Learning Exposure Mapping Functions for Inferring Heterogeneous Peer Effects
Authors: Shishir Adhikari
Sourav Medya
Elena Zheleva
Venue: ICLR 2026 Poster (2026)
Keywords: causal inference
peer effects
network interference
exposure mapping function
graph neural network
Primary area: causal reasoning
TL;DR: We propose GNN-based exposure mapping function learning to learn expressive peer exposure representation for robust heterogeneous peer effect estimation.

Abstract: Peer effect refers to the difference in counterfactual outcomes for a unit resulting from different levels of peer exposure, the extent to which the unit is exposed to the treatments, actions, or behaviors of its peers. In practice, peer exposure is typically captured through an explicitly defined exposure mapping function that aggregates peer treatments and outputs peer exposure. Exposure mapping functions range from simple functions like the number or fraction of treated friends to more sophisticated functions that allow for different peers to exert different degrees of influence. However, the true function is rarely known in practice and when the function is misspecified, this leads to biased causal effect estimation. To address this problem, the focus of our work is to move away from the need to explicitly define an exposure mapping function and instead introduce a framework that allows learning this function automatically. We develop EGONETGNN, a graph neural network (GNN), for heterogeneous peer effect estimation that automatically learns the appropriate exposure mapping function and allows for complex peer exposure mechanisms that involve not only peer treatments but also attributes of the local neighborhood, including node, edge, and structural attributes. We theoretically and empirically show that GNN models that use peer exposure based on the number or fraction of treated peers or learn peer exposure naively face difficulty accounting for such influence mechanisms. Our evaluation on synthetic and semi-synthetic network data shows that our method is more robust to different unknown underlying influence mechanisms when compared to state-of-the-art baselines.

Published as a conference paper at ICLR 2026
LEARNING EXPOSURE MAPPING FUNCTIONS FOR INFERRING HETEROGENEOUS PEER EFFECTS
Shishir Adhikari" Sourav Medya Elena Zheleva University of Illinois Chicago, Chicago, IL, USA {a (sadhik9, medya, sehelevaa@eic.een edu
ABSTRACT
Peer effect refers to the difference in counterfactual outcomes for a unit resulting from different levels of peer exposure, the extent to which the unit is exposed the treatments, actions, or behaviors of its peers. Peer exposure is typically captured through an explicitly defined exposure mapping function that aggregates peer treatments and outputs peer exposure. Exposure mapping functions range from simple functions like the number or fraction of treated friends to more sophisticated functions that allow for different peers to exert different degrees influence. However, the true function is rarely known in practice and when the function is misspecified, this leads to biased causal effect estimation. To address this problem, the focus of our work is to move away from the need to explicitly define an exposure mapping function and instead introduce a framework that allows learning this function automatically. We develop EGONETGNN, graph neural network (GNN), for heterogeneous peer effect estimation that automatically learns the appropriate exposure mapping function and allows for complex peer exposure mechanisms that involve not only peer treatments but also attributes of the local neighborhood, including node, edge, and structural attributes. We theoretically and empirically show that GNN models that use peer exposure based on the number or fraction of treated peers or learn peer exposure naively face difficulty accounting for such influence mechanisms. Our evaluation on synthetic and semi-synthetic network data shows that our method is more robust to different unknown underlying influence mechanisms when compared to kate-of-the-art basclines.
INTRODUCTION
In networked environments, the outcome of a unit can be influenced by the treatments or outcomes of other units, a phenomenon known as interference. For example, a contact network, the smoking habits of peers may affect an individual's 's respiratory health, and in social network the political affiliations of peers may influence one's stance on a policy issue like immigration. Peer effects capture this influence by comparing an individual's outcomes under different peer network conditions (e.g.. having no smoker peers versus some smoker peers, or observed peer political affiliations versus counterfactual, flipped affiliations). Peer effect estimation is important for policy-makinn and targeted intervention design in many domains, including healthcare (Barkley et al. (020), online advertisement (Nabi al. 2022 and education (Patacchini al. 2017).
Peer network conditions are typically captured through an explicitly defined exposure mapping function (Aronow and Samii 2017) that summarizes the peer treatments and peer network and outputs peer exposure, which is the equivalent to a composite peer treatment value. The peer effect is defined as the difference in outcomes under two distinct levels of peer exposure. Different peer exposure mapping functions capture different possible underlying influence mechanisms. Typically. domain experts define exposure mapping functions appropriate to the causal question and domain of interest. The advantage of exposure mapping functions is that they reduce the high dimensionality of peer network attributes and that they are invariant to irrelevant contexts (e.g., permutation of peers).
Figure I presents examples of prominent exposure mapping functions and the resulting peer exposure values for a toy peer network. The first graph shows Gaby's peer network along with the observed
"Currently affiliated Icahn School of Medicine Mount Sinai, NY, USA.

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Anna
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Treatment
Example exposure mapping functions
Peer exposure value
Factual
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Counterfactual treatments
Closed tradd: Count
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Illustration of different possible peer exposure representations for a node (Gaby) in a toy peer network. Red nodes represent peers in the treatment group, and blue nodes represent peers in the control group. Gray star node represents the node that has a fixed treatment.
Figure 1:
(i.e., factual) treatments for Gaby's peers. The second graph shows hypothetical (i.e., counterfactual) treatments for the peers. The peers in the treatment group (e.g., smokers) and control group (e.g.. non-smokers) depicted as red and blue nodes, respectively. The edge weights capture the tie strengths in the network. Binary peer exposure mapping is the simplest and it summarizes peer treatments 0 or 1, e.g., whether any peers have been treated (Bargagli-Store whether the weighted treatment of peers has reached linear threshold (Tran and Zheleval 2022).
et al.) 2025) or
Some exposure mapping functions assume that all peers influence equally (e.g., fraction of treated peers (Hudgens and Halloran 2008 Jiang and Sun 2022). while others consider that different peers can exert different degrees of influence (e.g., weighted fraction (Forastiere et al., 202I) or sum (Zhao) et al. 2024) of treated peers). Peer exposure has also been modeled with counts of different causal network motifs, i.e., recurrent subgraphs in a unit's 's's peer network with treatment assignments as attributes (Yuan et al. 2021). We discuss the related work in more detail in the Appendix A.2
A key challenge in peer effect estimation is that the true exposure mapping function is rarely known in practice and when the function is misspecified, this leads to biased causal effect estimation. The focus of this paper is to move away from the need to explicitly define an exposure mapping function and instead learn this function automatically from data. This has the advantage of reducing subjectivity and allowing for automated representation of peer exposure under unknown and complex peer influence mechanisms. More specifically, we study the problem of exposure mapping function learning in the context of heterogeneous peer effect estimation. Heterogeneous peer effects (HPE) denote variation in peer effects across individuals that may originate from personal attributes or from characteristics of their peer networks. For example, while having friend who smokes may have negative effect on health for some people, it may make no difference for others.
We propose EGONETGNN: novel graph neural network (GNN) architecture, that automatically learns relevant exposure mapping function under appropriate identifiability assumptions. EGONET- GNN allows for complex peer influence mechanisms that, in addition to peer treatments, can involve the local neighborhood structure, node, and edge attributes. Our work builds upon the success of utilizing neural networks (NNs) (Shllt et al. 2017 Im et al. 202 Shi al. 2019) and graph neural networks (GNNs) (Jiang and Sun 2022 Cai et al. 2023; Chen et al. 2024: Khatami et al.) 2024) for end-to-end learning of counterfactual outcome models causal effect estimators. Few studies have utilized GNNs to learn the exposure mapping function (Mao et al.) [005]] Wu et al.) 2025) or to derive peer exposure embedding by aggregating feature embeddings and peer treatments (Adhikari and Zheleva, 2025 Zhao et al. (2024). However, these works use off-the-shelf GNNs like GCN (Kipf and Welling 2016) or GIN (Xu et al. 2018) and prior work (Chen al.) 2020) has shown such architectures lack expressiveness for counting subgraphs with cycles and for capturing mechanisms involving local neighborhood structure. Yet counts of such subgraphs, like causal network motifs, are rich features for capturing local structural contexts (Yuan et al.) 202T), but they are expensive to compute, inflexible, and may not capture every local structural context (e.g., edge weights).
EGONETGNN captures exposure mapping functions from previous works, including relevant causal network motifs and scaling to higher-order motifs. To add robustness to the downstream peer effect estimation task, EGONETGNN is designed to learn the exposure mapping function to produce repre- sentation that is expressive to differentiate between different peer exposure conditions and invariant to irrelevant contexts. Moreover, EGONETGNN is designed to promote bounded representation with substantial coverage of possible peer exposure values. Figure 2 shows an overview of EGONETGNN. While most peer effect estimation frameworks contain feature mapping and counterfactual out-
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Figure 2: An overview of the proposed EGONETGNN model to learn exposure mapping function for peer effect estimation. EGONETGNN extracts ego networks, for each node with peer treatments along with feature embedding and ego edge attributes as peer features. Then, node-level aggregations are performed to capture local neighborhood contexts. These contexts are passed through a masked weight layer and encoded by an multi-layer perceptron (MLP) to learn relevant influence mechanisms and summarized with graph-level aggregation. The learned peer exposure embeddings (p.). along with the feature embeddings (c₂), and treatment (fz) are passed to a counterfactual outcome model that is used to infer peer effects. The graph transformation ensures expressiveness, while balance, coverage, entropy, and sparsity losses promote the robustness of the peer exposure representation.
come model component, the novel additional component in ours is the custom-designed exposure mapping function learning (marked in green in the figure). We design this component to excel in counting attributed subgraphs, such causal network motif's, enhancing its expressiveness to capture unknown underlying peer exposure mechanisms. We theoretically and empirically show that, unlike EGONETGNN, existing GNN- approaches that solely rely on homogeneous peer exposure or learn heterogeneous peer exposure naively lack expressiveness in capturing heterogeneous peer influence mechanisms based on local neighborhood structure.
CAUSAL INFERENCE PROBLEM SETUP
Notations. We represent the network undirected graph g (V,E) with set of [V] nodes,
a set of edges ε, node attributes X, and edge attributes Z. Let t =< fi,.... Ia,,.,, > be a random variable comprising the treatment variables 1, for each node v, random variable for 14 outcome. Let π =<
in the network and y; be Ts.... T. > be an assignment to t with n. E {0. 1} l. and . = π क. denote random variable and its value for treatment
assigned to t. Let t.: assignment to other units except which we refer to as peer treatments for convenience.
Peer exposure reflects how much unit is exposed to peer treatments and defined as follows.
Definition 1 (Peer exposure and exposure mapping function). Peer exposure for unit 0; is defined as P. E [0, 1jdh de ( X, Z), where &e is the exposure mapping function that maps high- dimensional contexts (T- G. X, Z) to d-dimensional peer exposure representation bounded be- tween 0 and such that y, (t.
t-, π. )) (G, X, Z y, n. P. p.)l(9, X,
Definition 1 maps peer treatments t.; -- and peer network contexts X9 Z} to peer exposure P; P; in terms of equivalence of counterfactual outcomes (4 ,, t-, = - and (t = n. P. = p:). Here, (t Ti Pi p.). captures that, in interference settings, the counterfactual outcome of a unit v; is influenced by both unit's treatment t, = n, and peer exposure p. p... Note that the exposure mapping function could map different contexts to the same peer exposure.
Peer effect refers the difference in counterfactual outcomes for different values of peer exposure. Heterogeneous peer effects (HPE) refers to different units having different peer effects dependent on their contexts. For any given unit v; its heterogeneous peer effect is described through its context, i.e., for peer exposures p; P; versus = p and unit's treatment I; conditioned on the unit's contexts C, it is defined as:
(1) where expectation is over units with similar contexts c., referred to as effect modifiers (c.g.. unit's degree node attributes), defined by feature mapping function of contexts (g. X, Z) from v; 's perspective, i.e., c = (5 G, x, Z). Substituting peer exposures pt and p with corresponding
8p p!) Ejy, (1. =
p:)]c.] E(y p: pp))c:].
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exposure mapping functions for two peer treatment assignments -; versus π² in Eq.
get:
_,( (ly,(t: n., P. ( -, 9, X, - (t: P. Φ (T G. X, c)) (2)
Causal identification. Now, we discuss the identification of peer effects that involves expressing counterfactual outcomes in terms of observational and/or interventional distributions. First, we make two commonly adopted assumptions network interference settings.
Assumption 1 (Pre-treatment network). The network along with node attributes X and edge attributes Z are measured before treatment assignments t = π and treatments are not mutable.
Assumption (Neighborhood Interference). The counterfactual outcome of a unit depends only on its immediate neighborhood treatments, i.e., (4 = t-; = --i)le; yy(4: = nT. t क = y; ni, de (T G, X, Z),,,, where N₁ (j (14 n2) E E), t- = fty 1 E MM., and j E AG denote neighborhood set, treatments, and assignments, respectively.
