}.$
(22)

In Figure 1 we can see the velocity variation of the learned GNN measured on
unseen data. Figure 1 validates the utility of the proposed method, as we are
able to learn GNNs utilizing Algorithm 1 that achieve a comparable performance
with the GNN trained on all the nodes. GNNs are that are trained with starting
number of nodes $n_{0}=\\{10,20\\}$, and that add $10$ agents per epoch (green
line) are able to achieve a similar performance when reaching $100$ agents
than the one they would have achieved by training with $100$ agents the same
number of epochs. This is the empirical manifestation of Theorem 2. Moreover,
adding less agents per epoch (orange and blue lines) still achieve the same
performance, but it takes more epochs to achieve. Overall, all the presented
configurations are able to obtain a comparable performance that the one
obtained with the full graph of $100$ agents while taking steps of graphs of
growing sizes.

##  5 Conclusion

In this paper we presented a method for learning GNNs on very large graphs by
growing the size of the graph as we train. Denoted learning by transference,
we exploit the fact that the norm of the gradient on WNN decreases as it
approaches a minima, and so we increase the precision at which we estimate it
as epochs increase. We provide a proof of convergence of our algorithm, as
well as numerical experiments on a multi-agent problem.

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##  Appendix A Proof of Theorem 1

Theorem 2 follows directly from Theorem [22, Theorem $1$], and [22, Lemma $3$]
with $\alpha$ as in the node stochasticity (cf. Definition 2), and $beta$ as
in the edge stochasticity (cf. Definition 3). This proof relies on [19,
Proposition $5$]. It is also needs to be noted that ${\mathcal{X}}_{1}$,
${\mathcal{X}}_{2}$, and ${\mathcal{X}}_{3}$ should be selected as $\sqrt{n}$.
The maximum in the final bound comes from fact that the bound is governed by
the maximum rate between $\beta$, and $\alpha$.

##  Appendix B Proof of Theorem 2

See [22, Theorem $2$].