Using Embeddings for Causal Estimation of Peer Influence in Social Networks
Authors: Irina Cristali
Victor Veitch
Venue: NeurIPS 2022 Accept (2022)
Keywords: causality
peer contagion
social networks
embeddings

Abstract: We address the problem of using observational data to estimate peer contagion effects, the influence of treatments applied to individuals in a network on the outcomes of their neighbors. A main challenge to such estimation is that homophily - the tendency of connected units to share similar latent traits - acts as an unobserved confounder for contagion effects. Informally, it's hard to tell whether your friends have similar outcomes because they were influenced by your treatment, or whether it's due to some common trait that caused you to be friends in the first place. Because these common causes are not usually directly observed, they cannot be simply adjusted for. We describe an approach to perform the required adjustment using node embeddings learned from the network itself. The main aim is to perform this adjustment nonparametrically, without functional form assumptions on either the process that generated the network or the treatment assignment and outcome processes. The key contributions are to nonparametrically formalize the causal effect in a way that accounts for homophily, and to show how embedding methods can be used to identify and estimate this effect.

Using Embeddings for Causal Estimation of Peer Influence in Social Networks
Irina Cristali
Victor Veitch
Department of Statistics, The University of Chicago Google Research
Abstract
We address the problem of using observational data to estimate peer contagion effects, the influence of treatments applied to individuals in network the out- comes of their neighbors. A main challenge to such estimation is that homophily- the tendency of connected units share similar latent traits-acts served confounder for contagion effects. Informally, it's hard to tell whether your friends have similar outcomes because they were influenced by your treatment, or whether it's due to some common trait that caused you to be friends in the first place. Because these common causes are not usually directly observed, they cannot be simply adjusted for. We describe an approach to perform the required adjustment using node embeddings learned from the network itself. The main aim is to perform this adjustment nonparametrically. without functional form assump- tions on either the process that generated the network or the treatment assignment and outcome processes. The key contributions are to nonparametrically formalize the causal effect in a way that accounts for homophily, and to show how embed- ding methods can be used to identify and estimate this effect. Code is available at http://ggttubb.oo//rnnniiiiiit -ow.eeenooaaaaooooo--eeeeeee
an unob-
Introduction
We are interested in estimating peer effects, the causal influence of individuals on their neighbors.
Example. We want to infer the effect that social pressure has on vaccination. Suppose we observe data from a population where each unit i is person in interconnected social network, and for each unit we know whether they were vaccinated at the beginning of the study period, T,, and whether they were vaccinated the end of the study period, Y, We are interested in estimating the effects of the treatment T; of person on the outcome Yj of person j. In addition to their vaccination status, each unit has attributes C that act as (proxies for) causes of both the particular network ties they form, and their vaccination behavior. For instance, C, may include age, race, education status, income level, political affiliation, and so forth.
The core challenge here is that we want to estimate a causal effect (c.g., what would happen if we intervened by vaccinating popular people?), but the variables C, Cj act confounders between the treatment Т, and outcome Y, The reason is that when defining a contagion effect from j to i we must condition and Cj. so the conditioning creates a dependency between these variables (it acts as a collider). For example, if we learn that Alice and Bob are friends then can infer that they likely live in the same city. Accordingly, the association between T, and Y; may be either due to contagion: the causal influence of T3 on Y, (Bob got vaccinated because Alice did), or due to homophily: the tendency of similar people to be connected in network (c.g., Alice and Bob got vaccinated because they both
the presence of edge between (i. The edge is causally influenced by both C,
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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live in a major city, and it's only due to chance that Alice got vaccinated before Bob), In general, homophily is confounded with contagion [STI1].
Now, if we observed the attributes C, we could formalize the contagion effect using standard causal tools and then identify the effect from observational data by adjusting for C. [EKB16; EB17; BDF20J. However, often such detailed knowledge of C' is unavailable. In this case straightforward causal identification and estimation is not possible.
In such situations, we might hope to make use of the following intuitive observation. The pattern of network ties itself carries information about each C.. Indeed, this is the root problem that causes the network itself obtain estimates C
the confounding issue. Accordingly, we might hope for the attributes of each node. Then, we could adjust for the estimated C₁ in some suitable causal estimation procedure. The aim of this paper is to clarify this type of procedure.
Shalizi and McFowland III [SM16] provide estimation strategy of this kind. They first assume that the network is generated by either stochastic block model [HLL83; LW19] or a latent space model [HRH02; RFR16]. They assume the attributes C correspond to the latent community identities or latent space positions. They then further assume that the outcome of each unit is defined by a particular linear structural equation, which includes term for both the average treatment of that node's neighbors, and a term for the effect of the attributes C.. Their procedure is to estimate C using the assumed network model, and then use the estimated C' in linear regression to determine the coefficient of the werage-neiighoor-reein treatmett term.
This approach, however, critically relies on the assumed parametric form of both the network model and the outcome model. Indeed, even the target of estimation is defined parameter of this assumed model. Accordingly, when the parametric network model or linear outcome model is mis- specified, the meaning of both the estimand and estimation procedure becomes unclear.
The goal of this paper is develop a non con-paraettric procedure in this vein. That is, the aim is causal estimation of contagion effects by adjusting for network-inferred attributes without relying on detailed parametric assumptions. Towards this end, the paper follows three main steps:
1. Formalize the target causal effect non oon-parametrically. The main challenge is that the estimand must depend on the network we are working with (because contagion requires knowing who is friends with whom) and the network must itself be modeled as a random variable which is a function of the unobserved confounders C' (to accommodate homophily).
2. Derive sufficient conditions for the estimated attributes to yield causal identification. The idea is that it is not necessary to exactly reconstruct C, but only extract the minimum information that will identify the causal effect- this turns out to be (plausibly) much easier task.
3. Give concrete method for contagion estimation using node embedding techniques to extract the information from the network that is relevant for predicting peer influence, and illustrate the practical performance of this technique.
2 Setup
Consider a network G, of n individuals, where connections between people are encoded through undirected edges between nodes. We define the degree of a node as the number of connections has. The neighbors of node are the nodes with which has ties. Each such link is captured by the network adjacency matrix A, where As = (i and j share a tie)- We take Ay = for all We also consider a vector of variables associated with each node: (Y,, C., T,). Here, Y, is the observed outcome, T; is the treatment, and C, are (unobserved) attributes that may causally influence Y, T and A.