Assumption D is general assumption in experimental and observational studies, and Assumption 2 is common simplifying assumption that presumes network influence is mediated by immediate neighbors but our work could be extended to consider interference from multiple-hop neighborhoods. For ease of exposition, we drop the superseript N₁ in neighborhood treatments and assignments.
For causal identification, we assume unconfoundedness, similar to previous work (Ma et al. 2022
Wu et al. )025):
Assumption 3 (Unconfoumdedness). For all unit treatment m, E {0, 1] and peer treatment assignments v- E (0, 13m-1, there exists feature mapping function o, E 2 and an exposure mapping function &e 4. such that the counterfactual outcome is independent of unit treatment and peer exposure conditions given the context c; = ( G, X, Z), i.e., (1 T. Pi ( X, Z)) 1 (t,, p:)
inmmpi implies that the observed network context is sufficient for controlling for confounding. and there are functions able to represent it compactly. Under this assumption, it is still possible to learn feature mapping and exposure mapping functions that do not approximate the true functions which leads to a misspecification error. Therefore, it is important to learn an expressive function (e.g., a GNN) that is able to capture wide range of possible functions. We also assume the standard consistency (Assumption] 4 and positivity (Assumption [)) described in more detail in Appendix A.. Next, we present the causal identification conditions and formally define the problem of exposure mapping function learning in the context of peer effect estimation.
Assumption
Proposition 1. With Assumptions observational data as
5 the HPE 8, Eq. A.3 can be estimated from experimental or
8. (T = Ely./t: m. p: (n G. X. Z), (5 X. Z)] -Ely, T. (. 9, X, Z), C, = 9, Z)).
The proof presented in Appendix A.3 stems from consistency and unconfoundedness assumptions. Problem 1 (Exposure mapping function learning). Given network contexts {g. X, Z), treatments t, and outcomes of n units, estimate the feature and exposure mapping functions Ø, and &e along with counterfactual outcome model fy such that mean squared error between true heterogeneous peer effect (HPE) 6 and estimated HPE 8, i... E (8, 6.)2, is minimized, where = fy (m. C:) (T,, .. Pl, ē.) with أ. = óe ( G, x, 2), p. = de (T'L,, G, X, Z), and é; = लy (v, 9, x, Z).
The true HPE is unknown, but due Proposition]] I the factual outcomes can be utilized to jointly estimate 6y, Se. and fr; as discussed in the next section.
EGONETGNN: LEARNING EXPOSURE MAPPING FUNCTION WITH GNNS
Figure 2 shows overview of the proposed EGONETGNN model simultaneously learn exposure mapping function &., feature mapping function éy, and counterfactual outcome model f, for peer effect estimation. We aim to learn exposure mapping function ó. with three key properties: I) expressiveness, 2) invariance, and 3) bounded and balanced representation. The expressiveness property ensures the peer exposure representation P; returned by the function & is unique for
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different relevant contexts, while the invariance property assures the representation Ps does not vary due to irrelevant contexts. For example, in Figure 1 if the underlying peer influence depends on clustering coefficients among treated, the function Φe is expressive if it can capture the first closed triad substructure. The standard message passing GNNs (e.g. GCN, GIN, etc) cannot capture essential causal network motifs like closed triads (i.e., triangular motifs) (Chen et al., (020). The graph transformation and automated exposure mapping function learning our EGONETGNN model are designed to ensure that the peer exposure representation is at least as expressive as or superior to the approach of feature extraction by counting causal network motifs. In the above example, the
function Φe is invariant to irrelevant contexts if the difference in other features like node attributes and edge weights do not change the learned representation p.. To satisfy the property of bounded representation, the learned representation p; should be bounded, e.g., between 0 and 1, to reflect no exposure and maximum exposure. Moreover, the representation P: should have substantial coverage, which means it should be distributed across the possible range of exposure. Next, we describe our feature mapping, exposure mapping, and counterfactual outcome model in detail.
3.1 ARCHITECTURE OF EGONETGNN
EGONETGNN first maps the attributed network to feature embedding using a MPGNN and extracts ego networks for each node v; incorporating peer treatments, node features, and edge attributes. It performs node-level aggregation to capture local context, which is processed through a masked weight layer and an MLP followed by graph-level aggregation to learn peer exposure representation.
Feature mapping MPGNN. The feature mapping module aims to capture contexts that are potentially confounders or effect modifiers. Let @ denote a multi-layer perceptron (MLP) and || denote a concatenation operator. The feature embedding is obtained for 1-th layer as:
C; (x()))) h² and h lu + Σ 2ony
(3)
JEN,
Xz1, and h 0 are initial conditions and MN denote neighbors of node u,. This MPGNN architecture incorporates edge attributes Z.y while disentangling the hidden representation of the unit's own attributes ()(X,) from that of aggregated peer and edge attributes hi.
where Ko
Ego network construction. To learn an exposure mapping function that is as least as expressive as or superior the approach of feature extraction by counting network motifs, transform the node regression task to graph regression by extracting ego networks for each unit. In the extracted network, the triangle structures involving an ego node transformed dyads, which mitigates the limitation of GNNs to capture closed triad motifs. The ego network G. G.V,. E.) is extracted from G(V, E) for each node V; such that node set V. consists neighbors of Uú i..., 17 = {uy & consists edges between neighbors of 0s i.e., & fezk eza E ay E V. A E V.).
: Biy E E E V) and edge set
Peer feature encoder and aggregation. Peer feature encoder module takes relevant peer feature embeddings and the distance between ego and peer feature embeddings, i.e., Cuj = 2 (eyfearc,)l(C (c)l((c, ),))), to capture peer influence mechanisms involving peer attributes and feature similarity between ego and peers. Then, we transform an ego wy edge attributes Z,, to node attributes, i.e., X, Zy, in the ego network (),,) Es) because the ego itself is not present in the ego network. The node aggregation for each node by in the ego network G. considers neighbors' node attributes Xx. feature encoding Cak. edge attributes Zzk. and neighbor treatments t.. and is defined for 1th layer as follows:
h
h
i-1 +
Σ h with h t X |/c. Zzz and h O. REN;
(4)
Masked weights and exposure encoder. Masked weights promotes representation that is invariant irrelevant contexts and feeds the concatenation of node attributes and hidden state after layers of node aggregation, i.e., h229 X,xXll [h, through masked fully connected layer as follows:
h mask ReLU(((( mask) W Wh bagg).
(5)
where ReLU and a are rectified linear unit and sigmoid activation functions, C indicates element- wise product, W mask and W agg are the weight matrices, and bagg is the bias vector. The masked hidden representation hmask is passed into an exposure encoder MLP to extract low dimensional em- bedding. The goal of this module is to capture complex mechanisms based on the local neighborhood and reduce dimensionality. Formally, the output embedding h::p is obtained as follows:
h (0s. ReIU (e (h]mask)) + 1))),
(6)
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Gene and Geap are two MLPs and In denotes log transformation that offers the benefit of rescaling features with large values that are significant in scale-free networks (e.g.. online social networks) and introduces inductive bias capture mechanisms involving ratios.
Graph readout. Finally, the peer exposure embedding p; for node v; is obtained by aggre- gating the representation h for all vj E Vi on the entire ego network &,),, pt
E, x //, h1 - (0, xx3")) We consider two aggregations such that the peer expo- sure embedding is bounded between 0 and 1, with 0 being the case of no peer exposure. The first aggregation captures proportion similar to the fraction of treated peers, but we weight each peer by h "P />, h eep learned by the preceding layer. The second aggregation captures scale and is analogous to the number of treated peers, except that each peer is weighted by h
3.2 END-TO-END LEARNING OF EGONETGNN
The resulting peer exposure embeddings (p:) and the feature embeddings (c,) from the above module along with unit treatment (x₂) are passed to counterfactual outcome model fy(t: = nú P;
Pi. C; c,) to obtain conditional counterfactual outcome (ty = Tg P: pi)lc = c,] =
Ely, mi P; = Ps; C2 eq] (Eq. ??). We adapt the Treatment Agnostic Representation Network (TARNet) and Counterfactual Regression (CFR) models (Shalit et al. (017) as the counterfactual
outcome model f.. The TARNet architecture consists of a single embedding MLP and two prediction heads to estimate counterfactual outcomes with unit treatment 4 = 1 and I; = 0, i.e.,
hmb = e (c)))., 8,(0) = yo (h.m²), 8,(1) = 9, (h.mm))
(7)
Our CFR architecture is similar except for autoencoder to produce the embeddings, i.e.. h = @ems lpp.) and h. = dec (hgmb)) ). Note that, unlike the original CFR, our CFR utilizes an autoencoder because it, along with reconstruction loss, helps mitigate the potential loss in ex- pressiveness while balancing representations across treatment groups The CFR or TARNet model
(T, Pi, c.) predicts outcome 5, (1) if x, = 1 and i. = (0) if T. 0. The unit-level factual prediction loss Ly, is defined as
Eyp loss(y; [L n. p: d. (T 9. x, Z: O c, = by (u., 9. x, O,)):
(
where loss is an appropriate loss function (e.g.. square error loss) based on data type of the outcome and @ (0., ,,y are learning parameters be optimized for exposure mapping function ée. feature mapping function dy. and counterfactual outcome model fy. respectively.
autoencoder reconstruction loss and the Integral Probability Metric (IPM) (Shalit al.) (017)) measure of distance between treatment and control groups using Wasserstein (Cutun and Doucet 2014 AArovsky et al. 2017). jointly referred to as balance loss, i.c.,
Balance loss. The CFR architecture
Lont = 10 x 1 (h -sill.)² + Abat x IPM( 4 = 1}. (hamb :4 = 0}),
(9)
71
where Abal 2 0 is hyperparameter and (PM(.) balances the distribution PP(c, p/t = 0) and P(c, 0), where P(c, p(t) is equivalent PP (cppt)P(clp,tt) t). Intuitively, Lbat balances peer exposure distribution between treatment groups and covariate distribution с across peer exposure conditions and treatment groups while maintaining expressiveness due the autoencoder component. Although the ideal objective would be to balance representation across any exposure conditions. i.c., P(clp,,) t) R P(cip', t'), our balancing technique is still a computation-frient ffrenndyy and useful heuristic. We discuss more on the practical infeasibility of implementing the ideal objective in the Appendix.
For the end-to-end learning of de. &y. and fy., we introduce three custom loss functions designed for EGONETGNN: coverage loss, sparsity loss, and entropy loss. These custom loss functions serve as priors to make the learned exposure mapping function stable and reliable.
Coverage loss. We use a prior that encourages the bounded peer exposure embedding to have substantial coverage. This loss function checks how far the learned peer embedding distribution is from a continuous uniform distribution between 0 and 1, i..., Looe = (mean(p) - - + (var (p) 1 + (range(p) 1)2. Here, we consider mean squared error of mean, variance, and range of learned embedding P against corresponding value of the uniform distribution.
Entropy loss and sparsity loss. Entropy loss encourages mask weights, i.c., p == ( mask) to take values toward or 1 and sparsity loss pushes for a few weights with high values. Formally, we define entropy loss and sparsity loss as Lent mean( (pphng()))) 10--pp p) - p)) and Cep = mean(p).

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Overall loss. We obtain the overall loss function L
C= -= E Cy. + Lbat Acov Leoe Aent x Cent + Aap x Lep + AL1 x |10 Pmgmllll Il,,
(10)
where Acous Aent. and Asp are the hyperparameters and @ grn denote overall parameters in â, and ge. and the last term is L1 loss to promote invariance to irrelevant contexts by preferring sparse weights.
Inference. The peer effeet is obtained 6, T'.)) f(T, -T 9,,, (. fLTT. P.. fu( pl, ei), where f is the end-to-end EGONETGNN.
3.3 THEORETICAL ANALYSES OF EGONETGNN
Expressiveness. We perform a theoretical analysis of the expressive power of graph neural networks (GNNs) in capturing the causal network motifs proposed in the Yuan et al. (202T) paper. Building on previous research regarding the capacity of GNNs to count substructures (Chen et al. 2020). we demonstrate that existing message-passing GNN methods are not expressive enough to capture all causal network motifs. contrast, our method expressive to capture relevant causal network motifs. We defer the detailed theoretical framework and results, along with the relevant background, to Appendix A.4 We state our main result here.
Proposition 2 (Expressiveness of EGONETGNN). EGONETGNN expressive enough to capture all dyad, open triad, closed triad, and open tetrad causal network motifs.
We sketch the proof by dividing the statement into following two claims. The details of the proofs of these claims are in Appendix A.4
Claim 1. EGONETGNN is as expressive as standard MPGNN in capturing dyad, open triad, and open tetrad causal network motifs.
Claim 2. EGONETGNN also captures closed triad causal network motifs.
Time complexity. Our analysis of runtime complexity included in Appendix A.4.3 shows our method is, roughly average, Ps aeg(d) times more computationally expensive than standard MPGNNs, where Pe is the average edge density and avg(d) the average degree.
Misspecification errors. We extend Shalit et al. (017)'ss analyses of theoretical counterfactual predic- tion error bounds for the CFR model to study misspecification errors in the end-to-end EGONETGNN using the sequential error decomposition trick in Appendix [...4 By focusing learning the expressive exposure mapping function, we are reducing its misspecificatiin error directly.
