Telling Peer Direct Effects from Indirect Effects in Observational Network Data
Authors: Xiaojing Du
Jiuyong Li
Debo Cheng
Lin Liu
Wentao Gao
XIONGREN CHEN
Ziqi Xu
Venue: ICML 2025 poster (2025)
Keywords: Estimate Causal Effects
Peer Effects
Causal Mediation Analysis
Graph Neural Networks
Primary area: general_machine_learning->causality

Abstract: Estimating causal effects is crucial for decision-makers in many applications, but it is particularly challenging with observational network data due to peer interactions. Some algorithms have been proposed to estimate causal effects involving network data, particularly peer effects, but they often fail to tell apart diverse peer effects. To address this issue, we propose a general setting which considers both peer direct effects and peer indirect effects, and the effect of an individual's own treatment, and provide the identification conditions of these causal effects. To differentiate these effects, we leverage causal mediation analysis and tailor it specifically for network data. Furthermore, given the inherent challenges of accurately estimating effects in networked environments, we propose to incorporate attention mechanisms to capture the varying influences of different neighbors and to explore high-order neighbor effects using multi-layer graph neural networks (GNNs). Additionally, we employ the Hilbert-Schmidt Independence Criterion (HSIC) to further enhance the model’s robustness and accuracy. Extensive experiments on two semi-synthetic datasets derived from real-world networks and on a dataset from a recommendation system confirm the effectiveness of our approach. Our findings have the potential to improve intervention strategies in networked systems, particularly in social networks and public health.

Telling Peer Direct Effects from Indirect Effects in Observational Network Data
Xiaojing Du Jiuyong Li! Debo Cheng'
Lin Liu
Wentao Gao
Xiongren Chen
Ziqi Xu
Abstract
intereating trealments directly indiractly [PIE] impact individual outcome?
Estimating causal effects crucial for decision- makers in many applications, but is particularly challenging with observational network data due to peer interactions. Some algorithms have been proposed to estimate causal effects involving net- work data, particularly peer effects, but they often fail to tell apart diverse peer effects. To address this issue, we propose a general setting which con- siders both peer direct effects and peer indirect effects, and the effect of individual's own treat- ment, and provide the identification conditions of these causal effects. To differentiate these ef- fects, we leverage causal mediation analysis and tailor it specifically for network data. Further- more, given the inherent challenges of accurately estimating effects in networked environments, we propose incorporate attention mechanisms capture the varying influences of different neigh- bors and to explore high-order neighbor effects using multi-layer graph neural networks (GNNs). Additionally, we employ the Hilhert-Schmidt In- dependence Criterion (HSIC) to further enhance the model's robustmess and accuracy. Extensive experiments on two semi-synthetic datasets de- rived from real-world networks and a dataset from recommendation system confirm the effec- tiveness of our approach. Our findings have the potential to improve intervention strategies in net- worked systems, particularly in social networks and public health.
Treatment onl
Outcome (Infection)
Social tiee
PDE
(STÉ)
Neighbor
Individuali
Neighborj
PIE
(b)
Figure I. (a) An example showing the relationship between an indi- vidual and their neighbors in network setting about vaccination. (b) A causal graph representing PDF, PIF, and STE. In the dia- gram, T; and Y; indicate the treatment (e.g., vaccination status) and outcome (c.g., infection condition), respectively. The subscript denotes individual and j + i denotes i's neighbors.
data is collected from sources such as social networks, com- munication networks, or biological networks, accurately estimating causal effects becomes challenging due to the interconnected nature of individuals (e.g., units, nodes). In network setting, individual's 's outcomes are influenced not only by their own treatment but also by the treatments and outcomes of their neighbors (Sinclair et al., 2012). In network data, as shown in Fig. there are different types of causal effects involved with respect to an outcome of interest, including individual outcomes influenced by their treatment (self-trealment effects, STE), and peer effects (arising from the treatments and outcomes of their neigh- bors), which consist of peer direct effects (PDE) and peer indirect effects (PIE). It is important to distinguish and esti- mate these different types of causal effects in network data, particularly the PDE and PIE.
1. Introduction
Causal effect estimation is important area of study, with the focus on determining cause-and-effect relationships be- tween variables (Imbens& & Rubin, 2010; Pearl, 2009). When
We use an example from epidemiology (VanderWeele et al., 2012) to show the importance of differentiating between PDE and PIE. Suppose that individual is unvaccinated against disease but can benefit from their peers' vaccina- tion in two ways. First, vaccination of i's peers reduces the likelihood of them transmitting the disease to others (PDE), thereby lowering i's risk of contracting the disease. Second, vaccinated peers are less likely to become infected,
Universityy of South Australia, Adelaide, Australia Sehool of Computing Technologies, RMIT University, Melbourne, Australia. Correspondence to: Jiuyong <jiuyong. wiiiuiii..euuuuuu Debo Cheng <dcbo. cheng @ @unisa.edu.au>.
Proceedings of the 42nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Telling Peer Direct Effeets from Indirect Effeett in Observational Network Data
which reduces i's exposure the disease (PIE). Thus, in the context of vaccination, it is essential to understand how vaccines directly reduce transmission (PDE) and how the effect of vaccination of peers decreases the contagion risk of an individual (PIE). To illustrate this point, consider an extreme case: suppose the vaccination of i's peers does not reduce the likelihood of spreading a disease (PDE is zero). However, if the vaccine reduces the potential of infection, hence reducing i's exposure the disease, then the PIE is non-zero. In this case, under limited resources, it is ad- visable to prioritize vaccinating individuals with a higher probability of infection to achieve herd immunity.
Table I. A comparison of problem settings across different meth- ods.
Method
PDE PIE STE Relationship Type
(VanderWeels et al., 2012)
v
One to. One
(Chiba, 2013)
One-to-One
(Cai al., 2021)
One-to-Oni
(Shpitser al,, 2017)
One-to-One
(Jinng & Sun, 2022)
Many-to-One
(Cai al. 2023)
Many to One
(Chen et al., 2024)
X
Many One
gDIS (Ours)
V Many- Io-One
A similar example can be found in product promotion cam- paigns. Let's assume that an individual has not exposed to an advertisement for a product, but i's peers have. i might decide to purchase the product because their peers received the advertisement, even if the peers did not buy it (PDE); or i might be influenced by the fact that their peers bought the product (PIE). Drawing an analogy to the vaccination exam- ple, if an advertisement is "infectious" (i.e., has high PDE), spreads quickly through the network, reaching untargeted individuals who purchase the product based on their peers' exposure. If an advertisement is "contagious" (i.e., has high PIE), individuals' purchasing behavior is influenced by their peers' purchases of the product. This distinction helps advertisers better understand the effectiveness of their campaigns and allows for more strategic planning of future advertisements.
rion (Pearl, 2009) to distinguish these effects, and provide their identification conditions to support the soundness of the estimation. Based on the theoretical analysis, we de- velop new algorithm, gDIS, for group-level PDE and PIE, PE and STE estimation with network data. To accurately capture the complexity of network effects, gDIS employs a multi-layer GNN to focus on high-order neighbor interfer- ence and leverages attention mechanisms (Niu al., 2021) to account for the varying influence weights of different neighbors on each individual. Furthermore, to fully uti- lize the structural information of graphs, we integrate the Hilbert-Schmin Independence Criterion (HSIC) (Ahmad et al., 2021) into the GNN, enhancing the model's 's robustness and accuracy.
The main contributions of this paper are summarized as follows:
The existing works do not effectively distinguish between PDE and PIE in network data. A summary of works in peer effect analysis of network data is shown in Table 1, where we see that the works by (Jiang & Sun, 2022; Cai et al., 2023; Chen et al., 2024) do not disentangle PDE and PIE. However, in many cases, particularly in infectious diseases and marketing, is not enough to simply know that overall peer effects. It is important to distinguish between PDE and PIE, as each has different practical implications. (Van- derWeele et al., 2012: Chiba, 2013; Shpitser et al., 2017; Cai et al., 2021) separately consider PDE and PIE, but they primarily focus on two-person transmission scenarios (i.c.. one-to-one interactions), with the first two studies assuming binary outcomes. The one-to-one interaction restricts their applicability to real-world networks where multi-to one re- lationships are prevalent. The details of the limitations of these methods are discussed in the Related Work section.
We propose a framework to distinguish peer effects in observational network data into direct and indirect components at the group level, and provide theoretical analyses of the identification conditions of the causal effects.
We propose novel algorithm gDIS to estimate effects in network data by utilizing GNNs and attention mech- anisms, enabling more nuanced and robust analysis of network data.
We validate the effectiveness and robustness of gDIS on two semi-synthetic datasets based on real-world networks, as well as on recommendation system data, demonstrating the ability of gDIS to accurately esti- mate different types of causal effects in complex net- work data.
To address these limitations, we consider a more general setting for observational network data, estimating the PDE and PIE at the group level, and additionally evaluating the STE within networks, where group is defined as a set of individuals connected through social ties, rather than treat- ing each individual in isolation. We apply the principles of mediation analysis (Pearl, 2014) and the backdoor crite-
2. Related Work
In this section, we review methods for estimating causal effects in network data that are related to our gDIS method. The methods for peer effect estimation in network data fall in two broad categories: methods which do not differentiate
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Telling Peer Direct Effects from Indirect Effeets in Observational Network Data
between direct and indirect peer effects, and methods which do.
x
W.