The data is generated according to the following structural equation model, adapted from Ogburn et al. [Ogb+17] 17] to model the network random variable.'
We simplify by removing the influence of neighbors' covariates outcomes and treatments.

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C. - folec.l: Ag fal(Ci)begl:
(2.1)
T. - ST]
Y fy (ST() A. 11), Cy,
The E variables represent exogenous noise, which we take to be identically distributed and indepen- dent of the network and of each other. The function Sy summarizes the neighbors' treatment e.g., Sy could be the average function, or the logical OR function. This means that the treatment assign- ment for each node depends only on its attributes C, the outcome assignment depends on the node attributes C; and the treatments of all its neighbors, and the network structure depends on all C. The structural functions f are fixed but unknown.
Formalizing the causal estimand
Consider setting T, t for each unit i. Our aim is to formalize the idea of "average influence of node's 's neighbors' treatments on its outcome"..
An intuitive choice is
full info
1 Σ B[V]jd((T==")) t*), (C.)., G.).
(3.1)
Here, do is Pearl's do notation, and indicates that the treatment (vector) of all nodes is intervened on and set t". Note that, following the structural model in equation 2.1, nodes' outcomes only affected by their neighbor's 's treatments. Accordingly, the estimand is the average outcome we would have seen had the treatment assignment been set T 1", keeping both the node attributes {Ci) and the network G. fixed. The interpretation of this effect is: the average outcome under the hypothetical treatment, applied the same of people connected by the same link structure.
Unfortunately, this estimand will not usually be identifiable. The reason that the node attributes C' are unobserved; e.g., we don't observe city of residency, and cannot perfectly reconstruct it from the network structure. To circumvent this, we instead define the version of this estimand that can (hope to) identify from the graph alone:
1 Σ TE do = =.",, G.]-
thr- ==
(3.2)
This new estimand can be understood as taking the estimand in (3.1) and marginalizing out the in- formation about C that cannot be inferred from G,, is the average outcome under the hypothetical treatment, applied to the same link structure and to a set of people consistent with the link structure. In other words, it represents the peer contagion effect the same graph with node properties only fixed to be consistent with the graph structure. Although this is somewhat less natural, we will see that it makes identification plausible.
There an apparent significant drawback of the formalizations presented in Equations (3.1) and (3.2): both estimands are fundamentally tied to the particular sample available in study, since both involve conditioning on the observed network GR. We may be nervous that the formal quantity is tied to idiosyneracies of the particular observed network. In Theorem 1, we show that kind of sample dependency is not a serious issue. The approach is to show that both the causal estimands full info and shz- introduced in Equations (3.1) and (3.2) converge in probability to fixed quantity as the size of the network G₂, increases. See Appendix A for the proof.
Theorem 1 (Generalizing the sample-based causal estimands). Consider observed social net- work G.,, and let Y, be the labels associated with each node i. Assume the the following
(i) There exists M > such that Var(Y.) S for all i;
(ii) For any pair (i,5) of nodes selected IP{ and share neighbor)) - 0, as x..
uniformly at random,
3

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full info Then, there exists fixed real number sb such that Ve
-> ab and tht- - s) in probability.
This theorem establishes that, provided the graph is sparse suitable sense, then both causal es- timands converge to the same fixed quantity which does not depend on the particular network Gn- This decouples the interpretation of the causal estimand from the particular network under consid- eration. Additionally, shows that the distinction between fuull info and tht- is not so important for practical applications-ffr for large networks, the values will anyway be very close. Note that this result is about proving that the estimand is well-defined, not about establishing convergence of some estimator.
4 Causal inference of peer effects using node embedding methods
Having formalized and motivated the target causal effect of interest, now turn to sufficient con- ditions for causally identifying it.
As previously discussed. if we could exactly reconstruct the latent C2 then identification would be straightforward. However, usually, this reconstruction is impossible. Happily, is not necessary: an imprecise proxy for C; might still suffice for causal identification. Our goal now is to find sufficient conditions for an inferred proxy to enable identification. The next theorem gives such a condition. Informally, the condition is that we need proxy X; that carries enough information about C; so that the graph itself carries no further information about Y, after conditioning on Aę. As we shall discuss in Section 5, node embedding methods can be viewed as tool for constructing such Aş-
We first define the following object, which plays substantial role the remainder of the paper.
Definition 1.
Let V; Sy As 1}}² be the aggregated treatment at node i, and ve its value under the hypothetical treatment intervention tt. For graph G, the vertex conditional outcome model is the function given by mG, (v Xs) = EY; vt Ae}.
Theorem (Sufficient conditions for identification). Let G, be the network, A its adjacency matrix, and Y, the node-associated labels. Suppose that for each node we have proxy for the latent attributes, x E RR, such that
i Y; L. Asyl(>;, ) for all i and j- P(V; = = 1x,) > 0 for all v": ii Xa is C.-measurable.
Then E[Y; =t); Gn;
mG. (ut, Az).
Proof. We sketch the proof and provide the full mathematical justification in Appendix B. Con- sider Figure I illustrating the core identification issue. The node-level causal effect E[{]/do(T' = the network structure, and, in particular, Doing so opens a backdoor path between Y, and T, (hence, also Vi) going through the unobserved C. The main idea of the theorem is that conditioning on A; removes the effect of conditioning the Aij, and thus prevents the backdoor path from being opened.
t"), Gn, Å,] relies conditioning
the edges Aij-
In Theorem 2, condition (i) gives notion of "sufficient amount of information about C" for causal adjustment, and is the main point of the theorem. Con- dition (ii) is the standard pos- itivity assumption required for causal identification. Finally, condition (iii) is the condition that X, is a partial reconstruction
A
Ay
Figure 1: Identification of causal peer effects using embeddings.
In general, the function Sy is set by the analyst for practical estimation purposes; this paper assumes Sy is the mean function.

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of the node-specific attributes C, This condition is necessary to ensure X; is not itself collider for the latent node attributes.
This result readily extends to a condition for the identification of the peer contagion estimand dh-- = E [E(; |oo(T = 1"), G.. introduced in Section 3. This is the content of the next result; Appendix C for a proof.
