# Superfast Quarks in Deuterium

Dmitriy N. Kim and Gerald A. Miller  Department of Physics, University of
Washington, Seattle, WA 98195-1560, USA

###### Abstract

We present a continuation to our previous study on Nuclear Parton Distribution
Functions (nPDFs) using Light-Front Holographic Quantum Chromodynamics
(LFHQCD) Kim and Miller (2022); Brodsky  _et al._ (2015). We focus on applying
the effects of nucleon motion inside the nucleus (Fermi motion/smearing) to
our results for deuterium, extending our nPDFs (and hence the DIS $F_{2}$
structure function for deuterium, $F_{2}^{d}$) to the superfast, $x>1$, region
Frankfurt and Strikman (1988). We find that our previous model, with no
additional new parameters, is in excellent agreement with deuterium EMC ratio
data obtained from the BONuS experiment Fenker  _et al._ (2008); Baillie  _et
al._ (2012); Tkachenko  _et al._ (2014); Griffioen  _et al._ (2015). Lastly,
looking beyond conventional nuclear physics, and in anticipation of ongoing 12
GeV experiments at Jefferson Lab, we use an ansatz provided by Mulders and
Thomas Mulders and Thomas (1984) to predict the contributions of an exotic
six-quark state to $F_{2}^{d}$ in the superfast region. Our results show that
a six-quark state has minimal impact in the $x<1$ region, while causing the
slope of $F_{2}^{d}$ to get closer to $0$ for $x>1$.

##  I Introduction

Parton distribution functions are ubiquitous in particle physics because they
describe the relationship between Quantum Chromodynamics’ (QCDs) basic degrees
of freedom, quarks and gluons (partons), and the physical observable states,
hadrons. This makes their nuclear counterparts, nPDFs, indispensable tools
towards understanding the emergence of intrinsic nuclear properties from the
dynamics of their constituent partons. Programs to extract nPDFs have been,
and are continued to be, supported with the operation of the Large Hadron
Collider, the 12 GeV upgrade at Jefferson Lab (JLab), and upcoming Electron-
Ion Collider (EIC). It is clear that understanding QCD interactions, how the
dynamics of partons leads to the emergence of nucleons and nuclei, is of great
interest and is a fundamental goal in nuclear science.

A domain where one can probe the interplay between nuclear physics and QCD’s
degrees of freedom lies in $x>1$, dubbed the superfast region Frankfurt and
Strikman (1988). Superfast quarks are inherently generated by the nuclear
environment. In other words, quarks with $x>1$ cannot be produced by the QCD
dynamics of a single free nucleon (unlike quarks inside a free nucleon, a
quark inside a nucleus can have a momentum fraction as large as
$x=m_{A}/m_{N}$, where $m_{A}$ and $m_{N}$ are the masses of the nucleus and
nucleon respectively). As such, investigations into superfast quarks are
foundational towards uncovering intersections between nuclear physics and QCD.

Nuclear deep inelastic scattering (DIS) at high-$x$ gives us an opportunity to
probe the superfast region. By extracting the DIS $F_{2}$ structure function
of the nucleus, $F_{2}^{A}$, we can study the $x>1$ momentum distributions of
quarks inside a nucleus. To date, three experiments have undergone such
investigations: The BCDMS Collaboration at CERN Benvenuti  _et al._ (1994),
the CCFR Collaboration at Fermi Lab Vakili  _et al._ (2000), and most recently
at JLab Fomin  _et al._ (2010). However, the trends of $F_{2}^{A}$ at
high-$x$, extracted from all three experiments, do not agree with each other.
The experiment at JLab originally reported results in agreement with BCDMS
Fomin  _et al._ (2010). However, a recent study by Freese et al. Freese  _et
al._ (2019) improved on the $Q^{2}$ evolution procedure used in the JLab
study, obtaining results slightly in favor of BCDMS, but overall not strongly
aligning with either of the two experiments.

It is clear that there is still much experimental work to be done in
extracting the behavior of $F_{2}^{A}$ in the superfast region. The 12 GeV
beam energy upgrade at JLab hopes to accomplish this by improving on its
predecessors results. The experiment aims to extract $F_{2}^{A}$ at an even
larger $x$ threshold, in kinematics where quasielastic contributions and
scaling violations in the cross section are minimized Arrington  _et al._
(2006); Hen  _et al._ (2014).

In contrast to experimental progress, there has been little theoretical
development in studying $F_{2}^{A}$ at $x>1$, this is the goal of this paper.
This study will focus on deuterium and the contents of this paper are as
follows: In Sec. II, we introduce the theoretical formalism that is used in
this study, providing a brief overview of the previous LFHQCD model used to
study the EMC effect Kim and Miller (2022); Brodsky  _et al._ (2015). In Sec.
III, we introduce the conventional Fermi motion procedure, slightly modify it
by incorporating results from the LFHQCD EMC study, and then apply the new
Fermi smearing procedure to obtain $F_{2}^{d}$ for $0<x<m_{A}/m_{N}$ — our
first steps into the superfast region. In Sec. IV, we present a six-quark PDF
ansatz used by Mulders and Thomas Mulders and Thomas (1984) for deuterium, and
incorporate it to our results in Sec. III. Our concluding remarks/discussion
are given in Sec. V.

##  II Light-Front Holographic QCD Model of the EMC Effect

To begin, we provide a brief overview of the phenomenological LFHQCD-based
model used in Ref. Kim and Miller (2022) — we will refer to it as the nuclear
LFHQCD (nLFHQCD) model for brevity. The model outputs closed-form expressions
for nPDFs that are dependent on two phenomenological parameters. The
parameters are proportional to the virtuality, a quantity that measures how
off-shell a nucleon is inside the nucleus. The nPDFs were used to construct
$F_{2}^{A}$, and numerical results for the phenomenological parameters were
obtained by fitting to EMC ratio data in the region $0.3\leq x\leq 0.7$ (EMC
region). The results of the fitting were successful in reproducing EMC ratio
data for a variety of nuclei. Having the correct behavior in the EMC region,
the nLFHQCD model is an excellent candidate to naturally extend to $x>1$.

The physical intuition for the nLFHQCD model was inspired by the QCD physics
of color transparency Frankfurt and Strikman (1985); Brodsky and de Teramond
(1988); Ralston and Pire (1988); Jennings and Miller (1993). We treated the
infinite number of quark-gluon configurations of the nucleon as two
configurations: a large-sized, blob-like configuration (BLC), consisting of
complicated configurations of many quarks and gluons, and a small-sized,
point-like configuration (PLC) consisting of three quarks. When placed inside
a nucleus, the BLC will feel the usual nuclear attraction, however, the PLC
feels far less by virtue of color screening. This decreases the BLC’s energy,
increasing the probability that the nucleon occupies that state, and
consequently suppresses the PLC probability.