Methods Which Do Not Differentiate between Direct and Indirect Peer Effects. Many methods have been developed for estimating peer effects without differentiating between direct and indirect peer effects. For example, Forastiere al. (2021) addressed this issue with a covariate adjustment method using generalized propensity score (PS) (Feng et al., 2012) to balance both individual and neighborhood covariates. Jiang & Sun (2022) introduced NetEst, frame- work that employs graph neural networks (GNNs) to capture feature representations of both individual nodes and their first-order neighbors. Cai et al. (2023) expanded on this by deriving generalization bounds and proposing a joint propensity score approach combined with representation learning via weighted regression. Chen al. (2024) inte- grated the targeted learning (Van der Laan & Rose, 2011) into neural network training, developing a causal effect esti- mator. Ma et al. (2022) developed HyperSCI, which lever- ages hypergraph neural networks (HGNNs) (Feng et al., 2019) to model interference using multilayer pereeptron (MLP) (Popescu et al., 2009) and hypergraph convolution. However, these methods do not differentiate between the various types of peer effects. It is crucial to differentiate between PDE and PIE, as these two components carry dis- tinct practical implications and inform different strategies for intervention and decision Aecision-making making.
W
(e)
(b)
Figure 2. (a) An illustration of the causal relationships considered in our work. Node 2 has nodes and 3 as neighbours in the network. The features, treatment, and outcome of node are represented by X1, Tú and Y,, respectively, (b) The summary causal graph where Wa. Wr, and Wa, represent the aggregated features, treatments, and outcomes of node i's neighbors.
in Table 3 of Appendix B.
3.1. Notations
Uppercase letters (e.g., Wt.) denote variables, and lower- case letters (e-g., wi;) their values. Bold uppercase letters (c.g., Wx) represent sets, vectors, matrices, while bold lowercase letters (e.g., w.) represent their values.
As illustrated in Fig. 2(a), a network consists of a collection of nodes (or entities) and edges (or connections) that repre- sent relationships between them. For example, in a social network, individuals serve as nodes, while their friendships or interactions form the edges connecting them. Two nodes are neighbors if they are connected by edge. For simplic- ity, node indices (e.g.. i) are used to represent nodes when the context is clear.
Methods differentiating between direct and indirect peer effects. This line of research focuses on estimating direct and indirect peer effects separately in network data. Van- derWeele et al. (2012) and Chiba (2013) used logistic re- gression and inverse probability weighting (IPW) methods, respectively, to study the effects of vaccination in small fam- ily units, such as the impact of child's vaccination on the mother. Shpitser et al. (2017) developed a chain graph-based method to decompose peer effects into unit-specific compo- nents. Cai al. (2021) evaluated contagion, susceptibility, and infectiousness effects in symmetric partnerships under infectious disease settings. The work presented in (Van- derWeele et al., 2012: Chiba, 2013; Shpitser et al., 2017; Cai et al., 2021) primarily focuses on scenarios with only two individuals the transmission unit, limiting its practi- cal applicability. Additionally, the studies in (VanderWeele al., 2012) and (Chiba, 2013) focus on binary outcomes only. In contrast, work focuses estimating both direct and indirect peer effects at the group level, making it more applicable to real-world scenarios.
The causal relationships this study are represented by the directed edges between the nodes (variables) in a causal directed acyclic graph (DAG), as shown in Fig. 2(b). The node set is defined as V = XUTUU Y, where X (X,,..., Xm) represents features, T (Ti...., } denotes binary treatments (c.g., vaccination), and Y (Y,,.... Ym) represents continuous outcomes (e.g.. im- munity levels). Each node i has a feature set X, = (Xa...., X, where E (1....,m).
3.2. Definitions of Causal Effects
We focus on estimating Peer Direct Effects (PDE), Peer Indirect Effects (PIE), and Self-Treatment Effects (STE) from network data, defined within the potential outcomes framework (Imbens & Rubin, 2010).
3. Preliminaries and Background
In the potential outcomes framework, Yi(!') is the potential outcome when treatment 7 takes the value t'; when there are two treatments, Y,t,,, th)) denotes the potential outcome under Ti = ti and T2 = t. The individual treatment effect (ITE) for unit is defined as ITE Y₁(t") - Y:)),, and the
This section outlines the notations, definitions, problem settings, and assumptions used in the paper. More details of background and notations are provided in Appendix A and
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Telling Peer Direct Effects from Indirect Effeets in Observational Network Data
PDE represents the effect of changing the treatment status of an individual's neighbors while keeping i's neighbors' outcome corresponding to a fixed treatment status, In the above definition of PDE, W (w/,) represents the potential outcome when We, takes the value w/.-- V.(we, W (we4, )) represents i's potential outcome when We, is set to we and W 's value corresponds to what would be if W,, were set to w/ The meanings of the remaining terms in the equation follow similar logic.
Figure 3. A classic causal mediation graph consists of three vari- ables: T; (treatment), M; (mediator), and Y; (outcome).
average treatment effect (ATE) is ATE
where l' represents the intervention (treatment) state and represents the no-intervention (control) state.
In the vaccination example, corresponding to Eq. (la): w/, means that neighbors of individual i are vaccinated, and Wt; means that the neighbors are not vaccinated. PDE indicates the reduction in the risk of neighbors' infectiousness to individual due to their vaccination directly.
When there is a mediator, people are interested in knowing the direct effect of the treatment on the outcome without passing the mediator and indirect effect via the mediator only (Xu et al., 2023), illustrated in Fig. and formally defined below, following the definitions in Pearl (2014). Definition 3.1 (Natural Direct Effect (NDE)). NDE (H, Mt) (t, (())..
In product promotion campaigns, corresponding to Eq. (Ib): wi, means neighbors of individual i are exposed to an advertisement, and Wt, means that neighbors are not exposed to an advertisement. PDE indicates an increase in the likelihood of individual i's purchasing a product due to its neighbors' exposure to an advertisement for the product. Definition 3.5 (Peer Indirect Effects (PIE)). PIE(w.) Y(W,,, W (w),)) Yi(w,, Wa, (we,
NDE quantifies the expected change in the outcome Y; when the treatment changes from T; = t to T; = t', keeping the mediator M, fixed at the value it would have taken under T, = t. In the example in Fig. 3, it is the effect via the causal pathway T; Y,
Definition 3.2 (Natural Indirect Effect (NIE)). NIE (t, (())) Y,tt, (t)).
(2a)
PIE(w:) = [Yi(wl, W,, (w(,)) Y:(w,, wo, (wr,
(2b)
NIE measures the expected change in Y; when the treatment is held constant T, while M; changes to the value it would have taken under T; t'. In the example in Fig. 3, it is the effect via the causal pathway T; + Mj + Y,
PIE represents the effect of changing the aggregated out- come of individual i's neighbors on individual i's outcome, while holding the neighbors' aggregated treatment constant.
Note that the choice of the baseline treatment level f can vary depending on the specific context.
In the vaccination example, corresponding to Eq. (2a), PIE captures the reduction of the risk of infection of individual from their neighbors due to the reduction of the risk of infection of the neighbors themselves resulting from their vaccination.
Definition 3.3 (Total Effect (TE)). TE
(t', (t)) (t, M()t)]]
TE measures the expected change in Y; when the treatment changes from T, = f T, t', while allowing the mediator M, to vary naturally under each treatment level.
In product promotion campaigns, corresponding to Eq. (2b), PIE reflects the increase of the chance of pur- chase of the individual i due to their neighbors' purchases resulting from their exposure in the promotional campaign.
We extend Definitions 3.1 and 3.2 to network data (Vander- Weele et al., 2012). Specifically, we instantiate these defi- nitions in two distinct scenarios: vaccination and product promotion campaigns. As shown in Fig. 2(b), let W., and Wy; represent the aggregated treatment and the aggregated outcome of i's neighbors respectively. We are interested in how the aggregated treatment directly and indirectly impacts i's outcome, i.e. PDE and PIE respectively.
Proposition 3.6 (Peer Effect Decomposition (Baron & Kenny, 1986)). The peer effect (PE) can be decomposed into the sum of the PDE and the PIE. That is,
PE EE [Y,(w), Ww, (w(,)) )) Y.(we W Wy.(wr)) (wx, PDE( (w)) + (IE))
= PDE((w.) (w.) + PIE (w;).
Definition 3.4 (Peer Direct Effects (PDE)).
This proposition shows that the total peer effect integrates both the direct and indirect pathways of peer influence. Definition 3.7 (Seat-Trea Effect (STE)).
PDE(w()= E [Y,,,, Wq, (w(,)) Yu(we, Wy, )), (la)
(DE(w;) = E [Y, We (wr)) Yu(,,, Wy (we,) ))]
(1b)
STE = E (Vi(t') - Yi(t))-
(4)
4

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Telling Peer Direct Effects from Indirect Effects in Observational Network Data
STE represents the effect when the treatment changes from I to 1' of individual i.
P(4 X,, (1,(X)))ENN)) 1 and 0 X, (X))) <1
P P( Wa:
In the examples of vaccination, STE indicates the average difference of potential outcomes of individuals if they had been vaccinated versus if they had not been vaccinated.
These assumptions collectively ensure that causal effects can be estimated from observational network data.
In the examples of product promotion campaigns, STE represents the average difference in individuals' potential outcomes between receiving the advertisement and not ceiving it.
4. The Proposed gDIS Method
In this section, we present the theoretical analysis of the identifiaillity of PDE, PIE, and STE, and introduce our gDIS model for estimating these effects from network data.
3.3. Problem Setting
4.1. Identifiability PDE, PIE, and STE
The goal of this work is to estimate the causal effects PDE, PIE, and STE from network data. We provide formal problem definition as follows.
To present our main theoretical results (Lemma 4.1 and Theorem 4.2), we need the following graphical criterion for identifying an appropriate adjustment set relevant the identifiability of the PDE and PIE.
Given observational network data with node features, treatments, and outcomes, and assuming the causal structure as in Fig. 2b, our goal is to obtain unbiased estimation of the following causal effects.
Problem Definition.
Criterion 1 (Sequential Ignorability (Pearl, 2014; Imai al., 2010)). A set of observed covariates W is said to satisfy sequential ignorability if il blocks all buck-door paths from M, to Y, that do not pass through T3 (Tr-avoiding backdoor paths), and all back-door paths from T₃ to either Mj or Y,, and no element of W a descendant of T.;
Peer Direct Effects (PDE): The causal effect of Wi, on Y; via the pathway Wa, + Y,, as formally defined in Definition 3.4.