Corollary 3 (Identification of the causal estimand th Consider the same set-up and notation as that of Theorem 2. Then, the causal estimand ih- can be identified as follows
n V1 Σ
4/--
Blma, (ot;A)))G.)
(4.1)
Corollary 3 gives, in principle, a strategy for identifying the causal effect using the observed A. Namely, learn mG, (u, A²) and P(A, | G,)) and plug in to (4.1). This result is exact, and applies to any network. However, learning P(): Gv) seems difficult-it's unclear how to model this distribution. To circumvent the need learn the unit-wise uncertainty about A; we now give another result that again relies on the validity of the law of large numbers derived in Theorem 1. Intuitively, when the law of large numbers holds, we can replace averaging over the per-node uncertainty by averaging over all of the nodes (that is, the mean converges to the unit-level expectation).
We now state the main identification result Theorem 4 below which is derived from both Theo- rem 2 and Theorem 1. We include its proof in Appendix D.
Theorem 4. Consider the same set-up as that of Theorem 2. Furthermore, assume that the two conditions of Theorem are also satisfied. Then lim,-sco n Ei=1 mG, (Vi, Ax) 1è, in probability,
full info and lim the v/ in probability.
where db is the fixed real number such that lim V/-
5 Estimation of causal peer influence effects
We have now established formal causal estimand vhe- and conditions node-attribute proxies that enable identification. The basis for our estimation procedure will be Theorem 4, which says that-for suitable À and sufficiently sparse Gn-thhe statistical estimand
1
the = ma., (v7, As)
(5.1)
approximates the causal estimand whe with the approximation becoming exact as + oc. Recall mG, (vf A:) E[[Y 1 Ar]. Our task is to estimate e from the observed data using the observed treatments and outcome (Ti, Y;) and the link structure G,,.
semi-supervised prediction models to learn R and ma. (v", Xx) =s E[Y; uz. Å.J. Then, we plug these estimates into (5.1) as m and x. For concrete- ness, describe particular approach based on Veitch al. [Vei+19] and Veitch et al. [VWB19]. The estimation procedure follows three main steps:
To achieve this, we use embedding
Step 1. We train model using relational empirical risk minimization 19] to jointly learn the embeddings ); and mec (v; Å.). To do this, first compute 5y (ft, Ai 1}) each vertex iie. we create a new aggregate treatment feature and attach it to each vertex.
Relational ERM works by minimizing risk defined as an expectation over randomly sampled sub- a sampling algorithm that returns a random
graphs of the larger network. Let Sample(Gn,) k) (Gn,k) subgraph of size l from G, (e.g., the subgraph induced by random walk of length k). Let va, be the set of vertices of the subgraph Gg. We then define the loss function on the Gş to be
L(G 7, 9
Σ(m ( m(1; (6)))))) 2
Σ
CrossEntropy (XT),) (5.2)
NERG
L.jEG, XVO,
where q E [0, and a is the sigmoid function. The second term a network reconstruction term that extracts node-level information from the network by requiring the embeddings to be predictive
5

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of the edge structure. The first term learns predictor m, parameterized by (global parameters shared across the network), that predicts y; from the aggregated treatment v; and the embedding A₃- In our experiments, binary, the binary cross-entropy loss is used instead of the MSE loss. Although the map from the em- beddings is linear, because the embeddings are unconstrained, the combined embedding- linear map can flexibly represent complex relationships between the network structure and trroommm/rremmett....mmm In Section 6 we see that the model ably captures relationship between Y and latent attribute C that is non-linear in C, where the relationship between the network and C' is complex.
will be a linear predictor, and the (parameters of the) linear map. When y is
Then, we train the model by fitting
A f = argmin A)
(5.3)
x,7
Step 2.
We define mG, (of, X²) = m(v A; where of is the value of es attained under the interventional treatment T = L". Then, for each unit vertex i in G, we compute ma,, (0% A²)
Step 3.
Finally, the peer influence estimate the average over the estimated vertex conditional outcomes ma, (v7, 2z):
dh- (Gn) 1 Enna
man (Ut Ad).
5.1 Validity of the estimation procedure
In this section, we informally discuss the validity of the estimation procedure described in the previous section. More specifically, recall that, in Theorem 1, we showed that both causal peer influence estimands proposed, v22
info and ehy- converge, in probability, to a fixed real num- ber vi, as n + DO. The question that remains is whether the estimator proposed, vi- = 1/n Σi= mG. (uf 2c). also converges to eb, in probability, as n + co. This will clearly hold as long as ma. (u] X²) + mc.. (vt >;) for some À satisfying the conditions of Theorem 2.
Why should one expect the above consistency property? First, consider in(v, A) + E[Y, v. Aj. The map m. is defined as the minimizer of the squared error empirical risk. The minimizer of the true risk is the required conditional expectation. Accordingly, this part of the learning is, essentially, the same as what occurs with non-network data and is plausible.
The trickier judgement is why can one expect that Å, = X for some satisfying the conditions of Theorem 2? That is, is it plausible that the learned embeddings behave as proxies for the node-level attributes (i.e., are C-measurable), and carry sufficient information to adjust for the homophily effect (i.e., Y 1 Ag (A,, ?))?
The C--mcaurabillity is just the requirement that each node-level embedding captures unobserved information specific to that node. This is exactly the aim of embedding methods, to leverage net- work structure in order to capture the latent properties of each node [HYL17]] Accordingly, it is reasonable to assume that the embedding-bassed method described in Section 5 satisfies this property (at least, asymptotically). With respect to whether the embeddings carry sufficient information to adjust for the homophily effect when estimating peer influence, we note the following: in order to satisfy this condition, suffices that the embeddings are predictive of the outcome or of the edge structure. To see why this is plausible, recall the expression of (5.3) used to fit the embeddings. The cross-entropy term encourages the embeddings to be sufficient (at the graph level) for predicting the edge structure. The squared error term encourages the embeddings to be sufficient for predicting the labels Y, Hence, the objective function is aimed at exactly the requisite information.
Unfortunately, despite the intuitive justification above, there is no firm guarantee that any given em- bedding method actually achieves its conceptual goal. In effect, by working with embedding meth- ods one is trading off the precise guarantees enabled by parametric models (exploited in (SM16]) in favor of techniques that have good empirical performance even in the (typical) case that the networks have structure that is inconsistent with parametric specification.