SSaend
EXPERIMENTS AND RESULTS
4.1 EXPERIMENTAL SETUP
Dataset. Similar to other works in causal inference, rely synthetic and semi-synthetic data. We consider three synthetic network models with a fixed number nodes (N = 3000) with different data generating parameters and edge densities: (1) the Watts Strogatz (WS) network (Watts and Strogatz 1998). which models small-world phenomena, (2) the Barabási Albert (BA) network (Albert and Barabási 2002), which models preferential attachment phenomena, and (3) the Stochastic Block Model (SBM) that models community structures. We control the density of edges for BA and WS networks and the number of communities in the SBM network. We also use two real-world social networks, BlogCatalog and Flickr, with more realistic topology and attributes to generate treatments and outcomes. We defer additional details on data generation to Appendix A..5
Evaluation metries. To evaluate the performance of heterogeneous peer effect (HPE) estimation, we use the Precision in the Estimation of Heterogeneous Effects (CrEhE) (Hill 2011) metric defined
EPERE V
(5,((TT E, T'.))
6.( -6 =²)))² where &(TT.,, is true HPE and 8<(TT_,,
the estimated HPE, where π', denotes counterfactual scenario where treatments of peers are flipped. ÉPEнE (lower better) measures the deviation of estimated HPEs from true HPEs. For each experimental result, we report the mean and standard deviation of εPEнE for 5 different simulations, ie., data generation with different seeds. Evaluation across multiple simulations aims to demonstrate robustness across various possible patterns of peer treatment assignments or exposure conditions. For most experiments semi-synthetic data, we use random model initializations for each simulation.
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FgoNeeGHNNTARMN (Ours)
INE-TARNeE
TNet Notestimator
GNN-TARNGC-OTTIS
HGNI
AEMNet
Mutual Connections
Clustering Coefficient
Attribute Similarity
Rolmmat Error cicic
Estimation Error
Khore (Log Scale)
10:
Edimation Error
(Log Scelo)
107
10F
100
10-2
10
Preferential attachment parameter (m)
Preferentiol otlachment parameter (m)
Preferential attachment parameter
Figure 3: Peer effect estimation error for Barabasi Albert network when true peer exposure depends on mutual connections, clustering coefficient, and attribute similarity. Our method shows robust performance across different underlying peer influence mechanisms and edge densities (m).
Table Mean and standard deviation (across data simulations with 3 random model initializations each) of peer effect estimation enror (εPEнε) for different methods in BlogCatalog dataset for four settings when true peer exposure mechanisms depend on clustering coefficients, connected components, mutual connections, and attribute similarity.
Mechanisms
Ours-
Ours-
GNN-
INE-
IGNN-
DWR
AEMNet
TNet
Netlist
CauGramer
TARNet
CFR
Motifs
TARNet
ISIC
Clus. Coef.
2.13
0.951aa
2.39
1.35
1.2 7.49146
7.58
9.52
4.58
6.1612a
Con. Comp.
1.47+0.9
4.98-16
4.78+1.1
6.78+19
7.68+1.
11.2740.
9.984a.
8,56.0.7
7,07+12
Mut. Con.
2.8621.
2.24
2.81a1a
2.50+0.
10.30g60
8.72gas
13.33gsa
11.17a85
5.34g1a
5.18ga0
Attr. Sim. 3,95
3.60,24
4.61a2.
3.59-1.
15,25+4.7
17.96a3.7
14.10asa
14.60as.
11.71a22
14.4gar
Table 2: Mean and standard deviation (across 5 data simulations with one model initialization) HPE estimation error (EPEHE metric) for three variants of our method (original, without mask, and without feature encoder and mask) in the BlogCatalog (BC), Barabasi Albert (BA), and Watts Strogatz (WS) datasets for three true peer exposure mechanisms.
Mechanism
Mutual Connections
Clustering Coefficient
Attribute Similarity BC
Network
BC
BA
WS
BC
BA
WS
BA
WS
Model Variants
Ours- Ours-TARNet
2.6le1a
0.20
0.30
1.71+12
0.90
1.18.aa
4.79.aa
1.28-oor
1.09
Ours (wlo mask)
2.97
0.21.04
0.351oa
2.54
1.0
1.20
5.18
2.37
1.29
Ours (wl feat&mask)
2.07=1.
0.27=0.2
0.31eoa
2.11e0s
0.97+a7
1.91+1s
3.18
13.73eaa
13.85e40
Table 3: Evaluation of exposure representation, in terms of absolute correlation, in Blog data with no effect modification. The results for the learned peer exposure representation by our method is better (higher is better). We use the fraction of treated friends as baseline and the dimension of
p. p' E [0, 1]d=2 with highest correlation is shown.
Con
Clus, Coef.
Con. Comp.
Mut. Con.
Att. Sim.
Con.
Clus, Coef.
Con. Comp.
Mut. Con.
Attr. Sim.
0.81
0.34
0.73+02
0.29
r(a
0.85 0,41
0.30
0.74
0.50gna 0.61
0.17
0.12
0.09...00
0.28
(
0.14
0.09
Baselines. We compare EGONeTGNN with state-of-the-art (SOTA) peer estimation methods. NetEst (Jiang and Sun| (022) and TNet (Chen et al. (024)) use the fraction of treated peers as peer exposure, but the estimator based on adversarial learning and the doubly robust method,
respectively, for robustness. DWR (Zhao et al. 2024) learns attention weights based on attribute similarity, and IGNN-HSIC (Ma and Tresp 2021) uses GNNs summarize peer treatments as heterogeneous contexts while using homogeneous exposure. We also use the recently proposed GNN- and autoencoder-basse automated exposure mapping approach (AEMNet) (Mao et al. 2025 and
GNN- and transformer-based CauGraner (Wu et al. 2025) baselines for estimating peer effects in our setup. We also consider INE-TARNet (Adhikari and Zheleva, 2025) adapted for peer effect estimation baseline. although it was developed for direct effect estimation. We include the GNN-TARNet-Motifs approach that considers manually extracted causal network motifs (Yuan et al. 202 as peer exposure and TARNet estimator (Shalit al. 2017) strong baseline. We discuss hyperparameter tuning and model selection in Appendix A.6
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4.2 RESULTS
Next, we present results for experimental setups designed to answer five research questions (RQs). RQ1. How well do methods for peer effect estimation perform when peer exposure mechanisms depend on local neighborhood conditions? In this setup, we evaluate the performance of peer effect estimators when the underlying peer exposure mechanism is unknown. We generate treatments and outcomes such that there is confounding due to a subset of node attributes and mean peer attributes. For the outcome generation, we consider five mechanisms for true peer exposure conditions where peer exposure is given by 1) the clustering coefficient between the treated peers, 2) the number of connected components among treated peers, and weighted fraction of treated peers with weights as 3) the square root of number of mutual connections, 4) attribute similarity, and 5) tie strength. Here, the unit's treatment acts as an effect modifier, where the peer exposure is doubled if the unit treated. Figure shows peer effect estimation error (y-axis), across five data simulations with fixed model initialization, when true peer exposure mechanisms depend on mutual connections, clustering coefficient, and attribute similarity in Barabasi Albert networks with three network generation parameters (x-axis), resulting in different edge densities (low to high). The preferential attachment parameter m = 1 produces sparse star-topology network, lacking cycles or triangular structures. In this setting, all methods perform relatively well when peer exposure mechanisms depend on local structure because MPGNNs expressive enough to capture star-shaped motifs. However, with increased edge density and more complex network topology, unlike our method, the baselines are not sufficiently expressive to capture underlying mechanisms and suffer significantly. The GNN- TARNet-Mottis (GTM) approach is expressive in capturing clustering coefficients, and both GTM and INE-TA -ARReet approximate mutual connections. This is reflected in the performance, where GTM is competitive for the clustering coefficient peer exposure mechanism. EGONETGNN-TARNY outperforms the baselines except for INE-TARNet, which is competitive in a setting with the peer exposure mechanism dependent attribute similarity. Figure 3 and other results in Appendix AA.7 show that for unknown peer exposure mechanisms, our method is as expressive as or superior to the strongest bascline with significantly better performance for denser networks.
RQ2. How reliable are the models for heterogeneous peer effect estimation in more realistic scenario? RQ2 investigates the performance of the models using more realistic semi-synthetic networks and node attributes. In addition to confounding and heterogeneous peer influence, there more complex peer effect modification depending on whether the unit is treated and the values of the unit's attributes. Table [ shows the mean and standard deviation of peer effect estimation error (EPEНE). across five data simulations with three model initializations each, for different methods in the BlogCatalog (BC) dataset for four settings when true peer exposure mechanisms depend on clustering coefficients, connected components, mutual connections, and attribute similarity. The results show the robustness of GGONETGNN in a more realistic setting. where the variants of EGONETGNN are mostly the best performing ones. The baseline INE-TARNet the most competitive, exhibiting slightly better performance than ours for the attribute similarity mechanism. However, like other methods, it still struggles when the underlying mechanisms involve complex local structures. In this setup, peer effects are heterogeneous due to the interaction of peer exposure conditions and effect modifiers, and our method is able to approximate them better than the baselines. Appendix A.88 presents additional experiments for this setup, including results for the Flickr dataset (T) which is more challenging for the baselines. Table the Appendix shows that the variance in the results is primarily due differences in data simulations rather than model initializations, as peer exposures resulting from some patterns of neighborhood treatment assignments can be easily captured by the models, while others cannot.
RQ3. How do the components of EGONETGNN contribute to its robustness in estimating peer effects? We conduct ablation studies to assess the contributions of masked weights and the feature encoder MLP. Table 2 displays the performance of three variants of EGONETGNN TARNet (original, without the masked weights, and without the feature encoder and masked weights) across BlogCatalog (BC), Barabási Albert (BA), and Watts-Strogatz (WS) datasets. The results show that excluding masked weights can bias peer effect estimates due to the model's sensitivity to irrelevant contexts. Removing the feature encoder MLP limits EgoNetGNN's ability to capture mechanisms based on attribute similarity. Interestingly, for the semi-synthetic network, removing features produced even better results, most likely due to homophily, which results in attribute similarity that is almost homogeneous. As expected, for the peer exposure mechanisms relying on local structures, the model performs better when irrelevant features are ignored. Overall, these findings demonstrate

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that the feature encoder MLP enhances expressiveness, while masked weights promote invariance to irrelevant contexts. Table g in the Appendix shows that the autoencoder component in our CFR module preserves expressiveness and promotes robustness by comparing its performance with that of the original CFR, which does not include an autoencoder. Additionally, we analyze the EGoNETGNN''s sensitivity the choices of peer exposure embedding dimension and noisy networks in Appendix A.9 and sensitivity to balance loss coefficient in Appendix AA.1 (Table
To mitigate the issue where peer exposure embedding (δe) captures correlated pattern rather than the underlying mechanism, we perform model selection based on prediction loss and coverage loss in 20% validation dataset. The idea is that choosing a correlated pattern rather than a true one akin to overfitting In Table TO we evaluate this model selection strategy against the one based on prediction loss only. The results show model selection utilizing coverage loss is more robust, which could be because the coverage loss aims to prevent the equivalent of mode collapse, where the distribution of output peer exposure representation is limited.
RQ4. How well are the underlying mechanisms captured by the learned exposure mapping function? In Table 3, we directly compare the (absolute) Pearson correlation coefficient (higher is better) between the learned peer exposure representation, P and p', and the actual peer exposure under four different mechanisms. Compared to the commonly used fraction of treated friends baseline, learned peer exposures are informative of true peer exposures for mechanisms involving local structure, For attribute similarity mechanism, the baseline approach maybe competitive due to homophily, where attributes of neighbors are homogeneous.
RQ5. How well does EGONETGNN perform under homogeneous exposure and imperfect conditions when the model assumptions are violated? First, we evaluate the models in the simplest setting, where all baselines make the correct exposure mapping function assumption, i.c., true peer exposure depends on the fraction of treated peers. Table IT shows that variants of our model remain superior to the baselines even when they make correct assumptions about the underlying peer exposure mechanisms, as they struggle in settings such as complex effect modifications and arbitrary counterfactual spaces (i.e., flipped counterfactuals). We show the trade-off between computation time, memory requirement, and performance for this simple setting in Table 12 to show how our method gains robustness with extra but manageable computation time. Second, the setting with censored or noisy features by randomly zeroing out 10% of the features and adding Gaussian noise. Finally, evaluate the models in the presence of confounding and interference from two-hop neighbors to examine how our model performs when its assumptions are violated. Tables [3] and [4 show that, although the magnitude of error increased, our methods are competitive with or better than the prominent baselines in such imperfect settings.
evaluate the models in
DISCUSSION, LIMITATIONS & FUTURE WORK
Our work motivates the problem of learning exposure mapping function for peer effect estimation and proposes EGONETGNN for addressing unknown peer influence mechanisms involving local neighborhood conditions. Our theoretical analysis and experimental results demonstrate increased ex- pressiveness of EGONETGNN to capture complex local neighborhood exposure conditions. We have designed EGONETGNN promote invariance irrelevant contexts, and output low-dimensional peer exposure embedding with bounded and balanced representation to partially mitigate issue of potential violation of the positivity assumption with continuous treatment exposure. The empirical results have shown the effectiveness of EGONETGNN in many peer effect estimation settings.