Peer Indirect Effects (PIE): The causal effect of W., on Y, via the pathway Wa, + Wy, + Y,, as formally defined Definition 3.5.
We present our first theoretical finding below.
Lemma 4.1. In the causal DAG represented in Fig. 2(b), the aggregated set of neighbor features W satisfies Crite- rion 1.
Self-Treatmer Effects (STE): The causal effect of T on Y; via the pathway T; - Y, as formally defined in Definition 3.7.
Proof. First, W.; blocks all backdoor paths from W W) to Yi that do not pass through Wt, In the causal DAG shown in Fig. 2(b), all W,, avoidiin backdoor paths from Wy, to Y; (W,, n, < Wa, + Y., Wy, + Wa, + T; + X; - Y.. and WT, y. + W.. - T2 + Y) are blocked by the set Wa. Next, from Fig. 2(b), there are the following backdoor paths from We, to W (W,, + W., 2: + W We, + T; Y, < W yy and We, + W.. + Ti + X; + Y, + W and from W.,
As PE can be obtained based on PDE and PIE following proposition 3.6, we do not include the estimation of PE in our problem definition.
3.4. Assumptions
Wa, + Y: +- W Mr W., +- W
To estimate PDE, PIE, and STE from observational network data, some assumptions are required (Jiang& & Sun, 2022). Assumption 3.8 (Network Unconfoundedness). The po- tential outcome is independent of both individual treat- ment and neighborhood exposure, given individual and neighbor features. i.e., Y,(4)) IL. T, X,, (X,beN and ",W,,,,,,) 1L We, X,, (X,),N,
to Y, (W,, + Wz, + Y, We + W + W + Y,, We, + W.. + T. + Y,, and We, + Wz. + T, + X, +.Y.). We can see that W., blocks all these backdoor paths. Finally, W., is not a descendant of W., in Fig. 2(b). Therefore, W., satisfies the Criterion I and thus serve an adjustment set for unbiased estimation of PDE and PIE.
Assumption 3.9 (Network Consisteney). The potential out- come equals the observed outcome when an individual is exposed to the same treatment and neighborhood exposure.
Based on Lemma 4.1 and Eq.13 in (Pearl, 2014), the PDE (la), PIE (2a), and STE (4) can be represented using do-expressions. Furthermore, using Pearl's back-door ad- justment formula and do-calculus rules (Pearl, 2009), these do-expressions can be further transformed into probability expressions. This result forms the basis of our theoretical finding in Theorem 4.2.
i..., Y; = (f44) and Y; = Yi(,,, Wp.) when the individual is exposed at & and (W,,, Wy.). respectively.
Assumption 3.10 (Network Overlap). Every treatment and neighborhood exposure combination (T,, Wo., Ww)
must have a positive probability of occurring; i.e., 0 <
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Telling Peer Direct Effects from Indirect Effects in Observational Network Data
Theorem 4.2. If P(T Y3 X,, W Z. W ye W is gener- ated from the causal DAG in Fig. 2(b), then the PDE, PIE, and STE can be identified from the data as follows:
(Y, TT Wa: W We: Wa.)ow, where Wa, represents the removing all edges with arrows emanating from Wi, (rule 2 of do calculus).
PDE- = [E(Y] | Wt. = w/, Wy yi =Wyy WA, E, -w,)
We apply rule 2 of do calculus again. P(W,, vi = w/ I do(W,, = =',,), Wa, = w,,) = P(W,, y, = w. W = w/, W.. Wx.) because (Wy. 1L Wt, W2,)Gw,
BY;; Wr, = We:: W y, =wyy W... N, wr:)]
x PWy. = =w'g 1 Wt. = w/, WR, -wz)
Up to this point, Eq. (8) has been reduced to Eq. (5). The proof for reducing Eq. (9) to Eq. (6) is similar, and we do not repeat the derivations here.
x P(Wz. = We:):
(5)
PIE = E(Y, | Wt = = W.: = yy Wz. = We:)
We now reduce Eq. (10) to Eq. (7). Based on the back-door adjustment formula (Pearl, 2009), IP(,, = ys | do = f)) is identifiable because all backdoor paths from T; to Y, are blocked by X, and W... e, Specifically, T3 < X, + Y, is blocked by X,, T. <- W z; + Wt - Y; is blocked by W TB f Wa, + Wt, + W,, - Y, is blocked by W.,, T. + W.. + W.. + Y, is blocked by W.., and T; + Wx. + W,, +- We; - Y, is blocked by Wa.- Hence,
x [P(W;: D: = =wy Wt, = w/o Wx, = )
-PW,, y. = Wy. Wt = W W2, = W..]
x xPWW.. = Wz.)
(6)
STE P(X; = x.),.Wx. = w [E(Y; T. -4, X, WA. - - BEY [ = = -, X; W
P(Y, y: do, = +()) = P(Y, = y4 T, = t, X,
= Wn.)]
(7)
X;, W z = W.,, JP(X = x: P(W,, W.;)
Proof. When Lemma 4.1 is satisfied, PDE, PIE and STE are
4.2. The Proposed gDIS Model
represented as the following in do-expression (Pearl, 2014).
The analogous representations for PDE (1b) and PIE (2b)
In this section, we present the gDIS model for estimating PDE, PIE and STE from observational network data by fol- lowing Eq. (5) to Eq. (7). Due to space limit the workflow of the model is shown in Fig. 6 in Appendix C. Calculating the posterior probability of W., and Y, is challenging us- ing traditional methods due to the non-linear relationships between Wa, and both Wg, and Y,, the continuous nature of Wy. and the high-dimensiona space of W W..- GNNs have demonstrated an outstanding capability to capture such non-linear relationships in network data (Scarselli al., 2008; Kipf & Welling, 2017). Thus, we propose a practical solution combining GNN with attention mechanisms and a fully connected layer. The GNN models node interactions and extracts higher-order structural information, while the fully connected layer captures non-linear relationships be- tween Wz, and Wg.. This design enables the gDIS model to estimate causal effects accurately in complex networks.
are omitted for brevity.
PDE EY; do,, = =,', Wy, ys =wy,): Wa, = W
- E[x do(W,, = W Wy, = =w wyy Wa, W
xP xP,W,,, 34 = w do(, =w/s), ) W x; = W..))
xPP(W,:- 22 W
PIE = E (Y, dW, W W y, Ws, 5. W..)
x [P(W, y, = Wy, WW, = w/)) W., = w
- P(W,, y. Wy, do = w.,), Wa, x: W..)) )
x P(W., x, W
(9)
STE= = E EY; | do,T; =
(T; = t.)) (10)
We now prove that Eq. (8) to Eq. (10) are identifiable, i.c.,
To estimate PDE, PIE and STE as identified in Eq. (5) to Eq. (7), respectively, we need to compute the aggregated variables: W., = -EJEN, WiyT), Wy. = EjEN, wiy Yj exple.,) and W.,, as = LGEN, WiyXj, where Wij = EEeN, explex.)* X, Cij = |II,IXIX,II and Nr represents the set of neighbors of node i in the network. Correspondingly, as shown in Fig. 6 in Appendix C and Algorithm 1 in Appendix D, gDIS consists of three key stages as detailed in the following.
the do-operators can be reduced to the do-free expression in
Eq. (5) to Eq. (7), respectively. The causal DAG is shown
in Fig 2(b). Based on rule 2 of do calculus (Pearl, 2009), we have PYY; y do doW,, W., w WT = ) W m; wx.) = PY; = ys do,, = w/,)) W. v. = wy. W 2,
= ) since (Y; TL Wy, WR, Wa;) Va.)Gww,w.. where W., represents the removing all edges with arrows pointing to W., and Wy, represents the removing all edges with arrows emanating from Wg..
Stage 1: To estimate the conditional probability P(W, n W Wz,), we employ a 2-layer GNN with atten- tion mechanisms and single fully connected layer as the es-
Continue the above reduction, we have P(Y; = 9i
do = w(,)), Wy: = wy. Wa, = Wa:) = P(Y: =
V: Wr, = w/,, Wy, = 21
W W2;- = Wx;)) because
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Telling Peer Direct Effects from Indirect Efffetts in Observational Network Data
to W ) W. designed to model this mapping by minimizing the mean squared error (MSE) (Chicco et al., 2021). The loss function is defined as:
timator. The estimator, denoted as /1 : (Wtp
log P(W..) x.))
W
Wzy 11?
log(
exp
)).
(16)
2mh
2h2
L1 = m
W.. - fi(,, w
(11)
where h is the bandwidth parameter in KDE, which controls the smoothness of the estimated density.
A After obtaining the estimation of the conditional probabili- ties and the marginal probabilities, PDE, PIE and STE can be estimated via Eq. (5) to Eq. (7) respectively.
Stage 2: To estimate the conditional probability P(Y|W,;; We Wa,), we use 2-layer GNN with atten- tion mechanisms and a 3-layer fully connected network as the estimator to refine predictions. The estimator, denoted as /2 (W.,, Wg. W.,) - Y, models this mapping by optimizing the total loss function, defined as:
5. Experiments
In this section, we first evaluate the performance of gDIS on real-world social network datasets and compare it with existing methods. Then, assess gDIS under different hy- perparameter configurations and analyze its time complexity. Finally. we conduct a case study using a recommendation dataset to evaluate the effectiveness of disentangling peer effects into direct and indirect components. For each ex- periment, we repeat it five times and report the average result and the standard deviation. Detailed experimental setup, parameters, and model architectures are provided in Appendix G.
m (W W.., )z.))²
C2 m i=1
Σ (V,
+ X m 1)2
( C).
(12)
The first term represents the MSE for prediction accuracy, and the second term is the Hilbert-Schmidt Independence Criterion (HSIC) regularization. The hyperparameter X balances the HSIC term's 's contribution. C= // 11= is
the centering matrix that removes the mean from the kernel matrices, is the m x m identity matrix, 1 is an m x vector of ones.