We used the intuition above to study $F_{2}^{A}$ in LFHQCD, a semi-classical
approximation to QCD that utilizes AdS/CFT correspondence Brodsky  _et al._
(2015). The formalism is able to produce closed-form expressions of proton
PDFs with excellent agreement to experiment de Teramond  _et al._ (2018).
Additionally, the LFHQCD construction has a nice interpretation of hadronic
PDFs as the sum of $\tau$-constituent Fock state PDFs, where $\tau$ is the
number of constituents. For the proton/neutron, the smallest $\tau$-particle
Fock state is three quarks, the nucleonic PDF in LFHQCD is given by:

$f^{p/n}(x)=\sum_{\tau=3}^{\infty}c_{\tau}^{p/n}q_{\tau}(x).$ (1)

Where $f^{p/n}(x)$ is the PDF of the proton or neutron, $q_{\tau}(x)$ is the
PDF of a state with $\tau$-constituent partons, and $c_{\tau}^{p/n}$ is a
constant that is related to the contribution of the $\tau$-particle Fock state
in the basis expansion. With this construction, we were able to incorporate
the PLC-BLC model for nucleons in nuclei. Using the LFHQCD model for nucleon
form factors and PDFs Sufian  _et al._ (2017); de Teramond  _et al._ (2018),
we only considered the first two components of the proton/neutron states,
$\tau=3$ and $\tau=4$, and identified them as the PLC and BLC respectively.
The $c_{\tau}$ constants were modified from the original construction in Refs.
Sufian  _et al._ (2017); de Teramond  _et al._ (2018) by incorporating two
phenomenological parameters: $\delta r_{p}(A,Z)$ and $\delta r_{n}(A,Z)$, one
for each nucleon, both dependent on $A$ and atomic number, $Z$; the physical
interpretation of $\delta r_{p}(A,Z)$ and $\delta r_{n}(A,Z)$ is they are
quantities related to the nuclear potential affecting the probabilities of
finding a spin up or down quark in a proton or neutron, breaking SU(6)
symmetry. The modified PDFs for the proton were,

$\tilde{u}^{p}(x)=\left(\frac{3}{2}-3\delta
r_{p}\right)q_{3}(x)+\left(\frac{1}{2}+3\delta r_{p}\right)q_{4}(x),$ (2)

$\tilde{d}^{p}(x)=\left(-3\delta r_{p}\right)q_{3}(x)+\left(1+3\delta
r_{p}\right)q_{4}(x).$ (3)

Where $\tilde{u}^{p}(x)$ and $\tilde{d}^{p}(x)$ are the medium-modified proton
up and down PDFs respectively. The neutron PDFs are obtained by replacing
$\tilde{u}^{p}(x)\rightarrow\tilde{d}^{n}(x)$,
$\tilde{d}^{p}(x)\rightarrow\tilde{u}^{n}(x)$, and $\delta
r_{p}\rightarrow\delta r_{n}$. $q_{\tau}(x)$ is given by:

$q_{\tau}(x)=\frac{\Gamma\left(\tau-\frac{1}{2}\right)}{\sqrt{\pi}\Gamma(\tau-1)}\big{(}1-w(x)\big{)}^{\tau-2}\,w(x)^{-1/2}\,w^{\prime}(x),$
(4)

$w(x)=x^{1-x}e^{-a(1-x)^{2}}.$ (5)

Where the flavor-independent parameter $a=0.531\pm 0.037$ and $q_{\tau}(x)$ is
evaluated at the matching scale between LFHQCD and pQCD, $\mu_{0}=1.06\pm
0.15$ GeV de Teramond  _et al._ (2018).

Noting that the medium modified proton wavefunction, $\tilde{\Psi}^{p}(x)$, is

$\tilde{\Psi}^{p}(x)\sim\tilde{u}^{p}(x)+\tilde{d}^{p}(x),$ (6)

we see that the BLC probability is enhanced and PLC is suppressed due to the
nuclear potential. From here, one can construct the $F_{2}^{A}$ structure
function, outlined in Ref. Kim and Miller (2022),

$F_{2}^{A}=Z\tilde{F}_{2}^{p}+N\tilde{F}_{2}^{n}=ZF_{2}^{p}+NF_{2}^{n}+\frac{5x}{3}(q_{4}-q_{3})(Z\delta
r_{p}+N\delta r_{n}).$ (7)

Where $F_{2}^{p}$ and $F_{2}^{n}$ are the DIS $F_{2}$ structure functions of
the proton and neutron respectively; they can be obtained by using Eqs. (2, 3)
and setting $\delta r_{p}$ and $\delta r_{n}$ to zero. Lastly, we would like
to note that Eq. (7) has a striking resemblance to the SRC-based model in Ref.
Schmookler  _et al._ (2019), further discussion and tabulated values of
$\delta r_{p}$ and $\delta r_{n}$ can be found in Ref. Kim and Miller (2022).

In our previous study, we did not include information about nucleons moving in
the nucleus (Fermi motion), Eq. (7) outputs results for $0<x<1$, and we only
focused on the EMC region $0.3<x<0.7$. Applying Fermi motion into our previous
model will extend its application beyond $x=1$ through the use of conventional
nuclear physics, the first step towards investigating the $x>1$, superfast
region.

##  III Fermi Motion

###  III.1 Conventional Fermi Motion Procedure

A derivation of the Fermi smearing expressions can be found in Refs. Jung and
Miller (1988, 1990). In this study, we will use the expressions in Refs. Jung
and Miller (1988, 1990), but use the flux factor correction given in Ref.
Frankfurt and Strikman (1987); the flux factor correction to the Fermi motion
procedure correctly accounts for nuclear baryon sum rules in the Spectral
function. Incorporating this correction yields the following Fermi smearing
expressions in the DIS limit:

$F_{2}^{A}(x)=\int_{x}^{\bar{A}_{N}}f^{A}_{N}(y)\,F_{2}^{N}(x/y)\,dy,$ (8)

$f^{A}_{N}(y)=\int_{-\infty}^{\infty}\delta(y-\bar{A}_{N}\,p^{+}/P^{+})\,\frac{\bar{A}_{N}\,(p^{0}+p^{3})}{P^{0}}\,S^{A}_{N}(p^{0},\vec{p})\,d^{4}p.$
(9)

We denote $p$ and $P$ as the four-momentum of the nucleon and nucleus
respectively, $A$ ($N$) denotes the nucleus (flavor of the nucleon), $m_{A}$
($m_{N}$) is the mass of nucleus $A$ (nucleon $N$), $\nu$ is the energy
difference between the incoming and scattered electron, $x=Q^{2}/2m_{N}\nu$ is
the longitudinal momentum fraction carried by the struck quark inside nucleon
N, $y=\bar{A}_{N}\,p^{+}/P^{+}$ is the fraction of the $+$ momentum carried by
the nucleon inside the nucleus multiplied by $\bar{A}_{N}=m_{A}/m_{N}$ (where
$+$ momentum is defined as: $p^{+}=p^{0}+p^{3}$), $F_{2}^{A}$ ($F_{2}^{N}$) is
the DIS $F_{2}$ structure function for nucleus $A$ (free nucleon $N$),
$f^{A}_{N}(y)$ is the light-front distribution for nucleon $N$ inside nucleus
$A$, and $S^{A}_{N}(p^{o},\vec{p})$ is the spectral function for nucleon N in
nucleus A. Note that we do not denote whether $x$ is $x_{p}$ or $x_{n}$ (same
for the variable $y$), it should be clear from context. For instance, if the
variable resides in $f_{p}^{A}(y)$, then $y=y_{p}$. In our calculations, we
used the nucleus rest frame, $P^{+}=P^{0}=m_{A}$.

The spectral function for deuterium in its rest frame is given by:

$S^{d}_{N}(p^{0},\vec{p})=\rho^{d}_{N}(p)\delta(p^{0}-m_{d}+\sqrt{m_{N}^{2}+\vec{p}\,^{2}}),$
(10)

where $\rho^{d}_{N}(p)$ is the momentum distribution of nucleon $N$ inside
deuterium (for deuterium, the proton and neutron momentum distributions are
the same). Furthermore, note that Eq. (8) has an implicit sum over nucleons,
$N$. Lastly, note that the upper-bound integration limit in Eq. (8) is
$\bar{A}_{N}=m_{A}/m_{N}$. Conventionally, most studies involving Fermi motion
use the mass number as an upper-bound in Eq. (8), this marginally affects
numerical results. Here, we use $\bar{A}_{N}$ to be kinematically correct.