Limitations & Future Work. Our main theoretical results are the expressiveness of the GNN to capture complex causal network motifs. We also discuss counterfactual prediction error bounds by decomposing to misspecification errors in features mapping, exposure mapping. and outcome models. Yet establishing thcoretical bounds and asymptotic properties with complex GNNs for heterogeneous causal effect estimation is still a developing research area (Khatami et al. 2024) and important future direction, and it is not within the scope of our current work. This work can be extended to incorporate other network effects like direct effects and total effects. The increased expressiveness and robust peer effect estimates of our model come with the trade-off of slightly longer runtime to process ego networks. Future work could consider relaxing the assumption of interference from immediate peers while addressing the scalability. Our work relies upon reliable attributed network as input, but future research should consider capturing expressive representations in noisy networks, Appendix A. discusses societal impacts, scalability, and plausibility of assumptions.
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REPRODUCIBILITY STATEMENT
To support reproducibility, we release the complete codebase and experimental procedures in the repository https: :tttts///gin //g //ggtttb...... com/ redgeslabl Secggelabb/goNeeeeee repeated them at least five times, with different data generation and/or model initialization. We have provided the details of the data generation process (Sec. [4.1]nn and Appendix AA.5). Also, the details of the configurations and setups for replicating our results are in Appendix and in the documentation of the parameters in the source code.
For all the experiments, we have
ACKNOWLEDGMENT
This research was supported in part by the U.S. National Science Foundation under Grant No. 2047899 and by the Defense Advanced Research Projects Agency (DARPA) under Contract No. HR001 121C0168. Any opinions, findings, and conclusions or recommendations expressed this material are those of the author(s) and do not necessarily reflect the views of the NSF, DARPA, or the U.S. Government.
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A APPENDIX
A.1 DISCUSSION
Societal impacts. The implications of our work include identifying unit-level peer effects and discovering subpopulations with heterogeneous peer effects. The potential societal impacts could include the development of targeted interventions or the identification of policies that enhance desired outcomes in social networks.
Plausibility of neighborhood interference assumption. Neighborhood interference (Assumption 2 in Sec. 2 is common simplifying assumption and can be realistic in situations where peer interference is mediated by immediate neighbors or diminishes quickly for non-immediate neighbors. However, there could be some situations where interference could occur between peers beyond immediate neighbors. If we assume such interference is mediated via immediate neighbors, then we could model it by stacking multiple exposure mapping function learning layers, where the subsequent layers would summarize the exposures of neighbors. Another alternative is to use the K-hop ego network with edge existence and/or hop distance as additional node features. The former approach may be more scalable than the latter one because the K-hop neighborhood can grow rapidly. Ideas from recent works to infer unknown interference structure (Wu al.. [025 Lin et al. 2024) could be adopted in conjunction with our approach of learning expressive peer exposure representations. Our method can be extended to integrate attention or transformer-be based mechanisms in either feature mapping learning or exposure mapping learning by replacing/modifying MPGNN architectures. While we assume a reliable network structure is provided as input, our experiments with noisy networks reveal that EGONETGNN performs reliably well with imperfect data.
Plausibility of unconfoundedness assumption. Following existing work in the intersection of causal reasoning and representation learning (Shalit et al., 2017: Shi al. [2119] Ma et al. 2022 Wu et al. 2025). we assume causal identification conditions are met and focus on expressive rep- resentation learning to mitigate model misspecification errors. Unconfoundness is strong and untestable assumption and requires sufficiency of observed network contexts and expressiveness of their representation. While we assume the sufficiency of observed contexts, we make effort to satisfy the expressiveness of representation by considering all network contexts, like node attributes, edge attributes, and network structure. If the presence of unobserved confounding cannot be ruled out, alternative causal identification approaches like proximal causal inference (Tchetgen et al. (0200) or double negative controls (Miao et al.] 2024). front-door criteria (Pearl) (009). and instrumental variables (Angrist et al. 1996) should be considered. Although randomized experiment can remove unobserved confounding between unit treatments and the outcome, peer exposure conditions may not be randomized directly, and confounding could exist even for experiments unless the unconfounded- ness assumption is made and observed network contexts are controlled for. So, an interesting future direction could be to explore alternative identification conditions.
Scalability. Although EGONETGNN is more expressive, it has additional computational costs. A few ways to address large runtime and/or memory usage could be sampling ego networks to reduce the training set or sampling the neighborhood within a K-hop ego network. In Appendix A.8 (Table 6), our experiments with randomly augmented network show that the performance does not degrade significantly for our method with the removal of edges. From an implementation point of view, we can parallelize our framework easily exploit the power of GPUs. More specifically, there are two components in our framework. The feature mapping GNN takes the entire network at once to learn an embedding with an L-layer GNN. Subsequently, EgoNeIGNN batches nodes with their neighbor nodes and a mapping of which edges belong to which node in the batch. This batching can be parallelized to improve the overall efficiency.
Representation balancing in CFR. We note that explicitly enforcing IPM balancing (using ap- proaches such as the Wasserstein distance) for the ideal condition, P(c P(c)p',t'), is non- trivial at best and computationally infeasible at worst. The challenges arise primarily because the peer exposures (p versus p') are multi-dimennional continuous values. There may be several exposure conditions with mostly limited samples, which complicates the calculation of the Wasserstein distance (optimal transport) and may render it difficult or unreliable, even when using modern methods. Our balancing technique is computation -rienndly and useful heuristic approach (as evidenced by the experiments), where for each stratum of peer exposure condition, we want balanced covariates across different treatment groups. It does not perform the ideal balancing scenario. However, even in
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network experiments where treatments are randomized, such balancing is not possible because peer exposure conditions depend not only on peer treatments but also on other covariates; therefore, the randomization for peer exposures is not achieved. We instead control for different possible network contexts (derived from the observational attributed network), similar to regression adjustment, to deal with the imbalance of peer exposure conditions.
Theoretical guarantees and confidence intervals with complex GNNs. While complex GNN and transformer architectures are very powerful in capturing confounding variables, effect modifications, and unknown influence mechanisms in complex network data, they lack interpretability and theoretical guarantees of consistency or convergence, As discussed in our paper, theoretical properties of complex GNNs have been shown in simpler settings where there is a homogeneous exposure mapping function and for average treatment effects (Khatami et al., 2024, Chen et al., 2024). Establishing theoretical guarantees in the setting with unknown exposure mapping functions and heterogeneous effects is an important future direction, but is outside the scope of our paper. Instead, our main theoretical results in this paper are on the expressiveness of the GNN to capture complex underlying influence mechanisms. In this work, we focus on point estimates without confidence intervals. For real-world data, techniques like bootstrapping and random model initializations could be used to obtain measure of uncertainity similar to confidence intervals. Like ours, other works have used point estimates present empirical evidence to support robustness in complex interference settings, such as using GNN on hypergraphs to model group interactions (Ma et al., KDD 2022) and using Graph Transformers to model unknown interference structure (e.g., CauGramer (Wu et al., ICLR 2025)), leaving theoretical results on error bounds with complex models to future work. We hope our work could spur future research directions and collaborations address these limitations. Our framework is flexible enough to be adapted to utilize unbiased estimators like the Horvitz- Thompson (HT) estimator or the Targeted Maximum Likelihood Estimation (TMLE) estimator for causal estimation, making it still appealing for practitioners.
A.2 RELATED WORK
Research in causal inference under interference has focused on estimating three main causal effects of interest, referred to as network effects: direct effects induced by unit's own treatment, peer effects induced by treatment of other units, and total effects induced by both the unit's and others'
treatment (Hudgens and Halloran 2008). These network effects estimated as average effects (e.g.. (Arbour et al. [016]] U Ugander et al. 2013)) for the entire population or as heterogeneous effects (e.g.. (Forastiere et al. [021 Bargagli-Stoffi et al. 2025)) for specific subpopulations or contexts. Our work focuses on heterogencous peer effect estimation. Most methods for estimating heterogeneous or individual-level causal effects under interference, including peer effects, assume peer exposure is binary (Bargagli-Stoffi et al. 2025) or homogeneous, e.g., based on fraction of
treated peers (Tiang and Sun] [022) (Oguunn et al. 2022] ( al. 2023] Chen et al., (024). These methods assume a homogeneous or known exposure mapping function and focus on enhancing network effect estimation by adapting techniques like adversarial training (Jiang and Sun 2022).
propensity score reweighting (Cai et al., 2023), double machine learning (Khatami et al. 2024
doubly robust estimation (Leung and Loupos| (0022). targeted maximum likelihood estimate (Ogburn al. 2022), and targeted learning (Chen et al. 2024).
has looked into more complex functions of peer exposure, allowing for heterogeneous peer influence, in which different peers can have varying degrees of influence. Some of these works refer to heterogeneous peer influence as heterogeneous interference (Qu et al.) 2021 Zhao et al. weighted fraction of treated peers using known edge attributes as weights. Lin et al. (202)) consider heterogeneity due to multiple entities types and Qu et al. (202I) considered heterogeneity due to known node attributes for defining peer exposure. [Tan and Zheleva (2022) studied peer effect estimation with linear threshold
Recent research research
2024 Lin al. (023). Forastiere et al. (2021) considered peer exposure
peer exposure model but different lnit- threshold could be vary for different units capturing heterogeneous susceptibilities to the influence. /Zhao et all](222)) used attention weights derived based on the similarities of the units' covariates to determine peer exposure as the weighted sum of treated peers. Yuan et al. (202 1) capture peer exposure with features based on counts of different causal network motifs, i.e., recurrent subgraphs in a unit's ' ego network with treatment assignments as attributes. Ma and Tresp (202]) consider homogeneous peer exposure based fraction of treated peers but they summarize the covariates of treated peers using a graph neural network (GNN) to capture heterogeneous contexts involving treatment assignments. Unlike our work, none of these
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studies has explicitly studied the issue of automatically learning the exposure mapping functions to define peer exposure representation while capturing the underlying influence mechanisms.
Ma and Tresp (2021) learn heterogeneous contexts based on peer treatments but not the exposure mapping function or the peer exposure representation, (Zhao et al. (2024) obtain single-dimension peer exposure embedding using weighted sum of treated peers with attention weights derived from the cosine similarity of feature embeddings. Although ZZha et al. (2024) use attention weights to define peer exposure, they assume specific exposure mapping function, and it cannot adapt according to the underlying peer influence mechanism. Adhikari and Zheleva (2025) use GNNs to learn peer exposure embedding by addressing unknown peer influence mechanisms, but their scope is limited to direct effect estimation, i.e., the effect of a unit's 's own treatment. AAecificallyyAAdhikaa and Zheleva (2025) learn multi-dimmnsional peer exposure embedding using a weighted fraction of treated peers with feature embeddings and second-order adjacency matrix as weights. |M et al., (2022) employ similar method like Ma and Tresp (202T) for hypergraphs to model heterogeneity due to model group interactions. The idea is to learn summary function and representation equivalent to the exposure mapping function and peer exposure using a hypergraph convolution network and attention mechanism. However, they assume the learned representation is expressive enough to capture the underlying influence mechanism. In this work, we do not make such an assumption and evaluate how well the learned peer exposure representation captures the underlying influence mechanisms.
AAdiikari
Neural networks (NNs) (Shalit et al. 2017 Im et al. 2021 Shi et al.
2019) and, recently, graph neural networks (GNNs) (Tiang and Sun| 2022] Cai et al. [202]] Chen et al. 2024 Khatami et al. (0224) have
been widely utilized for end-to-end learning of feature mapping function and counterfactual outcome model or effect estimator. A feature mapping function maps raw features to feature embedding to capture potential confounders and effect modifiers. A counterfactual outcome model (Shalit et al., 2017: Ma and Tresp (002)) predicts counterfactual outcomes for different levels of treatment, while an effect estimator (Shi et al., 2019); Chen et al. 2024) directly learns the causal effect of interest. Only a few studies have considered learning the exposure mapping function (Mao et al. 2025) or
peer exposure embedding (Adhikari and Zheleva 2025 Zhao et al. 2024). [in et al. (2024) consider a setting with an unknown network and interference structure and propose an approach to first infer network structure and represent peer exposure for direct effect estimation. Unlike their work, our settings focus on peer effect estimation with observed network structure but unknown peer exposure mechanisms that manifest due to local neighborhood contexts.