In fact, for the specific case of relational ERM we can say something further. Davison and Austern [DA21] study the asymptotics of the embedding learning procedure under the assumption that G, is

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Table The embedding- estimator the effectively adjusts for confounding and recovers true treatment effect. The ground truth value of peer contagion is 1. Zero, low, and high confounding levels correspond to B1 0, 1, and 10, respectively. For shr-, reported values represent the mean over 100 different global random seeds. The seed for the simulated treatment and outcome data is kept constant. For the Unadjusted and Parametric estimators, the reported values represent the respective estimated regression coefficients for the aggregated treatment used when predicting Y
district
age
join date
Conf. level
Zero
Low
High
Zero
Low
High
Zero
Low
High
Unadjusted
0.99
1.64
7.40
1.00
1.39
4.90
0.99
1.38
4.81
Parametric
0.99
1.41
5.28
1.00
1.33
4.20
0.98
1.28
4.00
dit- 0.84
0.96
1.17
0.94
0.94
1.11
1.01
1.03
1.10
generated by an exchangeable random graph, or a graphon (a broad family of models that includes, e.g., stochastic block models and latent space models [Bor+ 16; CCB16; CF17; CD18; J8)- Informally, Davison and Austern [DA21] show that, asymptotically, the embeddings are functions of node-specific attributes C, alone and that these embeddings capture the limiting distribution of
the network (see Davison and Austern [DA21] Thm. 7). Accordingly, this case, the learned embeddings satisfy the conditions of Theorem 2.
Experiments
To study the practical performance of the estimation procedure described Section 5, we conduct experiments using semi-synthetic data. We website Pokec [TZ12; LK14]. The data includes both links between users and limited node-level covariates. Our basic strategy will be use these node level covariates as the latent C,'s. To do this, we simulate treatment and outcomes for each unit in a manner that depends on C.- At estimation time, we hide the C₁ variables, and check how well we can recover the true causal effect using only the graph data and observed treatments and outcomes.
subset of data from the Slovakian social media
We extend the setup of Veitch al. [VWB19] who applied a similar relational empirical risk min- imization based technique in order to adjust for the network unobserved confounding effect when performing average treatment effect (ATE) estimation of Ti on Y. That work assumes pure ho- mophily, whereas this paper examines peer-based contagion effect estimation.
Following Veitch et al. [VWB19] 19] we analyze a sub- network of approximately 70000 users connected by roughly 1.3 million links, representing subset of the original Pokec data restricted to the districts Zilina, Cadca, and Namestovo, from the same region. The analysis consists of two parts:
1. estimating the peer contagion effect of binary treatments on continuous response variables, vary- ing both the source of confounding and its strength;
2. estimating the peer contagion effect of binary treatments assigned at time 4, the beginning of a study period, treatments assigned at subsequent time 12, the end of that study period.
See supplement (Appendix F) for additional experimental details.
6.1 Continuous outcome
For the first set of experiments, the treatment T and outcome Y values are generated from three variables (one
time), taken the unobserved confounders C: district, age, and Pokec join date. For each of these three confounders, we standardize, then bin each of them into three possible values: -1, 0, and 1. We then apply function g which transforms them into probabilities: g(C) 0.5 + 0.35 C, so that gCC) E (0.15, 0.5, 0.85). Then, we simulate according to:
T = Bern(g
V. Average(T, Agg 1); Y; = 3o V, + 31 g(C,) + Eis ~ (0.
(6.1)
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The parameter Bu controls the amount of peer influence, while 31 measures the amount of network confounding. In this simulation setup, the causal peer contagion effect of interest, dh- is precisely equal to 36. We fix 30- = 1 as the ground truth value of confounding, and let 31 vary in {0, 1, 103 corresponding to no, low, and high contagion, respectively.
Since we only observe T; Vi and Y, we test whether oh. the estimator proposed in Section 5, adjusts for the unobserved confounding due C, and accurately estimates f3g 1. For each combination of C, and strength level 31, we obtain simulated V; and Y; values, then we train the embedding- -based relational empirical risk minimization model described in Section 5, using a random walk sampler with negative sampling, and the default settings of Veitch et al. [VWB19]. Finally, we use the trained model to predict two sets of adjusted Y values:
Yj's resulting from the aggregated treatment uf. when all T, are set O, i.e. mn oou
2. Y'ss resulting from the aggregated treatment uy. when all T3 are set to 1, i.c. m(A, ofr1;=1))
The causal effect estimator
interest
is then 1/n m(de TT.q=1))
1/n E m()., (((1;=0))
We compare the results of this procedure against those of two baseline estimators, The first one the naive estimator which does not adjust for confounding, namely 1/n >, B|
1/n E; B[Y (u where expectations are the estimated values obtained via linear regression. The second baseline seeks to mimic the parametric peer contagion estimation procedure of Shalizi and McFowland III [SM16]. by fitting a mixxe-membersihh stochastic block model to the Pokec data, inferring 128 latent communities (128 is equal to the dimension of the embeddings), then predicting Y via linear regression on the aggregated treatment and inferred latent community values.
Table I summarizes the results of the embedding method compared to those of the above baselines. Whenever confounding was present, thz- yielded the most accurate results out of all three estima- tors, by significant margin. This confirms that the nonparametric method can leverage the network structure and obtain reliable peer contagion estimates. With no confounding present, all three esti- mators produced valid results, yet, when district and age were used as unobserved confounders, vh-- was slightly less accurate compared to the baselines. Indeed, if there is no unobserved confounding. then no network-based adjustment is necessary; the variation due to satheddiig-mode fitting causes errors that simpler methods do not suffer from. In Appendix E, Table 3, we provide error bars for the average estimated peer effects obtained in Table 1, which aim to illustrate the consistency of results. The errors reported for dhz- represent the standard errors computed over 100 different global random seeds, while the seed for the simulated treatment and outcome data is kept constant. For the Unadjusted and Parametric estimators, the reported error bars represent the standard errors of the estimated regression coefficients for the aggregated treatment used when predicting Y. The main caveat is that due to the relational, non-i.i.d. structure of our data, these errors do not repre- sent proper confidence bands, and should be interpreted with caution. To obtain valid confidence intervals, proper asymptotic normality results for the studied estimators are necessary.