###  III.2 Fermi Motion + nLFHQCD Model

Eq. (8) incorporates information about nucleon motion inside the nucleus and
Eq. (7) includes information about how the nuclear potential modifies the
nucleons, putting the two together will give us a description of moving,
medium-modified, nucleons inside a nucleus. This is done by modifying Eq. (8)
to,

$F_{2}^{A}(x)=\int_{x}^{\bar{A}_{N}}f^{A}_{N}(y)\,\tilde{F}_{2}^{N}(x/y)\,dy,$
(11)

where we have done the replacement $F_{2}^{N}\rightarrow\tilde{F}_{2}^{N}$.

###  III.3 Fermi Motion Applied to Deuterium in nLFHQCD

The nucleon momentum distribution in deuterium was obtained via the AV18
wavefunction Wiringa  _et al._ (1995). The momentum distribution was tabulated
for finite points of $p$, the magnitude of the total momentum, and an
interpolator was used. Fig. 1 presents the nucleon momentum distribution in
deuteron for a $p$-range of $0<p<10$. The tabulated data was in the range
$0<p<15$, anything outside this range was evaluated to be 0.

Figure 1: A plot of the interpolated AV18 nucleon momentum distribution in
deuterium in the region $0<p<10$.

Using the AV18 momentum distribution, we obtained the light-front distribution
by using Eq. (9). The light-front distributions for deuterium are presented in
Fig. 2; the dip in the distribution at $y\approx 1.65$ was identified to be
due to the interference between S and D states in the momentum distribution at
$p\approx 7.5$.

Figure 2: The proton (solid blue line) and neutron (dashed orange line) light-
front distributions for deuterium as a function of their respective $y_{N}$,
the momentum fraction of the nucleon in the nucleus weighted by $\bar{A}_{N}$.
The differences between the two distributions are minuscule.

Finally, using Eq. (11) we obtained $F_{2}^{d}$ evaluated at $Q^{2}=1.12$
GeV2, extended to $x>1$ through conventional nuclear physics (presented in
Fig. 3). The value used for both $\delta r_{p}$ and $\delta r_{n}$ was
$0.010\pm 0.003$, obtained from our previous study in Ref. Kim and Miller
(2022).

Figure 3: DIS $F_{2}$ structure functions on a logarithmic $y$-axis, evaluated
at $Q^{2}=1.12$ GeV2. The solid black line is the sum of the free proton and
neutron structure functions, the dot-dashed blue line was obtained by using
Eq. (8), and the solid red line was obtained by using Eq. (11). The dot-dashed
blue and solid red lines are on top of one another, indicating that the medium
modification for deuterium has a small impact on high-$x$ behavior.

Notice that for deuterium, the medium modification has a small impact on the
high-$x$ behavior. Furthermore, we identified that the small dip in
$F_{2}^{d}$ at $x\approx 1.1$ is consequence of the low-momentum behavior of
$\rho_{p/n}^{{}^{2}H}$.

Lastly, Fig. 4 presents the EMC ratio for deuterium, with Fermi smearing
applied to our nLFHQCD model using Eq. (11). All DIS $F_{2}$ quantities
evaluated at $Q^{2}=1.12$ GeV2. As we can see, there is very good agreement
with data obtained from the BONuS experiment, which extracted
$F_{2}^{n}/F_{2}^{d}$ by using a spectator tagging technique on semi-inclusive
electron-deuteron collisions Fenker  _et al._ (2008); Baillie  _et al._
(2012); Tkachenko  _et al._ (2014); Griffioen  _et al._ (2015). Notice that
simply applying Fermi smearing to $F_{2}^{p}+F_{2}^{n}$ does not yield a
sufficient deuterium EMC effect that captures the data.

Figure 4: All DIS $F_{2}$ quantities are evaluated at $Q^{2}=1.12$ GeV2. The
solid red line is the Fermi-Smeared nLFHQCD result using Eq. (11), the dot-
dashed blue line is Fermi smearing applied to $F_{2}^{p}+F_{2}^{n}$ using Eq.
(8), and the filled black points are experimental results obtained from the
BONuS experiment Fenker  _et al._ (2008); Baillie  _et al._ (2012); Tkachenko
_et al._ (2014); Griffioen  _et al._ (2015). The $1\sigma$ uncertainties are
presented in respective colored bands.

##  IV Addition of Six-Quark Cluster

With the Fermi-smeared $F_{2}^{d}$ result, we can now explore ideas outside
conventional nuclear physics. Motivated towards understanding intersections
between nuclear physics and QCD, we model the PDFs for an exotic six-quark
state to make predictions on its contribution to $F_{2}^{d}$. The deuteron can
occupy a six-quark state through quantum fluctuations, causing the proton and
neutron to overlap completely. This six-quark cluster/bag, compared to a bound
proton and neutron, allows for a greater sharing of momentum between the
quarks in deuterium, enhancing the distribution of high-momentum quarks
Arrington  _et al._ (2006). However, since most of deuteron’s properties can
be described by the picture of a bound proton and neutron, we expect the six-
quark bag probability, $P_{6q}$, to be very small. A previous study
investigated the contribution of a six-quark state to the $b_{1}$ structure
function of deuteron Miller (2014). The study used a six-quark probability of
$P_{6q}=0.15$% to match the experimental extraction of $b_{1}$ at $x=0.452$ by
the HERMES Collaboration Airapetian  _et al._ (2005). Since this value for the
probability was obtained from fitting to one point, we will use it as a
conservative upper-bound for $P_{6q}$.

Now we need to model the PDF of a six-quark hadronic state in order to get its
DIS $F_{2}$ structure function,

$F_{2}^{6q}=\frac{4}{9}\,x_{d}\,u^{6q}(x_{d})+\frac{1}{9}\,x_{d}\,d^{6q}(x_{d}).$
(12)

Where $u^{6q}$ and $d^{6q}$ are the up and down PDFs of the six-quark state
and $x_{d}=Q^{2}/2m_{d}\nu$. To include the contributions of a six-quark
cluster to $F_{2}^{d}$, we added Eq. (12) to Eq. (11), multiplying both terms
by a six-quark probability factor — not applying Fermi smearing to Eq. (12) as
a six-quark cluster does not have moving-nucleonic components.

$F_{2}^{d}(x)=(1.0-P_{6q})\int_{x}^{m_{d}/m_{N}}f^{d}_{N}(y)\,\tilde{F}_{2}^{N}(x/y)\,dy+P_{6q}F_{2}^{6q}(x).$
(13)

Now, we need an ansatz for the PDFs of a six-quark state in deuterium.