Sävje (2024) advocates for interpretable but possibly misspecified exposure mappings and character- izes causal estimation errors due misspecified exposure mappings, but follow-up research (Auer. bach et al. 2024) has highlighted the importance of capturing underlying interference mechanisms in policymaking More recently. |Mo et al. (2025) have explored the use of GNNs with autoencoders and clustering to learn discrete exposure conditions and their probabilities, aiming to estimate overall causal effects in networks. Similarly, Ww et al. (2025) utilize GNNs with Transformers to model unknown interference from K-hop neighborhood. Their identifiability assumption relies on captur- ing unit and peer covariates, while our identifiability assumption relies on capturing all attributed network contexts, including structure and edge attributes. These works use off- off-the-shelf message passing GNNs (like GCN and GIN) and lack expressiveness to capture mechanisms involving local neighborhood structure. Prior research (Xu et al., 2018 Chen et al.] 2020) on the expressiveness of GNNs has shown that popular GNN architectures lack expressiveness to count subgraphs. On the other hand, counts of subgraphs like causal network motifs are rich features that could capture underlying influence mechanisms due to local neighborhood structure (Yuan et al.) 2021). Counting such subgraphs can be computationally expensive, and they may not be able capture every local structure. We design EGONETGNN to excel in counting attributed triangle subgraphs, enhancing its expressiveness capture underlying mechanisms involving neighborhood contexts.
A.3 CAUSAL INFERENCE ASSUMPTIONS AND IDENTIFICATION OF PEER EFFECTS
A fundamental prerequisite for causal identification is the consistency assumption, which enables equivalence among counterfactual, interventional, and factual outcomes.
Assumption 4 (Consistency under interference). The underlying outcome generation is independent of the treatment assignment mechanisms (i.e.. hypothetical, experimental, or natural). For a unit 2, if
t म. and t_; ; -- then y, (t = mi, t- - = Yv-
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dyad (2-)
triad (3o-)
closed triad (3c-)
Examples higher-order network molifs that cannot be capturedby EgoNetGNN
Examples of higher-order network motifs that can be captured by EgoNetGNN
open tetrad (40-)
Figure 4: Example causal network mo- tifs considered by Yuan et al. (202T). Stars represent ego nodes and circles represent their peers. The red circles indicate treated nodes and blue cir- cles indicate control nodes, The gray shapes indicate nodes that could ei- ther be treated or control. Here, the characters in red indicate a particular causal network motif (e.g.. 3c-2 indi- cate closed triad with treated peers).
Figure 5: Examples of higher-order network motifs with four and five nodes. Stars represent ego nodes and circles represent their peers. The gray shapes indicate nodes with any treatment assignment. If the subgraph of a network motif, after removing edges connected to the ego node, forms tree, then our model is expressive enough to cap- ture the network motif and the corresponding causal net- work motifs. A network motif is a subgraph without any attributes, whereas causal network motif is a subgraph that includes peer treatment assignments as attributes.
Positivity is another standard assumption in causal inference that requires every unit e; to have non-zero probability of being assigned every possible unit treatment and peer exposure conditions.
Assumption 5 (Positivity). There is a non-zero probability of unit treatment and peer exposure conditions for all possible contexts C,, i.e., (t,, P.(c;) > for every level of and p,, where PP is the probability density function.
The proof of Proposition D
as follows.
Proof. Our causal estimand of interest (Eq. 2) is as follows:
iT T = ( = T,, = d. ( X. Z)) P;
[c)c
Due to unconfoundedness assumption (Assumption 3. unit treatment and peer exposure conditions are independent of counterfactual outcome conditioned on network contexts C, This allows us to rewrite the estimand as:
8,( + (t, n. P: 6.
Z) T. o. Z))
E Ety J.. P. = db 9. Z)) = m; o ( 9, X. Z)), ))) c.]-
Here, Assumption | ensures introducing new terms related to treatment and peer exposure in the conditional does not affect existing set of contexts because they are measured pre-treatment. Similarly, Assumption 2 makes the sufficiency of learned representation requirement in unconfoundedness assumption more plausible. Next, the consistency assumption allows replacing the counterfactual outcome with observed outcome, i.c.,
(T T Ely, T. P: (e Z)) Ely,t; m. p: (T 9, Z)).
Assumption T also ensures consistency assumption is satisfied because the treatments not mutable. This estimation above is tractable from observational or experimental data because of positivity assumption and the causal effects be identified.
A.4 THEORETICAL ANALYSES OF EGONETGNN
A.4.1 PRELJMINARIES
Causal network motifs. Yuan et al. (2021) proposed causal network motifs important features to capture peer exposure accounting for local neighborhood conditions. Causal network motifs are attributed subgraphs with peer treatments attributes. Figure 4 shows four categories of causal
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network motifs: dyads, open triads, closed triads, and open tetrads. In the figure, stars represent ego nodes and circles represent their peers. The red circles indicate treated nodes and blue circles indicate control nodes. The gray shapes indicate nodes that could either be treated or control.
Message passing graph neural networks (MPGNNs). The message-passingg graph neural network (MPGNN) is generic GNN model that incorporates several standard GNN architectures and relies on local aggregations of information within graphs (Chen et al.) 2020). For a graph G(V, E, X, Z), an MPGNN with layers is defined iteratively with aggregate function AGG and update function Ud as follows:
-' AAGG ( (2,))),
(11)
where M denotes neighbors of unit v and O' denote learnable parameters like multi-layer perceptron. To obtain the hidden state at the 1th layer, local aggregation of the previous layer's hidden states (h/-1 and h ') and, optionally, edge attributes Zij is performed and then combined with h The hidden states are initialized as node attributes, i.e., hς = X, Typically, in various GNN architectures, the update and aggregation functions chosen part of architecture design.
Expressiveness of MPGNNs in counting substructures. Here, we summarize the results obtained by CChn et al. (2020) that are relevant theoretical analysis. We list their findings after defining relevant concepts.
Definition (Subgraph). A subgraph G (Vls), ElSl)) of graph G(V, E)) consists of subsets of its nodes, i.e., v C and edges, i.e., Els C E.
Definition 3 (Induced subgraph). A induced subgraph (ls)((((s)) E of a graph G(V, consists of subset of its nodes, i.e., vis" C V and all edges between nodes Vis", i.e., Els E n Vis
All induced subgraphs are subgraphs but reverse is not true. For example, all causal network motfis are induced subgraphs (and subgraphs) of the original graph. An open triad motif is subgraph, but not an induced subgraph, of closed triad motif.
Definition 4 (Star-shaped pattern). A pattern GIPI( ElPl) is a star-shaped pattern it can be represented by tree structure.
Definition (Connected pattern). A pattern ((P(( ElPl) y is connected pattern if cannot be represented by tree structure.
For example, a closed triad motif is connected pattern and dyads, open triads, and open tetrads are star-shaped patterns.
Chen al. (2020) obtain the following results on the expressiveness of MPGNNS for counting substructures.
Corollary 3.4. Chen et al. 2020 MPGNNs cannot induced -ubuuced-ubbrraphccourt any connected pattern with 3 or more nodes.
Theorem 3.5. (Chen et 2020 MPGNNs can perform subgraph-count of star-shaped patterns.
A.4.2 EXPRESSIVENESS OF EGONETGNN
Here, demonstrate that standard MPGNNs lack the expressiveness to capture closed triad motifs, and our model addresses this limitation.
Without loss of generality, assume node attributes for each node U/ are < 1, and constant edge attributes < 1 >
Definition 6 (Expressiveness in counting causal network motifs). Let G be space of graphs. A representation by an MPGNN f is expressive in counting causal network motif G!Pl if, for all ego networks GiLl, G(2) E g, distinct counts, i.c., (G ,l!)) * CG GIPl), get distinct representations, i.e., (G + f(Gi?)), where C₁ returns unduced-subbraphe
of pattern G(P).
Restating Propusition 2 (Expressiveness of GNNN EGONETGNN is expressive enough to capture all dyad, open triad, closed triad, and open tetrad causal network motifs.
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Proof. We proceed the proof by dividing the statement into following two claims. Restating Claim 1: EGONETGNN is as expressive as standard MPGNN in capturing dyad, open triad, and open tetrad causal network motifs.
Proof. The dyad, open triad, and open tetrad causal network motifss are star-shaped patterns, and these patterns can be counted by standard MPGNNs (Chen et al. (2020)'s 's Theorem 3.5.). Our model employs MPGNN (refer Eq. 4 and Figure 2) transformed graph, where all edges connected to the ego node are removed, and the corresponding edge attributes from the removed edges are included as node attributes in the transformed graph. We need to show that this transformation preserves the expressiveness to capture dyad, open triad, and open tetrad causal network motif's. The dyad, open triad, and open tetrad causal network motif's are transformed into subgraphs with isolated one, two, and three nodes, respectively, in the transformed ego network. MPGNN in the transformed graph can perform a subgraph count of patterns with k isolated nodes because they are subgraphs of star-shaped patterns with an empty set of edges. Furthermore, the addition of new attributes does not affect the expressiveness because these attributes are added as additional feature dimensions. Hence, our model is as expressive as standard MPGNN for capturing dyad, open triad, and open tetrad causal network motifs.
Restating Claim 2: EGONETGNN also captures closed triad causal network motifs.
Proof. The closed triad causal network motifs are connected patterns of three nodes and these patterns cannot be counted by standard MPGNNs (Chen et al. | (2020)'s 's Corollary 3.4.). Due to the construction of the ego network, all the edges with the ego node are removed, and the closed triads are transformed to dyads in the transformed ego network. These dyads can be counted by node aggregation (refer Eq
which is an MPGNN employed in the ego network. Therefore, EGONETGNN captures closed
triad causal network motifs.
Higher-order causal network motifs and attributed causal network motifs. Here, we show how our model is superior to the approach of counting predetermined causal network motifs by discussing EGONETGNN's ability to capture relevant causal network motifs including higher-order and attributed causal network motifs. Proposition showed our model is as expressive as the approach of counting predetermined causal network motils considered by Yuan et al. (202T). In general, if the subgraph of a network motif, after removing edges connected to the ego node, forms a tree, then EGONETGNN is expressive enough to capture the network motif and the corresponding causal network motifs. Figure 5]depicts some examples of higher-order motif's with four and five nodes. EGONETGNN, with depths of L = 2 and L (refer Eq. 4). is expressive enough capture most higher-order motifs with four and five nodes, respectively. Only if the network motifs consist of cycle without the involvement of the ego node, then EGONETGNN is not expressive enough to capture Furthermore, compared to predetermined causal network motifs, EGONETGNN can accommodate motifs with additional node and edge attributes. Incorporating node and edge attributes will not reduce the expressiveness of counting original causal network motifs because these attributes are added as additional feature dimensions.
A.4.3 TIME COMPLEXITY OF EGONETGNN-TARNER
Typically, the complexity of a standard MPGNN (e.g. GCN), is O(N + L|E(F), where N. L, and F are the number of nodes, edges, GNN layers, and the dimensionality of feature embeddings. respectively (Blakely et al. [n.d.]. D. In our model, the feature mapping MPGNN (refer to Eq. 3 has
the time complexity of O N 2) for ego feature embedding module O, (X,). where de the depth of MLP and Fz. is the dimensionality of node feature embedding, and (Lde F I for peer feature embedding and aggregation, where F' F, + F, is the dimensionality of node and edge feature embeddings. For node aggregation (refer to Eq. 4), we extract ego network for each node and perform neighborhood aggregation. Therefore, the time complexity is O Emax FF), where |Emazl is the number of maximum edges in the ego network. For subsequent masking and exposure encoding MLP, the time complexity is O(Nd Emas!! F²?, where dMIP LP is the depth considering overall MI.Ps
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Assuming single-layer MPGNN with F <<N < [E), for simplicity, standard MPGNN scales linearly with the number of edges, i.c. O(E) or O(N x avg(D)), where (D) is the average degree. Similarly, for EGONETGNN the time complexity simplifies to ( x [Emarl)). In the worst case, |mmea max (D)2, where mar D) is the maximum degree in the network G(V, E). However, since networks generally sparse, the approximate runtime complexity for networks with uniform degree (c.g., Watts Strogatz network or Stochastic Block Model network) is O(N X P. X )vg( where Pe density edges. So, our method is approximately P. avg(D) times more computationally expensive than standard MPGNNs. On the other hand, the time complexity for counting predetermined causal network motifs with K nodes is O(Nmart Vmaa(D))) K assuming access to O(1) adjacency set and adjacency matrix. This approach scales poorly with higher-order motifs and EGONET mitigates the problem by capturing most higher-order motifs with the same computational cost.
A.4.4 COUNTRRFACTUAL OUTCOME PREDICTION ERROR BOUNDS FOR EGONETGNN
Our work utilizes Shalit et al. (217))s 's TARNet and CFR estimators, adapted to network settings, for estimating heterogeneous peer effects in both observational and experimental data. Their analysis shows the PEHÉ metric is bounded by factual (F), i.c., supervised learning and counterfactual (CF) prediction error, i.e., CPEHE (fu) S 2(ccr(ff (fy) cr(f)) 20 where of is the variance of the outcome. These prediction errors or biases incorporate Säyje (2024)'s definition of exposure mapping specification
along with feature representation errors and outcome prediction errors.