6.2 Binary outcome: model for peer influence and vaccination
Recall the motivating example in Section I: some proportion of the observed population gets vacci- nated time 4, while others get vaccinated at subsequent time (2. We now illustrate how to use node embeddings to study the peer influence of the former group on the latter. We use the following data setup:
1. Simulate the treatment by the same procedure as in Section 6. i.c. as a function of the unobserved confounders;
2. Randomly select 50% of the vertices, and set Y; = T;, but then delete T; for those respective nodes. The uncensored treatment values T; correspond to individuals who were vaccinated at t1, while the Y; values represent individuals vaccinated at tz;
3. Compute the causal effect of Vú on Y, only for the vertices where Y; is defined.
The above design matches the following scenario: whether or not an individual gets vaccinated depends on latent attributes such as their age, or location. Since Y, Tú, there is no causal peer
8

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The embedding- estimator the accurately recovers the true peer contagion effect of binary treatments other subsequent binary treatments. The ground truth peer contagion value is O. For vi... the reported values represent the mean over 100 different global random seeds. For Unadjusted and Parametric estimators, the reported values represent the respective estimated regression coefficients for the aggregated treatment used when predicting Y.
Table 2:
Peer influence on vaccination
district
age
join_date
Unadjusted Parametric L
2.03
0.12
0.68
1.30
1.03
0.98
0.09
0.11
0.22
influence effect of the neighbors j on node i. Despite the lack of a causal relationship, an association clearly present due to the dependence on latent confounders. The question is whether the proposed method can adjust for the unobserved causes and render the true causal effect
Using the same baseline estimators, and an analysis similar to that in Section 6.1, we obtain estimates for the peer contagion effect, summarized in Table 2. We note that, for all three hidden confounders, the embedding- "age" covariate, the naive, unadjusted estimate is very close to the embedding- that age may not be the optimal predictor for network tie formation, preventing the non-parametric method from appropriately adjusting for the network structure, and yielding our method similar to the naive one, in this situation. We also note that, when registration date is used confounder, all three estimates are less accurate, suggesting that "join date" may also not be the best predictor of the network edges. Nonetheless, in this scenario, the nonparametric estimator still has superior performance over the two baselines. In Appendix E, Table 4, we provide error bars for the variation induced by the choice of random seed, obtained in similar way to those corresponding to the continuous outcome scenario.
method produces the estimates closest to the ground truth value of O. For the
one, showing
hidden
In Appendix G we perform additional analyses further varying the values of 31 and noise level e, as well as illustrate the performance our method on the Wikipedia hyperlink network [Yin+17].
Related work
There is an active literature on estimating causal effects from networked data [ZA21].
Randomized controlled trials. To estimate causal effects under interference and network con- founding, several recent works rely on designing randomized experiments. Toulis and Kao [TK13] use potential outcomes and sequential randomization design estimate peer influence on ob- served networks. Eckles et al. [EKB] 6] develop an experimental design to study the impact of peer feedback on the behavior of Facebook users. Fatemi and Zheleva [FZ20b] propose an experimental design for estimating the effect of the treatment alone on the unit's outcome, isolating it from peer effects, while Fatemi and Zheleva [FZ20a] discuss an approach of minimizing selection bias when performing A/B testing on networks.
Causal network inference from observational data. Within observational settings, van der Laan [van]4], Ogburn et al. [Ogb+17]. Tchetgen Tchetgen et al. [TFS17), and Tran and Zheleva [T722] tackle the issue of causally estimating social contagion from observed networked data, yet they assume there are latent sources of network confounding which affect both the treatment and the outcome, by considering all node-related features as known. This is a significant limitation which this paper seeks to address.
Other works which account for latent confounding assume that the network (and, implicitly, the unobserved confounders) can be represented using parametric methods. Shalizi and McFowland III [SM16] and Shalizi and Thomas [STI1] use stochastic block models and latent space models in order to obtain consistent estimators of peer contagion. Building upon this work, Sridhar et al. [SBB22] adjust for the unobserved confounders in peer influence estimation by using Poisson Influence Factorization (PIF) models.

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Some of the recent studies which address nonparametric peer influence estimation from observa- tional data propose different upproaches which complement the node embedding based method pro- posed in this paper. Fckles and Bakshy [EB17] estimate peer effects on hundreds of millions of observations from Facebook data by using high-dimensional adjustments for covariates via propen- sity score models. One limitation of this work is that covariates are readily available, whereas this paper seeks to cover the scenario in which they are unobserved. Egami and Tchetgen Tchetgen [ETT21], on the other hand, tackle the situation of unobserved confounding, by using negative con- trol outcomes and exposure variables to estimate contagion effects. This approach complements the methods of this paper. While not directly related to peer influence estimation, Guo et al. [GLL20] also propose a nonparametric technique of adjustment for the unobserved features that affect net- work treatment and outcome. They propose using "graph attentional layers" in order to map the network information to a partial representation of the hidden confounders and use it to perform counterfactual evaluation.
8
Discussion
This paper studies the problem of causally estimating peer influence from observational data in the presence of unobserved confounding. The main contributions of this work are
formalizing and justifying two causal estimands for contagion while accounting for homophily (Section 3);
2. giving sufficient conditions for network embeddings to enable causal identification (Theorem 2); illustrating how embeddings can be used to estimate peer contagion and showing their practical performance is overall better than that of standard naive and parametric estimation techniques (Section and Section 6).
There are several avenues for further work. First, one current methodological limitation is the lack of an asymptotic normality result for the peer influence estimator shz- defined in Section Character- izing the asymptotic normality of the main estimator would allow one to construct valid confidence intervals around the estimated values, desired significance level. Secondly, this paper only fo- cused network embeddings learned via subsampling as proposed nonparametric method for peer contagion estimation. An important future work direction is exploring other nonparametric techniques (e.g. graph neural networks [Zho+18; Wu+19; ZCZ22; Ma+22; Hus+21J) which lever- age the homophily present social networks and are able to learn meaningful node representations to be used downstream for peer influence estimation.
9 Acknowledgements
This work was partially supported by Open Philanthropy. We thank Steve Yadlowsky for helpful comments and feedback. We would also like thank the Research Computing Center (RCC) The University of Chicago for computing resources.
10

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References
[Bor+16]
C. Borgs, Chayes, H. Cohn, and N. Holden. "Sparse exchangeable graphs and their limits via graphon processes". In: Journal of Machine Learning Research (2016).
