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Uniform Inference on Quantile Effects under Network Interference
\title{Uniform Inference on Quantile Effects under Network Interference\footnote{
We thank Jianfei Cao, Jizhou Liu, Yuya Sasaki, and Yichong Zhang for their helpful comments and discussions.}}
\author{
Zequn Jin\thanks{School of Economics, Shanghai University of Finance and Economics. Emails: [email removed]}; [email removed]}.}
\and
Gaoqian Xu\thanks{Tippie College of Business, University of Iowa. Email: [email removed]}.}
\and
Zixin Yang\thanks{School of Statistics and Data Science, Shanghai University of Finance and Economics. Email: [email removed]}.}
\and
Zhengyu Zhang\footnotemark[2]
}
\date{\today}
\maketitle
\vspace{0.5cm}
\begin{abstract}
This paper studies quantile treatment and spillover effects in network experiments. Average spillover effects reveal how treating a unit’s neighbors affects its outcome on average, but mask the heterogeneity of these effects across the outcome distribution. We define structural quantile effects that compare outcome quantiles between exposure states, characterizing how own treatment and exposure to treated neighbors affect different parts of the outcome distribution. Building on \citet{leung2020treatment}, we first establish the weak convergence of the estimated quantile-effect process under conditions requiring the stabilization of the degree distribution and the network-dependent covariance structure. Our main contribution is to propose uniform confidence bands (UCBs) based on Gaussian approximations conditional on the realized network, avoiding these stabilization requirements. The proposed method is evaluated through extensive
simulation studies and an empirical application to a randomized savings-account experiment in Nepal \citep{prina2015banking}.
\end{abstract}
\vspace{12pt}
\textit{JEL codes}: C14, C31, C54
\vspace{3pt}
\textbf{Key Words}: Network data, quantile effects, Gaussian approximation, uniform confidence bands.
\newpage
\section{Introduction}
Network experiments have become increasingly prevalent in economics and the social sciences, with applications to information diffusion, technology adoption, microfinance, school-based social interventions, and input-subsidy programs \citep{BandieraRasul2006, BakshyEtAl2012, BanerjeeEtAl2013, cai2015social, PaluckEtAl2016, carter2021subsidies}. In such experiments, an individual’s outcome may depend not only on their own treatment assignment but also on the assignments of other units in the network, thereby violating the stable unit treatment value assumption (SUTVA). A growing literature has therefore developed methods for causal inference under network interference, typically by summarizing the high-dimensional assignment vector through low-dimensional exposure mappings such as own treatment status and the fraction of treated neighbors \citep{manski2013identification,aronow2017estimating,leung2020treatment,leung2022causal}.
To our knowledge, most existing work focuses on average treatment and spillover effects. Yet average effects may mask economically important heterogeneity across the outcome distribution. For example, \citet{carter2021subsidies} find positive network spillover effects of agricultural input subsidies on maize yields in Mozambique, where maize is a major source of household income. However, their average-effect analysis does not reveal how these gains are distributed. In particular, it does not reveal whether farmers at the lower tail of the yield distribution benefit from technology diffusion through social networks. Motivated by this question, this paper studies quantile treatment and spillover effects under network interference and develops uniform inference for the entire quantile-effect curve.
Building on the structural quantile model of \cite{matzkin2003nonparametric} and \cite{chernozhukov2005iv}, we assume that an individual’s potential outcome depends on their own treatment, the number of treated neighbors, the number of friends (node degree), and unobserved characteristics. Our parameters of interest are comparisons of quantile response functions between exposure states at each quantile level. These comparisons capture how variations in own treatment or friend exposure change the entire distribution of economic outcomes. In an experimental setting, the quantile response functions can be identified by the conditional quantile of observed outcomes among units with the same own treatment status, number of treated friends, and node degree, following \citet{leung2020treatment}.
The identification leads to a simple estimator based on conditional quantile regression (QR). The inferential targets are quantile effects at different quantile levels, so valid inference requires uniform control over the quantile effect curve. This is a nontrivial extension of large-network inference for average treatment and spillover effects in \cite{leung2020treatment}. Moreover, relative to the existing literature on quantile treatment effect (QTE), our inference problem is nonstandard for two reasons. First, the conditioning variables in our QR are network-dependent. For example, the number of treated friends is tied to node degree and mechanically depends on neighboring treatment assignments. Second, unobservable characteristics may exhibit network dependence, since linked individuals or individuals with common neighbors may share common information and local shocks. Therefore, standard weighted or multiplier bootstrap procedures developed for QR and QTE inference \citep{belloni2019conditional,qu2019uniform,chiang2019robust,zhang2020quantile} do not directly apply here without modification.
A natural route for asymptotic inference is to embed the observed network in a growing sequence of graphs and derive a limit distribution for the estimator along that sequence \citep[e.g.,][]{leung2020treatment,leung2022causal,kojevnikov2021limit}. To obtain such a limit, this route requires the graph sequence and the induced dependence to stabilize asymptotically. For example, \citet{leung2020treatment} assumes convergence of the empirical node-degree distribution and a probability limit for the network-dependent asymptotic variance.\footnote{Assumption 2 in \citet{leung2020treatment} imposes convergence of the degree distribution, and Assumption 6 assumes the existence of a probability limit for the network-dependent asymptotic variance.} He estimates the average treatment and spillover effects in a single large network, establishes their asymptotic normality, and conducts inference using consistent estimators of the asymptotic variance that account for network dependence. As a benchmark, \cref{subsec:nonparametric-QTE} follows this route and extends it to quantile effects under network interference, establishing the weak convergence of quantile effect estimators. However, this approach requires convergence of the degree distribution and the network-dependent covariance structure, both of which are difficult to verify from a single observed network.
Our main contribution is the construction of uniform confidence bands (UCBs) for quantile effects under network interference, using Gaussian approximations conditional on the realized network. A key advantage of this approach is that the procedure does not require the QR process to converge to a fixed Gaussian process, nor does it require the studentized supremum statistic to have a fixed limiting distribution. To compute the critical values for the UCBs, we adopt the Gaussian approximation framework of \citet{chernozhukov2013gaussian,chernozhukov2014anti,chernozhukov2014gaussian} to approximate the distribution of the supremum of the $t$-statistic process under network dependence. Through the Bahadur linearization of the QR estimator, this problem reduces to approximating the distribution of the maximum of a high-dimensional vector formed by sums of network-dependent random variables \citep{fang2021high,chang2024central}. We then simulate the corresponding Gaussian maximum using the estimated network-conditional covariance structure and use its empirical quantiles as critical values.
We assess the finite-sample performance of the proposed UCBs through Monte Carlo simulations. The results show that the bands achieve coverage probabilities close to the nominal levels, with coverage improving as the sample size increases.
\subsection*{Related Literature}
First, this paper contributes to the literature on causal inference under interference. One common framework is partial interference, which assumes that the sample consists of many independent clusters and rules out interference across clusters; see \cite{hudgens2008toward,liu2016inverse,liu2019doubly} and \cite{qu2026semiparametric}. Within this framework, \cite{cheng2026nonparametric} study network quantile causal effects and develop efficient inference. However, any empirical settingsinvolve a single connected network rather than independent clusters. In such settings, researchers often impose a low-dimensional exposure mapping, under which potential outcomes depend on the high-dimensional treatment assignment vector only through exposure states such as own treatment status and the fraction of treated neighbors; see \cite{manski2013identification,aronow2017estimating,leung2020treatment,li2022random} and \cite{vazquez2023identification}. Existing work in this setting has primarily focused on average treatment and spillover effects. To our knowledge, this is the first study of quantile effects in a single large network.
Among the existing literature, our paper is closely related to \cite{leung2020treatment}. This paper starts from an average structural function (ASF), which describes the mean potential outcome under a counterfactual treatment status and number of treated neighbors. The ASF is identified using network experiments, and average exposure effects are defined as differences between ASFs evaluated at distinct exposure states. Our paper follows this structural perspective, which differs from design-based approaches to exposure effects \citep{leung2022causal,gao2025causal}.
Second, our paper contributes and builds on the large literature on quantile treatment effects (QTE) and quantile regression (QR), where QR serves as the main estimation and inference tool for QTE \citep{koenker2002inference,angrist2006quantile,belloni2019conditional}. QTEs have been studied extensively under the SUTVA framework in both experimental and observational settings \citep{chernozhukov2005iv,chernozhukov2006instrumental,chernozhukov2008instrumental,firpo2007efficient}. Recent work develops uniform inference for QTEs under a variety of (quasi-)experimental designs without interference \citep{chiang2019robust,qu2019uniform,zhang2020quantile,hou2026nonparametric}. We extend this line of work to network experiments.