###  IV.1 Mulders and Thomas Ansatz

The original Mulders and Thomas six-quark ansatz is motivated by the
assumption that all six quarks are in the same single-particle orbital. Also,
the argument used in the PDFs of their ansatz was $x/2$, here we will use
$x/\bar{A}_{N}$ to be kinematically correct. Modifying the original Mulders
and Thomas ansatz Mulders and Thomas (1984), we now have:

$u^{6q}(x)=\frac{1}{\bar{A}_{p}}u^{p}(x/\bar{A}_{p})+\frac{1}{\bar{A}_{n}}u^{n}(x/\bar{A}_{n}),$
(14)

$d^{6q}(x)=\frac{1}{\bar{A}_{p}}d^{p}(x/\bar{A}_{p})+\frac{1}{\bar{A}_{n}}d^{n}(x/\bar{A}_{n}).$
(15)

Where the up and down PDFs for the six-quark cluster are simply the sum of the
free proton and neutron PDFs. Notice that the arguments of the proton and
neutron PDFs are $x/\bar{A}_{N}$, this ensures that the up and down PDFs are
now extended all the way up to $x=\bar{A}_{N}$, where
$\bar{A}_{N}=m_{d}/m_{N}$ (the correct notation for deuteron should be
$\bar{d}_{N}$, but is written as $\bar{A}_{N}$ to avoid confusion with the
down PDF). Furthermore, the factor of $1/\bar{A}_{N}$ is there to ensure
proper normalization of the PDFs.

By combining Eqs. (12, 14, 15) we obtained:

$F_{2}^{6q}=\frac{1}{\bar{A}_{p}}F_{2}^{p}(x/\bar{A}_{p})+\frac{1}{\bar{A}_{n}}F_{2}^{n}(x/\bar{A}_{n})$
(16)

Where we used LFHQCD inputs for the up and down proton PDFs. Fig. 5 presents
the predictions of the Mulders and Thomas six-quark model to our Fermi-smeared
results from Section III, evaluated at $Q^{2}=1.12$ GeV2.

Figure 5: DIS $F_{2}$ structure functions on a logarithmic $y$-axis. All
$F_{2}$ quantities are evaluated at $Q^{2}=1.12$ GeV2. The black line is the
sum of the free proton and neutron structure functions and the filled-in red
volume displays the predicted contributions of a six-quark cluster. The lower
boundary of the volume is $P_{6q}=0\%$ and the upper boundary is
$P_{6q}=0.15\%$. The red dotted and dashed lines display $P_{6q}=0.05\%$ and
$P_{6q}=0.005\%$ respectively, and are shown to clarify the trend of
$F_{2}^{d}$ with varying $F_{2}^{6q}$ contributions.

Fig. 6 presents the same information as Fig. 12, but evaluated at $Q^{2}=20$
GeV2 — kinematics that current JLab experiments are proposed to reach
Arrington  _et al._ (2006); Hen  _et al._ (2014). To accomplish this, the PDFs
were evolved to a higher scale with the Dokshitzer-Gribov-Lipatov-Altarelli-
Parisi (DGLAP) equations Altarelli and Parisi (1977); Dokshitzer (1977);
Gribov and Lipatov (1971) using the APFEL package Bertone  _et al._ (2014).
The inputs for DGLAP evolution, such as the initial scale, renormalization
scheme, and heavy quark thresholds, are identical to the ones used to study
PDFs in LFHQCD in Ref. de Teramond  _et al._ (2018).

Figure 6: DIS $F_{2}$ structure functions on a logarithmic $y$-axis. All
$F_{2}$ quantities are evaluated at $Q^{2}=20$ GeV2. The black line is the sum
of the free proton and neutron structure functions and the filled-in red
volume displays the predicted contributions of a six-quark cluster. The lower
boundary of the volume is $P_{6q}=0\%$ and the upper boundary is
$P_{6q}=0.15\%$. The red dotted and dashed lines display $P_{6q}=0.05\%$ and
$P_{6q}=0.005\%$ respectively, and are shown to clarify the trend of
$F_{2}^{d}$ with varying $F_{2}^{6q}$ contributions.

##  V Discussion/Conclusion

The results presented in this paper are an extension to our previous study in
Ref. Kim and Miller (2022). Focusing on the deuteron, we applied Fermi motion
effects to extend the nLFHQCD model beyond $x>1$. In doing so, we found that
our previous model turned out to be in excellent agreement to BONuS data for
the EMC ratio Fenker  _et al._ (2008); Baillie  _et al._ (2012); Tkachenko
_et al._ (2014); Griffioen  _et al._ (2015). On top of conventional nuclear
physics, we implemented the contributions of an exotic six-quark state to
$F_{2}^{d}$ using an ansatz from Mulders and Thomas Mulders and Thomas (1984).
We found that the six-quark distribution enhances the $x>1$ region, while
minimally affecting $x<1$ — displaying correct qualitative behavior as a six-
quark cluster allows for a greater sharing of momentum between quarks,
enhancing the high-momentum behavior of $F_{2}^{d}$, while minimally affecting
the low-momentum region. We displayed the predictions of the six-quark ansatz
to $F_{2}^{d}$ for $0<P_{6q}<0.15\%$ in Figs. 5 and 6, with the upper-bound in
$P_{6q}$ motivated by Ref. Miller (2014). We found that the predicted six-
quark distribution leads causes the slope of $F_{2}^{d}$ to get closer to $0$
in the superfast, $x>1$, region (of course, it is important to note that the
results in Figs. 5 and 6 are displayed a log plot). Furthermore, we find that
the slope of $F_{2}^{d}$ in the superfast region increases with increasing
$P_{6q}$. Thus if such a behavior is seen in experiment, this is a way to
determine the six-quark probability in $F_{2}^{d}$. With the current $12$ GeV
experiments happening at JLab, we hope to test these predictions against
experimental data in the near future.

##  VI Acknowledgements

D. N. Kim and G. A. Miller would like to thank John Arrington for motivating
this study and Adam Freese for useful discussions. This work was supported by
the U. S. Department of Energy Office of Science, Office of Nuclear Physics
under Award Number DE-FG02-97ER-41014.

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# Enhancing Educational Outcome with Machine Learning: Modeling Friendship
Formation, Measuring Peer Effect and Optimizing Class Assignment

Lei Bill Wang  Department of Economics, Ohio State University  Om Prakash
Bedant  Department of Electrical and Computer Engineering, Ohio State
University  Haoran Wang  Department of Economics, Ohio State University
Zhenbang Jiao  Department of Statistics, Ohio State University  Jia Yin
Department of Electrical and Computer Engineering, University of Pittsburgh

##  1 Introduction

Peer effects refer to the impact that individuals within the same social
circles have on each other. These effects can manifest in various aspects of
our lives, including our behavior, attitudes, and performance. Due to its
ubiquity and importance, a groundswell of studies have delved into the realm
of peer effects (See Guryan et al. (2009), Carrell and Hoekstra (2010),
Carrell et al. (2018) and Golsteyn et al. (2021)), exploring how individuals
or their specific characteristics influence their peers. These studies
predominantly utilize the linear-in-means model, assuming homogeneity in
spillover effects among cohorts. Though linear-in-means model is easy to
implement and has a clear interpretation, it suffers from two major drawbacks.

Firstly, linear-in-means model assumes a uniform strength of connection
between any pair of individuals. Consequently, it does not shed light on
network formation and can hardly capture differential peer effects individuals
receive from their cohorts. Though under assumptions specified by Manski
(1993), linear-in-means model successfully identifies some parameters related
to the average peer effect, researchers may be interested in the entire
distribution of peer effect, or at least some aspects of the distribution that
is not fully pinned down by the average. To uncover the distribution of peer
effect, consideration of dynamic network formation is necessary. Linear-in-
means model ignores intricacies of network development and therefore,
struggles to provide insights on distribution of peer effect. Secondly,
neglecting the network/friendship development hinders the formulation of
effective policies that can address issues arising from peer effects. This
deficiency in capturing the nuances of network dynamics poses a challenge when
optimizing group or class assignments. Therefore, exploring alternative models
that account for the complexities of network formation is imperative for more
comprehensive learning about peer effect and more effective design of group
assignment policy.