Moreover, Shalit et al. (017)) show that the bound for counterfactual prediction error (which cannot be measured in the real world) depends the Integral Probability Metric (IPM) measure of distance between treatment and control group distribution, which implies (FEHE(fy) S (f))
6=00 (fy) + az (.IPMM(Mh (hgmb; f = 1}, (hgmb, 4 = 0}) - 220 ). where 4 = x; denotes conditioning. hemb Démb(Cillp) and denotes concatenation. To study how misspecification errors of EgoNeIGNN propagate the factual prediction error, we substitute the oracle values and estimated values (denoted with hat) and further decompose the errors by using sequential error decomposition trick, i.c.
F
Cp 4= =(( - )
y, y, [y pi: c.) [ Ps; c:)
81 y; cy + Ef, where Ey captures enror due to learned outcome prediction module using learned representations, i.e.,
Cy :- fy ( E,),
ce captures error due exposure mapping misspecification using learned feature representation but true outcome prediction module, i.e.,
E = [U(TT, Pi, é.) - (TT; Pi, dz),
and, finally, captures error due to feature mapping misspecification but true exposure and outcome prediction function, i.e..
Ef := fy p., é - fy,,, C:).
By plugging these decomposed errors in the factual prediction loss, we get,
C h= (fa) = El(c, + c2 {
F + E(-) + Fl + 2(F(e + F|roeyl +Feeel)
By automatically learning relevant exposure mapping function, we aim to directly minimize the error terms involving & and the downstream error cy. Other estimators (e.g., Doubly robust orthogonal learning after handling unknown exposure mapping function) can be employed in future work for more tight error bounds.
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A.5 DATASET GENERATION
For the Barabasi Albert (BA) model, the preferential attachment parameter m E [1,5, 10] is used to generate sparse to dense networks, where new node connects to Phu existing nodes to form the network. For the Watts Strogatz (WS) model, we set mean degree parameters k E {0.002N, 0.005.V, 0.0LN} with fixed rewiring probability of 0.5, similar to prior works (Yuan) et al. 202T Adhikari and Zheleva, 2025). For the Stochastic Block Model (SBM) model, we use the number of blocks parameters b E {500, 200, 100) with randomly generated edge probabilities within and across communities. We also use two real-world social networks BlogCatalog and Flickr with more realistic topology and attributes to generate treatments and outcomes. We use LDA (Blei et al. 2003 to reduce the dimensionality of raw features to 50.
Treatment model. The treatment assignments could depend on the unit's covariates as well as peer covariates and some edge attribute. We generate treatment T, for a unit ey T, ~ ((a(Te Wr X
EJEN, X
+ (1 - Te) WT X")), where 0 denotes Bernoulli distribution, a : R ++ [0, 1] is an
Ejex, Z5z
activation function, To E [0, 1] controls spillover influence from unit 0; ss peers, X C X is a subset of node attributes, Ze E Z is an edge attribute, and Wr is a weight matrix.
Outcome model. The outcomes depend on unit's treatment, peer treatments based on the local neighborhood condition, the confounders, and the effect modifiers. We generate outcome Y, for unit U as:
Y; = (deap + dem x T:) x
(12)
(Ta + Tem X Dem Lem((.. (G, 2)) x T. + g Zo: +E.
captures peer effects, where (G, X, 2_;-) captures true peer exposure that depends on local neighborhood condition (e.g.. the number of mutual connections between treated peers and ego unit or attribute similarity) and deep and dem are coefficients controlling magnitude/direction of peer effects. The term g(Xe, Ze, G) captures confounding and c ~ (N 1) is random noise. The remaining term captures direct ef- feet due unit's own treatment with effect modification by some contexts. For semi-synthetic data, to generate heterogeneous peer effects, we use additional effect modification due to a unit's covariates, i.e., 8em T: x (X,m,,) where X.m C X and do is a weighted mean function with randomly generated weights. Please refer to the source code in the repository for detailed implementation of data generation. When the underlying mechanism depends on the number of mutual connections between treated peers and ego unit or attribute similarity between peers and the ego unit, the ground truth exposure mapping de is defined as (G, T-T = "i,2 where Emy, why- cardinality {ux : eis E E A ezk E E}) for the former mechanism and Waj rbf(x,, x,), where rbf is Gaussian radial basis function. For mechanisms involving clustering coefficient and connected components between treated peers, corresponding graph function called in a subgraph with treated peers, i.e., (G, X, :-) = graph func unc(subgroph(uu U/e EG A & = 1)).
Here, the first term (6erp + dem X T:) x X,
A.6 ADDITIONAL EXPERIMENTAL SETTINGS
Model implementation, hyperparameters, and model selection. We have used Aent Asps Abal, Acov: λ71 as hyperparameters in our loss function. For a set of hyperparameters, we choose reasonable values and for the rest we use Python's "Ray Tune" framework for hyperpa- rameter tuning. Although ground truth causal effects are unavailable to truly tune hyperparameters, our error analysis (extended from Shalit et al. (2021)'s work) shows that error in factual outcome prediction (and IPM distance metric) can be used as a proxy for hyperparameter tuning.
First, we describe the choice of values for regularization hyperparameters Aent Aaps and λL1-
Entropy regularization coefficeint Aent = 1 to promote mask weights W Wmask [0. 1] approaching 0 or such that average entropy is low. An extremely low value does not enable the intended behavior of the soft switch for enabling or disabling certain features, and large value could interfere with other loss terms.
Sparsity regularization coefficient App 0.1 to encourage sparse mask weights i.e., few weights approaching 1. In conjunction with entropy loss, a value that is too high could lead all weights toward zero, and value that is too low could produce non-sparse weights.
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LI --regularizztion coefficient λ7ι 1 for low-dimensional synthetic data encourage highly sparse model parameters and λL1 = 0.1 for comparatively higher-dimensionn semi-synthetic data to encourage sparse model parameters.
We tune the coverage parameter Acoe E {0.01, 0.1} } for semi-synthetic data but choose a conservative Acce 0.01 for synthetic data for efficiency. We choose the oovariat-ballnccin hyperparameter Abal 0.8 based on the unalysis of the original paper (Shalit et al., 2017). We set the output embedding dimension of the exposure encoder MLP giving a 6-dimensional peer exposure repre- sentation. We use layer deep MPGNNs for feature and exposure mapping functions. Moreover, we perform grid search hyperparameter tuning by varying GNN learning rate {0.1, 0.04, 0.02, 0.01) }, and setting TARNet learning rate to 0.01. We use Adam optimizer with weight decay of 10 and the learning rate is decayed by 50% after 50 epochs. 20% held-out dataset is used for model selection, where model with lowest outcome prediction loss Ly, is chosen for reporting. We employ model checkpointing every other epoch to select the best performing model in a total of 100 epochs. Our implementation is similar to Adhikari and Zheleva (2025'ss TARNet (also known as IDE-Net in original paper) in terms of MLP with residual network architecture, parameter tuning and model selection, and data generation.
The baselines INE-TA MEETARNeet and GNN-TARNet-Moide are also tuned similarly to our method by conducting grid search of the GNN's learning rate with (0.2, 0.02) and variance smoothing regular- ization hyperparameter with {0.1, 1}, keeping TARNet's learning rate 0.02 and other hyperparameters default. DWR is calibrated for epochs balance representation. For other baselines, we use default hyperparameters.
Implementation of baselines. We use publicly available code shared for the baselines INE- TARNet (Adhikari and Zheleval (025)) TNet (Chen et al. (2024), NetEstimator (Jiang and Sun 2022), and CauGramer (Wu al. (025). We adapt the code provided by authors extend for peer effect estimation for AEMNet (Mao et al., 2025). We implement IGNN-HSIC (Ma and Tresp 2021) and DWR (Zhao et al. 2024) ourselves following the paper as closely as possible.
MAN-TTANee-MOTIISS is available baseline of INE-TARNet.
Computational resources. All the experiments are performed in a machine with the following
resources.
CPU: AMD EPYC 7662 64-Core Processor (128 CPUs)
Memory: 256 GB RAM
Operating system: Ubuntu 20.04.4 LTS
GPU: NVIDIA RTX A5000 (24 GB)
CUDA Version: 11.4
A.7 SYNTHETIC DATA EXPERIMENTS AND RESULTS
Figures to 10 show the performance our method and baselines for three synthetic networks when the underlying peer exposure mechanisms depend on clustering coefficient, connected components, number of mutual connections, tie strengths, and attribute similarity. The results discussed in the main paper apply to additional peer exposure mechanisms and data generation conditions.
A.8 SEMI-SYNTHETIC DATA EXPERIMENTS AND RESULTS
We present results for RQ2, that investigates performance more realistic settings, for the larger benchmark network dataset Flickr in Table EGONETGNN-CRE in all settings. EGONETGNN-TARGE TARGNN is either better than all the baselines or competitive to stable baselines TARNet and GNN-Motifs. For mechanisms involving network structures, sophisticated models like CauGramer and TNet have poor performance as they are unable to generalize or converge for some data generation and model initialization settings.
is the best performing model
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EgodeeC/MMAAANN (Ours)
INE TARhet
DWR AEMNeC
That NetEstimator
MOTIPS
1GNN
Estination Eave
Barabasi Albert
Watts Strogatz
Estiration Error Fwee
Stochastic Block Model
107
Fstimationn Frror
(Lng Scale)
(Lop Scalle]
10-
10
m-1/0.2005 20077)
m- 510.003331 00331
m-10dd g06a)
C- 001;
6-11 0C5)
In- 1-10/0.01
b-55000 (302) b-200u0 0205) b- b-10010 D1D01
Preferentaa attachment parameter (m))
Mean degree parameter (k)
Number of bicceslcommnties (b)
generating parameter corresponding average edge densities
higher)
Figure 6: Peer effect estimation error when true peer exposure depends on clustering coefficient among treated peers. Our method is better than competitive to baseline using predetermined causal network motif counts when the underlying peer exposure mechanism can be explained by causal network motif counts.
EgoNetENN TARNet (Ours)
INE TARNet
DWR
Thet NetEstimator
1GNN,
AEMNeC
Estrmation bax
Barabasi Albert
Watts Strogatz
Estimation Emarr frow
Stochastic Block Model
10:
Ealimatton Fiver
(L9g Scale)
10:
Log Scale)
20:
102
(Lay
m=1/0.200 1/0,200571 (w550.003311
Lw610 002)
1m1150 005)
kw0000.0111
b-50010 2302) be200k0 0205) b=10310.00 02001 Number of Dicesscommunieee (b)
Preferential attachmert parameter (m)
Mean degree par armelerr (R)
generating parameter corresponding average edge densities higher)
Figure 7: Peer effect estimation error when true peer exposure depends on number of mutual connections with the ego. Our method significantly outperforms all baselines showing its capability to count closed triad network motifs (i.e., triangle substructures) in the ego network.
FgeNetC/NAAAAmN (Ours)
INF TARNH
DWR
Thet
IGNN
Ettimtton Car
Albert
Stregatz
Stochastic Model
10-
Extimaation Froo
(19) Scales
10:-
Extimation Ertor (L9g Scele)
10a
1.0
101
1-610.002
Preferential attachment parameter
Maan degg wil (K)
Number Dicciskammunnies
Network generating parameter and corresponding average edge densities (lower to higher)
Figure 8: Peer effect estimation eтror when true peer exposure depends on connected components among treated peers. Our method performs well compared all baselines when underlying peer exposure mechanism cannot be explained totally with causal network motif structures only.
FgekeetCNNATAINe MOTTIS
GXN MOTTFS
1GNN HSIC
AEMNe:
NetEstimator
Ettimation Cass
Barabasi Albert
Edtiaation Error Sapss
Watts Strogatz
Edtimation Gase
Stochastic Block Model
101
[Log Scalel
1000
1000
2.00
Preferential
parameter
Network generating parameter and correspanding average edge densities (lower to higher)
Figure 9: Peer effect estimation when true peer exposure depends tie strengths between ego and treated peers. Our method consistently outperforms all baselines because it can incorporate edge attributes and leam if those attributes relevant for underlying peer exposure mechanisms.
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EgeNeeCCMMATAANI (Ours)
INE TARhet
DWR AEMNeC
That NetEstimator
GNN, WAM_TARNeR_MON MOTIPS
IGNN_HSE HSIC
Eame
Barabasi Albert
Edtination Expe
Watts Strogatz
Estination Gane
Stochastic Block Model
Estination Saall)
101
201
20c
201
[Loe
(Lop
m- 1:0 (0007)
m--10 0033)
m- 1016, 000e)
k-6,0.2001
L--500 0051
2-3000.03)
B-50000
222001000055 b- 10010 bact)
Preferential attachment parameter (m)
Mean degree par parameter (k)
Number of blockstcammniiiee (b)
generating parameter correspending average edge densities (lower higher)
Figure 10: Peer effect estimation error when true peer exposure depends on attribute similarity between ego and treated peers. Our method consistently outperforms all baselines because it can capture and learn attribute similarity are relevant for underlying peer exposure mechanisms.
Table 4: Mean and standard deviation (across data simulations with 3 random model initializations each) of peer effect estimation (CEEHE) Æ)) for different methods in Flickr dataset for four settings when true peer exposure mechanisms depend on clustering coefficients, connected components, mutual connections, and attribute similarity.
Mechanisms
Ours-
Ours-
GNN.
INE-
IGNN- HSIC
DWR
AEMNet
TNet
NetEst
CauGramer
TARNet
CFR"
Motifs
TARNet
Clus. Coef.
7.38
3.71
5.05
3.64
13.361sa
9.95
7.45g1.
19.12_19.
22. 22.50_35. 7.86
10.57
Con. Comp.
16441.
1.19:15
2.34.o.r
2.38
19.16:2e.
3.32
11,50
3.77
22.57=200.
Mut. Con.
2.72+1.7
2.20
2.61-1.1
2.41.1a
3.22=1.3
3.67+1.
5.72+41
1163-224.3.19:
31.27=36.5 18.581m.
Attr. Sim.
8.27
6.97
11.37
10.66
21.12 25.29
13.56112.1
10551
13.70_11.
A.9 ABLATION STUDIES AND HYPERPARAMETER SENSITIVITY
Table 5 shows the performance of EGONETGNN for different output dimension of peer exposure embedding P; for four settings when true peer exposure mechanisms depend on clustering coefficients, connected components. mutual connections, and attribute similarity. As seen in the results, lower- dimensional peer exposure embeddings could lose expressiveness, while higher dimensions could introduce variance due irrelevant contexts violations of positivity. Lower-dimensional peer exposure embedding has better performance for simpler peer exposure mechanism like clustering coefficient and higher-dimensional peer exposure embedding has better performance for complex peer exposure mechanism like attribute similarity.
Table shows the performance of EGONETGNN and top baselines in the BlogCatalog Data when the network is augmented to make it noisy by randomly removing or adding 10 and 20 percent of edges. We expect the models to perform worse with higher noise but we want to test whether EGONETGNN is stable or whether it degrades compared to the top baseline models. The results show that for different noisy settings, our model is consistently better than the baselines, specifically when the underlying peer exposure mechanism depends complex structures like connected components and clustering coefficients. The results show that our model is more susceptible to noise compared to baselines for peer exposure mechanisms with attribute similarity.
A. ADDITIONAL ANALYSES
Study of variance due to data simulation and random model initializations. We study the variation in performance our models and prominent baselines for BlogCatalog data, where true peer exposure depends on mutual connections. We use three different random initializations of the models for the same data- data-genent generating process simulation to see the variance due to model initializations. As expected, Table 7 shows that there is low variance due to model initializations and more variance due to data simulation. This is because peer exposure resulting from some configurations of neighborhood treatment may be inherently more difficult capture than others. In this experiment, although explicit
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Table Performance of EGONETGNN-CFR in BlogCatalog Data for different output dimension of peer exposure embedding P; for four settings when true peer exposure mechanisms depend clustering coefficients, connected components, mutual connections, and attribute similarity.
Output Dimension
Mechanism
Clustering Coefficient
7.04
0.95
0.97
Connected Components
3.24+18
1.50
1.45
Mutual Connections
7.8242.7
2.24
1.66.0.7
Attribute Similarity
4.64
3.65-2.4
3.81
Table 6: Performance of EGONETGNN and top baselines in the BlogCatalog Data when the network augmented make noisy by randomly removing or adding certain percentage of edges.
Edge Augmentation
-20%
-10%
0%
10%
20%
Mechanism
Estimator
Attribute Similarity
EgoNeIGNN-CFR+
4.55+24
4.54
3.78
2.89
3.85
WNN_MOTIFS
5.40=2.9
5.77
4.53+1.
4.82+2.0
3.7942.9
INE.TARNet
4.1
3.67=2.0
3.60
4.15g2.1
4.7642.2
Clustering Coefficient
FgoNetGNN-CFR+
1.60
1..1
1.06
2.39
1.61
GNN_MOTIFS
2.56=12
2.17
2.34
2.27
2.5941.1
INE.TARNet
2.43
2.63
2.28
2.52+1.0
2.4740.0
Connected Components
EgoNetGNN-CFRK
3.08
2.56
1.93
2.54
3.3641.3
GNN-MOTIFS
4.14
4.34
4.96+1.
4.8941.3
4.74+1.5
TARNTe
4.3
4.43
4.53
4.52+a.7
4.62+0.8
Mutual Connections
EgoNeIGNN-CFR+
3.06
2.09
2.14
3.0622.1
4.78+3..
ENN_MOTIFS
3.28
3.22+114
2.61
2.5841.0
3.7942.9
INE_ TARNet
3.02=1.2
2.57+1.0
2.3640.9
2.74
2.70+0.7
Table 7: Performance (in BlogCatalog Data in terms of PEHE) of our models and prominent baselines, with true peer exposure mechanism depending on mutual connections, for each simulation (i.e., data generating process seed) and three random model initializations.
Simulation EgoNerGNN- TARNet EgoNeIGNN-CFF
GNN-MOTIFS INE.TARNet
CauGramer
2.58-10.46
2.2310.55
4.65+11.04
2.1100.34
5.60-12.83
1.63++1.27
0.72++0.11
2.03++0.12
1.98+0.06
3.83=+0.05
3.45
1.06
3.07-10.53
3.16:10.23
5.38-12.22
3.49-10.46
3.51
3.04-10.16
3.68:10.99
5.85+2.22
3.16::2.36
0.70=10.10
1.26=10.07
1.25++0.03
5.25++1.22
Overall
1.86
2.24:11.55
2.81
2.5011092
1.18
counting of graph motifs performs better in some simulations, FgoNetGNN-CFF exhibits the most robust performance across all simulations.
Study of the contribution of the autoencoder variant of CFR compared to the original CFR. Our EGONETGNN-CFTS architecture implements an autoencoder architecture with reconstruction loss in addition to IPM balance loss, as this helps mitigate the potential loss in expressiveness while balancing representations across treatment groups. To test this, we run an experiment to compare the original CFR without autoencoder to two variants of our method and the best baseline. The results in Table show our variant with the autoencoder has almost always better performance than the original CFR without autoencoder. The balancing approach without autoencoder seems
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Table 8: Performance (in BlogCatalog Data in terms of <PEHE) of three variants of outcome models
in our method: EgoNetGNN-TARNet (without balancing), EgoNeIGNN-CFR (original CFR with balancing), and EgoNerGNN-CFR* (CFR w/ autoencoder). Experiments are conducted for five different data simulations, each with three random model initializations.
Mcchanism
EgoNerGNN- EgoNorGNN-CFR*
EgoNerGNN-CFR (Original)
INE-TARNct
Clustering Coefficient 2,134:1.88
0.95:10.54
2.03=1.56
2.35.10.71
Structural Diversity
1.47=00.90
1.50::0.68
1.57=0.67
4.78:1..00
Mutual Connections
2.86
2.24
4.32.12.32
2.50.10.92
Attribute Similarity
3.95-12.66
3.65:12.40
4.14=1.84
3.59
degrade the performance, even compared to our model without balancing, because the expressive representation learned by earlier modules may not be preserved.
Study of sensitivity to balance loss coefficient. In our experiments, we selected the covariate- balancing hyperparameter Abat = 0.8 based on the analysis presented the original paper (Shalit) et al. 2017). We analyze the sensitivity of the Abat hyperparameter favor values of hyperparameter less than 1, but there is variance in the performance. Therefore, computation resources and time are available, it is best to hyperparameter tuning and model
performance. The results
selection.
Table Sensitivity analysis hyperparameter Abal in BlogCatalog Data with attribute similarity as underlying peer influence mechanism. Experiments are conducted for five different data simulations, with fixed model initialization.
Abal
0.2
0.4
0.6
0.8
1.0
Attribute Similarity
3.75
2.29
3.94
3.78
4.33
Study of the contribution of coverage loss in model selection for robust counterfactual prediction.
Table 10: Comparison of selection strategies using outcome-only features (I,y) outcome
covariates (Ly + Leou).
Simulation
Selection (Ly)
Selection (Ly + Leor)
0
2.23::0555
2.23+0.55
5.43
0.72++0.31
2
10.31
1.06
3.70+:0.69
3.51
4.10:15.39
0.70+10.10
Study of performance under homogeneous peer exposure. We ran additional experiments to show the performance of our methods and baselines in the BlogCatalog Dataset when the true peer exposure mechanism depends on the fraction of treated peers. This is the simplest setting in which all baselines make the right exposure mapping function assumption. When the baselines know what the true mechanism is, our method still performs better than or competitive with the baselines. Interestingly. even though the peer exposure mechanism simple, the baselines suffer heavily. To investigate this further, we remove effect modification and flipped counterfactual settings and find that these factors impact the performance of the baselines. The variants of our model are still better even in the simple setting.
Study empirical runtime and memory requirements for the benchmark data. Appendix A.4 discussed the relative worst-case runtime complexity compared to approaches based on message passing GNN and an approach involving counting causal motifs. We include comparisons of rough time taken and memory usage to experiments the benchmark dataset BlogCatalog (5,196 nodes and 171,743 edges). We note that the implementations of our method and other baselines are not optimized for efficient memory usage, as they designed with reproducibility and simplicity in
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Table 11: Performance in BlogCatalog Dataset in terms of PEHE metric (five data simulations with one model initialization) when true peer exposure mechanism depends on the fraction of treated peers. Comparison of various models with/without effect modifications and with flipped counterfactuals or without flipped. i.e.. no neighbor treated, counterfactuals.
Ous TARNet
GNN MOTIFS
INE TARNeT
JGNN HSIC
DWRS
CEM
TNet
NedEstmater
Coutrumnr
W/ EM
2.4082.74
1.35-182
4.25:22.22
1.75
13 13,00:2.00
1665
14.3633.1
13.4::2
11.76:2.20
5.48:6.12 17
Wif EM4
LI558883
3.88:362
2.17:2.00
2.472234
12. 12,50:4377
1531:241
118221.82
12.58-412
10.58
4.9=11.54
with FM wio Hp
0.26
150 90
2,2242.55
3,00-166
1.96
1.4-2.00
1.W+2.40
08911.26
287-199
1.00-088
mind (not production use). The results below support the theoretical analysis, where runtime is slightly more than MPGNN-based approaches, but is compensated for by a performance gain.
Table 12: Runtime and memory requirements for experiment in BlogCatalog Data with homogeneous exposure underlying mechanism.
midiing time insiate) GPU memory (UBB)
1D44
10.78
12.
6.12
0,47
1.59
9.11
4.78
1.28
24
25
222
4.3
13.022294
11,76:2.20
Study of performance under censored and noisy features. We have already performed an experi- ment (Table 8 in the Appendix) with noisy networks by augmenting edges (i.e, randomly adding and removing 10% and 20% edges in the overall network), and our method's 's performance consistently better than the top- sep-erfformig baselines. For the situation where some node features are missing, we expect the performance will be stable as long as these features are unrelated to the underlying mechanism of peer exposure confounding. Similarly, if these missing features are correlated to other available features or network conditions, the performance is expected to have less of an impact. We performed an experiment with randomly censoring (setting zero) 10% of the features and adding Gaussian noise (0, 0.05) to the features in semi-synthetic BlogCatalog data with attribute similarity as the underlying mechanism to study the sensitivity to small measurement errors and missing features.
Table 13: Sensitivity to noisy and censored network attributes in terms of сPEHE metric in the BlogCatalog Data for attribute similarity as true peer exposure mechanism. Here, 10% node features are randomly set to 0 to simulate missing data and noise is added to simulate measurement error.
MgoNeetiNN-TARNA EgoNetGiNN CFR+ UNN,TARNee,MOTI
INR_TARNeI CauCiramer
Attribute Similarity
1.59
3.54:2.14
4.19::1.75
3.69 10.77:41.71
Study of performance under violation of assumptions. We add an experiment to test the sensitivity to violation of the confounding and interference assumption by generating data where there exists confounding and interference from two-hop neighbors. For confounding, both treatment assignments and outcome generation rely on aggregated neighbor attributes, with first-hop neighbors having a greater weight than second-hop neighbors. To model 2-hop interference, the outcome is influenced by the treatments of 2-hop neighbors, with the degree of influence depending on attribute similarity between the ego node and the neighbors. In the results below, although the magnitude of error has increased, our methods are competitive with or better than the prominent basclines.
Table 14: Heterogeneous peer effect estimation error (εPEHE) when both confounding and inter- ference from two-hop neighbors are present in BlogCatalog Data, when the true peer exposure mechanism depends on attribute similarity. The reported results are for data simulations with one fixed model initialization seed.