Third, our paper builds on a seminal line of work on Gaussian approximation \citep{chernozhukov2013gaussian,chernozhukov2014anti,chernozhukov2014gaussian,chernozhukov2017central}. This literature provides Gaussian approximations to the laws of maxima of sums of high-dimensional vectors, without the existence of a fixed limit distribution. While \Cref{subsec:nonparametric-QTE} establishes weak convergence of the quantile-effect estimators by embedding the observed network in a growing network sequence, that route requires the covariance kernel, which depends on the underlying graph sequence, to converge to a fixed limit uniformly over all quantile levels. This motivates the uniform inference based on Gaussian approximation in \Cref{sec:dependent bootstrap}, which avoids characterizing a limit covariance kernel or an explicit limit distribution for the supremum statistic. To accommodate network dependence, we draw on recent Gaussian approximation results for dependent high-dimensional data \citep{zhang2017gaussian,zhang2018gaussian,fang2021high,chang2024central}.
\paragraph{Organization of the paper.}
The remainder of the paper is organized as follows. \Cref{sec:framework} introduces the econometric model, defines the structural quantile effects, and establishes their identification in randomized network experiments. \Cref{subsec:nonparametric-QTE} proposes the QR estimator and establishes the weak convergence of the estimated quantile-effect process under conventional large-network asymptotics, which require the stabilization of the degree distribution and the network-dependent covariance structure. \Cref{sec:dependent bootstrap} develops UCBs based on Gaussian approximations conditional on the realized network and establishes their asymptotic validity without requiring either of these stabilization conditions. Finally, \Cref{section: Numerical_Examples} reports simulation results and presents an empirical application to the randomized savings-account experiment in Nepal \citep{prina2015banking}.
\paragraph{Notation.}
For any $n\in\mathbb N^+$, let $[n]\equiv\{1,\ldots,n\}$.
We use
$\mathds{1}\{\cdot\}$ to denote the indicator function.
For $a,b\in\mathbb R$, let
$a\vee b=\max\{a,b\}$ and $a\wedge b=\min\{a,b\}$.
We use $\operatorname{Bern}(p)$ denote the Bernoulli distribution with $p \in (0,1)$.
For a matrix $\boldsymbol{Q}$, define
$\|\boldsymbol{Q}\|_{\max}=\max_{j,k}|\boldsymbol{Q}_{jk}|$, and let
$\|\boldsymbol{Q}\|_{\mathrm{op}}$ denote its operator norm.
We write $\boldsymbol{Q}\succeq0$ when $\boldsymbol{Q}$ is symmetric and positive semidefinite. For a network with adjacency matrix $\boldsymbol A_n \in \mathbb{R}^{n \times n}$, let
$\ell_{\boldsymbol A_n}(i,j)$ denote the shortest path
between $i$ and $j$, with
$\ell_{\boldsymbol A_n}(i,i)=0$.
We write
$\mathbb P_{\boldsymbol A_n}(\cdot)
=\mathbb P(\cdot|\boldsymbol A_n)$,
and use $\mathbb E_{\boldsymbol A_n}$, $\operatorname{Var}_{\boldsymbol A_n}$ and
$\operatorname{Cov}_{\boldsymbol A_n}$ analogously. The abbreviations i.i.d.\ and PSD stand for ``independent and
identically distributed'' and ``positive semidefinite,'' respectively.
\section{Econometric Model}\label{sec:framework}
We consider a randomized controlled trial (RCT) with $n$ units. The population of $n$ units is embedded in a network, characterized by a symmetric adjacency matrix $ \boldsymbol{A}_n \in \{0,1\}^{n \times n}$, where $A_{ij}=1$ indicates a link between units $i$ and $j$. For each unit $i \in [n]$, let $Y_i \in \mathbb{R}$ denote the observed post-treatment outcome, and $D_i \in \{0,1\}$ denote the treatment assignment, where $D_i \sim \mathrm{Bern}(p)$ for some $p \in (0,1)$. The vector of treatment assignments is denoted by $\boldsymbol{D}_n \equiv (D_i)_{i=1}^n \in \{0,1\}^n$. Under the Neyman-Rubin causal model \citep{imbens2015causal}, the potential outcome for unit $i$, denoted by $Y_i(\boldsymbol{d})$, depends on the entire treatment assignment vector $\boldsymbol{d} = (d_i)_{i=1}^n \in \{0,1\}^n$.
\subsection{Main Specification}\label{section: structural_quantile_3odel}
To restrict the dimensionality of arbitrary interference, we assume local interference via exposure mappings \citep{manski2013identification, aronow2017estimating}. Specifically, we assume that a unit's outcome is determined by its own treatment status and those of its neighbors. Let $|N_i| = \sum_{j=1}^n A_{ij}$ denote the degree of unit $i$, and $T_i(\boldsymbol{d}, \boldsymbol{A}_n) = \sum_{j=1}^n A_{ij} d_j$ represent the number of its treated neighbors under the treatment assignment $\boldsymbol{d}$. Moreover, let $W_i(\boldsymbol{d}, \boldsymbol{A}_n) = \left( d_i, T_i(\boldsymbol{d}, \boldsymbol{A}_n), |N_i| \right)$. When the context is clear, we may omit the explicit dependence on $\boldsymbol{A}_n$ and simply write $T_i(\boldsymbol{d})$ and $W_i(\boldsymbol{d})$.
\begin{assumption}\label{assumption:Interference}
For each unit $i \in [n]$, the potential outcome $Y_i(\boldsymbol{d})$ is determined by
\begin{equation}\label{eq: Interference}
Y_i(\boldsymbol{d}) = q \left( W_i(\boldsymbol{d},\boldsymbol{A}_n), U_i \right) = q \left(d_i, T_i(\boldsymbol{d}), |N_i|, U_i \right),
\end{equation}
where $q\left(\cdot\right)$ is an unknown function, and $U_i \in (0,1)$ captures unobserved heterogeneity.
\end{assumption}
\cref{assumption:Interference} imposes a structural restriction. We implicitly rule out treatment-induced changes in the network. For the fixed pre-treatment network, the potential outcome
\(Y_i(\boldsymbol d)\) depends on \(\boldsymbol d\)
only through $d_i$ and $T_i(\boldsymbol{d})$.
The degree \(|N_i|\) is not itself a causal treatment. It is instead viewed as a
conditioning/control variable, as the intervention does not change the interference network. By consistency, the observed outcome is $Y_i = q(D_i, T_i, |N_i|, U_i)$, where $T_i = \sum_{j=1}^n {A}_{ij} D_j$ is the number of treated neighbors. Following \cite{manski1993identification,manski2013identification} and \cite{leung2020treatment}, we refer to $(D_i, T_i)$ as the effective treatment. To interpret $q(\cdot)$ as a
structural quantile function (SQF), we impose conditions under which \(U_i\) can be normalized as a latent rank in the degree-conditional outcome distribution.
{assumption:Interference}{assumption:normal}
\begin{assumption}\label{assumption:normal}
The function $q: \{0,1\} \times \mathbb{N}^2 \times (0,1) \rightarrow \mathbb{R}$ and the unobservables \(\{U_i\}_{i=1}^n\) satisfy the following conditions:
\begin{assumpenum}
\item\label{assumption:nor} Conditional on $|N_i|$, the unobservable $U_{i}$ are uniformly distributed, i.e.,
$ U_i\mid |N_i| \sim \mathrm{Unif}(0,1)$.
\item\label{assumption:mono} For any $w \equiv (d, t, l) \in \{0,1\} \times \mathbb{N}^2$ with $l \geq t$, the function
\(\tau \mapsto q(w,\tau)\) is continuous and strictly increasing.
\end{assumpenum}
\end{assumption}
\cref{assumption:nor} is a natural normalization in view of the Skorohod
representation \citep{chernozhukov2005iv,chernozhukov2006instrumental}.
In randomized network experiments, the treatment assignment changes a unit's own treatment
status and the number of treated neighbors, but not the interference network.
Thus, \(U_i\) can be viewed as a latent rank determined by unobserved
factors that do not vary across counterfactual treatment assignments. Meanwhile, these underlying unobserved factors may be related to network
position, which motivates normalizing the latent rank conditional on degree
\(|N_i|\). \cref{assumption:mono} imposes monotonicity and
continuity of the function $q(w, \tau)$ in $\tau$. Hence, \cref{assumption:normal} implies
that, conditional on degree \(l\), the potential outcome distribution under the effective treatment $(d, t) $ is characterized by \(q(w,\tau)\), where $w = (d,t,l)$. In particular, \(q(w,\tau)\)
is the \(\tau\)-quantile of the potential outcome for units with degree \(l\)
when the effective treatment is $(d,t)$.
\begin{remark}
Assumptions \ref{assumption:Interference} and \ref{assumption:normal} essentially assume a correctly specified exposure mapping and rank invariance \citep{doksum1974empirical,heckman1997making}. The latter can be relaxed to rank similarity without affecting the results of this paper \citep{chernozhukov2005iv,torgovitsky2015identification}.\footnote{Under rank similarity, the analogous versions of Assumptions \ref{assumption:RCT}, \ref{assumption:CE}, and \ref{assumption:local dependency} can be imposed on the family of exposure-specific ranks \(\left\{U_i(w): w\in\mathcal{W}\right\}\), where \(\mathcal{W}\) denotes the set of feasible exposure states.} To formalize this relaxation, retain the assumption that $Y_i(\boldsymbol d)=Y_i(\boldsymbol d')$ whenever $W_i(\boldsymbol d)=W_i(\boldsymbol d')$ for all $i$ and $\boldsymbol d,\boldsymbol d'\in\{0,1\}^n$. Rank similarity allows the latent rank \(U_i(w)\) to vary across exposure states while preserving its marginal distribution. Accordingly, conditional on $W_i(\boldsymbol{d}) = w$, the potential outcome $Y_i(\boldsymbol{d})$ admits the following representation:
\[
Y_i(\boldsymbol d) =q\left(w,U_i(w)\right), \quad \text{where} \quad U_i(w)\sim\mathrm{Unif}\left(0,1\right).