[BDF20] Y. Bramoulle, H. Djebbari, and B. Fortin. "Peer effects in networks: survey". In: An- nual Review of Economics (2020).
[CCB16] D. Cai, T. Campbell, and T. Broderick. "Eddg-exchanggabble graphs and sparsity (nips 2016)". In: arXiv: Machine Learning (2016).
[CF17]
Caron and E. Fox. "Sparse graphs using exchangeable random measures", In: Journal of the Royal Statistical Society: Series B (Statistical Methodology) (2017).
[CD18]
H. Crane and W. Dempsey. "Edge exchangeable models for interaction networks". In: Journal of the American Statistical Association (2018).
[DA21]
A. Davison and M. Austern. "Asymptotics of network embeddings learned via subsam-
pling". In: arXiv:stat. ML (2021).
[EB17]
D. Eckles and E. Bakshy. "Bias and high- -ihh-iimeniioall adjustment in observational stud- ies of peer effects". In: Journal of the American Statistical Association (2017).
[EKB16]
D. Eckles, R. Kizilcec. and E. Bakshy. "Estimating peer effects in networks with peer encouragement designs". In: Proceedings of the National Academy of Sciences (2016). N. Egami and E. Tchetgen Tchetgen. "Identification and estimation of causal peer ef- fects using double negative controls for unmeasured network confounding". In: arXiv: stat.ME (2021).
[ETT21]
[FZ20a]
Z. Fatemi and E. Zheleva. "Minimizing interference and selection bias in network ex- periment design".. In: Proceedings of the Fourteenth International AAAI Conference Web and Social Media (ICWSM 2020). 2020.
[F720b]
Z. Fatemi and E. Zheleva. "Network experiment design for estimating direct treatment effects". In: KDD International Workshop on Mining and Learning with Graphs (MI.G 2020). 2020.
[GLL20]
R. Guo, J. Li, and H. Liu. "Counterfactual evaluation of treatment assignment functions with networked observational data". In: 2020.
[HYL17] W. Hamilton,
Ying, and Leskovec. "Representation learning on graphs: methods and applications". In: IEEE Data Engineering Bulletin (2017).
[HRH02]
P. D. Hoff, A. Raftery, and M. Handcock. "Latent space approaches to social network analysis". In: Journal of the American Statistical Association (2002).
[HLL83]
P. Holland, K. B. Laskey, and S. Leinhardt. "Stochastic blockmodels: first steps". In: Social Networks (1983).
[Hus+21]
H. Hussain, Duricic, E. Lex. Helic, and Kern. "The interplay between commu- nities and homophily in semi-supervised classification using graph neural networks". In: Applied Network Science (2021).
[Jan18]
S. Janson. "On edge exchangeable random graphs". In: Journal of Statistical Physics (2018).
[LW19]
C. Lee and D. Wilkinson. "A review of stochastic block models and extensions for graph clustering". In: Applied Network Science (2019).
[LK14]
J. Leskovec and A. Krevl. "(Snap datasets): {stanford} large network dataset collec- tion". In: 2014.
[Ma+22]
Y. Ma, X. Liu, N. Shah, and J. Tang. "Is homophily a necessity for graph neural net- works?" In: International Conference on Learning Representations. 2022.
[0gb+17]
E. L. Ogburn, O. Sofrygin, I. Diaz, and M. J. van der Laan. "Causal inference for social network data". In: arXiv e-prints (2017).
[RFR16]
R. Rastelli, N. Friel, and A. Raftery. "Properties of latent variable network models". In:
Network Science (2016).
[SM16]
C. Shalizi and E. McFowland III. "Estimating causal peer influence in homophilous social networks by inferring latent locations". In: arXiv e-prints (2016).
[ST11]
C. R. Shalizi and A. C. Thomas. "Homophily and contagion are generically confounded in observational social network studies".. In: Sociol. Methods Res. (2011).
[SBB22] D. Sridhar, C. D. Bacco, and D. Blei. "Estimating social influence from observational
data". In: First Conference on Causal Learning and Reasoning. 2022.
11

---

[TZ12]
L. Takac and M. Zābovský. "Data analysis in public social networks". In: International Scientific Conference and International Workshop Present Day Trends of Innovations (2012).
[TFS17]
E. J. Tchetgen Tchetgen, I. Fulcher, and I. Shpitser. nuuto---computation of causal effects on network". In: arXiv e-prints (2017).
[TK13]
P. Toulis and E. Kao. "Estimation of causal peer influence effects". In: Proceedings of the 30th International Conference on Machine Learning. 2013.
[TZ22]
C. Tran and E. Zheleva. "Network experiment design for estimating direct treatment effects". In: Proceedings of the 2022 AAAI Conference Artificial Intelligence (AAAI 2020). 2022.
[van14]
M. van der Laan. "Causal inference for population causally connected units". In: Journal of Causal Inference (2014).
[Vei+19] V. Veitch, M. Austern, W. Zhou, M. Blei, and Orbanz. "Empirical risk minimiza-
tion and stochastic gradient descent for relational data". In: Proceedings of the Twenty- Second International Conference on Artificial Intelligence and Statistics. 2019.
[VWB19] V. Veitch, Y. Wang, and D. M. Blei. "Using embeddings to correct for unobserved con- founding in networks". In: Advances in Neural Information Processing Systems. 2019.
[Wu+19] Z. Wu, S. Pan, F. Chen, G. Long, C. Zhang, and P. S. Yu. "A comprehensive survey on graph neural networks". In: IEEE Transactions on Neural Networks and Learning Systems (2019).
[Yin+17]
H. Yin, A. R. Benson, J. Leskovec, and D. F. Gleich. "Local higher-order graph cluster- ing". In: Proceedings of the 23rd ACM SIGKDD International Conference Knowl- edge Discovery and Data Mining. 2017.
[ZCZ22]
Z. Zhang, P. Cui, and W. Zhu. "Deep learning on graphs: survey". In: IEEE Transac- tions on Knowledge & Data Engineering 01 (2022).
[ZA21]
E. Zheleva and D. Arbour. "Causal inference from network data". In: Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery
Data Mining. 2021.