In this paper, we look at one type of the group assignment problems: school
principal’s class assignment decision. In the context of class assignment, a
principal seeks to optimize a certain objective through the peer effect
channel. The closest study to our paper is Carrell et al. (2013). The authors
conduct an experiment of class assignment aiming to maximize the positive
influence of the high ability students on the bottom 33% students’ GPA. In
their study, the experiment result turns out to be surprising as the designed
class assignment policy does not improve the bottom 33% students’ performance.
The authors reconcile this counterintuitive result by arguing that linear-in-
means model does not consider dynamic friendship formation which aligns with
aforementioned criticism of linear-in-means model. In contrast, modeling
friendship formation is an integral part of our research. The friendship
formation model allows us to predict the friendship intensity between
different individual students after assigning them into classrooms. Replacing
the linear-in-means assumption, we measure peer effect with a more reasonable
friendship-weighted assumption. The ultimate goal is to design a class
assignment policy that maximizes the peer effect of secondary school
classmates who have performed well in elementary school on bottom quartile
students’ cognitive ability.

Our research will address these questions by (1) predicting friendship
formation probability based on predetermined characteristics, (2) measuring
peer effects through weighted averages with an instrumental variable approach
and (3) simulating counterfactual scenarios to optimize class assignments. The
methodologies that we develop in this paper can be easily extended to other
group assignment problem when researchers observe either the full network or
some aggregated relational data with respect to the network. We will describe
in greater detail the 3 components of our research methodologies in later
sections of the paper. Next, we summarize in high level some of the important
findings we discover in this study.

The findings from the friendship formation prediction reveal a strong
inclination for students of the same gender to form connections, with girls
exhibiting more influence on boys than vice versa. Notably, a significant
number of girls predominantly receive peer influence from other girls.
Comparative testing against the linear-in-means model underscores the
importance students place on their friends’ gender in our model, as evidenced
by significantly lower mean square errors out of sample. This result can be
perceived as a special case for the well known homophily effect in network
formation literature. Our result emphasize that in the context of Chinese
secondary school education, gender homophily effect is a dominant factor in
friendship formation and should be considered when it comes to using class
assignment policy as a tool to optimize peer effect. The other major finding
is that students who performed well academically in their elementary school
tend to be considered friends by more classmates. Coupled with the gender
homophily effect, this finding points at the potential benefit of dividing
students into gendered classrooms.

The instrumental variable (IV) estimation outcomes highlight a meaningful
positive impact of peers’ sixth-grade rank on current cognitive scores,
indicating that a 10% increase in friends’ average class rank results in a
0.1253 point increase in cognitive scores which are measured on a scale from
-2 to 3.

Additionally, when classes are divided by gender, the counterfactual analysis
demonstrates that, post-assignment to all-boy classes, average boy students’
realized peer effect increase by 2.19%. Similarly, after assignment to all-
girl classes, average girl students see an enhanced realized peer effect
improving by 2.12%. These results underscore the nuanced dynamics of peer
influence in the context of gender-based class assignments, shedding light on
potential optimal class assignment policy for students’ cognitive outcomes.

Our research makes several noteworthy contributions to the understanding of
peer effects and class assignments. Firstly, it leverages neural network
method to approximate the functional relationship between students’ own
characteristics and their friendship preferences, providing valuable insights
into the intricate processes of how students form connections. This approach
allows for an examination of friendship dynamics beyond what traditional
linear models can offer. Secondly, replacing the linear-in-means assumption
with a more reasonable friendship weighted assumption, we conduct a
comprehensive analysis of heterogeneous peer effects based on students’
characteristics. This allows us to discern the varying impacts of peer
influence across different demographic and behavioral dimensions, contributing
to a deeper understanding of peer effect distribution. Lastly, through
counterfactual analysis, we go beyond describing observed patterns and
actively identify optimal class assignment strategies that maximize students’
cognitive scores. This practical dimension of our research holds implications
for educational policies and practices, offering actionable insights for
educators and policymakers seeking to enhance student outcomes through
thoughtful class assignments.

##  2 Data description

We use China Education Panel Survey (CEPS) data to conduct this research. The
data is open to public and ensures that our project is replicable.

CEPS surveyed 7649 students (and their family members and teachers) from 179
classes. If a class is selected to participate in CEPS, all students from the
class were surveyed. The survey was conducted in two rounds, once in 7th
grade, once in 8th-grade. Both rounds of surveys asked all participants
(student, family member and teacher) their demographic information (such as
age, gender, occupation, income, education level and studying hours per week).
In total, more than 2000 variables are collected. The richness of CEPS helps
extend our research into many future research directions. In our paper, we
investigate a very specific channel of peer effect. Future research can
leverage the model that we propose in this paper and look into other
mechanisms of the peer effect.

Students and family members were asked about the students’ 6th grade
information. This feature of CEPS is extremely useful for the class assignment
policy design question that we aim to answer in our project. We exclusively
use students’ 6th grade information to predict friendship formation as it
aligns with the principal’s class assignment problem in which principal has to
divide students into classrooms upon their school entry. The only information
that the principal has at the time of making class assignment decision is
students’ 6th grade information.

Students were asked to provide aggregated relational data of their friends.
The linkage data, though surveyed in section C20 as depicted in Figure 1, is
redacted from CEPS data. Instead, aggregated relational data of friendship
network (See section C21 from Figure 1) is provided. Theretofore, we have
information on “out of the five best friends of a student, how many of them
?” where  can be filled by any question from section C21. We use the
aggregated relational data to uncover friendship formation pattern.

Figure 1: The friendship data is aggregated relational. The identification of
network formation parameter with such data is studied by Breza et al. (2020).
We focus on friendship formation prediction instead of inference on the
preference parameters.

In order to study the causal impact of friends’ predetermined characteristics
on students’ academic outcome, we exploit the random class assignment. From
the total sample of 7649 students in CEPS, 5860 students are randomly assigned
to 139 classes conditional on school choice. Leverage on the random assignment
data structure, we construct valid instruments to address endogenous network
formation issue.

##  3 Model setup

We solve the class assignment optimization problem with the following three
steps: (1) predicting friendship formation probability based on predetermined
characteristics, (2) measuring peer effects through weighted averages with an
instrumental variable approach and (3) simulating counterfactual scenarios to
optimize class assignments.

###  3.1 Friendship formation neural network

We build a novel neural network architecture to predict friendship formation
with aggregate relational data. The neural network model can be divided into
four stages: (1) friendship market, (2) linearized propensity score, (3)
softmax normalization and (4) simulation-based estimator. As compared to more
conventional neural network architectures, our model is micro-founded and
interpretable (can be explained as a discrete choice model in economics). The
overall architecture is summarized in Figure 2. Next, we will describe the
four stages separately.

Figure 2: Illustration of the neural network algorithm for our friendship
formation model

Stage I: Friendship market

In our model, we assume that students have preference for friends and those
preference parameters $DD$ ($N\times K$) are functions of their observed
characteristics. We refer to these functions as $f$. In short,
$DD_{i}=f(X_{i})$ where $DD_{i}$ is the preference parameters of student $i$,
it is a K-dimensional vector. $DD_{ik}$ indicates how much student $i$ prefers
to have a friend with characteristic $k$. We use example 3.1 to illustrate the
concept of preference parameters. We take an agnostic view of the function
that maps student input information, $X$ to preference parameters $DD$. Hence,
we use feedforward neural network to approximate this function.