GgoNeiGNN-TARNG
EgoNeIGNN-CFR+
GNN.TARNeL
TARNee
CauGramer
Attribute Similarity
10.41::4.71
9.74::4.07
10.47=4.28
10.22=4430
11.86:2..88
27


Reviews:
[{"review_id": "Pf2QrzASOI", "reviewer": "Reviewer_gdhX", "summary": "This paper proposes a new method for estimating peer effects in the presence of interference. Instead of relying on assumptions about the exposure mapping function, their method is able to learn this function from data. The authors show the performance of their method on a variety of network datasets.", "questions": "Why do you only evaluate based on peer effect estimation performance? If I understand the method correctly, I would think that you could also learn direct and total effects?\n\nOn line 260-261: \"Masked weights promotes representation that is invariant to irrelevant contexts ...\" Why do masked weights promote this?\n\nCould you explain why the two aggregations for $\\rho$ are enough and relevant for all different exposure mapping functions? Are there any exposure mappings that the method cannot learn?\n\nHow well does your method perform compared to the baselines when the baselines make the right assumption about the exposure mapping function? I think this would provide additional insights into the performance of the different methods.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- The authors propose a novel method for an important and interesting problem.\n- The authors extensively validate their method on synthetic data and a wide variety of different exposure mapping functions", "weaknesses": "In Section 3.2 the authors explain TARNet and CFR and how they apply balancing to their objective. As I understand it, in the original paper [1] the architecture of these two methods is the same; the difference is that CFR uses representation balancing with an IPM metric. In section 3.2, however, it seems like these two methods have a different architecture. It would be helpful if the authors could clarify this, and maybe also change the confusing naming of their two variants.\n\nRelated to this, the authors write that IPM balances the distribution $P(c,p|t)$. However, I am not sure I understand why this would be desirable.  In CFR, the representations are being balanced with respect to the treatment of interest. Here, given that the goal is to estimate peer effects, the representations are being balanced to a variable that is not the exposure of interest (which would be $\\rho$). Could the authors give some more explanation as to why balancing with respect to the individual treatment would give better peer effect estimates?\n\nIt is unclear how the authors select the hyperparameters, especially the loss weights $\\lambda$. Hyperparameter selection is a very important problem in treatment effect estimation because you cannot optimize them based on the loss of interest (which contains unobservable counterfactuals) [2]. Given that hyperparameter selection can have a very large impact on results, I would like the authors to discuss this in the paper.\n\nOn line 361 the authors write that they report PEHE with respect to peer effect estimation for 5 different simulations. What exactly is meant by simulations in this case? Does this mean that the data is always different in each simulation? Would it make more sense to use the same DGP, but with different random initializations of the model?\n\n[1] Shalit, U., Johansson, F. D., & Sontag, D. (2017, July). Estimating individual treatment effect: generalization bounds and algorithms. In International conference on machine learning (pp. 3076-3085). PMLR.\n\n[2] Curth, A., & Van Der Schaar, M. (2023, July). In search of insights, not magic bullets: Towards demystification of the model selection dilemma in heterogeneous treatment effect estimation. In International conference on machine learning (pp. 6623-6642). PMLR.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "Tv3fZwE9BK", "reviewer": "Reviewer_EFAX", "summary": "This paper tackles the problem of estimating heterogeneous peer effects under network interference when the true exposure mapping function is unknown or misspecified.\nThe authors propose EGONETGNN, a graph neural network framework that learns the exposure mapping function automatically from ego-net structures and contextual features.\nComprehensive experiments on synthetic and semi-synthetic datasets demonstrate the effectiveness of the proposed method.", "questions": "1. What would happen if the unconfoundedness or neighborhood-interference assumptions (Assumptions 2–3) are violated? In particular, the neighborhood-interference assumption restricts interference to first-order neighbors, which may not hold in real-world networks. How sensitive is the method to such violations?\n\n2. The model is computationally about ρε​×avg(d) times more expensive than a standard MPGNN, which may limit its scalability to large networks. \nRegarding scalability, the authors mention that sampling or parallelization could alleviate the computational burden. Could the paper provide quantitative comparisons of runtime and memory usage?\n\n3. How does EGONETGNN perform in noisy networks or when some node features are missing? Does it still outperform the baselines under such imperfect conditions?\n\n4. What are the key differences and advantages of EGONETGNN compared to recent methods such as AEMNet (ICASSP 2025) and CauGramer (ICLR 2025)? Could the framework potentially be extended to integrate attention or transformer-based mechanisms?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. The experiments are comprehensive.\n2. The paper provides the source code.", "weaknesses": "1. The framework still relies on the unconfoundedness and neighborhood-interference assumptions (Assumptions 2–3), which may limit its causal interpretability in real-world settings. If these assumptions are violated, the model’s learned exposure representations and effect estimates could become biased, as the identification no longer holds.\n\n2. The abstract and introduction emphasize theory as part of the contribution, but the main text does not explicitly present any novel theoretical results.\n3. Though scalability is discussed (Appendix A.4), the runtime grows with ego-network size, and large-graph applicability remains an open concern.\n4. The paper only validates the model on synthetic and semi-synthetic datasets, without real-world intervention data.\n5. Appendix A.4 provides runtime measurements only for EGONETGNN itself but does not include baseline comparisons. A relative runtime or scalability comparison with the baseline methods would make the analysis more complete.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "n4nCGwUVee", "reviewer": "Reviewer_KRe6", "summary": "The paper addresses an important and underexplored problem: *learning exposure mapping functions automatically* in causal inference under interference.\nIts motivation is strong, and the proposed framework is technically original and relevant to both the causal inference and graph learning communities.\nThe theoretical ambition (Sec. 3.3) and experimental comprehensiveness (Sec. 4) are commendable.\n\nHowever, under **top-tier conference standards**, several aspects are insufficiently rigorous:\n\n1. The causal identification relies on strong, untested assumptions.\n2. Theoretical guarantees (e.g., consistency, identifiability) are not formally proven.\n3. Experimental evidence conflates representational expressiveness with causal validity.\n4. Interpretability and computational scalability remain weakly supported.", "questions": "1. **On causal identification and assumptions**\n\n   * You rely on *unconfoundedness* and *neighborhood interference* (Assumptions 2–3) for identifiability.\n     Could you clarify **how these assumptions are justified or tested** in your semi-synthetic datasets?\n     For instance, are confounders explicitly simulated and controlled?\n     If not, how do you ensure that violation of these assumptions does not bias the estimated heterogeneous peer effects (HPE)?\n   * Have you considered or tested *partial interference* (interference beyond immediate neighbors)? If so, could you show how EGONETGNN’s performance changes under that violation?\n\n2. **On the learned exposure mapping function ( \\hat{\\phi}_e )**\n\n   * Can you provide **interpretability or visualization** of the learned peer exposure embeddings ( \\rho )?\n     For example, do certain dimensions correspond to structural motifs (e.g., triangles, degree centrality, clustering coefficients) or node attributes?\n   * How do you ensure that ( \\hat{\\phi}_e ) captures *causal influence* rather than merely *correlated structural patterns*?\n     Have you tried using random edge rewiring or attribute permutation tests to probe causal validity?\n\n3. **On theoretical soundness**\n\n   * The paper claims improved expressiveness compared to standard MPGNNs (Sec. 3.3). Could you provide **formal statements or empirical ablations** verifying that EGONETGNN can indeed represent specific causal network motifs (e.g., closed triads) that GCN/GIN cannot?\n   * Is there any **consistency guarantee** that the learned exposure mapping converges to the true one as sample size grows? If not, can you at least discuss conditions under which your end-to-end loss (Eq. 11) provides an unbiased estimator of the HPE?\n\n4. **On training stability and loss design**\n\n   * The overall loss combines several terms (Eq. 11). How sensitive are results to the hyperparameters ( \\lambda_{\\text{cov}}, \\lambda_{\\text{ent}}, \\lambda_{\\text{sp}} )?\n     Could you include a sensitivity plot or table showing their impact on εPEHE or embedding coverage?\n   * How does the *coverage loss* quantitatively affect the learned exposure distribution—does it prevent mode collapse or just rescale embeddings?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "* **Novel problem definition:**\n  The paper formalizes *exposure mapping function learning* as a distinct subproblem, bridging a gap in causal inference with interference (Sec. 1–2).\n\n* **Architectural innovation:**\n  EGONETGNN combines ego-network extraction, masked weighting, and exposure encoding, with coverage-based losses designed for invariant and expressive exposure learning (Sec. 3.1–3.2).\n\n* **Comprehensive experiments:**\n  The study spans multiple network types and exposure mechanisms, demonstrating robustness across topologies and influence structures.\n\n* **Theoretical ambition:**\n  Expressiveness analysis (Sec. 3.3; Appendix A.4) connects to prior results on GNN motif-counting power (Chen et al., 2020) and causal motifs (Yuan et al., 2021).\n\n* **Transparent ablations:**\n  Ablation studies (Table 2) clearly show that the feature encoder and masked weights contribute meaningfully to performance.\n\n* **Reproducibility commitment:**\n  Code release and detailed configuration documentation (Appendix A.6) facilitate replication.", "weaknesses": "1. **Strong and untested causal assumptions.**\n   The identification (Prop. 1) depends on *unconfoundedness* and *neighborhood interference* (Assumptions 2–3). These are unrealistic in most observational networks and are not empirically validated.\n\n2. **Lack of formal theoretical guarantees.**\n   The expressiveness argument shows motif-counting capability but does not establish *consistency* or *generalization bounds* for ( \\hat{\\phi}_e ) or ( \\hat{f}_y ). No identifiability result ensures that learned exposures converge to true exposures.\n\n3. **Unclear causal vs. representational validity.**\n   Reduced εPEHE may result from representation fitting rather than correct causal identification. There are no counterfactual or ablation diagnostics that isolate causal correctness.\n\n4. **Limited interpretability.**\n   The learned exposure embeddings ( \\rho ) lack transparency. Table 3 reports correlations but provides no qualitative insight into which node/edge structures the model captures.\n\n5. **Scalability not demonstrated.**\n   EGONETGNN’s per-node ego-network extraction and masked MLP operations increase complexity by roughly ( \\rho_E \\times \\text{avg}(d) ) over MPGNNs (Sec. 3.3), yet no runtime or memory benchmarks are reported.\n\n6. **Statistical reliability and variance.**\n   The paper reports mean εPEHE but omits confidence intervals, variance decomposition, or sensitivity to hyperparameters ( \\lambda_{\\text{cov}}, \\lambda_{\\text{ent}}, \\lambda_{\\text{sp}} ).\n\n7. **Incomplete comparison to modern baselines.**\n   Proximal and double-negative-control causal estimators (Tchetgen et al., 2020; Miao et al., 2024) are cited but not evaluated. Transformer-based causal GNNs are missing.\n\n8. **Potential endogeneity in joint learning.**\n   Because ( \\phi_e ) is optimized via outcome loss ( y ), exposure representations may absorb post-treatment bias. No regularization or identification constraint ensures causal directionality.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "gES83QMi7X", "reviewer": "Reviewer_DGdQ", "summary": "The paper proposes a new, custom GNN architecture, EgoNetGNN, for the network interference problem. The idea is to learn the underlying exposure mapping behind the network interference.  EgoNetGNN is more expressive than standard GNN architectures that have been used for this task in previous works, allowing EgoNetGNN to exhibit superior counterfactual estimation performance compared to such competing baselines, which is validated on synthetic and semi-synthetic interference settings.", "questions": "Could the authors provide further significance for the practical relevance of exposure mechanisms that they demonstrate superior performance over? For example, empirical works motivating the given exposure mechanism, etc. \n\nWould it be possible to compare the performance of EgoNetGNN to the baselines under exposure mechanisms that more of the baselines actually are expressive enough to handle? Basically if practitioners were to use your method not knowing wha the true underlying exposure mapping should be, and it ends up being one that other methods could also express, what kind of relative performance might we expect?\n\nIf a practitioner wanted to EgoNetGNN but had missing graph information, would there be any recommended modifications or imputation methods?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Developing a novel GNN architecture that can automatically learn exposure mappings for heterogeneous peer effect estimation is a significant contribution over using out of the box GNNs. The authors also motivate this novel architecture by showing, theoretically, a gap in expressivity. The definitions and claims around these results are clearly stated. The architecture of EgoNetGNN is clearly stated as well as the training procedure. Within the given experimental settings, EgoNetGNN exhibits generally superior performance to the baseline methods, and in scenarios where that it is not the case it is explained that the baseline method is able to express the given specific exposure, while a strength of EgoNetGNN lies in expressiveness across each of eg. mutual connections, clustering coefficient, attribute similarity. This reflects EgoNetGNN being able to learn the appropriate exposure mapping in general while baseline methods may be more limited. Experimental details are presented well and ablation studies are included, strengthening the empirical results.", "weaknesses": "The weaknesses of the paper do not rest in whether it adequately addresses the scope that it claims to cover, but rather the significance of the scope and setting compared to previous baselines. In particular, at least as written, it felt that exposure mechanisms were chosen not because of practical relevance but because this is where the improvement compared to baselines emerges.\n\nEven beyond the scope of GNN based approaches, as a method for estimating counterfactuals under network interference, EgoNetGNN relies on the, at least in some settings, strong assumption of known network structure and features. While reasonable for some setting such as perhaps a social media platform, this limits the scope of the work to the broader network interference problem.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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:

Decision: Accept (Poster)