[Zho+18] Zhou, G. Cui, Z. Zhang, C. Yang, Z. Liu, and M. Sun. Graph Neural Networks: A Review of Methods and Applications. cite arxiv: 812.04434. 2018
12

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Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's
contributions and scope? [Yes]
(b) Did you describe the limitations of your work? [Yes] Please see Section 5.1 which discusses the limitations of the proposed method, light of the required assumptions.
(c) Did you discuss any potential negative societal impacts of your work? [N/A] This paper mainly focuses proposing new causal estimation procedure for peer influ- ence, and illustrating the performance of this procedure on social network data; this purely methodological paper which does not have any negative societal impacts. (d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
2. If you are including theoretical results...
(a) Did you state the full of assumptions of all theoretical results? [Yes] The full model assumptions are detailed in Section 2. All theoretical results (i.e. Theorem 1, Theorem 2, Corollary 3, and Theorem include full assumptions in their statements. (b) Did you include complete proofs of all theoretical results? [Yes] Full proofs of all theoretical results presented in Appendix A, Appendix B, Appendix C, and Ap- pendix D which can be found in the "Appendix.pdf" file within the folder "SUPPLE- MENTARY.zip"
3. If you ran experiments..
(a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material as a URLY? [Yes] The full code and data are included in the supplemental material within the folder "SUPPLEMEN- TARY.zip". This folder also includes the instructions needed to reproduce the main experimental results, which can be found in the file "instructions.md"
(b) Did you specify all the training details (e.g., data splits, hyperparameters. how they were chosen)? [Yes] Please see Section 6, Appendix F, as well as the code attached in the supplementary materials.
(c) Did you report error bars (e.g., with respect the random seed after running experi- ments multiple times)? [Yes] We present error bars for all our results in Appendix together with a full description of how they were obtained.
(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Appendix F.
4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
(a) If your work uses existing assets, did you cite the creators? [Yes]
(b) Did you mention the license of the assets? [N/A] All assets used are publicly available
online.
(c) Did you include any new assets either in the supplemental material or as URL? [Yes] The full code and data used are attached in the supplementary material.
(d) Did you discuss whether and how consent you're re using/curating? [N/A] The data used in experiments is publicly available online https://snap.staate www.sooo..edddaaatttee wwwpe//d//saaao/ooooooo Pokec.html
obtained from people whose data
(c) Did you discuss whether the data you are using/curating contains personally identifi- able information or offensive content? [N/A] The data used does not contain person- ally identifiable information or offensive content.
5. If you used crowdsourcing conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]
(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, applicable? [N/A]
(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]
13


Reviews:
[{"review_id": "hc3ylt3lz67", "reviewer": "Reviewer_muDH", "summary": "The authors propose a model and estimator for the \"contagion\" effect of treatments applied to a node's immediate neighbors in a graph structure. They control for homophily by conditioning on the node's graph embedding. Identifiability is formalized and the method is validated on semi-synthetic datasets.", "questions": "Please refer to the concerns above. I would be happy to amend my score if any of the major weaknesses are addressed in the rebuttal.\n\nMaybe further expanding on the role of the aggregation function can illuminate the novelty of this approach.", "limitations": "I would suggest for the authors to briefly discuss the fundamental limitations in using a social-network structure to draw conclusions about social contagion in humans, which is really driven by many other factors.", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n* The idea is clever and simple, yet highly relevant to the field of causal inference (on graphs in particular.)\n* Most of the paper is clear and easy to read.\n* The theorems are significant and their accompanying explanations are intuitive. This is especially the case for Theorem 2.\n\nWeaknesses:\n* I am a bit hesitant about the novelty of this work, as the method is quite similar to that introduced in [VWB19]. *Operationally*, the only major difference appears to be the aggregation operator that defines $V_i$ (Definition 1), which, coincidentally, is kept on rather vague terms until much later in the paper. The latter issue can be a source of confusion.\n* The results are limited to synthetic outcome models. In the first experiment (6.1), the outcome's parametrization is linear with i.i.d Gaussian error, which matches the proposed loss function perfectly. The formulation of the second (6.2) is quite simple, which is compelling, but not a comprehensive evaluation. It would be helpful to explore the performance under minor model mismatch.\n\n\nSmaller concerns:\n* The `do` calculus should be introduced before using it in Equation 3.1. The notation $(T=t^*)$ should more clearly reflect the fact that it is a vector assignment. Also, the context preceding that equation is confusing. How can it be viewed as an intuitive choice for capturing the \"average influence of a node's neighbors' treatments on its outcome\" if there is no explicit incorporation of neighborhoods? That specification comes a whole section later, leaving the reader in the dark. \n* The justification given in footnote #2 for considering the embeddings nonparametric is a bit of a stretch. Some distributional assumptions appear to be made implicitly, through the outer product and sigmoid in the loss function. Please clarify this claim.\n* It would be helpful for the reader to clarify that $m_G$ is used to model a node's *potential outcomes*.\n* In Section 7, the first paragraph on RCTs is outside the scope of this topic. Perhaps some of this space can be allocated for a conclusion section.", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "Qc68JRTTT83", "reviewer": "Reviewer_4Cyd", "summary": "This paper present a method that use node embedding to causally estimate the peer influence in social networks. In this task, the key challenge is to distinguish the similarity of linked users and peer influence. This paper claims that for a graph that is sparse enough, the hidden attributes can be ignored if we can provide sufficient information on graph structure. And then under this theoretic results, they propose that jointly adding a term relugarizing the node embedding to help reconstruct the graph structure can help us distinguish the peer similarity and peer influence. Experimental results on a semi-synthetic show that the proposed method can bring a significant boost.", "questions": "How is the q in Eq. 5.2 decided if we do not have ground truth causal effect?", "limitations": "See the summary.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1. The topic of this paper is interesting. \n2. The presentation of this paper is good.\n3. The idea of this paper is easy to understand, yet with theoretical support.\n\nWeakness:\n1. In this framework, how should we remove the bias brought by the correlation between treatment and covariates? In traditional frameworks, such a challenge is usually addressed by inverse probability of treatment weights (IPTW). But I do not think that computing IPTW in the proposed framework is trivial.