Baselines. We compared our model with six baselines: (1) CFR+(N) (Shalit et al., 2017), which estimates effects and incorporates peer exposure using integral probability metrics (IPM) for distribution balancing.
The HSIC regularization measures the dependence between node features X and the learned embeddings in Reproduc- ing Kernel Hilbert Space (RKHS), ensuring the embeddings do not retain spurious correlations from the original features, thereby enhancing model robustness (Berlinet & Thomas- Agman, 10. HSIC computed by first defining similarity matrices for the feature space (H) and the embedding space (H') using Gaussian kernels:
(2) TARNet+(N) (Shalit et al., 2017), a variant of CFR+(N) that does not use IPM. (3) NetDeconf (Guo al., 2020), an adaptation of CFR for network data that uses GNNs to encode confounders. (4) 1-GNN (Ma & Tresp, 2021), a GNN- causal ef- fect estimator for network interference. (5) NetEst (Jiang & Sun, 2022), which applies adversarial learning to bridge graph machine learning and causal effect estimation. (6) TNet (Chen et al., 2024) integrates target learning to im-
H: 1 )
(13) prove the accuracy of effect estimation. (7) gDIS(-HSIC), our gDIS method without the HSIC module.
(KH)a = exp(
272
Metrics. We evaluate the algorithms using MSE (Chicco et al., 2021) and PEHE (Grimmer et al., 2017). MSE (EMSE = m EE Y2)²) measures the bias in effect estimation, while PEHE (EPEHE =
H H/l
(K = cxp(-
(14)
27²2
where y is the Gaussian kernel bandwidth (Kakde et al., 2017), set to the median of input feature distances.
E. (8,(()) P,(0) - (7,(t) Y((0)) quanti- fies the precision of the effect estimation. m is the number of samples, and Y; and Y, denote the predicted and true outcomes for individual i, respectively. Lower values corre- spond better performance.
Stage 3: Kernel density estimation (KDE) is applied to den- sity estimation in high-dimensional settings (Botev et al., 2010; Duong & Hazelton, 2003). We employ KDE to estimate the probability density functions of P(X.) and P(W..).
X,
X )),
log P(X,) = log(
exp(-
m
/2mh
2h2
k=1
(15)
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Telling Peer Direct Effects from Indirect Effeets in Observational Network Data
Table Causal effect estimation results. The EPEnE error is reported. The top-performing results are emphasized in bold, and the second- best results are underlined. Note that "/" /" indicates the model is not applicable for this effect.
Data
Effects
CFR(+N)
ND(+N)
TARNET(+N)
L-GNN
NetEst
TNet
gDIS
gD15
peer
0.7412
0.7545.00121
1.699
a.6770.
0.6729
0.659 Lomss
0.2478aamer
0.2093mum4
BC
peer
0.1456,0m:29
0.1265
(within-sample)
peer indirect
0. 0.194.0mu
0.135%,a.me
self-treatment
0 gur05
0,4648200853
0.4282200005
O. 3000
0.462gn0077
0.0073ga0002
0.0588
0.7399
0.7549200122
0.6094200122
O. 686600007
0.6823
0.657an
0,2476
0.2034:a.1124
BC
pear direct
0. 0.66000011
0.110
(out-of sample) peer indirect
0.1979amm
0.1312.a.a00v
self-treatment
0. 0.6062amms
0.4588
0.4240
0. 0.3672.n011
0
0,4584
0 0,0656
0.0545
peer
0.9736.a.mm
0.9820.oo116
0.975L.corr
0.9800.noro
m.9650
0.9757.09410
0.1312
0.1262.a.awer
Flickr
poer direct
0,11 18 18
0.1118:a.0013
(within-sample) peer indirect
0.0933 0,093 00.000
0.0901 20,0000
teel-treatment
0.5249.amus
0.5
0.4740.000
0.3
0.3588
0.1599.a.m
0.1578
peer
m.9845mmms
0.9830u0012
0.9848
0. 2811
0.9651
m.9762
0.1307 ao
0.1206,a.007
Flickr
peer
0,1 0011
0.1008
(out-of-sample)
poer indiroct
0,0898 10
0,086
self-treatment
0.5242a0.119
0. 0.52843 2000
05800201112
O. 3850
0.3409
O. 003703
0. 0.154% 0.1543200003
0,15022a.008
BC samplel
BC lout-ots
Filckr sample
Riekr lout mple)
3.34
Secton
0.30
1.44
0.51
2 3.20
3.20
130
200
0.5
315
030
6:22
Flp Ratc
Figure 4. The results illustrating the relationship between the estimation bias (cMSS)) and the percentage of units with treatment flip.
5.1. Performance Evaluation on gDIS with Social Network Data
the HSIC component (denoted as gDIS(-HSIC) in Table 2). From Table 2, we observe that the full version of gDIS out- performs the variant without HSIC. This result implies that HSIC can promote independence between node features and learned embeddings, thereby enhaneing model robustness and mitigating overfitting.
We applied gDIS to two semi-synthetic real-world datasets, BlogCatalog (BC) and Flickr (Li et al., 2015), to evaluate its effectiveness on estimating PDE, PIE and STE. Due to space limitations, details about the data preprocessing and generation process are provided in Appendix E. We evaluate the method using "within-sample" estimates training networks and "out-of-sample estimates on testing networks.
Bias Analysis with Simulated Treatment Flip. To evaluate the robustness of gDIS under treatment assignment noise, we follow (Jiang & Sun, 2022) and simulate outcomes with varying rates of treatment flips (0.25, 0.5, 0.75, 1). As shown in Fig. 4, higher flip rates introduce more noise, increasing MSE. gDIS shows lower MSE on Flickr than that on BC, likely due the richer connectively in the Flickr dataset. Across all flip rates, gDIS consistently outperforms other methods, demonstrating strong resilience to noisy treatment assignments.
Causal Effect Estimation Results. Table 2 shows the ÊPEHE results on the BC and Flickr datasets. The com- plete gDIS model demonstrates superior performance in both within-sample and out-of-sample scenarios, outper- forming existing baselines. This advantage arises from our explicit modeling of neighbor influences under natural net- work conditions. By effectively leveraging graph structure and disentangling peer effects into direct and indirect com- ponents, gDIS provides a more comprehensive framework for causal inference in social networks.
Hyperparameter Impact. We analyzed the impact of the HSIC regularization coefficient X, varying from 0 to 0.5, on the error сPEНE, as shown in Fig. 5. We found that À 0.3 produced the best overall performance. When À 0.3, it begins to over-penalizz the feature representations, leading to a decline in performance.
The role of HSIC. To validate the effectiveness of HSIC regularization, we implemented variant of gDIS without
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Telling Peer Direct Effects from Indirect Effeett in Observational Network Data
rect user targeting should receive the majority of marketing resources, while the substantial network effect proportion indicates social influence strategies remain important. The fact that PIE exceeds PDE highlights that users are more influenced by their peers' actual purchasing behaviors than by peer ad exposure alone. Recommendation strategies should therefore focus not only on expanding ad reach but also on optimizing early adopters' satisfaction to maximize cascading effects through social networks. Leveraging these patterns can significantly enhance campaign effectiveness by strategically amplifying indirect peer effects.
250
033
875
1807 T00AANN
1900 peeture
611-
614
610
6. Conclusion
Summary of Contributions. this work, we address the novel problem of differentiating and estimating the three types of causal effects PDE, PIE and STE in observational network data. Through theoretical analysis, we establish the identifiability conditions for estimating these causal ef- fects from the data and provide corresponding proof' for the identifiability of these causal effects. Supported by the identification results, we propose the gDIS algorithm for ac- curately estimating PDE, PIE, and STE in observational ner- work data. To capture complex network interactions, gDIS employs a multi-layer GNN with attention mechanisms and incorporates HSIC to effectively reduce dependencies be- tween node features. We have validated the effectiveness and robustness of gDIS on two semi-synthetic social net- work datasets and a recommendation system dataset. gDIS has demonstrated strong performance even in complex net- work environments.
Figure Impact of HSIC regularization (A) on EPEнE for BC and Flickr datasets across different effect types.
Time Complexity. The time complexity analysis is pro- vided in Appendix H.
5.2. A Case Study of gDIS for Recommendation Systems
Understanding the impact of PDE and PIE is crucial for de- signing effective interventions in recommendation systems. To highlight the importance of separating PDE from PIE, we conducted experiments using the Coat Dataset', which simulates MNAR (missing not random) data from online coat purchases. As described in (Schnabel al., 2016), the dataset includes user features and 5-point ratings, with I being the lowest preference. Each user-item interaction with one of the top 10 rated products is marked as ad expo- sure, ie., treatment = 1, and a rating of 4 or higher implics purchase. The user- treatment is calculated as the proportion of interactions with promoted items.
Limitations & Future Work. Although supported by the theoretical and empirical results, gDIS has a major limita- tion. It relies on the network unconfoundedness assumption, which may be violated practice, despite being commonly seen in the literature. Future work will focus on relaxing this assumption to enhance its applicability by exploring approaches like instrumental variables (IVs) to mitigate hid- den confounders, network representation learning to capture latent structures, and causal discovery methods to identify potential confounders.
User relationships are constructed using pairwise cosine similarity between feature vectors, which represents the strength of the relationships. This network structure enables us to measure how ad exposure spreads through social con- nections and influences purchasing behavior. The estimated causal effects this network are as follows: PDE 0.0127, PIE = 0.0212, PE 0.0339, and STE = 0.0635.
Acknowledgements
To better understand the relative importance of different causal mechanisms, we amalyzed the proportional contribu- tions of STE, PDF, and PIE to the total effect, i.e., PE + STE 0.0974. STE contributes 65.20% of the total effect, while network effects account for 34.80% including 21.76% from PIE, notably larger than the contribution of PDE (13.04%). This distribution provides key insights for recommendation system design. The dominance of STE suggests that di-
This work has been partially supported by the Australian Research Council (grant number 02230101122).
Impact Statement
This work addresses the critical challenge of differentiating and estimating the three types of causal effects PDE, PIE, and STE in observational network data. Through theoretical analysis, establish identifiability conditions for these
httpps:// htttos://wwwwcoo cs. cornel schnattshhttt/////
mnar/
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Telling Peer Direct Effeets from Indirect Effeett in Observational Network Data
causal effects and provide corresponding proofs, laying solid foundation for causal inference in networks. Building on these results, propose the gDIS framework, which in- tegrates graph neural networks, attention mechanisms, and HSIC regularization to effectively estimate PDE, PIE, and STE. This contribution is potentially useful for optimizing intervention strategies in various applications and domains such as public health, marketing, and social influence analy- sis.
Chiba, Y. A simple method of measuring vaccine effects on infectiousness and contagion. Open Journal of Statistics, 3(4):7, 2013.
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Telling Peer Direct Effects from Indirect Effeets in Observational Network Data
Supplement to "Telling Peer Direct Effects from Indirect Effects in Observational Network Data"
A. Background on Causal Inference
To aid readers' understanding, in the following, define the terms causal DAG, path, d-separation, Structural Causal Model, and Back-door Criterion, which are used throughout this paper.
Causal DAG. Let G = (V, E) be directed acyclic graph (DAG), where V - (Vi...... .., Vm and E S V x V are the set of nodes and edges of the DAG respectively. G is causal DAG each directed edge V; - V3 E E encodes a direct causal relationship of V; on V.
Path. A path ı between nodes and Vj in 9 is any sequence of distinct nodes or variables (Vi...., Vj) such that each consecutive pair is adjacent (i.e., there is edge between the pair of nodes). If every edge points toward Vj, then is called directed (or causal) path; in this case V; is ancestor of V, and V, descendant of V,.
d-Separation. In the DAG G = (V, (V,E), path π between V; and Vj is said to be blocked (or d-separated) by a set M C. V if (i) π contains a chain Va - V + Ve or fork Va < V, - Ve with V/ E M, and (ii) contains a collider V. + V, +- V. such that neither V,, nor any its descendants is in M. Otherwise V; and Vj are d-connected given M. We write V, TL VG M when in G every path between VÚ and Vj is blocked by M.
Structural Causal Model. A Structural Causal Model is a tuple M V, where U {UI...., Um3 are exogenous variables, V (Vi...., Vn are endogenous variables, and F {f:) is collection of structural functions f:: Pa(V)) U, + V. where Pa(V,) is the set of direct causes of V,
The Back-door Criterion (Pearl, 2009). In DAG G = E), for an ordered pair (V, Vj). set M C V (V, Vÿ} is called a back-door set relative to (V, VG) if no variable in M a descendant of V; and M blocks every path from V/2 to V, that begins with an arrow into V,,
B. Symbols
A summary of the key symbols used in the paper is presented in Table 3.
Table 3. Key symbols used in the paper
Symbol
Description
X
Features vector of individual Treatment of individual i
Y
Outcome of individual
W
Aggregated neighbor features for individual i Aggregated neighbor treatment exposure for individual
W.
Wo
Aggregated neighbor outcome exposure for individual
W Vur(w())
The potential outcome of Wg, when the treatment W., takes value wi, The potential outcome of Y, when W,, takes value wt, and W is fixed what it would be when its treatment takes value w/. Dimension of the feature vector of node Neighbors set of individual Sample number
Y(WWEs> Wy, (w()))
k
N
m
C. Workflow of the gDIS Framework
Fig. 6 illustrates the complete workflow of the gDIS model for estimating PDE, PIE, and STE in network data. The process consists of three main stages: the first stage focuses on estimating W y the second stage incorporates HSIC regularization
12

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Telling Peer Direct Effects from Indirect Effeets in Observational Network Data
GNN Layer + Allectionn
Layer
W
f
Stage
W
W,
Pg0, HSSCC
Stage 2
W
X
logP(X)
KDF
Stage 3
W
IngP(W))
Figure 6. The workflow of our gDIS model for estimating PDE, PIE, PE and STE within network data.
for response prediction, and the third stage computes the probability density functions P(X;) and IP(W,,) using kernel density estimation.
D. The gDIS Algorithm
The pseudo-code of gDIS for estimating PDE, PIE and STE presented in Algorithm
E. Datasets
Each unit i has only one observed treatment T; and outcome Y; (the factual outcome), making direct causal effect estimation challenging due to unobservable counterfactuals. Since evaluating causal inference methods requires ground truth ITE (Individual Treatment Effect) values (which are impossible to obtain from observational data alone), following previous works (Jiang & Sun, 2022; Chen et al., 2024), we create semisynthetic datasets where the network structure (features and topology) is real, but treatments and potential outcomes are simulated.
The semisynthetic datasets generated from two real-world social networks:
BlogCatalog' (BC): BlogCatalog is a social network where users publish blogs. In this dataset, each unit represents blogger, and each edge represents friendship link between bloggers. The features are represented as keywords extracted from the bloggers' posts. This dataset provides a rich network structure, with nodes connected by social relationships that capture the peer exposure.
Flickr": Flickr is a social network for sharing images and videos. Here, each unit is user, and each edge represents social connection between Each user's features represented by list of tags that indicate their interests, forming high-dimensional feature space. More details on these datasets are summarized in Table 4.
Table Detailed information datasets. A unit (node) represents user, and an edge represents a social connection between users.
BlogCatalog
Flickr
# of Users
5,196
7,575
# of Features
8,189
12,047
# of Links
171,743
239,738
hhttps://www http://wwwwwwcccaooooooooo com/ httpps: h//ww, ick,, com/
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Telling Peer Direct Effects from Indirect Effeets in Observational Network Data
Algorithm 1 gDIS (group-level PDE, PIE, and STE)
1: Input: X. T, Wt. Wy. W.. Y; learning rate lr; HSIC regularization parameter X; adjacency matrix of data A; number of epochs for Stage 1, or number of epochs for Stage 2; number of samples.
2: Output: Estimated PDE, PIE, and STE.
3: Initialize: /1 - -NN f2 4- GNNz-Aun-FCa.
4: > fi is for estimating P(W, W 2
5: f2 is for estimating P(Y; W We, W W.,).
6: for l....,, do
7: SSSagge 1: estimate parameters for
8: Wn. fi,,, )
9: Compute loss L1 via Eq. (11).
10: Update /i by descending V VILi- L1
11: end for
12: for e e l......, 0 do
Stage 2: estimate parameters for f2
14:
Y; f2(W,, ;. W1, Wz,)
15: Compute loss Le via Eq. (12).
16: Update f2 by descending VaL22. C2.
17: end for
18:
for prri= 1...., m do
19:
>> Stage 3: density estimation
Estimate densities P(X,) and P(W,.) 2.) via Eq. (15)-(16).
end for
22: Compute PDE, PIE and STE via Eq. (5)- respectively.
23:
Return: PDE, PIE, and STE.
Given the high-dimensional and sparse nature of the original features, we follow the approach (Jiang & Sun, 2022; Chen et al., 2024) and apply Latent Dirichlet Allocation (LDA) (Blei et al., 2003) reduce the feature dimension k to 10. To partition the network, we utilize the METIS (Karypis & Kumar, 1998) graph partitioning tool, which efficiently handles large-scale graphs. Further details and case studies regarding METIS can be found in the official documentation at https: /met a. readthedocs i en /1 atteatteetteee /index html #modul eeroduleeteee e-met i s. To reduce memory consumption and streamline the analysis, we construct a new graph by removing isolated nodes that do not contribute to the network structure. The resulting partitioned data distribution is shown in Table 5.
Table Data distribution after graph partitioning. Each tuple (m, k) indicates the number of nodes (m)) and the feature dimension (k).
Dataset
Train
Validation
Test
BlogCatalog
(1722, 10)
(1733, 10)
(1731, 10)
Flickr
(1557, 10)
(2526, 10)
(1829, 10)
F. Simulation
Treatments and potential outcomes are simulated according to the underlying causal DAG, as shown Fig. 2(a), allowing for the ground truths of causal effects in terms of evaluating our causal inference method.
Treatments Simulation. The features act as pre-treatment variables (i.e., features are parent nodes of the treatment) and influence the treatment. When simulating treatment generation, to account for the varying interference between node features, we use cosine similarity divergence to measure the similarity between connected units.
We calculate each individual's probability pTT, X,, (X) (iEN()) of receiving treatment. X, and (X, {X,lien() denote the feature vestor of individual and the feature vector of their neighbor j. respectively.
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Telling Peer Direct Effects from Indirect Efffetts in Observational Network Data
Algorithm 2 Gibbs Sampling Algorithm
1: Input: Initial values W yo = 0, Yo- = 0; number of iterations m.
2: Output: Wy Wy" y
3: for !
m do
4: for to do
5: Compute Wy W - -EjENN Wij V
6: Compute Y = f(W X T,TT,
7: end for
8: Check for convergence using Y vectors.
9: if convergence is achieved then
10: break
11: end if
12: end for
p(; = X,, (XI)iEN) = ((((aowoX,) + ta
Σ wiy witwiiX)) JEA
(17)
where Wo and W1 are feature weight vectors drawn from the uniform distribution 1,) Wij is the weight indicating the influence of on i, and the sigmoid function. We define am the fixed weight for individual features, and a1 0.5 the fixed weight for neighbor features.
The treatment data is generated by the Bernoulii distribution (Chen & Liu, 1997):
T, Bernoulli (p(T, X,, (Xy))ex))
(18)
Potential Outcomes Simulation. The potential outcome Y; is influenced by individual features X, treatment status T, neighbor exposure (Ty,ex, neighbor features {Xy) and neighbor outcomes {Yihee, Following (Tchet- gen Tchetgen et al., 2021; Zhao et al., 2024), we employ Gibbs sampling to iteratively generate (YG)ieN; and Y, as follows:
PY; x,, Ti, (XiijeN, (Ti)eN;, (Yy)se,,) = (Bow22X,) + (B1
Σ
WzWl;X,))
JEN:
(19)
B.T. + BB
WijT T
+ 34 Σ
wiyY, +
JEM
JEN;
where e, is the noise term, and 3o. 31, 32. 33. and B4 are values randomly sampled from a uniform distribution within the interval [o, 1). denoted Be ((, 1) for c 0, 2,33,,3, 4. W2 and W3 are feature weight vectors drawn from the uniform distribution 4), The iteration continues until the difference between Y in successive steps is less than
1 x 10-5, indicating that the generated data have reached stable state. The Gibbs sampling algorithm is shown in Algorithm 2. Specifically, we first simulate the treatment of each node T; based on its own features X, and its neighbors' features (X, 1KiieN, via Eq. (17)-(18). Then, for every possible neighbor treatment configuration Wt, and its corresponding neighbor outcome configuration Wy. we apply Gibbs sampling according to Eq. (19) to generate the potential outcome Y, (ta, Wt: Wy.). ). thus obtaining Y, under all scenarios (f,, Wt, Wy.) Finally, we plug these values into Definitions 3.4 and 3.5 to obtain the ground-truth PDE and PIE, and into Definition 3.7 to obtain the ground-truth STE.
G. Experimental Setup and Configuration
In this section, we provide additional details about the experimental setup and configurations. Our proposed model is implemented using the Python libraries TensorFlow (Abadi et al., 2016) and NetworkX (Hagberg et al., 2008). We performed a grid search on the validation set to select the key parameters (learning rate, hidden dimensions, and regularization strength) that yielded the best performance, as shown Table Table summarizes the parameter settings used for peer effect analysis on BlogCatalog and Flickr, Key parameter descriptions are as follows:
15

---

Telling Peer Direct Effeets from Indirect Effeetts in Observational Network Data
Reps: The number of repetitions for each experimental group ensure reliable results.
Epoch: An epoch represents one complete forward and backward pass through the neural network.
GNN Layers: The number of layers in the GNN.
FC layers: The number of fully connected layers the model.
Hidden Dimension: The dimensionality of the hidden layers.
Ir: The learning rate.
A: The regularization parameter for the HSIC.
h: The bandwidth parameter used in kernel density estimation.
Table 6. The parameter settings for the three stages datasets.
Stage
Parameter
BlogCatalog (BC)
Flickr
Reps
5
Epoch
200
200
Stage
GNN Layers
2
FC Layers
1
Hidden Dimension
8
Ir
0.001
0.001
Reps
5
Epoch
150
180
Stage
GNN Layers
2
FC Layers
3
Hidden Dimension
32
32
lr
0.01
0.001
A
1 x 0.3
x 0.3
Stage h
0.5
0.5
H. Time Complexity
We analyze the time complexity of gDIS under different GNN structures and attention layers, as shown in Table 8. We observed that the runtime of the GAT layer (i.e., the graph-attention and GNN components) is significantly higher than that of the fully connected layer, indicating that the graph attention mechanism incurs substantial computational overhead for feature aggregation. Sparse graphs like Flickr exhibit lower runtime second stage despite their larger size, benefiting from reduced average degree. Furthermore, Py Porch Geometric optimizes sparse graph processing, enabling efficient computation even for large-scale datasets.
16

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Telling Peer Direct Effeets from Indirect Effeetts in Observational Network Data
Table 7. Hyperparameter search space for our method (gDIS).
Stage
Parameter
Search Space
Ir
{le-4, 5e-4, 1e-3, Se-3, lc-2)
GNN Layers
{1, 2,
First Stage
FC Layers
{1,2)
Hidden Dimension
{8, 16, 12)
Epochs
{100, 150, 180, 2003
Ir
(le-4, 5e-4, le-3, 5e-3, le-2) [1,2]
GNN Layers
FC Layers
{1, 212,3,4)
Second Stage
Hidden Dimension
{8. 16, 32, 64)
Epochs
[100, 150, 180, 2003
Third Stage
KDE Bandwidth (h)
{0.3, 0.5, 0.7)
Table 8. Layer Runtime on BC and Flickr Datasets: Training vs. Validation (seconds)
Dataset & Stage
Layer
Runtime (s)
Training
Validation
GAT Layer
0.00099
0.00100
-irs Stage
GAT Layer
0.00100
0.00199
Fully Connected Layer
0.00007
0.00007
GAT Layer
0.00300
0.00700
BC Second Stage
GAT Layer
0.00400
0.00701
GAT Layer
0.00200
0.00100
Flickr First Stage
GAT Layer
0.00400
0.00400
Fully Connected Layer
0.00007
0.00007
GAT Layer
0.00200
0.00200
Flickr Second Stage
GAT Layer
0.00100
0.00200
Note: The timing measurements are taken for individual layers separately rather than for the entire model forward pass.
17


Reviews:
[{"review_id": "eodtjDH6wd", "reviewer": "Reviewer_igcq", "summary": "The paper addresses the challenge of estimating treatment effects in observational network data with network interference. The authors propose a framework to decompose peer effects into direct and indirect peer effects and provide theoretical analyses of the identification conditions. Additionally, the paper introduces gDIS, a novel algorithm that leverages graph neural networks and attention mechanisms to estimate effects in network data.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the valuable comments. Detailed responses to each specific comment are provided below.\n\n**W1:** *The effectiveness of the proposed approach depends on strong assumptions, which may be challenging to satisfy in real-world observational data.*\n\nOur method is based on three key assumptions: Network Unconfoundedness, Network Consistency, and Network Overlap. These assumptions are commonly assumed in causal inference in networked data [1].\n\nIn this paper, our focus is on the challenge of distinguishing and estimating different types of causal effects in network data (i.e., the *Peer Direct Effect (PDE)*, *Peer Indirect Effect (PIE)*, and *Self-Treatment Effect (STE)*), and we adopt the assumptions commonly used in the network causal inference literature. \n\nWe have included discussions regarding the limitations of assumptions in the \"Limitations & Future Work\" section.\n\n- [1] Jiang, Song, and Yizhou Sun. \"Estimating causal effects on networked observational data via representation learning.\" *Proceedings of the 31st ACM International Conference on Information & Knowledge Management*, 2022.\n\n**Q1:** *Are there any specific requirements for the network structure in order to ensure the validity of the proposed approach?*\n\nOur proposed approach does not impose structural requirements on the network (e.g., specific topology or connectivity patterns). However, we do assume the causal relationships are as in Figure 2(b), which are realistic.", "claims_and_evidence": "Yes, the paper provides the theoretical proofs and empirical evidence to support the claims.", "methods_and_evaluation_criteria": "The authors validated the proposed approach on simulated and semi-synthetic data and compared the model performance with six baselines.", "theoretical_claims": "Yes", "experimental_designs_or_analyses": "I did not find particular issues with the experimental design.", "relation_to_broader_scientific_literature": "This paper fits well into the classic literature on estimating peer effects on networks for observational studies.", "essential_references_not_discussed": "Essential references were discussed in this work.", "other_strengths_and_weaknesses": "**Strengths:**\n1. The paper tackles a significant challenge in estimating peer effects under network interference. The paper employs GNN with attention mechanisms to capture network structure.\n2. Theoretical justification and empirical results of comparison with other classic estimators are provided.\n\n**Weaknesses:**\n1. The effectiveness of the proposed approach depends on strong assumptions, which may be challenging to satisfy in real-world observational data.", "other_comments_or_suggestions": "No", "questions_for_authors": "1. Are there any specific requirements for the network structure in order to ensure the validity of the proposed approach?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "9bFRgOIYi5", "reviewer": "Reviewer_GreV", "summary": "This paper focuses on differentiating between various types of causal effects in network data: peer-direct effects (PDE), peer-indirect effects (PIE), and self-treatment effects (STE). The authors propose a general setting to identify and estimate these effects, with theoretical identification conditions and proofs. They developed a method called gDIS (group-level Direct and Indirect effects estimator), which leverages graph neural networks (GNNs) with attention mechanisms and Hilbert-Schmidt Independence Criterion (HSIC) regularization to estimate these effects.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their insightful comments and recognition of our work's importance.\n\n**W1:** Network unconfoundedness assumption... more discussion needed\n\nWe plan to explore: a) Instrumental Variable methods [1] to introduce variables influencing treatment but not outcomes; b) Hidden confounder modeling [2] to leverage graph-based representation learning for capturing unobserved confounding; c) Causal discovery [3] to employ data-driven methods for identifying adjustment sets and uncovering hidden confounding pathways.\n1. Angrist, J.D., et al. Identification of causal effects using instrumental variables.\n2. Louizos, C., et al. Causal effect inference with deep latent-variable models.\n3. Colombo, D., et al. Learning high-dimensional directed acyclic graphs with latent and selection variables.\n---\n**W2:** Multiple components (GNN, attention, HSIC) - each component contributions\n\nWe'll add to Section 4.2:\n\n>**GNN Layers:** GNNs capture network interactions. Without them, our model couldn't leverage network structure for modeling peer effects, leading to oversimplified estimations that ignore relational dependencies.\n\n>**Attention Mechanism:** Enables assigning different weights to neighbors based on feature similarity, useful where peer influence isn't uniform.\n\n> **HSIC Regularization:** Mitigates overfitting by encouraging independence between node features and embeddings, reducing spurious correlations.\n---\n**W3:** More detailed sensitivity analysis of data generation processes needed\n\nGibbs sampling provides a practical method for approximating causal effects with complex interdependencies in networks by iteratively sampling from conditional densities [4]. In our data generation process, we varied noise levels and found that—since our iteration continues until the difference between values of $Y$ in successive steps is less than $1 \\times 10^{-5}$—the data exhibits high stability across different noise levels. This indicates our process effectively captures complex network interactions. We plan to explore additional data generation methods and conduct further sensitivity analyses.\n\n4. Zhao, Z., et al. Learning individual treatment effects under heterogeneous interference in networks. \n---\n**W4:** Extension to non-binary treatments would increase practical utility\n\nOur framework is flexible and does not rely on the treatment must be binary. When computing PDE and PIE, the neighbor treatment exposure variable $W_{t_i}$ supports continuous, multi-valued, or binary treatments. For STE, we've followed standard binary treatment approaches, but as a CATE (conditional average treatment effect) problem, it's extendable to continuous/multi-valued treatments [5-6].\n\n5. Hirano, K., et al. The propensity score with continuous treatments.\n6. Feng, P., et al. Generalized propensity score for estimating the average treatment effect of multiple treatments.\n---\n**W5:** Scalability discussion needed on large-scale datasets\n\nWe'll add the following to Appendix I:\n\n> Although attention mechanisms introduce additional overhead compared to standard GNNs, we optimize sparse graph processing using PyTorch Geometric [7], enabling efficient computation even on large-scale datasets. Additional strategies can enhance scalability: sampling-based techniques like those in GraphSAGE [8] can limit neighbors sampled for each node, reducing computational demands while preserving structural information. Approximation or sparsification techniques for the attention layer can also alleviate computational burdens [9].\n\n7. Fey, M., et al. Fast graph representation learning with PyTorch Geometric.\n8. Hamilton, W., et al. Inductive representation learning on large graphs.\n9. Child, R., et al. Generating long sequences with sparse transformers.\n---\n**Q1:** Robustness to violations of network unconfoundedness\n\nWe're currently analyzing violations of the network unconfoundedness assumption. Our preliminary expectation is that robustness depends on factors like the strength of association between hidden confounders and treatment/outcome. Future work will include comprehensive sensitivity analyses to quantify how these factors affect performance.\n\n---\n**Q2:** Extension to continuous treatments or complex outcomes\n\nYes, gDIS can handle continuous treatments and complex outcome structures. Our framework is flexible with no binary treatment requirement. When computing PDE and PIE, the neighbor treatment exposure variable $W_{t_i}$ accommodates any treatment type. For STE, our implementation follows standard causal inference with binary treatment effects, but as STE estimation is a typical CATE problem, our approach can extend to continuous or multi-valued treatments using existing techniques [6].\nFor multivariate outcomes, the framework can be adapted using joint modeling approaches (e.g., multi-output regression [10]) to capture dependencies among multiple outcomes.\n\n10. Sener, O., et al. Multi-task learning as multi-objective optimization.", "claims_and_evidence": "The claims regarding the identification of causal effects appear well-supported by the theoretical results. The empirical claims about the superior performance of gDIS are backed by comprehensive experimental results comparing against several baselines.", "methods_and_evaluation_criteria": "The proposed methods appear appropriate for the problem. The evaluation of semi-synthetic datasets with ground truth is a standard approach in causal inference research, and the metrics used (MSE and PEHE) are appropriate for evaluating causal effect estimates.", "theoretical_claims": "I reviewed the theoretical claims related to identification conditions in Section 4.1, including Lemma 4.1 and Theorem 4.2. The proofs appear sound, with appropriate application of do-calculus and backdoor adjustment principles.", "experimental_designs_or_analyses": "The experimental design comparing against multiple baselines and analyzing performance under various conditions (e.g., treatment flip rates, hyperparameter sensitivity) is thorough. The simulation of treatments and outcomes follows established practices in the causal inference literature.", "relation_to_broader_scientific_literature": "The work clearly relates to the literature on causal inference in networks and mediation analysis, properly contextualizing its contributions in relation to existing approaches.", "essential_references_not_discussed": "No major omissions were found, though recent work on continuous treatment effects in networks could be relevant.", "other_strengths_and_weaknesses": "Strengths:\n1. The paper addresses an important gap in the literature by differentiating between direct and indirect peer effects in network settings, which is crucial for many real-world applications such as public health interventions and marketing campaigns.\n2. The theoretical foundation is solid, with clear identification conditions and proofs that provide guarantees for the proposed methods. \n3. The use of causal mediation analysis principles to differentiate between different types of peer effects is elegant and well-executed.\n\n\nWeaknesses:\n1. The assumption of network unconfoundedness (Assumption 3.8) is quite strong and may be violated in many real-world settings. While the authors acknowledge this limitation in the conclusion, more discussion on potential approaches to relax this assumption would strengthen the paper.\n2. The complexity of the model with multiple components (GNN, attention, HSIC) makes it potentially difficult to implement and tune for practitioners. A more detailed analysis of the relative contributions of each component could help guide practitioners on which aspects are most crucial.\n3. The simulated outcomes using Gibbs sampling may not fully capture the complex interdependencies in real network data. A more detailed sensitivity analysis of different data generation processes would strengthen the validity of the results.\n4. The paper focuses primarily on binary treatments, while in many real-world scenarios, treatments might be continuous or multi-valued. Extending the approach to handle such cases would increase its practical utility.\n5. While the paper mentions the time complexity analysis in the appendix, a more thorough discussion of scalability to very large networks would be beneficial given the computational demands of GNNs with attention mechanisms.", "other_comments_or_suggestions": "None", "questions_for_authors": "1. How robust is the approach to violations of the network unconfoundedness assumption? Have you conducted any sensitivity analyses to assess this?\n2. Could the gDIS framework be extended to continuous treatments or more complex outcome structures (e.g., multivariate outcomes)?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "rAk61ztfBQ", "reviewer": "Reviewer_dnoa", "summary": "This paper studies the causal effect estimation problem without SUTVA assumption. Specifically, the authors identify the overlooked problem that existing methods cannot distinguish between peer (in)direct effects and self-treatment effects. The authors propose a method called gDIS to estimate these estimands in the network scenarios. The proposed method is based on the back-door criteria, containing three stages to estimate the required density functions. The experimental results show its effectiveness.\n\n## update after rebuttal\nAll my concerns are well addressed. I will keep my positive score.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the valuable comments. Detailed responses to each specific comment are provided below.\n\n**Weakness 1:** *Lack of comparison with PIE/PDE estimators.*\n\nThe PIE/PDE estimators cited in our introduction (e.g., VanderWeele et al., Shpitser et al.) are designed for One-to-One interference settings, assuming dyadic interactions (e.g., household or pairwise influence). Our work focuses on the more general and realistic Many-to-One network setting. As such, these estimators cannot be directly or fairly compared with our method. Instead, we compare with recent network-level causal inference methods (e.g., NetEst, TNet, 1-GNN), which are designed to handle complex interference and represent the state-of-the-art in our setting.\n\n---\n**Other Comments Or Suggestions: Typos**\n\nWe thank the reviewer for the careful reading. We have corrected all.\n\n---\n\n**Question 1:** *Since it is \"Many-to-One\" interference, units will affect each other. For the summary causal graph of all units, is it reasonable to use the DAG tool to represent the causal relationship for their interference?*\n\nThe DAG shown in Figure 2b illustrates the unit-level causal relationships from the perspective of a single unit $i$, depicting how $i$'s outcome is affected by their own treatment and neighbors’ treatments and outcomes. This type of localized DAG is commonly used in the literature on causal inference with interference [1].\n\n[1] Jiang, Song, and Yizhou Sun. \"Estimating causal effects on networked observational data via representation learning.\" *Proc. ACM Int. Conf. Inf. & Knowl. Manag.* 2022.\n\n---\n\n**Question 2:** *There is still a lack of comparison with existing peer direct/indirect effect methods cited in the introduction. Is it because they are of the \"One-to-One\" type and cannot be applied to this scenario?*\n\nYes. The methods cited in the introduction (e.g., VanderWeele et al., Shpitser et al.) are designed for One-to-One settings and assume dyadic interactions. Our setup considers Many-to-One interference, which is more general and realistic in network data. These earlier methods cannot be directly or compared in our setting. Hence, we have compared with recent methods (e.g., NetEst, TNet, 1-GNN) that support many-to-one relationships.\n\n---\n\n**Question 3:** *The potential outcome simulation is through an iteration process. How do you obtain the ground true direct/indirect effects?*\n\nAs stated in [2-3], Gibbs sampling provides a practical method for approximating the causal effects of interest in the presence of complex interdependencies among individuals in a network. This is achieved by iteratively sampling from the conditional densities of each individual. Our iteration continues until the difference between the values of $Y$ in successive steps is less than $1 \\times 10^{-5}$, thereby ensuring that the generated data has reached a stable state. Then, using a given structural causal model (Equation 19 in the Appendix F), we can obtain the potential outcomes under different treatment conditions $T$. The ground true direct/indirect effects can be obtained from the potential outcomes.  \n\n[2] Zhao, Ziyu, et al. \"Learning individual treatment effects under heterogeneous interference in networks.\" *ACM Trans. Knowl. Discov. Data* 18.8 (2024): 1-21.  \n[3] Tchetgen Tchetgen, Eric J., Isabel R. Fulcher, and Ilya Shpitser. \"Auto-g-computation of causal effects on a network.\" *J. Am. Stat. Assoc.* 116.534 (2021): 833-844.\n\n---\n**Question 4:** *Could you provide more experimental results with a wider range of $\\lambda$? The paper claims $\\lambda = 0.3$ is optimal, but it seems only suboptimal as the curves in Figure 5 are still descending as $\\lambda$ increases.*\n\nIn our experiments, we evaluated $\\lambda$ values ranging from 0 to 0.5 (results available at the [anonymous link](https://anonymous.4open.science/r/icmlSupp-4556/2.jpg)). We found that $\\lambda = 0.3$ produced the best overall performance. When $\\lambda$ exceeds 0.3, it starts to over-penalize the feature representations, leading to a decline in performance.\n\n---\n\n**Question 5:** *Could you clarify how to choose the other hyperparameters shown in Table 6 and provide the hyperparameter space of the compared baselines?*\n\nFor our method, we performed a grid search on the validation set to select the key parameters (learning rate, hidden dimensions, and regularization strength) that yielded the best performance (shown in Table 6, Appendix H). Following this comment, we have updated the search space for these key parameters in Appendix H, as shown in [anonymous link](https://anonymous.4open.science/r/icmlSupp-4556/table1.png). For benchmark methods, we followed the recommended settings (as shown in [anonymous link](https://anonymous.4open.science/r/icmlSupp-4556/table2.png)) from their original papers and publicly available implementations.", "claims_and_evidence": "The claims in this paper are supported by clear proofs and experimental results.\n\nHowever, I am a bit confused by the DAG presented in Figure 2b.\n\nQ1: Since it is 'Many-to-One' interference, units will affect each other. For the summary causal graph of __all units__, is it reasonable to use the DAG tool to represent the causal relationship for their interference?", "methods_and_evaluation_criteria": "The proposed gDIS method makes sense to the network effect estimation problem.", "theoretical_claims": "I checked all proofs roughly and did not find any problems.", "experimental_designs_or_analyses": "The experimental designs and analyses are sound overall. However, I still have the following questions:\n\nQ2: There is still a lack of comparison with existing peer direct/indirect effect methods cited in the introduction. Is it because they are of the 'One-to-One' type and cannot be applied to this scenario?\n\nQ3: The potential outcome simulation is through an iteration process. How do you obtain the ground true direct/indirect effects?", "relation_to_broader_scientific_literature": "The studied problem is an overlooked problem, and this paper provides a practical and reasonable solution to estimating PDE/PIE and STE.", "essential_references_not_discussed": "The key related works are cited in the paper.", "other_strengths_and_weaknesses": "Strengths\n\n1. The setting is interesting.\n2. The proposed estimator is reasonable and effective.\n\nWeaknesses\n\n1. Lack of comparison with PIE/PDE estimators.", "other_comments_or_suggestions": "typos:\n\n1. Line 042: ana -> and\n2. Line 087: method which do not. \n3. Line 109: methods not -> methods do not\n4. Line 198: frrom -> from", "questions_for_authors": "Please see Q1-Q3 in the above discussion.\n\nQ4: Could you provide more experimental results with a wider range of $\\lambda$? The paper claims $\\lambda=0.3$ is optimal, but it seems only suboptimal as the curves in Figure 5 are still descent as $\\lambda$ increases.\nQ5: Could you clarify how to choose the other hyperparameters shown in Table 6 and provide the hyperparameter space of the compared baselines?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ZHZQJ7VFYY", "reviewer": "Reviewer_oyya", "summary": "Estimating causal effects in observational network data is challenging due to peer interactions. Existing methods struggle to distinguish different types of peer effects. To address this, the proposed approach defines a general setting that considers peer direct effects, peer indirect effects, and individual treatment effects, along with their identification conditions. Using causal mediation analysis tailored for network data, the method differentiates these effects. It incorporates attention mechanisms to capture varying neighbor influences and employs multi-layer graph neural networks (GNNs) to explore high-order neighbor effects. Additionally, the Hilbert-Schmidt Independence Criterion (HSIC) enhances model robustness. Extensive experiments on semi-synthetic and real-world recommendation datasets validate the approach, with potential applications in social networks and public health interventions.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely appreciate the reviewer's insightful comments and the high recognition of the value and importance of our work. \n\n**Comment:** *Baseline models details needed: Which peer and self-treatment effects do they evaluate?*\n\nIn our simulation, the baseline models provide estimates of the overall peer effects (PE) and self-treatment effects (STE). Specifically, the baseline methods estimate:\n\n- **Peer Effects (PE):** This represents the aggregated influence of peers on an individual's outcome. These models capture the total effect that the treatments and outcomes of an individual's neighbors have on that individual but do not distinguish between the direct influence (Peer Direct Effects, PDE) and the indirect influence (Peer Indirect Effects, PIE).\n\n- **Self-Treatment Effects (STE):** This measures the effect of an individual's own treatment on their outcome.\n\nThus, while the baseline models evaluate the combined peer effect and the self-treatment effect, they do not explicitly decompose the peer effects into PDE and PIE. Our proposed gDIS framework, on the other hand, is designed to disentangle these two components, offering a more detailed analysis of the peer influences within network data.\n\n\n**Q1:** *Product promotion example to main paper?*\n\nWe plan to include the product promotion campaign example in the introduction, alongside the existing example from epidemiology. We will move the proof to the appendix to make space for the example.\n\n**Q2:** *$W_{t_i}$, $W_{y_i}$ notation appears to be ambiguous. Do they depend on the realized values of $T$ and $Y$ for individual $i$?*\n\n$W_{t_i}$ and $W_{y_i}$ do not depend on the realized values of $T_i$ or $Y_i$. They represent summary of the treatments and outcomes of unit $i$'s neighbors. $W_{t_i}$ and $W_{y_i}$ are illustrated in Figure 2b and explained in Line 313 (left column), and Appendix B.\n\n**Q3:** *Definition 3.6 as proposition?*\n\nWe agree that Definition 3.6 may be more appropriately stated as a proposition as it presents a reformulation and decomposition of the peer effects introduced in Definitions 3.4 and 3.5.\n\nOur intention was to highlight this decomposition as a central conceptual component of our framework, which is why we initially presented it as a \"definition.\" However, we appreciate the reviewer's point and will revise the manuscript to present it as a proposition.\n\n**Proposition** (Peer Effect Decomposition).\nThe peer effect (PE) can be decomposed into the sum of the PDE and the PIE. That is,\n\n\n\\\\begin{aligned}\n\\\\text{PE}(w' _{t _i}) &= \\\\text{PDE}(w' _{t _i}) + \\\\text{PIE}(w' _{t _i}),\n\\\\end{aligned}\n\n\\\\begin{aligned}\n\\\\text{PE}(w _{t _i})  &= \\\\text{PDE}(w _{t _i}) + \\\\text{PIE}(w _{t _i}). \n\\\\end{aligned}\n\n\nThe peer effect (PE) can be decomposed into the sum of the peer direct effect (PDE) and the peer indirect effect (PIE). That is,\n\nThis proposition shows that the total peer effect integrates both the direct and indirect pathways of peer influence. Equation (3a) and Equation (3b) correspond to the vaccination and product promotion examples, respectively.\n\n**Q4:** *Alignment with previous work?*\n\nOur definition of peer effects aligns with prior works in the one-to-one setting, except that our peer treatment and outcomes are summaries of the treatments and outcomes of the neighbors of $i$, respectively.\n\n**Q5:** *Total effect representation?*\n\nYes, you are right when there is no interaction between peer treatment $W_{t_i}$ and individual treatment $T_i$. We have assumed this in our work. This is a very reasonable assumption, as individual $i$'s treatment and peer treatment do not affect each other directly, and the reason for an individual to take the same treatment as their peers is because the individual and their peers have similar features (characteristics).\n\n**Q6:** *Baseline vs. gDIS differences?*\n\na) In Table 2 of the paper, we made a mistake in summarizing the results for gDIS. We have updated the table and present the revised version at the following link (due to space limitations): https://anonymous.4open.science/r/icmlSupp-4556/table3.png\n\nb) The baseline methods' estimates of PE (peer effects) and STE (self-treatment effects) are formally consistent with our model; however, they do not explicitly decompose the PE into PDE (peer direct effects) and peer indirect effects (PIE).\n\nc) Using the given structural causal model (Equation (19) in Appendix F), we can obtain the potential outcomes under different treatment conditions.\n\n**Q7:** *Error computation for counterfactuals?*\n\nThe potential outcomes with respect to $T_i$ for each $i$ are known since the structural causal model is assumed to be known for the synthetic dataset. Hence, the ground truth values of STE, PE, PDE, and PIE can be derived. Subsequently, compute the estimation error using the PEHE (Precision in Estimation of Heterogeneous Effect) metric, by comparing the estimated effect with the ground truth causal effect, i.e., the difference in potential outcomes.", "claims_and_evidence": "This paper is well-structured and supported by evidence.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "Yes. The identifiability of PDE, PIE, and STE is a natural extension of previous works.", "experimental_designs_or_analyses": "Yes.", "relation_to_broader_scientific_literature": "This contribution enhances our ability to optimize intervention strategies in public health, marketing, and social influence analysis. Peer effects is an emerging topic in causal inference.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "Strength.\nThey introduced the concepts of self-treatment effects, peer effects, and both direct and indirect peer effects within a many-to-one framework. Additionally, they proposed a novel gDIS algorithm, which demonstrates strong performance.\n\nWeakness.\nSee my questions.", "other_comments_or_suggestions": "In simulation, more details regarding the baseline models are required. Specifically, what types of peer and self-treatment effects do they evaluate?\nTypos: Line 041 (right)", "questions_for_authors": "1. Do you plan to move the motivating example of the product promotion campaign to the main body of the paper?\n2. The notation W_{t_i} and W_{y_i} appears to be ambiguous. Do they depend on the realized values of  T  and  Y  for individual  i ?\n3. You have defined peer effects through Definitions 3.5 and 3.6. However, I believe that Definition 3.6 is not a definition but rather a proposition.\n4. Does your definition of peer effects align with previous works in the case of one-to-one relationships?\n5. What does the total of PIE, PDE, and STE represent? Does it correspond the totoal effect of w_{t_i}' to w_{t_i} and 0 to 1 for individual i on outcome?\n6. In simulations, why do the estimates for baselines and gDISs differ significantly? Do baseline estimates different peer and self-treatment effects? Is it possible to compute the ground truth values of PDE, PIE, and STE?\n7. How did you compute error for each estimator? The values of counterfactuals in each effect are unobserved.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: All reviewers are leaning weakly positive on this work and agree it addresses an interesting topic on the decomposition of peer effects in causal inference with network interference. Some concerns remain around the strong assumptions underlying the method.
Decision: Accept (poster)