\]
\end{remark}
\subsection{Parameters of Interest}
Building on the structural quantile model under network interference introduced in \cref{section: structural_quantile_3odel}, our main object of interest is the structural quantile effect. Specifically, conditional on the degree, we aim to measure how the quantiles of potential outcomes respond to counterfactual manipulations of the effective treatment.
For two exposure states $w=(d,t,l)$ and $w^\prime=(d^\prime,t^\prime,l)$ with $t,t^\prime \leq l$, we define the structural quantile effect at level $\tau \in (0,1)$ as:
\begin{equation}\label{eq: def_SQE}
q(w,\tau) - q(w^\prime, \tau),
\end{equation}
where the SQF $q(\cdot)$ is defined in \cref{assumption:Interference}. Throughout this paper, the quantile effects compare only exposure states with the same degree, which avoids conflating exposure effects with degree-related heterogeneity.
Interest often centers on quantile treatment and spillover effects. First, the quantile direct
effect measures the effect of changing a unit's own treatment status while
holding fixed the number of treated neighbors and the degree:
\[
q \left(1, t, l, \tau \right) - q \left(0, t, l, \tau \right).
\]
Second, the quantile spillover effect measures the effect of changing the
number of treated neighbors while holding fixed the unit's own treatment
status and degree:
\[
q\left(d, t, l, \tau\right) - q\left(d, t^\prime, l, \tau \right) .
\]
By varying \(\tau\), these estimands describe how direct treatment and
spillover effects differ across the outcome distribution.
\subsection{Identification}\label{subsection:identification}
When treatments are completely randomized,
structural quantile effects can be easily
identified from network experiment data. Since identifying these effects is a direct implication of identifying the SQF, we focus on the latter in this subsection. We first require that the treatment vector $\boldsymbol{D}_n$ is independent of both the interference network $\boldsymbol{A}_{n}$ and the latent ranks $\boldsymbol{U}_{n} \equiv (U_i)_{i=1}^n$.
\begin{assumption}\label{assumption:RCT}
The treatment assignments $D_i \overset{\mathrm{i.i.d.}}{\sim} \mathrm{Bern}(p)$ for a fixed $p \in \left(0,1\right)$. Moreover, $\boldsymbol{D}_n$ is independent of both $\boldsymbol{A}_{n}$ and $\boldsymbol{U}_{n}$, that is, $\boldsymbol{D}_n \ \rotatebox[origin=c]{90}{$\models$} \left(\boldsymbol{A}_{n}, \boldsymbol{U}_{n}\right)$.
\end{assumption}
\cref{assumption:RCT} formalizes the Bernoulli randomized network
experiment \citep{cai2015social,prina2015banking,carter2021subsidies}.
The network is treated as a pre-treatment object, and the treatment assignment
vector is assumed to be independent of both the interference network and the
latent individual ranks. As a result, given the pre-treatment network,
variation in a unit's own treatment status and in the number of treated neighbors
is generated solely by the randomized assignment. In particular, the probability
that a unit realizes a specific effective treatment is known from the experimental
design.
\cref{assumption:RCT} rules out selection into treatment, but still allows
latent ranks to be correlated with the pre-treatment network. To allow for such
network-related unobserved heterogeneity while preserving identification, we
impose the following conditional exogeneity condition.
\begin{assumption}\label{assumption:CE}
For any unit $i\in [n]$, $U_{i} \ \rotatebox[origin=c]{90}{$\models$} \ \boldsymbol{A}_{n}\mid \left|N_i\right|$.
\end{assumption}
\cref{assumption:CE} allows latent ranks to be related to network position
through degree, but requires that, conditional on degree, the network carries no further information about \(U_i\). For example, if
\(U_i\) captures unobserved sociability or communication skill, more sociable
individuals may have more friends, so \(U_i\) may be correlated with
\(|N_i|\). The assumption permits this dependence through the number of friends,
while ruling out additional dependence on who those friends are, conditional on
the number of friends. Together, \cref{assumption:RCT,assumption:CE} imply that
$U_i \ \rotatebox[origin=c]{90}{$\models$} \left(D_i,T_i\right) \mid |N_i|$,
which is the key condition for identifying the SQFs
from observed conditional outcome distributions.
For a prespecified exposure state $w=(d,t,l)$, the SQF $q(w,\tau)$ can be
identified only if the observed network contains units with degree $l$. We therefore impose the following degree support condition.
\begin{assumption}\label{assumption:overlap} For the prespecified common degree $l \in \mathbb{N}$,
there is a constant $c_l >0$ such that
\[
\liminf_{n\rightarrow \infty} \ \frac{1}{n} \sum_{i=1}^n \mathds{1}\left\{ |N_i| = l \right \} > c_{l} , \quad \text{a.s.}
\]
\end{assumption}
Under \cref{assumption:overlap}, once the network contains units with
degree $l$, the Bernoulli design assigns positive probability to every feasible
effective treatment $(d,t)$ with $d\in\{0,1\}$ and $0\leq t\leq l$.
\begin{proposition}\label{prop:id}
Suppose \cref{assumption:Interference,assumption:normal,assumption:RCT,assumption:CE}
hold. For all $\tau \in (0,1)$ and any feasible exposure state $w=(d,t,l)$ whose degree $l$ satisfies
\cref{assumption:overlap}, the SQF $q(w,\tau)$ is identified by
\[
\mathbb P\left[Y_i\le q(w,\tau)\mid W_i=w\right]=\tau.
\]
\end{proposition}
\section{Estimation and Asymptotic Theory}\label{subsec:nonparametric-QTE}
This section studies estimation of the structural quantile effects
over a range of quantile levels. Let $\mathcal T$ be a compact subset of $(0,1)$,
and fix two feasible exposure states \(w\) and \(w'\) with the same degree.
Recall from \cref{eq: def_SQE} that the quantile-effect function is defined as
\begin{equation}\label{eq: QTE_Def}
q(\tau) \equiv q\left(w,\tau\right)-q\left(w',\tau\right), \quad \tau\in\mathcal T .
\end{equation}
We propose to estimate the SQF by a conditional quantile regression. Let $\mathds{1}_i(w^\dagger )=\mathds{1}\{W_i=w^\dagger \}$ where $W_i \equiv (D_i, T_i, |N_i|)$, and let the check function be given by
$\rho_\tau(u)=u\left(\tau-\mathds{1}\{u<0\}\right)$. Following \cref{prop:id}, we estimate $q_w(\tau) \equiv q(w,\tau)$ by
\begin{equation}\label{equation: Normalized_estimator}
\widehat{q}_w(\tau) = \mathop{\operatorname{argmin}}_{q \in \mathbb{R} } \sum_{i=1}^n \mathds{1}_i (w) \rho_\tau( Y_i - q ) .
\end{equation}
Similarly, $q_{w^\prime}(\tau) \equiv q \left(w^\prime,\tau\right)$ can be estimated by $\widehat{q}_{w^\prime}(\tau)$. Recall $q(\tau) = q_w( \tau ) - q_{w^\prime}( \tau )$, and we then estimate $q(\tau)$ by
\[
\widehat{q}\left( \tau \right) = \widehat{q}_w( \tau ) - \widehat{q}_{w^\prime}( \tau ).
\]
The asymptotic analysis for $\widehat{q}\left( \tau \right)$ is nonstandard because effective treatments are
network-induced and latent ranks may be locally dependent. We therefore derive the
limit distribution of the estimated quantile-effect process under local
dependence of latent ranks, stabilization of the empirical degree distribution,
and high-level smoothness and
limit conditions stated below.
\subsection{Bahadur Representation}
Throughout the remainder of this paper, we maintain the assumptions in
\cref{sec:framework}, including \cref{assumption:overlap} for the
exposure states \(w\) and \(w'\). We impose the following regularity
conditions.
\begin{assumption}\label{ass:degree}
There is a universal integer $K_{\max} > 0$ such that $ \max_{1\le i\le n} |N_i| \leq K_{\max}$, almost surely.
\end{assumption}
Under the assumptions in \cref{sec:framework}, the conditional distribution
of \(Y_i\) given \(W_i=w^\dagger\) is common across units \(i\). For
\(w^\dagger\in \left\{w,w' \right\}\), let \(F_{w^\dagger}(\cdot)\) and
\(f_{w^\dagger}(\cdot)\) denote this common conditional distribution function
and density, respectively.
\begin{assumption} \label{assumption: density}
For each $w^\dagger \in \{w, w^\prime\}$, the conditional density $f_{w^\dagger}(\cdot)$ is bounded, bounded away from zero, and Lipschitz continuous on an open neighborhood of $\{ q_{w^\dagger}(\tau) : \tau \in \mathcal{T} \}$.
\end{assumption}
To accommodate network-correlated unobservables while keeping the dependence tractable, we allow latent ranks to exhibit local network dependence,
as in \citet{leung2020treatment} and \citet{viviano2025policy}.
\begin{assumption}\label{assumption:local dependency}
Conditional on the network $\boldsymbol{A}_n$, the sets of random variables
$\left\{U_i : i \in B_1 \right\}$ and $\left\{U_j : j \in B_2 \right\}$ are independent for any two disjoint
sets $B_1, B_2 \subset [n]$ such that
$\ell_{\boldsymbol{A}_n}(i,j) > 2$ for any $(i,j) \in B_1 \times B_2$.
\end{assumption}
\begin{remark}
\cref{assumption:local dependency} captures the idea that nearby units may
share unobserved local shocks or information.
Conditional on the network, latent ranks may be dependent for
directly linked units and for units sharing common neighbors, but are
independent for units whose network distance exceeds two. Our results continue to hold for any fixed local dependence radius $r \geq 3$.\footnote{The estimation procedure is unchanged. For the construction of the UCBs, it suffices to redefine $\boldsymbol{\Omega}_n(i,j)=\mathds{1}\{\ell_{\boldsymbol{A}_n}(i,j)\leq r\}$, while leaving the remainder of the inference procedure unchanged.}
\end{remark}
We next establish the uniform Bahadur representation of the estimated quantile-effect function $\widehat{q} \left(\tau\right)$ over $\tau \in \mathcal{T}$. Define
\begin{equation}\label{eq:influence_function}
\widetilde{\psi}_i(\tau)
=
\frac{
\mathds{1}_i(w)
\bigl(\tau-\mathds{1}\{Y_i\le q_w(\tau)\}\bigr)
}{
\pi_n(w)\,
f_w\bigl(q_w(\tau)\bigr)
}
-
\frac{
\mathds{1}_i(w')
\bigl(\tau-\mathds{1}\{Y_i\le q_{w'}(\tau)\}\bigr)
}{
\pi_n (w^\prime)\,
f_{w'}\bigl(q_{w'}(\tau)\bigr)
},
\end{equation}
where $\pi_n( w^\dagger) = n^{-1} \sum_{i=1}^n \mathbb{P}\big(W_i = w^\dagger | \boldsymbol{A}_n \big)$ for $w^\dagger \in \left \{ w, w^\prime \right\}$. For simplicity, the dependence of $\widetilde{\psi}_i(\tau)$ on $n$ is suppressed, but it depends on $\boldsymbol{A}_n$ through
$\pi_n(w)$ and $\pi_n(w')$.
\begin{lemma}\label{lemma:Bahadur representation}
Suppose \cref{ass:degree,assumption: density,assumption:local dependency} hold. Conditionally on $\{ \boldsymbol{A}_n \}_{n\geq 1}$, it follows that
\[
\sup_{\tau \in \mathcal{T}}\left|\sqrt{n} \left( \widehat{q}(\tau) - q(\tau) \right) - \frac{1}{\sqrt{n} } \sum_{i=1}^n \widetilde{\psi}_i(\tau) \right| = O_P\left(r_{\mathrm{B},n}\right),
\]
where $r_{\mathrm{B},n} = n^{-1/8}(\log n)^{1/4}$.
\end{lemma}
\subsection{Asymptotic Normality}\label{section: nonpara-estimation}
This subsection studies the weak convergence of $\{ \widehat q(\tau):\tau \in \mathcal{T} \}$.
The Bahadur representation established in
\cref{lemma:Bahadur representation} alone does not guarantee convergence to a fixed Gaussian process, because the scores $\widetilde{\psi}_i(\tau)$ and their covariance structure may vary along the network sequence $\{ \boldsymbol{A}_n \}_{n\geq1}$.
We therefore impose additional stabilization conditions on the empirical degree distribution and the network-conditional covariance kernel.
\begin{assumption}\label{assumption: DD_limit}
There exists a probability measure $\mu$ on $ \{0,1,\ldots, K_{\max} \}$ such that, for each
$0\leq k \leq K_{\max}$,
\[
\frac{1}{n}\sum_{i=1}^n \mathds{1}\{|N_i|=k \}
\overset{ \mathrm{a.s.} }{\longrightarrow}
\mu(k).
\]
Moreover, for the common degree \(l\) of the exposure states \(w\) and \(w'\),
we assume \(\mu(l)>0\).
\end{assumption}
Under \Cref{assumption:RCT,assumption: DD_limit}, for $w^\dagger = \big(d^\dagger, t^\dagger, l \big)$, both $\frac{1}{n}\sum_{i=1}^n \mathds{1}_i(w^\dagger)$ and $\pi_n \big(w^\dagger\big)$ converge to $\pi(w^\dagger)$ in probability, where
\begin{equation}\label{eq: pi_w}
\pi\big(w^\dagger\big) = \mu(l) \times \binom{l}{t^\dagger} p^{t^\dagger+d^\dagger}(1-p)^{l-t^\dagger+1-d^\dagger} > 0.
\end{equation}
We further define the score $\psi_i(\tau)$ by substituting $\pi(w^\dagger)$ for $\pi_n(w^\dagger)$ in \eqref{eq:influence_function}, for $w^\dagger \in \{w, w^\prime\}$. Combining the result $\left|\pi_n(w^\dagger) - \pi(w^\dagger)\right| = o_P(1)$ with
\cref{lemma:Bahadur representation} yields the following corollary.
\begin{corollary}\label{corollary: fixed_Representation}
Suppose \cref{ass:degree,assumption: density,assumption:local dependency,assumption: DD_limit} hold. Conditionally on $\{ \boldsymbol{A}_n \}_{n\geq 1}$, it follows that
\[
\sup_{\tau \in \mathcal{T}}\left|\sqrt{n} \left( \widehat{q}(\tau) - q(\tau) \right) - \frac{1}{\sqrt{n} } \sum_{i=1}^n \psi_i(\tau) \right| = o_P(1).
\]
\end{corollary}
To derive a fixed Gaussian limit for $\sqrt{n}\left(\widehat{q}\left(\tau\right) - q(\tau) \right)$,
we require the network-dependent covariance kernel to stabilize along the
graph sequence, analogously to Assumption 6 of \cite{leung2020treatment}. The network-conditional covariance kernel $ \mathbb V_{\boldsymbol A_n}(\tau,\tau^\prime)$ is defined as
\begin{equation}\label{eq: con_variance_kernel}
\mathbb V_{\boldsymbol A_n}(\tau,\tau^\prime)
\equiv \frac{1}{n}
\mathrm{Cov}\left(\sum_{i=1}^n\psi_i(\tau),\sum_{i=1}^n\psi_i(\tau^\prime)\Big|\boldsymbol A_n\right), \quad \tau, \tau^\prime \in \mathcal{T}.
\end{equation}
\begin{assumption}\label{ass:final_variance}
There exists a nonrandom, symmetric, and PSD kernel $\Sigma :\mathcal T\times \mathcal T \to \mathbb R$ such that
\[
\sup_{\tau,\tau' \in \mathcal T}
\left|
\mathbb V_{\boldsymbol A_n}(\tau,\tau')
-
\Sigma (\tau,\tau')
\right|
=o_P(1).
\]
\end{assumption}
\begin{proposition}\label{proposition:NP_quantile_process_convergence}
Suppose the conditions of \cref{corollary: fixed_Representation} hold. Under the additional \cref{ass:final_variance}, we have
\begin{equation}\label{eq:limiting_distribution}
\sqrt{n} \left( \widehat{q}\left(\tau\right) - q\left(\tau\right) \right) \rightsquigarrow \mathbb{G}(\tau) \quad \mathrm{in} \ \ell^\infty(\mathcal{T}),
\end{equation}
where $\mathbb{G}(\cdot)$ is a tight, mean-zero Gaussian process with covariance kernel $\Sigma(\tau, \tau')$.
\end{proposition}
\begin{remark}
\cref{ass:final_variance} is used mainly to derive \eqref{eq:limiting_distribution}, which establishes the limiting distribution of the quantile process $\sqrt{n} \left( \widehat{q}(\tau) - q(\tau) \right)$.
\end{remark}
\cref{proposition:NP_quantile_process_convergence} derives a Gaussian limit under the large-network asymptotics of \cite{leung2020treatment}, where the observed network is embedded into a growing sequence of graphs. This fixed Gaussian limit result requires
the network sequence to stabilize in two respects. In particular, \cref{assumption: DD_limit} requires
the empirical degree distribution to converge along the graph sequence, while \cref{ass:final_variance} assumes uniform convergence of the
network-conditional covariance kernel of the linearized quantile process. These conditions are restrictions on the underlying graph sequence and
cannot be directly verified from a single observed network.
\section{Uniform Inference}\label{sec:dependent bootstrap}
This section develops the uniform inference for the quantile-effect function. In particular, for the two fixed exposure states $w$ and $w'$, we construct uniform confidence bands (UCBs) for $q(\tau) \equiv q_w(\tau) - q_{w'}(\tau)$ for $\tau \in \mathcal{T}$. We do not rely on the fixed Gaussian limit shown in
\cref{proposition:NP_quantile_process_convergence}, we approximate the distribution of the studentized
supremum statistic by the maximum of a Gaussian vector
with an estimated network-dependent covariance \citep{chernozhukov2013gaussian,chernozhukov2014anti,chernozhukov2014gaussian}. Consequently, our procedure does not require the stabilization of either the
empirical degree distribution or the network-conditional covariance kernel
imposed in \cref{assumption: DD_limit,ass:final_variance}.
A standard i.i.d. multiplier bootstrap is inappropriate in our setting, since it perturbs the estimated score functions independently across units and therefore does not reproduce the local dependence.\footnote{A standard i.i.d. multiplier bootstrap would use
$\widehat{\mathbb G}^{\prime}_{n}(\tau)
=
n^{-1/2}\sum_{i=1}^n \xi_i\widehat\psi_i(\tau)$,
where $\xi_i \overset{\mathrm{i.i.d.}}{\sim} \mathrm{N}(0,1)$ and the multipliers are generated independently of the data.} As a result, it ignores the off-diagonal covariance terms between units that are directly connected or share common neighbors, which may distort the resulting standard errors and critical values.
\subsection{Construction of UCBs}\label{subsec:UCB}
Let $\boldsymbol{\Omega}_n$ be an $n \times n$ matrix with entries $\boldsymbol{\Omega}_n(i,j) = \mathds{1}\{\ell_{\boldsymbol{A}_n}(i,j) \le 2\}$. For each $i\in[n]$, we estimate the score $\widetilde{\psi}_i(\tau)$ defined in \cref{eq:influence_function} by
\begin{equation}\label{eq: feasible_score_UCB}
\widehat{\psi}_i(\tau)
=
\frac{\mathds{1}_i(w)\bigl(\tau - \mathds{1}\{Y_i \le \widehat{q}_w(\tau)\}\bigr)}
{\widehat{\pi}(w)\,\widehat{f}_w\bigl(\widehat{q}_w(\tau)\bigr)}
-
\frac{\mathds{1}_i(w')\bigl(\tau - \mathds{1}\{Y_i \le \widehat{q}_{w'}(\tau)\}\bigr)}
{\widehat{\pi}(w')\,\widehat{f}_{w'}\bigl(\widehat{q}_{w'}(\tau)\bigr)},
\end{equation}
where $\widehat{\pi} \big(w^\dagger\big) = \frac{1}{n} \sum_{i=1}^n \mathds{1}_i( w^\dagger)$ and $\widehat{f}_{w\dagger}(\cdot)$ denote the estimator of conditional density of $Y_i$ given $W_i =w^\dagger$. We define the network-dependent covariance kernel estimator
$\widehat{\mathbb V}_n:\mathcal T\times\mathcal T\to\mathbb R$ as
\begin{equation}\label{eq:variance_estimator}
\widehat{\mathbb{V}}_n(\tau, \tau^\prime) = \frac{1}{n} \widehat{\boldsymbol{\psi}}(\tau)^\top \boldsymbol{\Omega}_n \widehat{\boldsymbol{\psi}}(\tau^\prime),
\end{equation}
where $\widehat{\boldsymbol{\psi}}(\tau) = \bigl( \widehat{\psi}_1(\tau), \dots, \widehat{\psi}_n(\tau) \bigr)^{\top}$. A 100$(1-\alpha)$\% Gaussian bootstrap UCB for $\{ q(\tau): \tau \in \mathcal{T} \}$ is constructed as
\[
\tau \mapsto \left[ \widehat{q}(\tau ) - \frac{z_{1-\alpha}^\star \widehat{\sigma}_n(\tau) }{\sqrt{n}}, \ \widehat{q}(\tau ) + \frac{ z_{1-\alpha}^\star \widehat{\sigma}_n(\tau) }{\sqrt{n}} \right],
\]
where $\widehat{\sigma}_n^2(\tau) = \widehat{\mathbb{V}}_n(\tau, \tau)$, and $z_{1-\alpha}^\star$ is a bootstrap-based critical value to be defined below.
To compute the critical value, we approximate the supremum over
$\mathcal T$ by a finite grid. Let
$\mathcal{T}_n\equiv \{\tau_1,\ldots,\tau_{p_n}\}\subseteq\mathcal{T}$
be a pre-specified grid. We write $\boldsymbol\tau_n=(\tau_1,\ldots,\tau_{p_n})$ and define its mesh size $\delta_n$ of $\mathcal{T}_n$ by
\[
\delta_n
\equiv
\sup_{\tau\in\mathcal T}
\min_{1\le j\le p_n}
|\tau-\tau_j|.
\]
The number of grid points $p_n$ is required to increase with
$n$ so that $\delta_n\to0$. Let $\widehat{\boldsymbol V}_n\in\mathbb R^{p_n\times p_n}$ denote the
estimated covariance matrix on the grid, with $(i,j)$-th entry $\widehat{\mathbb{V}}_n(\tau_i,\tau_j)$. Let $\widehat{\boldsymbol \Gamma}_n = \widehat{\boldsymbol{\Lambda}}_n^{-1} \widehat{\boldsymbol V}_n \widehat{\boldsymbol{\Lambda}}_n^{-1} $, where $\widehat{\boldsymbol{\Lambda}}_n = \mathrm{diag} \left( \widehat{\sigma}_n(\tau_1),\ldots, \widehat{\sigma}_n (\tau_{p_n}) \right) \in \mathbb{R}^{p_n\times p_n}$. Since $\widehat{\boldsymbol\Gamma}_n$ may not be positive semidefinite (PSD) in
finite samples, we replace it by a nearest correlation matrix. Specifically, let
\begin{equation}\label{eq: PS_programming}
\widehat{\boldsymbol{ \Gamma}}^+_n \in \mathop{\operatorname{argmin}}_{ \boldsymbol{\Gamma} \in \mathbb{R}^{p_n \times p_n} } \left\{ \big \| \widehat{\boldsymbol \Gamma}_n - \boldsymbol{\Gamma} \big \|_{\max} : \boldsymbol{\Gamma} \succeq 0, \mathrm{diag}( \boldsymbol{\Gamma} ) = 1 \right\}.
\end{equation}
Conditional on the observed data and the realized network, draw a mean-zero
Gaussian vector
\[
\widehat{\boldsymbol{Z}}_n \equiv \big( \widehat{Z}_n(\tau_i): 1\leq i \leq p_n \big) \sim \mathrm{N}\big(0, \widehat{\boldsymbol{ \Gamma}}^+_n\big).
\]
The Gaussian bootstrap maximal statistic over the grid $\mathcal{T}_n$ is defined as
\[
\widehat{T}_n^\star
=
\sup_{1\leq i \leq p_n}
\left|
\widehat{Z}_n(\tau_i)
\right| .
\]
Finally, the critical value is given by the conditional $(1-\alpha)$-quantile of $\widehat{T}_n^\star$:
\[
z_{1-\alpha}^\star
=
\inf
\left\{
s\in\mathbb R:
\mathbb P_\star\big(
\widehat{T}_n^\star\le s
\big)
\ge 1-\alpha
\right\},
\]
where $\mathbb P_\star$ denotes the conditional probability, given
$\left(Y_i,W_i \right)_{i=1}^n$ and $\boldsymbol A_n$.
\subsection{Validity of UCBs}
We now establish the validity of the UCBs via Gaussian bootstrap. Recall the score $\widetilde{\psi}_i(\tau)$
defined in \cref{eq:influence_function}, and define its network-conditional
covariance kernel by
\begin{equation}\label{eq: network_conditional_covariance_kernel}
\widetilde{\mathbb V}_{\boldsymbol A_n}(\tau,\tau')
\equiv
\frac{1}{n}
\mathrm{Cov}\left(
\sum_{i=1}^n \widetilde\psi_i(\tau),
\sum_{i=1}^n \widetilde\psi_i(\tau')
\,\Big|\,\boldsymbol A_n
\right).
\end{equation}
By Assumptions in \cref{sec:framework} and \cref{assumption:local dependency}, it is easy to see that
$\mathbb{E} \big[\widetilde\psi_i(\tau)|\boldsymbol A_n \big]=0$, and
\[
\widetilde{\mathbb V}_{\boldsymbol A_n}(\tau,\tau')
=
\frac{1}{n}
\sum_{i=1}^n
\sum_{\ell_{\boldsymbol A_n}(i,j)\le2}
\mathbb E\left[
\widetilde\psi_i(\tau)\widetilde\psi_j(\tau')
|\boldsymbol A_n
\right].
\]
\begin{assumption}
\label{assumption:density_est_convergence} There is a interval $[a,b]$ such that $\{q_{w^\dagger}(\tau) : \tau \in \mathcal{T}\} \subseteq [a, b]$ for $w^\dagger \in \{w, w^\prime\}$. Moreover, there exist density estimators $\widehat{f}_{w^\dagger}$ for $w^\dagger \in \{w, w^\prime\}$ such that
\[
\sup_{a \leq y \leq b} \big|\widehat{f}_{w^\dagger}(y) - f_{w^\dagger}(y)\big| = O_P \big( r_{f,n} \big).
\]
\end{assumption}
\begin{remark}
We take an agnostic view on how the conditional density estimator $\widehat{f}_w$ is obtained, with \cref{assumption:density_est_convergence} imposing only high-level conditions on its convergence rate. Under sufficient regularity and an appropriate choice of bandwidth, estimators satisfying these rate conditions can be constructed using kernel density estimation.
\end{remark}
\begin{lemma}\label{lemma: feasible_variance_convergence}
Suppose \cref{ass:degree,assumption: density,assumption:local dependency,assumption:density_est_convergence} hold. Then, conditional on the \(\{ \boldsymbol{A}_n \}_{n\ge1}\),
\[
\sup_{\tau, \tau^\prime \in \mathcal{T}}\left|\widehat{\mathbb{V}}_n(\tau, \tau^\prime) - \widetilde{\mathbb V}_{\boldsymbol A_n}(\tau,\tau')\right| = O_P\left(\bar{r}_{\mathbb{V}, n} \right),
\]
where $\bar{r}_{\mathbb{V}, n} = r_{f,n} \vee n^{-1/4}$.
\end{lemma}
\begin{assumption}\label{assumption: smallest_variance_diagonal}
Let $\sigma_n^2(\tau) = \widetilde{\mathbb V}_{\boldsymbol A_n} (\tau,\tau)$.
There are constants $c_\sigma, c_\sigma^\prime >0$ such that
$c_\sigma \leq \sigma_n(\tau) \leq c_\sigma^\prime$ for all $\tau \in \mathcal{T}$.
\end{assumption}
Recall that \(\bar{r}_{\mathbb{V}, n} \) is defined in
\cref{lemma: feasible_variance_convergence}, and define
\[
\eta_n
=
n^{-1/8}(\log n)^{1/4}
+
\sqrt{\delta_n\log(e/\delta_n)}
+
n^{-1/2}\log(e/\delta_n)
+
\delta_n ,
\]
where $\delta_n$ is the mesh size of the grid $\mathcal T_n$. The following theorem shows that the conditional distribution of the
Gaussian maximum $\widehat{T}_n^\star$ approximates that of the studentized
supremum statistic.
\begin{theorem}
\label{theorem: feasible_gaussian_approximation}
Suppose \cref{ass:degree,assumption: density,assumption:local dependency,assumption:density_est_convergence,assumption: smallest_variance_diagonal} hold. Further assume that $\delta_n = o(1)$, $\left(\log p_n\right)^7/n\to0$, and $
\bar{r}_{\mathbb{V}, n} \log^2 p_n + \eta_n \sqrt{\log p_n} = o(1)$. Then, conditional on \(\{ \boldsymbol{A}_n \}_{n\ge1}\),
\[
\sup_{s\in\mathbb R}
\left|
\mathbb P_{\boldsymbol A_n}
\left[
\sup_{\tau\in\mathcal T}
\left|
\frac{\sqrt n\left(\widehat q(\tau)-q(\tau)\right) }
{\widehat\sigma_n(\tau)}
\right|
\le s
\right] - \mathbb P_\ast
\big(
\widehat{T}_n^\star
\le s
\big)
\right|
=o_P(1),
\]
where \(\mathbb P_\ast\) denotes probability conditional on $\left(Y_i, W_i\right)_{i=1}^n$ and $\boldsymbol A_n$.
\end{theorem}
\cref{theorem: feasible_gaussian_approximation} immediately yields the
asymptotic validity of the UCB constructed in
\cref{subsec:UCB}.
\section{Numerical Examples}\label{section: Numerical_Examples}
\subsection{Monte Carlo}
In this section, we examine the finite sample performance of the proposed estimators. Treatment is independently assigned according to $D_{i} \sim \text{Bern}\left(0.5\right)$.
The network $\boldsymbol{A}_n$ is generated following \cite{leung2020treatment}, with the parameter $\kappa$ set to $1.55$ to yield an expected degree of 3 in the simulated network.
Let $\left\{\nu_i \right\}_{i=1}^n$ be \text{i.i.d.} random variables drawn from $\mathrm{N}\left(0,1/4\right)$, and $V_i = \nu_i + \vert{}N_i\vert{}^{-1/2} \sum_{j=1}^{n} A_{ij} \nu_j$. We then set $U_i = \Phi\big(\sqrt{2}\, V_i \big)$, where $\Phi(\cdot)$ denotes the standard normal cdf, so that $U_i$ is marginally uniform on $(0, 1)$. The observed outcome $Y_i$ is generated by
\[
Y_i = \beta_0\left(U_i\right) + \beta_{\mathrm{D}}\left(U_i\right) D_i + \beta_{\mathrm{M}}\left(U_i\right) M_i,
\]
where $M_i = T_{i}/|N_i|$ is the fraction of treated neighbors, and $\beta(\tau) \equiv \left(\beta_0(\tau), \beta_{\mathrm{D}}(\tau), \beta_{\mathrm{M}}(\tau) \right)$ is given by
\[
\beta_0(\tau) = \Phi^{-1}(\tau)/\sqrt{2} , \quad
\beta_{\mathrm{D}}(\tau) = \Phi\left( \beta_0(\tau) \right),\quad
\beta_{\mathrm{M}}(\tau) = \frac{1.5 \exp\left( \beta_0(\tau) \right)}{1 + \exp\left( \beta_0(\tau) \right)}.
\]
\paragraph{Implementation.} We estimate and perform uniform inference for the quantile spillover effect $q(\tau) = q_w(\tau) - q_{w^{\prime}}(\tau)$ over $\mathcal{T} = [0.1,0.9]$. Specifically, as described in \cref{subsec:nonparametric-QTE}, we compute the estimators $\widehat{q}\left(\tau\right)$ over a grid of $p_n = 30$ equally spaced quantile levels, i.e., $0.1 = \tau_0 < \tau_1 < \ldots < \tau_{p_n} = 0.9$. Following Section 2.2 of \cite{belloni2019valid}, the estimates $\widehat{f}_{w^{\dagger}}(\widehat{q}_{w^{\dagger}}(\tau_k))$ are computed as
\begin{equation*}
\widehat{f}_{w^{\dagger}}(\widehat{q}_{w^{\dagger}}(\tau_k)) = \frac{\tau_k^+ - \tau_k^-}{\widehat{q}_{w^{\dagger}}(\tau_k^+) - \widehat{q}_{w^{\dagger}}(\tau_k^-)},
\end{equation*}
where $\tau_k^+ = \min\{\tau_k + h_n, 0.9\}$, $\tau_k^- = \max\{\tau_k - h_n, 0.1\}$, and $h_n \propto n^{-1/5}$. We construct our UCBs for $\left\{q(\tau) : \tau \in \mathcal{T} \right\}$ following the procedure outlined in Section \ref{subsec:UCB}, with critical values computed via $B = 200$ replications.
In order to provide more details on the quantile effect estimates, we report 95\% pointwise confidence intervals (CIs), the UCB developed in \cref{subsec:UCB}, and a UCB based on the network-dependent wild bootstrap. First, given the estimated covariance kernel $\widehat{\mathbb{V}}_n\left(\cdot, \cdot\right)$, the pointwise CI for $q(\tau)$ is constructed by
\[
\left[\widehat{q}\left(\tau\right) - \frac{1.96\cdot \widehat{\sigma}_n(\tau)}{\sqrt{n}} , \ \widehat{q}\left(\tau\right) + \frac{1.96\cdot \widehat{\sigma}_n(\tau)}{\sqrt{n}} \right],
\]
where $\widehat{\sigma}_n^2(\tau) = \widehat{\mathbb{V}}_n(\tau, \tau)$.
Alternatively, we implement a network-dependent wild
bootstrap following \cite{shao2010dependent}, \cite{conley2023bootstrap} and \cite{kojevnikov2021bootstrap}, referred to as ``DWB'' hereafter. This procedure introduces auxiliary multipliers whose covariance structure is designed to mimic the network dependence of the original data. The resulting UCB follows the similar construction as that in \cref{subsec:UCB}, using the same point estimates and pointwise standard errors, but differs in how the critical value is computed. Recall that the kernel matrix $\boldsymbol{\Omega}_n $ with
\(
\boldsymbol{\Omega}_n(i,j) = \mathds{1}\{\ell_{\boldsymbol{A}_n}(i,j) \le 2 \}
\), which may not be PSD. Following \cite{gao2025causal}, we adjust $\boldsymbol{\Omega}_n$ by truncating the negative eigenvalues. Specifically, let $\boldsymbol{Q}_n \boldsymbol{\Phi}_n \boldsymbol{Q}_n^{\top}$ be the eigen-decomposition of $\boldsymbol{\Omega}_n$, and $\boldsymbol{\Omega}^+_n = \boldsymbol{Q}_n \max\{\boldsymbol{\Phi}_n,0\} \boldsymbol{Q}_n^{\top}$ be the eigenvalue-truncated version of $\boldsymbol{\Omega}_n$. Let $\boldsymbol{\xi}_n = \left(\xi_1,\ldots, \xi_n \right) \sim \mathrm{N}( 0, \boldsymbol{\Omega}_n^+ )$, and the bootstrap score process is defined as $ \widehat{ \mathbb{G}}_n^\star(\tau)
=
\frac{1}{\sqrt n}
\sum_{i=1}^n
\xi_i\widehat\psi_i(\tau)$,
where $\widehat\psi_i(\tau)$ is defined in \cref{eq: feasible_score_UCB}. Define \[
\widehat T_n^{\mathrm{DWB}}
=
\sup_{\tau\in\mathcal T_n}
\left|
\frac{\widehat{ \mathbb{G}}_n^\star(\tau) }
{\widehat\sigma_n(\tau)}
\right|,
\]
where $\mathcal{T}_n = \left\{ \tau_k: 1\leq k \leq p_n \right\}$. Letting $c_{1-\alpha}^{\star}$ denote the empirical $(1-\alpha)$-quantile of $\widehat{T}_n^{\mathrm{DWB}}$, the resulting uniform confidence band is given by
\[
\tau
\mapsto
\left[
\widehat q(\tau)
-
\frac{ c_{1-\alpha}^{\star} \widehat\sigma_n(\tau)}{\sqrt n},
\
\widehat q(\tau)
+
\frac{ c_{1-\alpha}^{\star}\widehat\sigma_n(\tau)}{\sqrt n}
\right].
\]
\paragraph{Results.} We conduct 3,000 Monte Carlo (MC) replications and implement the proposed estimation and uniform inference procedures for
\( \left\{q(\tau): \tau \in [0.1,0.9]\right\}\).
\cref{fig:ucb_plot} plots the estimated quantile spillover-effect functions together with the true effect curves, 95\% pointwise CIs, and the 95\% UCBs proposed in \cref{subsec:UCB}. The three panels correspond to the quantile-effect functions $q(\tau) = q_w( \tau) - q_{w^\prime}(\tau)$ for $w^\prime=(0,0,3)$ with $w=(0,1,3)$, $(0,2,3)$, and $(0,3,3)$, respectively. As illustrated in \cref{fig:ucb_plot}, the true quantile-effect functions are seen to lie inside our UCBs.\footnote{To avoid confirmation bias in the visual presentation, the empirical realization plotted in each sub-panel is independently selected by minimizing the absolute deviation of its mean integrated squared error (MISE) from the median MISE across 300 independent replications with $n=3,000$.} The results of this MC experiment are presented in \cref{tab:simultaneous_coverage_results}. The pointwise CIs substantially under-cover the entire quantile-effect functions. While the UCBs constructed using the network-dependent wild bootstrap (DWB) tend to be conservative, our proposed UCBs achieve coverage rates that are much closer to the nominal levels, especially in larger samples.
\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{figures/SimulationUCB/UCB_np2.pdf}
\caption{Estimated quantile spillover-effect functions for $n=3{,}000$, together with 95\% pointwise CIs and 95\% UCBs constructed using our method and DWB. The three panels correspond to $w=(0,1,3)$, $(0,2,3)$, and $(0,3,3)$, respectively, with $w'=(0,0,3)$ serving as the baseline exposure state.}
\label{fig:ucb_plot}
\end{figure}
\begin{table}[htbp]
\centering
\begin{threeparttable}
\caption{Coverage probabilities of pointwise CIs and UCBs for quantile effects.}
\label{tab:simultaneous_coverage_results}
\small
\setlength{\tabcolsep}{5.2pt}
\begin{tabular}{cc ccc ccc ccc}
\toprule
& & \multicolumn{3}{c}{90\% Cover} & \multicolumn{3}{c}{95\% Cover} & \multicolumn{3}{c}{99\% Cover} \\
\cmidrule(lr){3-5} \cmidrule(lr){6-8} \cmidrule(lr){9-11}
Comparison & $n$ & PW & DWB & Ours & PW & DWB & Ours & PW & DWB & Ours \\
\midrule
\multirow{3}{*}{\makecell[c]{$(0,1,3)$ \\ vs \\ $(0,0,3)$}}
& 1,000 & 0.574 & 0.935 & 0.878 & 0.718 & 0.959 & 0.921 & 0.897 & 0.983 & 0.961 \\
& 2,000 & 0.600 & 0.966 & 0.911 & 0.750 & 0.983 & 0.948 & 0.923 & 0.996 & 0.983 \\
& 3,000 & 0.596 & 0.966 & 0.914 & 0.740 & 0.987 & 0.948 & 0.923 & 0.996 & 0.988 \\
\midrule
\multirow{3}{*}{\makecell[c]{$(0,2,3)$ \\ vs \\ $ (0,0,3)$}}
& 1,000 & 0.594 & 0.937 & 0.881 & 0.730 & 0.965 & 0.926 & 0.902 & 0.987 & 0.970 \\
& 2,000 & 0.607 & 0.963 & 0.914 & 0.762 & 0.980 & 0.948 & 0.928 & 0.993 & 0.982 \\
& 3,000 & 0.579 & 0.970 & 0.918 & 0.747 & 0.985 & 0.955 & 0.928 & 0.994 & 0.986 \\
\midrule
\multirow{3}{*}{\makecell[c]{$(0,3,3)$ \\ vs \\ $(0,0,3)$}}
& 1,000 & 0.592 & 0.906 & 0.849 & 0.727 & 0.937 & 0.890 & 0.873 & 0.970 & 0.946 \\
& 2,000 & 0.643 & 0.965 & 0.921 & 0.795 & 0.979 & 0.952 & 0.931 & 0.994 & 0.983 \\
& 3,000 & 0.630 & 0.973 & 0.929 & 0.785 & 0.989 & 0.961 & 0.937 & 0.997 & 0.990 \\
\bottomrule
\end{tabular}
\begin{tablenotes}[flushleft]
\item \textit{Note.} PW, DWB, and Ours denote pointwise CIs, UCBs using dependent wild bootstrap, and our UCBs, respectively. Coverage probabilities are calculated over $3,000$ replications. The average effective sample sizes, $\sum_{i=1}^{n}\mathds{1}_i( w)$, for $w=(0,0,3)$, $(0,1,3)$, $(0,2,3)$ and $(0,3,3)$ are $11.4$, $33.9$, $34.0$, and $11.3$ when $n=1,000$. These effective sample sizes increase to $25.3$, $75.6$, $75.4$, and $25.1$ for $n=2,000$, and to $36.7$, $109.4$, $109.3$, and $36.3$ for $n=3,000$.
\end{tablenotes}
\end{threeparttable}
\end{table}
\paragraph{Linear QR.} Given the simplicity and interpretability of linear QR,
we next examine the finite-sample performance of the estimators and UCBs for the linear QR coefficient. Details of the linear
QR and the uniform inference procedure under
network interference are provided in \cref{appendix: linear}. Specifically, we
consider a correctly specified linear model of the form
\[
q(d,t,l,\tau)
=
\beta_0(\tau)
+
\beta_{\mathrm{D}}(\tau)d
+
\beta_{\mathrm{M}}(\tau) (t/l),
\]
where $\beta_{\mathrm{D}}(\tau)$ and $\beta_{\mathrm{M}}(\tau)$ represent the direct and spillover quantile effect coefficients, respectively.
Our goal is to estimate and construct UCBs for $\beta_{\mathrm{D}}(\tau)$ and $\beta_{\mathrm{M}}(\tau)$ over $\mathcal{T} = [0.1, 0.9]$. Further implementation details are provided in \cref{appendix: linear}. Following \cite{powell1986censored} and \cite{angrist2006quantile}, we estimate the Jacobian matrix $J(\tau)$ as $\widehat{J}_n(\tau) = (2 n h_n)^{-1} \sum_{i=1}^n \mathds{1}\{ \vert{}Y_i - X_i^\top \widehat{\beta}(\tau)\vert{} \leq h_n \} X_i X_i^\top$, where $X_i = (1, D_i, M_i)^\top$ and the bandwidth $h_n(\tau) \propto n^{-1/5}$ as suggested by \cite{bofinger1975estimation}.
\cref{fig:ucb_coef_plots} plots the estimated linear QR coefficients $\widehat{\beta}_{\mathrm{D}}(\tau)$ and $\widehat{\beta}_{\mathrm{M}}(\tau)$ alongside their true counterparts $\beta_{\mathrm{D}}(\tau)$ and $\beta_{\mathrm{M}}(\tau)$, along with the 95\% pointwise CIs and 95\% UCBs for sample size $n=1,000$. \cref{tab:linear_coverage_rates} summarizes the MC simulation results evaluated over 3,000 replications. While the pointwise CIs exhibit severe under-coverage and the DWB UCBs tend to be conservative, our proposed UCBs deliver coverage rates that are much closer to the nominal levels across all sample sizes and significance levels.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.84\textwidth]{figures/SimulationUCB/UCB_lr3.pdf}
\caption{Estimated linear QR coefficients for $n=1,000$, together with 95\% pointwise CIs and 95\% UCBs constructed using our proposed method and DWB.
}
\label{fig:ucb_coef_plots}
\end{figure}
\begin{table}[htbp]
\centering
\begin{threeparttable}
\caption{Coverage probabilities of pointwise CIs and UCBs for linear QR.}
\label{tab:linear_coverage_rates}
\small
\setlength{\tabcolsep}{5.2pt}
\begin{tabular}{cc ccc ccc ccc}
\toprule
& & \multicolumn{3}{c}{90\% Cover} & \multicolumn{3}{c}{95\% Cover} & \multicolumn{3}{c}{99\% Cover} \\
\cmidrule(lr){3-5} \cmidrule(lr){6-8} \cmidrule(lr){9-11}
Coef. & $n$ & PW & DWB & Ours & PW & DWB & Ours & PW & DWB & Ours \\
\midrule
\multirow{3}{*}{$\beta_{\mathrm{D}}(\cdot)$}
& 1,000 & 0.437 & 0.953 & 0.883 & 0.633 & 0.980 & 0.934 & 0.888 & 0.995 & 0.984 \\
& 2,000 & 0.453 & 0.959 & 0.890 & 0.649 & 0.983 & 0.942 & 0.898 & 0.997 & 0.985 \\
& 3,000 & 0.449 & 0.965 & 0.898 & 0.661 & 0.985 & 0.947 & 0.904 & 0.998 & 0.988 \\
\midrule
\multirow{3}{*}{$\beta_{\mathrm{M}}(\cdot)$}
& 1,000 & 0.521 & 0.925 & 0.883 & 0.693 & 0.964 & 0.932 & 0.904 & 0.993 & 0.980 \\
& 2,000 & 0.523 & 0.941 & 0.897 & 0.708 & 0.970 & 0.945 & 0.922 & 0.993 & 0.985 \\
& 3,000 & 0.526 & 0.936 & 0.895 & 0.710 & 0.965 & 0.941 & 0.916 & 0.993 & 0.982 \\
\bottomrule
\end{tabular}
\begin{tablenotes}[flushleft]
\item \textit{Note.} PW, DWB, and Ours denote pointwise CIs, UCBs based on the dependent wild bootstrap, and our proposed UCBs, respectively. Coverage probabilities are estimated using $3{,}000$ Monte Carlo replications.
\end{tablenotes}
\end{threeparttable}
\end{table}
\subsection{Empirical Application}
To illustrate our methods, we revisit the randomized savings-account experiment studied by \cite{prina2015banking} and \cite{comola2021treatment}. In this experiment, female household heads in 19 slums surrounding Pokhara, Nepal's second largest city, were randomly offered fee-free savings accounts through public lotteries conducted within each community. The dataset consists of 915 households together with the network of regular financial support among these households. The network is constructed from survey responses on repeated financial exchanges: two households are linked if either household identifies a member of the other household as a regular financial-support partner. The resulting network is highly sparse, containing a total of $113$ undirected links and exhibiting a network density of $2.71 \times 10^{-4}$. Node degrees range from $0$ to $4$, with an average degree of $0.25$.
We focus on three household outcomes: monetary assets, total assets, and education expenditures. For each original outcome $Y_i^{o}$, we use $Y_i = \log(1+Y_i^{o})$ in our analysis. We estimate the following linear QR model:
\begin{equation}\label{empirical model2}
Q_\tau (Y_i\vert{}X_i) = \beta_0(\tau) + \beta_{\mathrm{D}}(\tau)D_i + \beta_{\mathrm{M}}(\tau) M_i + C_i^{\top}\gamma(\tau),
\end{equation}
where $X_i = (D_i, M_i, C_i)$, $D_i$ denotes the binary treatment whether household $i$ was offered a savings account, $M_i = T_i/|N_i|$ is the fraction of treated neighbors, and $C_i$ contains baseline household characteristics, village dummies, and a zero-neighbor indicator $\mathds{1}\{|N_i| = 0\}$ to properly account for isolated households. The parameter processes are evaluated over quantile domains $\mathcal{T}$: we set $\mathcal{T} = [0.10, 0.75]$ for monetary and total assets, and $\mathcal{T} = [0.40, 0.90]$ for education expenditures to avoid the point mass at zero concentrated at lower quantiles.
Figure \ref{fig:expenditure} reports the estimated coefficients $\beta_{\mathrm{D}}(\tau)$ and $\beta_{\mathrm{M}}(\tau)$, along with their 90\% pointwise CIs and UCBs constructed using the method in \cref{subsec:UCB}.
For monetary assets, the estimated coefficient $\widehat{\beta}_{\mathrm{D}}(\tau)$ is positive and generally declines with $\tau$. The 90\% UCB remains entirely above zero over
$\tau\in[0.1,0.75]$, suggesting that the offer of a savings account generated positive direct gains in monetary assets for households across the lower and middle quantiles. The estimated coefficients $\widehat{\beta}_{\mathrm{D}}(\tau)$ for total assets and educational expenditures exhibit a similar but less pronounced pattern, with positive estimates concentrated in the lower quantiles. Moreover, the 90\% UCBs for $\beta_{\mathrm{M}}(\tau)$ include zero
throughout the quantile ranges considered for all three outcomes, providing
no uniform evidence of spillover effects. The wide bands reflect
substantial sampling uncertainty of
$\widehat{\beta}_{\mathrm{M}}(\tau)$, which may partly arise because many
households have no recorded financial-support links.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{figures/Prina/empirical_linear_qte_results.png}
\caption{Estimated quantile regression coefficient processes with 90\% pointwise CI and UCBs.}
\label{fig:expenditure}
\end{figure}
\section{Concluding Remarks}
This paper develops a framework for identifying and estimating quantile treatment and spillover effects in randomized experiments with network interference. Based on a structural quantile model, we construct QR estimators for these effects. Our main contribution is a novel uniform inference procedure using Gaussian approximations conditional on the realized network. By directly approximating the distribution of the studentized supremum statistic, we construct the UCBs without requiring the stringent stabilization conditions on the degree distribution and the network-dependent covariance structure typically needed for a fixed Gaussian limit.
Our analysis relies on Assumptions \ref{assumption:Interference} and \ref{assumption:local dependency}, that is, a correctly specified exposure mapping and conditional independence of the latent ranks for units whose network distance exceeds two. \cref{assumption:Interference} is needed for the causal interpretation. When the exposure mapping is misspecified, the QR estimator constructed in \cref{subsec:nonparametric-QTE} still compares conditional outcome quantiles between the specified exposure states, but these contrasts may no longer represent structural quantile effects. \cref{assumption:local dependency} is needed for the validity of our uniform inference in \cref{sec:dependent bootstrap}. If the outcome dependence extends beyond network distance two, the proposed UCBs may be invalid.
However, \cref{assumption:local dependency} can be relaxed to allow for finite-range network dependence with any fixed radius, and our UCBs remain valid once the covariance-kernel estimator is adjusted to account for the expanded dependency graph. Another extension is to replace finite-range dependence with $\psi$-dependence on the QR process \citep{kojevnikov2021limit,leung2022causal}. Recent advances in high-dimensional Gaussian approximations for network-dependent data \citep{zhenggaussian2026} make this extension feasible. Finally, misspecification of the exposure mapping is a more fundamental issue, as it invalidates the structural interpretation of the estimand in our setting. Defining estimands that capture distributional heterogeneity robust to misspecified exposure mappings, alongside valid inference procedures, is left for future research.
\newpage