\n2. The experiments are conducted on a synthetic dataset with real-world graph structure. I am wondering if it is possible to run the model on a complete real-world dataset. I understand that on real-world dataset, it is hard to get ground-truth labels. However, we can try to use the model to identify some social effect that are found by social scientists.", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "KUUubFQ-50g", "reviewer": "Reviewer_9be6", "summary": "This paper presents a formulation for casual estimation of treatment effects in the presence of network influence. The proposed formulation allows for the covariates affecting both network formation and treatment adoption to be unseen and estimated from an observed network. More specifically, the proposed method uses network embeddings as a means of accounting for network influence in a causality model. ", "questions": "In addition to the points raised in the weaknesses, one other question may be helpful for future work. Can the proposed method be extended to the case where there is more than one network observed, maybe with some assumptions on independence between the networks? For scenarios like social media, there are often several networks present (following, friendship, replying, etc.) all of which may have confounding effects on treatment.", "limitations": "See the strength and weaknesses section about limitations for use in practical settings. The authors have addressed any limitations with regard to potential negative societal impacts.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "___Strengths___: The paper presents a clear, relatively novel method that is well supported theoretically. The use of network embeddings is a straightforward extension of incorporating network structure into a causality model based upon previous works that used techniques like stochastic blockmodels (SBMs) or latent space models to incorporate network information. The proposed method, in contrast to previous ones, does not require as many assumptions about the nature of the network being incorporated into the model as other techniques, like SBMs do. The paper is also attacking an important problem, that of determining causality in treatments in network settings, which has important implications for very serious problems like understanding vaccine choices. \n\t\nThe paper also has strong theoretical support for the proposed method. The convergence of the various forms of the estimand only requires mild, often true assumptions like the relatively large size of the network and sparsity of the network. Also, the incorporation of a loss function into the ERM embedding that supports the causal inference model is also noteworthy and makes the method more complete. \n\t\nFinally, the article's presentation is clear and the writing is generally free of errors. \n\n___Weaknesses___: The paper does not present a thorough empirical validation and has some practical limitations that are not addressed. The paper only considers one empirical data set and artificially crafts the treatment based on observed covariates. A more convincing empirical validation would feature more than one data set and, ideally, one that would have observed treatment effects from a real-world scenario, like vaccine uptake or voicing of political support of a candidate, or something else similar. Additionally, the paper does not investigate the nature of the embedding procedure on its results; the paper only considers one type of deep learning-based embedding technique. Thus it's not clear how much the embedding technique can affect the results. \n\t\nThe paper also does not address some important real-world limitations of the proposed model. For one, not all node attributes contribute to link formation in any given network. For example, the attribute of gender is usually inversely related to link formation in something like a sexual contact network. Additionally, this problem can be further compounded by non-assortativity in networks, which is another occurrence in real-world social networks. Furthermore, within the empirical results the paper identifies that when there is zero confounding, the performance of the proposed method suffers. So, it would seem there is a significant limitation in this method’s usage that can have deleterious effects on results in actual usage. \n", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "ByyorDZibMd", "reviewer": "Reviewer_6sfM", "summary": "The authors tackle the well-known challenging problem in social networks that influence and homophily effects are not trivially distinguishable. In particular, this problem is more challenging because confounding factors are typically unobserved. The proposed method uses the embeddings to represent latent confounding factors and adjusts such confounding factors by adding a network regularization term in the estimation of causal influence effects. Authors show the validity of the proposed method by providing enough theoretical analysis. Also, empirical analysis under semi-synthetic experimental setting verifies the benefits of the proposed method as opposed to baseline estimations.", "questions": "In general, experimental part is the weakest part throughout the manuscript. Here are some suggestions regarding the experiments.\n\n- While experimental results can be affected by network structure and the relationships between networks and node attributes, authors need to present richer experimental results on various datasets and simulation models. In particular, since contagion is simulated, the proposed experiment should be applicable for many public datasets.\n- Also, variations on the parameter values of \\beta's would be great to check. Currently, only \\beta_0 = 0 or \\beta_0 >= \\beta_1 cases are present. However, from the indecisive results for \"Zero\" case, we may think that the proposed method might be better when \\beta_0 is large enough compared to \\beta_1 but not necessarily good enough for small \\beta_0 case. Further experiments can check this out.\n- Moreover, the fixed \\beta_1 value does not show the performance over SNR (signal-to-noise ratio; here it should be \\beta_0 over the standard deviation of the noise). It would be interesting to see whether SNR impacts the performance more or \\beta_{1} / \\beta_{0} impacts more.\n- Isn't the formula in page 8 eventually \\beta_{0}? If so, it would be great to explicitly describe so in order to prevent any confusion. Before that formula, everything is described with respect to \\beta, but suddenly it seems as if a new estimator were introduced. Actually, I would suggested introducing this in the earlier part of the paper, instead of in the experiment section.\n\nHere are some other questions.\n- Formula (5.2) and experimental setup is based on the continuous variable regression. However, the introductory example is based on the binary outcome. Should Formula (5.2) remain the same regardless of the type of Y values? How about the simulation process in Page 7? Without some management, Y_{i} cannot be guaranteed to be binary.\n- In Page 4, line 128, authors mention that the identification of causal effects would be straightforward if we know the latent C_i's. However, the identification does not seem trivial when we do not know the impact of each C_i's on the final response variable. Authors need to provide explanation on this argument.\n- As described above, the choice of q value is not straightforward. Authors need to provide some guidance.\n- Authors describe that ", "limitations": "Authors somehow present the limitation of the proposed method through the discussion about experimental results. However, it would be great if authors can summarize the limitation of the proposed methods.", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths\n- Authors address a very well-known but critical problem, which fits well into the conference. The final estimation method is straightforward and easily understandable.\n- Authors provide enough theoretical support when developing each step of the proposed method. Such support helps understanding without any big logical gap.\n- Experimental results show that the proposed method outperforms baseline methods, particularly when the confounding factors become problematic in the estimation of peer influence effect.\n\nWeaknesses\n- While notations are clearly defined in theory, it is not easy to follow the meaning or condition (e.g. what kind of value that each variable can have) of  some variables. Some examples would be helpful for straightforward understanding.\n- Experimental setup needs to be enforced because the current results show limited aspect of the proposed method. Please take a look at the suggestions below.\n- Unlike the typical ML problem, the parameter q is not trivially tunable through AutoML or any hyperparameter tuning because q balances two non-comparable objectives. Authors need to provide some guidance.", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept