Using Embeddings for Causal Estimation of Peer Influence in Social Networks
Anonymous Author(s)
AffiliationAddress email
Abstract
We address the problem of using observational data to estimate peer contagioneffects, the influence of treatments applied to individuals in a network on theoutcomes of their neighbors.
A main challenge to such estimation is that homophily —the tendency of connected units to share similar latent traits—acts as anunobserved confounder for contagion effects.
Informally, it’s hard to tell whetheryour 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 thefirst place.
Because these common causes are not usually directly observed, theycannot be simply adjusted for.
We describe an approach to perform the requiredadjustment using node embeddings learned from the network itself.
The main aimis to perform this adjustment nonparametrically, without functional form assumptions on either the process that generated the network or the treatment assignmentand outcome processes.
The key contributions are to nonparametrically formalizethe causal effect in a way that accounts for homophily, and to show how embedding methods can be used to identify and estimate this effect.
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 observedata from a population where each unit i is a person in an interconnected social network, and foreach unit we know whether they were vaccinated at the beginning of the study period, T i andwhether they were vaccinated at the end of the study period, Y i .
We are interested in estimating theeffects of the treatment T i of person
i on the outcome Y j of person j .
In addition to their vaccinationstatus, each unit has attributes C i that act as (proxies for) causes of both the particular network tiesthey form, and their vaccination behavior.
For instance, C i 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 (e.g., what would happen if weintervened by vaccinating popular people?), but the variables C i , C j act as confounders between thetreatment T j and outcome Y i .
The reason is that when defining a contagion effect from j to i wemust condition on the presence of an edge between ( i, j ) .
The edge is causally influenced by both C iand C j , so the conditioning creates a dependency between these variables (it acts as a collider).
Forexample, if we learn that Alice and Bob are friends then we can infer that they likely live in the samecity.
Accordingly, the association between T j and Y i may be either due to contagion : the causalinfluence of T j on Y i (Bob got vaccinated because Alice did), or due to homophily : the tendency ofsimilar people to be connected in a network (e.g., Alice and Bob got vaccinated because they bothlive in a major city, and it’s only due to chance that Alice got vaccinated before Bob).
In general,homophily is confounded with contagion [ST11].
Now, if we observed the attributes C , we could formalize the contagion effect using standardcausal tools and then identify the effect from observational data by adjusting for C [EKB16; EB17;
BDF20].
However, often such detailed knowledge of C is unavailable.
In this case straightforwardcausal identification and estimation is not possible.
In such situations, we might hope to make use of the following intuitive observation.
The patternof network ties itself carries information about each C i .
Indeed, this is the root problem that causesthe confounding issue.
Accordingly, we might hope to use the network itself to obtain estimates ˆ C ifor the attributes of each node.
Then, we could adjust for the estimated ˆ C i in some suitable causalestimation procedure.
The aim of this paper is to clarify this type of procedure.
Shalizi and McFowland III [SM16] provide an estimation strategy of this kind.
They first assume thatthe network is generated by either a stochastic block model [HLL83; LW19] or a latent space model[HRH02; RFR16].
They assume the attributes C correspond to the latent community identitiesor latent space positions.
They then further assume that the outcome of each unit is defined by aparticular linear structural equation, which includes a term for both the average treatment of thatnode’s neighbors, and a term for the effect of the attributes C i .
Their procedure is to estimate Cusing the assumed network model, and then use the estimated ˆ C in a linear regression to determinethe coefficient of the average-neighbor-treatment term.
This approach, however, critically relies on the assumed parametric form of both the network modeland the outcome model.
Indeed, even the target of estimation is defined as a parameter of thisassumed model.
Accordingly, when the parametric network model or linear outcome model is misspecified, the meaning of both the estimand and estimation procedure becomes unclear.
The goal of this paper is to develop a non-parametric procedure in this vein.
That is, the aim iscausal estimation of contagion effects by adjusting for network-inferred attributes without relyingon detailed parametric assumptions.
Towards this end, the paper follows three main steps:
Formalize the target causal effect non-parametrically.
The main challenge is that the estimandmust depend on the network we are working with (because contagion requires knowing who isfriends with whom) and the network must itself be modeled as a random variable which is a functionof the unobserved confounders C (to accommodate homophily).
Derive sufficient conditions for the estimated attributes to yield causal identification.
The idea isthat 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) a much easier task.3.
Give a 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.
Setup
Consider a network G n of n individuals, where connections between people are encoded throughundirected edges between nodes.
We define the degree of a node as the number of connections ithas.
The neighbors of node i are the nodes with which i has ties.
Each such link is captured by thenetwork adjacency matrix A , where A ij = 1 { i and j share a tie } .
We take A ii = 0 for all i .
Wealso consider a vector of variables associated with each node: ( Y i , C i , T i ) .
Here, Y i is the observedoutcome, T i is the treatment, and C i are (unobserved) attributes that may causally influence Y , Tand A .
The data is generated according to the following structural equation model, adapted from
Ogburn et al.
[Ogb+17] to model the network as a random variable.
[Equation]
The ε variables represent exogenous noise, which we take to be identically distributed and independent of the network and of each other.
The function S Y summarizes the neighbors’ treatment—e.g.,
S Y could be the average function, or the logical OR function.
This means that the treatment assignment for each node i depends only on its attributes C i , the outcome assignment depends on the nodeattributes C i and the treatments of all its neighbors, and the network structure depends on all C .
Thestructural functions f are fixed but unknown.
Formalizing the causal estimand
Consider setting T i ← t ∗ i for each unit i .
Our aim is to formalize the idea of “average influence of anode’s neighbors’ treatments on its outcome”.
An intuitive choice is
[Equation]
This estimand is the average outcome we would have seen had the treatment assignment been set to
T ← t ∗ , keeping both the node attributes { C i } and the network G n fixed.
The interpretation of thiseffect is: the average outcome under the hypothetical treatment, applied to the same set of peopleconnected by the same link structure .
Unfortunately, this estimand will not usually be identifiable.
The reason is that the node attributes
C are unobserved; e.g., we don’t observe city of residency, and we cannot perfectly reconstruct itfrom the network structure.
To circumvent this, we instead define the version of this estimand thatwe can (hope to) identify from the graph alone:
[Equation]
This new estimand can be understood as taking the estimand in (3.1) and marginalizing out the information about C that cannot be inferred from G n .
It is the average outcome under the hypotheticaltreatment, 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 on the same graph with node properties onlyfixed to be consistent with the graph structure.
Although this is somewhat less natural, we will seethat it makes identification plausible.
There is 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 our study,since both involve conditioning on the observed network G n .
We may be nervous that the formalquantity is tied to idiosyncracies of the particular observed network.
In Theorem 1, we show thatkind of sample dependency is not a serious issue.
The approach is to show that both the causalestimands ψ full info t ∗ and ψ t ∗ introduced in Equations (3.1) and (3.2) converge in probability to afixed quantity as the size of the network G n increases.
See Appendix A for the proof.
Theorem 1 (Generalizing the sample-based causal estimands) .
Consider an observed social network G n , and let Y i be the labels associated with each node i .
Assume the following
(i)
There exists M > 0 such that Var ( Y i ) ≤ M , for all i ;
(ii)
For any pair ( i, j ) of nodes selected uniformly at random,
P ( { i and j share a neighbor } ) → 0 , as n → ∞ .
Then, there exists a fixed real number ψ such that ψ full info
t ∗ → ψ and ψ t ∗ → ψ ,
in probability.
This theorem establishes that, provided the graph is sparse in a suitable sense, then both causal estimands converge to the same fixed quantity which does not depend on the particular network G n .
This decouples the interpretation of the causal estimand from the particular network under consideration.
Additionally, it shows that the distinction between ψ full info t ∗ and ψ t ∗ is not so importantfor practical applications—for large networks, the two values will anyway be very close.
Note thatthis result is about proving that the estimand is well-defined, not about
Causal inference of peer effects using node embedding methods
Having formalized and motivated the target causal effect of interest, we now turn to sufficient conditions for causally identifying it.
As previously discussed, if we could exactly reconstruct the latent C i then identification would bestraightforward.
However, usually, this reconstruction is impossible.
Happily, it is not necessary: animprecise proxy for C i might still suffice for causal identification.
Our goal now is to find sufficientconditions for an inferred proxy to enable identification.
The next theorem gives such a condition.
Informally, the condition is that we need a proxy λ i that carries enough information about C i so thatthe graph itself carries no further information about Y i after conditioning on λ i .
As we shall discussin Section 5, node embedding methods can be viewed as a tool for constructing such λ i .
We first define the following object, which plays a substantial role in the remainder of the paper.
Definition 1.
Let V i :=
S Y ( { T j : A ij = 1 } ) 2 be the aggregated treatment at node i , and v ∗ i itsvalue under the hypothetical treatment intervention T = t ∗ .
For a graph G n , the vertex conditionaloutcome model is the function m given by m G n ( v ∗ i , λ i ) :
=
E [ Y i | v ∗ i , λ i ] .
Theorem 2 (Sufficient conditions for identification) .
Let G n be the network, A its adjacency matrix,and Y i the node-associated labels.
Suppose that for each node i we have a proxy for the latentattributes, λ
i ∈
R k , such that
i Y i ⊥⊥ A ij | ( λ i , v ∗ i ) , for all i and j .
ii P ( V i = v ∗ | λ i ) > [-
-]0 for all v ∗ ;iii
λ i is C i -measurable.
Then E [ Y i | do( T = t ∗ ) , G n , λ i ]
= m G n ( v ∗ i , λ i ) .
Proof.
We sketch the proof and provide the full mathematical justification in Appendix B. Consider Figure 1 illustrating the core identification issue.
The node-level causal effect E [ Y i | do( T =t ∗ ) , G n , λ i ] relies on conditioning on the network structure, and, in particular, on the edges A ij .
Doing so opens a backdoor path between Y i and T j (hence, also V i ) going through the unobserved
C i .
The main idea of the theorem is that conditioning on λ i removes the effect of conditioning onthe A ij , and thus prevents the backdoor path from being opened.
In Theorem 2, condition (i) gives anotion of “sufficient amount of information about C ” for causal adjustment, and is the main point ofthe theorem.
Condition (ii) is thestandard positivity assumption required for causal identification.
Finally, condition (iii) is the condition that λ i is a partial reconstruction of the node-specific attributes
C i .
This condition is necessary toensure
λ
i is not itself a collider forthe latent node attributes.
This result readily extends to acondition for the identification ofthe peer contagion estimand ψ t ∗ = 1 n (cid:80)
ni
=1 E
[ Y i | do( T = t ∗ ) , G n ] introduced in Section 3.
This isthe content of the next result; see Appendix C for a proof.
Corollary 3 (Identification of the causal estimand ψ t ∗ ) .
Consider the same set-up and notation asthat of Theorem 2.
Then, the causal estimand ψ t ∗ can be identified as follows
[Equation]
[Equation]
Corollary 3 gives, in principle, a strategy for identifying the causal effect using the observed λ .
Namely, learn m G n ( v ∗ i , λ i ) and P( λ i | G n ) and plug in to (4.1).
This result is exact, and appliesto any network.
However, learning P( λ i | G n ) seems difficult—it’s unclear how to model thisdistribution.
To circumvent the need to learn the unit-wise uncertainty about λ
i , we now give anotherresult 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 byaveraging over all of the nodes (that is, the mean converges to the unit-level expectation).
We now state the main identification result as Theorem 4 below which is derived from both Theorem 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 twoconditions of Theorem 1 are also satisfied.
Then lim n →∞ 1 n (cid:80) ni
=1 m G n ( v ∗ i , λ i ) =
ψ , in probability,where ψ is the fixed real number such that lim n →∞ ψ full info t ∗ and lim n →∞ ψ t ∗ =
ψ , in probability.
Estimation of causal peer influence effects
We have now established a formal causal estimand ψ t ∗ , and conditions on node-attribute proxies thatenable identification.
The basis for our estimation procedure will be Theorem 4, which says that—forsuitable λ and sufficiently sparse G n —the statistical estimand
[Equation]
[Equation] approximates the causal estimand ψ t ∗ , with the approximation becoming exact as n → ∞ .
Recallm G n ( v ∗ i , λ i ) :
=
E [ Y i | v ∗ i , λ i ] .
Our task is to estimate (cid:101) ψ t ∗ from the observed data using the observedtreatments and outcome ( T i , Y i ) i =1: n and the link structure G n .
To achieve this, we use embedding-based semi-supervised prediction models to learn ˆ λ ∈ R k andˆ m G n ( v ∗ , ˆ λ i )
≈ E
[ Y i | v ∗ i , ˆ λ i ] .
Then, we plug these estimates into (5.1) as m and λ .
For concreteness, we describe a particular approach based on Veitch et al.
[Vei+19] and Veitch et al.
[VWB19].
The estimation procedure follows three main steps:
Step 1.
We train a model using relational empirical risk minimization [Vei+19] to jointly learn theembeddings
ˆ λ i and ˆ m G n ( v i , ˆ λ i ) .
To do this, we first compute v i =
S Y ( { t j : A ij = 1 } ) at eachvertex i —i.e., we create a new aggregate treatment feature and attach it to each vertex.
Relational ERM works by minimizing a risk defined as an expectation over randomly sampled subgraphs of the larger network.
Let Sample ( G n , k ) be a sampling algorithm that returns a randomsubgraph of size k from G n (e.g., the subgraph induced by a random walk of length k ).
Let v G k bethe set of vertices of the subgraph G k .
We then define the loss function on the G k to be
L ( G k , λ, γ ) =
q · (cid:88)
[Equation]
where q ∈ [0 , 1] and σ is the sigmoid function.
The second term is a network reconstruction termthat extracts node-level information from the network by requiring the embeddings to be predictiveof the edge structure.
The first term learns a predictor m , parameterized by γ (global parametersshared across the network), that predicts y i from the aggregated treatment v i and the embedding λ i .
In our experiments, m will be a linear predictor, and γ the (parameters of the) linear map.
3 When yis binary, the binary cross-entropy loss is used instead of the MSE loss.
Then, we train the model by fitting
[Equation]
[Equation]
Step 2.
We define ˆ m G n ( v ∗ i , λ i )
= m ( v ∗ i , λ i ; ˆ γ ) , where v ∗ i is the value of v i attained under theinterventional treatment T = t ∗ .
Then, for each unit vertex i in G n we compute ˆ m G n ( v ∗ i , ˆ λ i ) .
Step 3.
Finally, the peer influence estimate is the average over the estimated vertex conditional outcomes ˆ m G n ( v ∗ i , ˆ λ i ) :
[Equation]
5.1 Validity of the estimation procedure
In this section, we informally discuss the validity of the estimation procedure described in theprevious section.
More specifically, recall that, in Theorem 1, we showed that both causal peerinfluence estimands proposed, ψ full info t ∗ and ψ t ∗ , converge, in probability, to a fixed real number ψ , as n → ∞ .
The question that remains is whether the estimator proposed, ˆ ψ t ∗ =1 /n (cid:80) ni
=
ˆ m G n ( v ∗ i , ˆ λ i ) , also converges to ψ , in probability, as n → ∞ .
This will clearly holdas long as ˆ m G n ( v ∗ i , ˆ λ i )
→ m G n ( v ∗ i , λ i ) for some λ satisfying the conditions of Theorem 2.
Why should one expect the above consistency property?
First, consider ˆ m ( v, λ ) → E [ Y i | v, λ ] .
Themap m is defined as the minimizer of the squared error empirical risk.
The minimizer of the truerisk is the required conditional expectation.
Accordingly, this part of the learning is, essentially, thesame as what occurs with non-network data and is plausible.
The trickier judgement is why can one expect
that ˆ λ i
≈
λ
i for some λ satisfying the conditions of
Theorem 2?
That is, is it plausible that the learned embeddings behave as proxies for the node-levelattributes (i.e., are C -measurable), and carry sufficient information to adjust for the homophily effect(i.e.
, Y ⊥⊥ A ij | ( λ i , v ∗ i ))?
The C i -measurability is just the requirement that each node-level embedding captures unobservedinformation specific to that node.
This is exactly the aim of embedding methods, to leverage network structure in order to capture the latent properties of each node [HYL17].
Accordingly, it isreasonable to assume that the embedding-based method described in Section 5 satisfies this property(at least, asymptotically).
With respect to whether the embeddings carry sufficient information toadjust for the homophily effect when estimating peer influence, we note the following: in order tosatisfy this condition, it suffices that the embeddings are predictive of the outcome or of the edgestructure.
To see why this is plausible, recall the expression of (5.3) used to fit the embeddings.
Thecross-entropy term encourages the embeddings to be sufficient (at the graph level) for predicting theedge structure.
The squared error term encourages the embeddings to be sufficient for predicting thelabels Y i .
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 embedding method actually achieves its conceptual goal.
In effect, by working with embedding methods one is trading off the precise guarantees enabled by parametric models (exploited in [SM16]) infavor of techniques that have good empirical performance even in the (typical) case that the networkshave 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 n isgenerated 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; Jan18]).
Informally, Davison and Austern [DA21] show that, asymptotically, the embeddings are functionsof node-specific attributes C i alone and that these embeddings capture the limiting distribution ofthe network (see Davison and Austern
[DA21] Thm. 7).
Accordingly, in this case, the learnedembeddings satisfy the conditions of Theorem 2.
Experiments
To study the practical performance of the estimation procedure described in Section 5, we conductexperiments using semi-synthetic data.
We use a subset of data from the Slovakian social mediawebsite Pokec [TZ12; LK14].
The data includes both links between users and limited node-level
[Table]
covariates.
Our basic strategy will be to use these node level covariates as the latent C i ’s. To do this,we simulate treatment and outcomes for each unit in a manner that depends on C i .
At estimationtime, we hide the C i variables, and check how well we can recover the true causal effect using onlythe graph data and observed treatments and outcomes.
We extend the setup of Veitch et al.
[VWB19] who applied a similar relational empirical risk minimization based technique in order to adjust for the network unobserved confounding effect whenperforming average treatment effect (ATE) estimation of T i on Y i .
That work assumes pure homophily, whereas this paper examines peer-based contagion effect estimation.
Following Veitch et al.
[VWB19] we analyze a sub-network of approximately 70000 users connectedby roughly 1.3 million links, representing a subset of the original Pokec data restricted to the districtsˇZilina, Cadca, and Namestovo, from the same region.
The analysis consists of two parts:
estimating the peer contagion effect of binary treatments on continuous response variables, varying both the source of confounding and its strength;
estimating the peer contagion effect of binary treatments assigned at time t 1 , the beginning of astudy period, on treatments assigned at a subsequent time t 2 , 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 threevariables (one at a time), taken as the unobserved confounders C : district, age, and Pokec joindate.
For each of these three confounders, we standardize, then bin each of them into three possiblevalues: − 1 , 0 , and 1 .
We then apply a function g which transforms them into probabilities: g ( C ) =0 .
5 + 0 .
35 · C , so that g ( C ) ∈ { 0 . 15 , 0 . 5 , 0 . 85 } .
Then, we simulate according to:
[Equation]
The parameter β 0 controls the amount of peer influence, while β 1 measures the amount of networkconfounding.
In this simulation setup, the causal peer contagion effect of interest, ˆ ψ t ∗ is preciselyequal to β 0 .
We fix β 0 = 1 as the ground truth value of confounding, and let β 1 vary in { 0 , 1 , 10 }corresponding to no, low, and high contagion, respectively.
Since we only observe T i , V i and Y i we test whether ˆ ψ t ∗ , the estimator proposed in Section 5, adjustsfor the unobserved confounding due to C , and accurately estimates β 0 = 1 .
For each combination of
C , and strength level β 1 , we obtain simulated V
i and Y i values, then we train the embedding-basedrelational empirical risk minimization model described in Section 5, using a random walk samplerwith negative sampling, and the default settings of Veitch et al.
[VWB19].
Finally, we use the trainedmodel to predict two sets of adjusted Y values:
Y i ’s resulting from the aggregated treatment v ∗ i , when all T i are set to 0 ,
i
e. ˆ m (ˆ λ i , v ∗{ i,t ∗ i =0 } ) ;
Y i ’s resulting from the aggregated treatment v ∗ i , when all T i are set to 1 , i.e. ˆ m (ˆ λ i , v ∗{ i,t ∗ i =1 } ) .
The causal effect estimator of interest is then {+1 /n (cid:80) ni =
+}
{+ˆ m (ˆ λ i , v ∗{ i,t ∗ i =1 } ) −1 /n (cid:80) ni =
 ˆ m (ˆ λ i , v { i,t ∗ i =0 } ).+}
We compare the results of this procedure against those of two baseline estimators.
The first oneis the naive estimator which does not adjust for confounding, namely 1 /n (cid:80) i
E [ Y i | v ∗{ i,t ∗ i =1 } ]
−1 /n (cid:80) i
E [ Y i | v ∗{ i,t ∗ i =0 } ] , where expectations are the estimated values obtained via linear regression.
The second baseline seeks to mimic the parametric peer contagion estimation procedure of Shaliziand McFowland III [SM16], by fitting a mixed-membership stochastic block model to the Pokecdata, inferring 128 latent communities ( 128 is equal to the dimension of the embeddings), thenpredicting Y via a linear regression on the aggregated treatment and inferred latent communityvalues.
Table 1 summarizes the results of the embedding method compared to those of the above baselines.
Whenever confounding was present, ˆ ψ t ∗ yielded the most accurate results out of all three estimators, by a significant margin.
This confirms that the nonparametric method can leverage the networkstructure and obtain reliable peer contagion estimates.
With no confounding present, all three estimators produced valid results, yet, when district and age were used as unobserved confounders, ˆ ψ t ∗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 embedding-model fitting causeserrors that simpler methods do not suffer from.
In Appendix E, Table 3, we provide error bars forthe average estimated peer effects obtained in Table 1, which aim to illustrate the consistency ofour results.
The errors reported for ˆ ψ t ∗ represent the standard errors computed over 100 differentglobal 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 errorsof the estimated regression coefficients for the aggregated treatment used when predicting Y .
Themain caveat is that due to the relational, non-i.i.d.
structure of our data, these errors do not represent proper confidence bands, and should be interpreted with caution.
To obtain valid confidenceintervals, proper asymptotic normality results for the studied estimators are necessary.
6.2 Binary outcome: a model for peer influence and vaccination
Recall the motivating example in Section 1: some proportion of the observed population gets vaccinated at time t 1 , while others get vaccinated at a subsequent time t 2 .
We now illustrate how to usenode embeddings to study the peer influence of the former group on the latter.
We use the followingdata setup:
Simulate the treatment by the same procedure as in Section 6.1,
i
e. as a function of the unobservedconfounders;
Randomly select 50% of the vertices, and set Y i
=
T i , but then delete T i for those respective nodes.
The uncensored treatment values T i correspond to individuals who were vaccinated at t 1 ,while the Y i values represent individuals vaccinated at t 2 ;3.
Compute the causal effect of V i on Y i only for the vertices where Y i is defined.
The above design matches the following scenario: whether or not an individual gets vaccinateddepends on latent attributes such as their age, or location.
Since Y i
= T i , there is no causal peerinfluence effect of the neighbors j on node i .
Despite the lack of a causal relationship, an associationis clearly present due to the dependence on latent confounders.
The question is whether the proposedmethod 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 estimatesfor the peer contagion effect, summarized in Table 2.
We note that, for all three hidden confounders,the embedding-based method produces the estimates closest to the ground truth value of 0 .
For the“age” covariate, the naive, unadjusted estimate is very close to the embedding-based one, showingthat age may not be the optimal predictor for network tie formation, preventing the non-parametricmethod from appropriately adjusting for the network structure, and yielding our method similarto the naive one, in this situation.
We also note that, when registration date is used as a hiddenconfounder, all three estimates are less accurate, suggesting that “join date” may also not be the bestpredictor of the network edges.
Nonetheless, in this scenario, the nonparametric estimator still has [Table] superior performance over the two baselines.
In Appendix E, Table 4, we provide error bars for thevariation induced by the choice of random seed, obtained in a similar way to those corresponding tothe continuous outcome scenario.
In Appendix G we perform additional analyses further varying the values of β 1 and noise level ε , aswell as illustrate the performance of 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 confounding, several recent works rely on designing randomized experiments.
Toulis and Kao [TK13]use potential outcomes and a sequential randomization design to estimate peer influence on observed networks.
Eckles et al.
[EKB16] develop an experimental design to study the impact of peerfeedback on the behavior of Facebook users.
Fatemi and Zheleva [FZ20b] propose an experimentaldesign for estimating the effect of the treatment alone on the unit’s outcome, isolating it from peereffects, while Fatemi and Zheleva [FZ20a] discuss an approach of minimizing selection bias whenperforming A/B testing on networks.
Causal network inference from observational data.
Within observational settings, van der Laan[van14], Ogburn et al.
[Ogb+17], Tchetgen Tchetgen et al.
[TFS17], and Tran and Zheleva
[TZ22]tackle the issue of causally estimating social contagion from observed networked data, yet theyassume there are no latent sources of network confounding which affect both the treatment and theoutcome, by considering all node-related features as known.
This is a significant limitation whichthis paper seeks to address.
Other works which account for latent confounding assume that the network (and, implicitly, theunobserved confounders) can be represented using parametric methods.
Shalizi and McFowland III[SM16] and Shalizi and Thomas [ST11] use stochastic block models and latent space models in orderto obtain consistent estimators of peer contagion.
Despite the attractive mathematical properties ofthese models, often times they fail to capture real-world network properties.
Some of the recent studies which address nonparametric peer influence estimation from observational data propose different approaches which complement the node embedding based method proposed in this paper.
Eckles and Bakshy [EB17] estimate peer effects on hundreds of millions ofobservations from Facebook data by using high-dimensional adjustments for covariates via propensity score models.
One limitation of this work is that covariates are readily available, whereas thispaper 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 control outcomes and exposure variables to estimate contagion effects.
This approach complements themethods of this paper.
Finally, 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 network treatment and outcome.
They propose using “graph attentional layers” in order to mapthe network information to a partial representation of the hidden confounders and use it to performcounterfactual evaluation.