######  Example 3.1.

Use two predictors: gender and whether a student did well in 6th grade to
predict friendship formation. We have the data matrix (Table 2) for a
classroom consisting of five students.

|  gender |  did well in 6th grade
---|---|---
Bill |  1 |  1
Om |  1 |  1
Jack |  1 |  0
Haoran |  0 |  1
Lei |  0 |  0
Table 1: $SS=X$ contains students’ predetermined characteristics: gender and
whether the student did well in 6th grade

|  gender |  did well in 6th grade
---|---|---
Bill |  1 |  0.5
Om |  1 |  0.5
Jack |  0.5 |  -0.5
Haoran |  -1 |  0.5
Lei |  -0.5 |  -0.5
Table 2: Preference parameter $DD$ indicates students’ preference for making
friends with classmate with certain characteristics

Since both input variables are binary, there are four types of students. (1)
boy who did well (2) boy who did not do well (3) girl who did well (4) girl
who did not do well. Assume that the four types of students have the following
preferences:

  * •

Boys who did well in elementary school prefers to make friends with boys over
girls and they prefer to make friends with those who did well in elementary
school.

  * •

Boys who did not do well in elementary school prefers to make friends with
boys over girls and they prefer to make friends with those who did not do well
in elementary school.

  * •

Girls who did well in elementary school prefers to make friends with girls
over boys and they prefer to make friends with those who did well in
elementary school.

  * •

Girls who did not do well in elementary school prefers to make friends with
girls over boys and they prefer to make friends with those who did not do well
in elementary school.

A possible $DD$ matrix to describe such preference pattern is Table 2. Note
that $DD_{i}$ is a function of $X_{i}$ and since Bill and Om share the same
$X$ values, their preference parameters are identical.

∎

Stage II and III: Linearized propensity score and normalization

With those preference parameters and observed characteristics of all students
in a classroom, we are able to calculate the deterministic part of the utility
of student $i$ picking student $j$ as best friend (i.e. linearized propensity
score for student $i$ consider student $j$ to be his best friend),
$\Upsilon_{ij}=\sum_{k=1}^{K}DD_{ik}SS_{jk}=(DD\cdot SS^{\prime})_{ij}$, where
$SS=X$. Utility for student $i$ to consider student $j$ as best friend is
subject to a type I extreme value distribution,
$U_{ij}=\Upsilon_{ij}+\xi_{ij}$, where $\xi_{ij}\overset{iid}{\sim}$ Gumbel
distribution. In our model, student cannot make friend with himself.
Therefore, we set $U_{ii}$ as negative infinity. Since student $i$ picks his
best friend based on who gives him the highest utility, the probability of
student $i$ picking student $j$ as best friend is given by softmax function

$W_{ij}^{*}=\frac{\exp{(\Upsilon_{ij})}}{\sum_{j^{\prime}}^{N}\exp{(\Upsilon_{ij^{\prime}})}},$

where $\Upsilon_{ii}=-\infty$ and $N$ is the number of students in the
classroom where student $i$ is from. $W^{*}$ denotes the probability all
students considering each of their classmate as best friend. We continue
example 3.1 to demonstrate how Stage II and III work.

######  Example 3.2.

We compute the outer product of $DD$ and $SS$ from example 3.1 to compute the
linearized propensity score $\Upsilon_{ij}$.

$\begin{pmatrix}1&0.5\\\ 1&0.5\\\ 0.5&-0.5\\\ -1&0.5\\\
-0.5&0.5\end{pmatrix}\begin{pmatrix}1&1&1&0&0\\\
1&1&0&1&0\end{pmatrix}=\begin{pmatrix}1.5&1.5&1&0.5&0\\\ 1.5&1.5&1&0.5&0\\\
0&0&0.5&-0.5&0\\\ -0.5&-0.5&-1&0.5&0\\\ 0&0&-0.5&0.5&0\\\
\end{pmatrix}=\Upsilon$

We then set the diagonal entries of $\Upsilon$ to be negative infinity such
that the probability of a student making friend with himself is 0. After that,
we apply softmax function to each row of $\Upsilon$.

$\displaystyle\begin{pmatrix}1.5&1.5&1&0.5&0\\\ 1.5&1.5&1&0.5&0\\\
0&0&0.5&-0.5&0\\\ -0.5&-0.5&-1&0.5&0\\\ 0&0&-0.5&0.5&0\\\
\end{pmatrix}\overset{\Upsilon_{ii}=-\infty}{\implies}$
$\displaystyle\begin{pmatrix}-\infty&1.5&1&0.5&0\\\ 1.5&-\infty&1&0.5&0\\\
0&0&-\infty&-0.5&0\\\ -0.5&-0.5&-1&-\infty&0\\\ 0&0&-0.5&0.5&-\infty\\\
\end{pmatrix}$ $\displaystyle\overset{\text{rowwise softmax}}{\implies}$
$\displaystyle\begin{pmatrix}0&0.455&0.276&0.167&0.102\\\
0.455&0&0.276&0.167&0.102\\\ 0.277&0.277&0&0.169&0.277\\\
0.235&0.235&0.143&0&0.387\\\ 0.235&0.235&0.143&0.387&0\\\ \end{pmatrix}=W^{*}$

∎

Stage IV: Simulation-based estimation

CEPS provides aggregated relational data. We design Stage IV in accordance
this data feature. If researchers have network linkage data, this stage can be
simplified.

Given a $N\times N$ matrix $W^{*}$ whose rows are multinomial distributions.
$W_{ij}^{*}$ is the probability of students $i$ considering student $j$ as a
friend. The diagonal of $W^{*}$ is 0. Each row of $W^{*}$ sums to 1. If we
knew the linkage data, we could do a classical maximum likelihood estimation.
In the case of aggregate relational data, we can randomly draw friends for all
students based on $W^{*}$ without replacement. Algorithm 1 describes a typical
drawing without replacement procedure.

Algorithm 1 An algorithm for drawing best friends of student $i$

procedure DF($B_{i},W_{i}^{*}$)

$W_{i}\leftarrow W_{i}^{*}$

$V_{i}\leftarrow\mathbf{0_{N}}$ $\triangleright$ $V_{i}$ is a placeholder for
whom are considered by student $i$ as best friends

for $b\leftarrow 1$ to $B_{i}$ do

Sample $bf$ from $W_{i}$ $\triangleright$ Sample $b^{th}$ best friend for
student $i$

$V_{i}\leftarrow V_{i}+bf$

$W_{i}\leftarrow W_{i}-bf\odot W_{i}$ $\triangleright$ $\odot$ indicates
elementwise multiplication

$W_{i}\leftarrow\frac{W_{i}}{1-bf\cdot W_{i}}$ $\triangleright$ $\cdot$ refers
to matrix product or dot product for vector multiplication

end for

return $V_{i}$

end procedure

$V_{i}$ is an N-dimensional vector, the entry for the drawn friends for
student $i$ will take on a value of 1, the rest of the entries are 0. We have
both aggregate relational data on how many friends of student $i$ engages in
different behaviors such as study hard, have a relationship, skip classes and
etc, $A^{*}$ and whether each classmate engages in different behaviors such as
study hard, have a relationship, skip classes and etc, $Y^{*}$. Therefore, we
can measure how well $W^{*}$ predicts friendship formation by the following
loss function

$\sum_{i}^{N}\sum_{q}^{Q}(A_{iq}^{*}-g(V_{i}\cdot Y_{q}^{*},B_{i}))^{2}$

where $q$ is the question number in section C21 from Figure 1 and function $g$
is defined by Table 3.

| $V_{i}\cdot Y_{q}^{*}=0$ | $V_{i}\cdot Y_{q}^{*}=1$ | $V_{i}\cdot Y_{q}^{*}=2$ | $V_{i}\cdot Y_{q}^{*}=3$ | $V_{i}\cdot Y_{q}^{*}=4$ | $V_{i}\cdot Y_{q}^{*}=5$
---|---|---|---|---|---|---
$B_{i}=1$ | $\\{1\\}$ | $\\{2,3\\}$ | $\times$ | $\times$ | $\times$ | $\times$
$B_{i}=2$ | $\\{1\\}$ | $\\{2\\}$ | $\\{2,3\\}$ | $\times$ | $\times$ | $\times$
$B_{i}=3$ | $\\{1\\}$ | $\\{2\\}$ | $\\{2,3\\}$ | $\\{3\\}$ | $\times$ | $\times$
$B_{i}=4$ | $\\{1\\}$ | $\\{2\\}$ | $\\{2\\}$ | $\\{3\\}$ | $\\{3\\}$ | $\times$
$B_{i}=5$ | $\\{1\\}$ | $\\{2\\}$ | $\\{2\\}$ | $\\{3\\}$ | $\\{3\\}$ | $\\{3\\}$
Table 3: Students’ responses to friendship questionnaire may cause ambiguity.
Given $B_{i}$ and $V_{i}\cdot Y_{q}^{*}$, we cannot uniquely pin down
students’ responses to the friendship questionnaire. We randomly break tie in
those ambiguous cases. Comparing our predicted responses and the true
responses from the student generate the ultimate loss function which completes
the neural network architecture construction. Over 90% of the students report
5 friends, so we do not anticipate much impact from the random tie-breaker.

So far, we describe the loss function for one class. CEPS provides information
about multiple classes, so we should sum up the loss function over all
classes. As a result, we obtain loss function

$\min_{f}\sum_{c}^{C}\sum_{i\in I_{c}}\sum_{q}^{Q}(A_{ciq}^{*}-g(V_{ci}\cdot
Y_{cq}^{*},B_{ci}))^{2}.$

where $c$ is the index for a classroom. $I_{c}$ is the set of students who are
in classroom $c$.

Algorithm 1 corresponds to a micro-founded model where students continuously
pick their best friends. Right before taking the survey, student $i$ decides
that he will write down $B_{i}$ best friends’ names (we do not claim to
provide micro-foundation for this behavior). At the moment of taking the
survey, the student stochastically pick his best friend and write down that
best friend’s name on section C20 of Figure 1. After writing the first name,
the student then randomly draw another best friend, if it repeats with
previous best friend choice, then he discard that draw and draw until he gets
a new best friend. Student $i$ stops this iterative procedure when he writes
down $B_{i}$ best friends’ names.

###  3.2 Instrumental variable peer effect measurement

The conventional econometric model for peer effect measurement is the linear-
in-means model

$y_{ics}=\beta\underbrace{\frac{\sum_{j=1,j\neq
i}^{N_{cs}}z_{jcs}}{N_{cs}-1}}_{\text{Exclusive
means}}+X_{ics}\gamma+\underbrace{\theta_{s}}_{FE}+\underbrace{\mu_{cs}}_{RE}+\epsilon_{ics}$

where $y_{ics}$ is student $i$’s 8th grade cognitive ability in class $c$
school $s$; $z_{jcs}$ is classmate’s 6th grade class rank in class $c$ school
$s$; $X_{ics}$ is additional controls for student $i$ such as gender, age,
ethnic group and parents’ education; $\theta_{s}$ is school fixed effects and
$\mu_{cs}$ is class random effects. We can rewrite linear-in-means model as
the $y_{cs}=\beta
W_{cs}z_{cs}+X_{cs}\gamma+\theta_{s}+\mu_{cs}+\epsilon_{ics}$ where

$W_{cs}=\begin{pmatrix}0&\frac{1}{N_{cs}-1}&\frac{1}{N_{cs}-1}&\dots&\dots&\frac{1}{N_{cs}-1}\\\
\frac{1}{N_{cs}-1}&0&\frac{1}{N_{cs}-1}&\dots&\dots&\frac{1}{N_{cs}-1}\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
\frac{1}{N_{cs}-1}&\frac{1}{N_{cs}-1}&\frac{1}{N_{cs}-1}&\dots&\dots&0\end{pmatrix}.$

Linear-in-means model implicitly assumes that all students exert the same
amount of peer influence to all their classmates as evidenced by $W$ matrix.
In our model, we assume that momentarily, a student receives peer effect
exclusively from his chosen best friend. Therefore, the peer effect a student
receives from one of his classmate is proportional to the probability of him
considering that classmate to be his best friend. We replace the linear-in-
means assumption with a friendship-weighted assumption (i.e. replace $W$ with
$W^{*}$ obtained from the neural network model as shown in Eq(2)).

However, we face endogeneity issue with $W^{*}$ matrix. We illustrate the
issue with endogeneity with example 3.3.

######  Example 3.3.

Students may choose to make friends with those of similar IQ as themselves. IQ
also has an impact on students’ 8th grade cognitive ability test score. Then,
if we run regression based on Eq(2), we may overestimate peer effect $\beta$.

∎

We resort to classical instrumental variable approach to address the
endogeneity. The instrument that we use is the average 6th grade class rank
for classmates that are randomly assigned to the students. A valid instrument
needs to fulfill two properties: exclusion constraint and relevance
constraint. We continue example 3.3 to illustrate why both properties are
satisfied.

######  Example 3.4.

We continue with example 3.3 where IQ is an endogenous variable.

Exclusion constraint requires the instrument (average 6th grade class rank for
all classmates) to be uncorrelated with the unobserved confounder IQ. This
requirement is trivially satisfied since all classroom assignment is random
conditional on school choice which is a covariate included in Eq(2).

Relevance constraint requires the instrument and the endogenous variable to be
correlated. Both average 6th grade class rank ($W_{cs}z_{cs}$) and friends’
weighted average 6th grade class rank ($W_{cs}^{*}z_{cs}$) are dependent on
all classmates’ 6th grade class rank ($z_{cs}$). Therefore, the relevance
constraint is satisfied. ∎

$\displaystyle\underbrace{{\sum_{j=1}^{N_{cs}}W_{ij}^{*}z_{jcs}}}_{\text{Friends'
average 6th grade ranking}}$
$\displaystyle=\pi_{1}\underbrace{\frac{\sum_{j=1,j\neq
i}^{N_{cs}}z_{jcs}}{N_{cs}-1}}_{\text{Classmates' average 6th grade
ranking}}+X_{ics}\delta+\phi_{s}+\eta_{ics}$ (1)
$\displaystyle\underbrace{y_{ics}}_{\text{Student i's 8th grade cognitive test
score}}$
$\displaystyle=\beta{\sum_{j=1}^{N_{cs}}W_{ij}^{*}z_{jcs}}+X_{ics}\gamma+\underbrace{\theta_{s}}_{FE}+\underbrace{\mu_{cs}}_{RE}+\epsilon_{ics}$
(2)

We estimate Eq(2) and Eq(1) with two-stage least square estimator.

###  3.3 Counterfactual analysis of class assignment

With the friendship formation neural network model and instrumental variable
estimate, we are able to simulate different class assignment policy and
predict all students’ peer effect. There are two possible scenarios to select
the best class assignment policy.

####  3.3.1 Finite number of possibilities

If a principal is contemplating a finite number of possible class assignment
policies, then he can use our model to predict peer effect for all students
and choose the one among the finitely many candidate policies that optimizes
the objective function. We adopt this approach to optimize class assignment
policy.

####  3.3.2 Combinatorial optimization

Class assignment is a combinatorial optimization problem. In principle, there
are finitely many possible class assignment policies and therefore, we can use
the approach described in section 3.3.1 to pick the best class assignment
policy. However, the approach is computationally infeasible. We use example
3.5 to illustrate this problem.

######  Example 3.5.

A principal needs to divide 100 students into two classrooms, each consisting
of 50 students. There are $\frac{100!}{50!}$ ways to assign students into
classrooms. In the context of Chinese secondary school education, most
principals face a much larger student population than 100 and need to divide
the students into more than 2 classrooms. Therefore, the brute force
enumeration method does not work. ∎

We resort to heuristic optimization methods such as genetic algorithm and
simulated annealing.

##  4 Results

###  4.1 Friendship formation pattern

Our friendship formation neural network model finds that gender is the single
dominant factor in terms of predicting friendship formation. Boys almost
exclusively make friends with boys and girls almost exclusively make friends
with girls.

(a) Heat map of $W^{*}$ for classroom 1

(b) Heat map of $W^{*}$ for classroom 235

Figure 3: Heat maps for two classrooms. Classroom 1 is the first class in our
dataset, classroom 235 is the biggest class in our dataset.

Figure 3 plot the heat of matrix $W^{*}$ for two classrooms. In Figure 3,
darker color indicates higher probability of student $i$ considering student
$j$ as his best friend. $(i,j)$ are the row and column number of the heat map
(matrix) respectively. The students are group into two gender groups first,
and within each group, boys are ordered from the best student to the to the
worst student in terms of 6th grade class rank; girls are ordered from the
worst student to the best student in terms of 6th grade class rank. Therefore,
the best students are placed at the two ends of the heat map and the worst
students are placed in the middle.

Two patterns emerge. (1) As mentioned previously, gender is the most powerful
predictor for friendship formation. There is a very clear discontinuity in the
color patterns in both Figures 3(a) and 3(b). The discontinuity happens
exactly as we move from the worst boy student to the worst girl student (both
in rows and in columns). This indicates that the probability for a boy to
consider a girl to be his best friend is much lower than than him considering
a boy as his best friend and vice versa. This aligns with the cultural context
in Chinese secondary school education where teenage romantic relationship is
high stigmatized and discouraged by parents and educators. (2) Students who
did well in elementary school are more popular than those who did not do well.
This is evidenced by darker color of the columns at left and right sides of
the heat map. This observation suggests a gender-based classroom assignment
policy. We compare this policy with the existing random class assignment
policy in section 4.3.

We also compare $W^{*}$ with $W$ in terms of predicting friendship formation.
In CEPS data, there are 1789 students that are not randomly assignment to 40
classrooms. We use them as test data. Note that there is a data shift as the
students in the test set are no longer randomly assigned to classrooms.
Nevertheless, using these 1789 students as test data is still fitting for our
problem because the ultimate goal for our research is to devise optimal class
assignment, which would be a nonrandom class assignment policy.

Figure 4: Our model is able to predict friendship formation more accurately
than the classical linear-in-means model commonly adopted in the peer effect
literature.

Figure 4 shows that our model is able to predict the gender pattern in
friendship formation much better than the linear-in-means model. However, for
the other questions in Figure 1, our model can only perform comparably as the
linear-in-means. One possible explanation is that while $A^{*}$ is student
$i$’s perception of his own best friends, $Y^{*}$ is classmates’ self-
perception. These two perceptions may not align with each other for question
like “How many of your best friends are in a romantic relationship?”. Student
$i$ might consider his best student $j$ to not be in a relationship whereas
student $j$ is secretly in a romantic relationship since teenage romance is
highly stigmatized in China. Nevertheless, the large gain in prediction
accuracy for gender pattern in friendship formation is a strong argument for
researchers to consider friendship formation when it comes to measuring peer
effect.

###  4.2 Positive peer effect from “good students”

We estimate Eq(1) and confirm the presence of positive peer effect from “good
stundet” (students who did well in elementary school). In particular, if
student $i$’s best friends’ weighted average 6th grade class quantile improves
by 10%, his 8th grade cognitive ability test score improves by 0.1253 on a
scale of 5.

Table 4: Holding everything else fixed, when a student’s friends’ 6th grade
class rank improves by 10%, the student’s cognitive ability test score in 8th
grade improves by 0.1253 point (on a scale of 5 total points).

###  4.3 Some discussions on “optimal” class assignment

The positive peer effect from good student informs the principals that in
order to maximize peer effect, he should place those students who are more
likely to make friends with a particular good student in the same classroom as
that good student. In other words, the principal should design class
assignment policy that maximizes the probability of other students considering
“good students” as best friends.

####  4.3.1 Gender-based class assignment

There are two major findings from friendship formation model (1) gender is a
dominant factor in predicting friendship (2) good students are more
influential than bad students. These two observations suggest a gender-based
class assignment can potentially maximize peer effect from good students. We
run our simulation two classrooms from the same school (classroom 234 and
235). We divide the students from these two classrooms into all-boy classroom
and all-girl classroom, predict their friendship formation $\hat{W}^{*}$,
calculate their realized peer effect $\hat{\beta}\hat{W}^{*}z$.

(a) After being assigned to all-boy classes, average boy students’ realized
peer effect improves from 0.9086 to 0.9285.

(b) After being assigned to all-girl classes, average girl students’ realized
peer effect improves from 0.9446 to 0.9646.

The results align with our expectation. In Figure 5(a), we plot histogram of
the realized peer effect for boys. The blue bars indicate the realized peer
effect for boys when they are randomly assigned to classrooms; the pink bars
indicate the realized peer effect for all boys when they are assigned to a
all-boy classroom. Their average peer effect improves from 0.9086 to 0.9285
when they are moved to a all-boy classroom. We plot the same histogram for
girls in Figure 5(b). Girls’ average peer effect improves from 0.9446 to
0.9646 when they are moved to a all-girl classroom.

####  4.3.2 Extreme mixing class assignment

Figure 6: We divide students into three groups. Top quartile, middle 50%,
bottom quartile in terms of 6th grade classroom quantile and put the top and
bottom quartile students into one class. The simulation setup is similar to
the experimental setup in Carrell et al. (2013). Bottom quartile students’
average realized peer effect improves from 0.7055 to 0.7842.

We mix the the top and bottom quartile students in terms of their 6th grade
class rank into one classroom. The model predicts that the realized peer
effect for the bottom quartile students improves from from 0.7055 to 0.7842.
We plot the realized peer effect for bottom quartile students in Figure 6.
Though such a classroom assignment policy does hurt top quartile students as
their realized peer effect drop, this type of extreme mixing class assignment
policy might be appealing to principals who are particularly concerned about
educational outcome equity. Alternatively, this policy can be used as an
initial point for heuristic optimization. For example, we can use genetic
algorithm and swap students until the class assignment policy no longer hurts
good students in hope that the resulting policy can still benefit bottom
quartile students substantially.

##  5 Conclusion

This paper builds a friendship formation prediction model and estimates peer
effect using a modified version of the linear-in-means model. From the results
of the friendship formation prediction and peer effect parameter estimates, we
are able to simulate counterfactual peer effect for all students. We compare
some class assignment policies to random assignment and find out that gender-
based class assignment helps improve average peer effect and extreme mixing
class assignment helps with bottom quartile students.

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