# An Approach to Causal Inference over Stochastic Networks

Duncan A. Clark  University of California - Los Angeles, Los Angeles, USA
<EMAIL_ADDRESS>Mark S. Handcock  University of California - Los Angeles,
Los Angeles, USA

###### Abstract

Claiming causal inferences in network settings necessitates careful
consideration of the often complex dependency between outcomes for actors. Of
particular importance are treatment spillover or outcome interference effects.
We consider causal inference when the actors are connected via an underlying
network structure. Our key contribution is a model for causality when the
underlying network is unobserved and the actor covariates evolve
stochastically over time. We develop a joint model for the relational and
covariate generating process that avoids restrictive separability assumptions
and deterministic network assumptions that do not hold in the majority of
social network settings of interest. Our framework utilizes the highly general
class of Exponential-family Random Network models (ERNM) of which Markov
Random Fields (MRF) and Exponential-family Random Graph models (ERGM) are
special cases. We present potential outcome based inference within a Bayesian
framework, and propose a simple modification to the exchange algorithm to
allow for sampling from ERNM posteriors. We present results of a simulation
study demonstrating the validity of the approach. Finally, we demonstrate the
value of the framework in a case-study of smoking over time in the context of
adolescent friendship networks.

###### keywords:

Causality, Social Networks, Network models, Spillover, Contagion,
Interference, Gibbs measures

##  1 Introduction

Causal inference is difficult, especially in systems with partially known and
likely complex structure. There is an extensive literature on causal inference
methods for so called “network settings” from a variety of perspectives
(Hudgens and Halloran, 2008; Shalizi and Thomas, 2011; Ogburn and VanderWeele,
2014; van der Laan, 2014; Sofrygin and van der Laan, 2017; DeAmour, 2016;
Aronow and Samii, 2017; Shpitser et al., 2021; Tchetgen Tchetgen et al., 2020;
Ogburn et al., 2020; Ogburn et al., 2020).

There are recent empirical studies claiming strong causal results in network
settings that have been controversial. For example, claims about social
contagion (Christakis and Fowler, 2007, 2008, 2010) with corresponding
criticism from others (e.g. Ogburn et al., 2020). Such results are
controversial due not only to their surprising and perhaps provocative
substantive nature, for example, statements such as “obesity is socially
contagious", but also due to the strong assumptions required to justify the
methodology.

Much of the problem stems from the unknown underlying social processes. For
example, as explicitly noted in Shalizi and Thomas (2011), contagion is often
confounded with homophily. They consider Directed Acyclic Graphs (DAGs)
(Pearl, 1995; Spirtes et al., 2000) and demonstrate that contagion can be
confounded when latent homophily is present.

In this paper we consider social structures represented by networks with
stochastic links and covariates that stochastically evolve over time. We
present a generalized chain graph approximation to a credible social process
DAG, which allows for a dependence structure that we believe to be compatible
with such problems. We then frame causal inference in terms of network
equilibrium potential outcomes, that is, potential outcomes derived from the
chain graph structure, and use an augmented variable Markov Chain Monte Carlo
(MCMC) algorithm to sample from their posterior distributions. The key
contribution of our approach is to allow for uncertainty in the network
structure and for codependence of edges and nodal covariates in the underlying
social process over time.

Chain graphs have been posited as a possible representation of this causal
structure (Tchetgen Tchetgen et al., 2020; Shpitser et al., 2021). By
considering a DAG of a network generating process over time, Ogburn et al.
(2020) suggest that estimating causal effects in this setting is not viable
unless fine grained temporal data is available and the causal structure is
simple.

We consider the situation where we observe a single network, considered a
realization of a random social process over both nodes and edges. Causal
inference in this setting cannot be reduced to an independent and identically
distributed (IID) problem in the same fashion as Sofrygin and van der Laan
(2017) and the chain graph approximation employed in Tchetgen Tchetgen et al.
(2020) and Ogburn et al. (2020) cannot be used as it does not allow for
stochastic connections. There have been efforts to allow for network
uncertainty (Toulis and Kao, 2013; Kao, 2017). However these methods require
unconfoundedness of the edges and the random nodal covariates, given enough
fixed information about the nodes. In not making such an assumption we require
a joint probability model for the random edges and nodal covariates in the
graph.

We note that the literature of network causal inference is at present sharply
disconnected from the literature concerning generative models for social
networks. As most approaches have considered the networks as fixed, there has
been little interest in placing a probability distribution over the space of
possible networks. Typically generative models for social networks for example
the commonly used Exponential-family Random Graph Models (ERGM) (Frank and
Strauss, 1986) consider the edges as the random variables to be modelled and
nodal covariates as fixed. There has been much sophisticated work on
understanding such models (Handcock, 2003; Robins et al., 2007; Schweinberger
and Handcock, 2015), and well developed MCMC based fitting procedures
(Snijders, 2002; Handcock, 2002) with associated complex software (Hunter et
al., 2008). However due to the assumed fixed nature of the nodal covariates,
these are of limited use for causal inference on nodal outcomes. Markov random
field (MRF) models treat the nodal covariates as stochastic but the
connections between nodes fixed. Encompassing both model classes are the novel
Exponential-family Random Network Model (ERNM) (Fellows and Handcock, 2012).
ERNM are a class of exponential family models that encompasses ERGMs and MRF
as special cases. ERNM allows for the edges and nodal covariates to be
stochastic, thus the nodal covariates and the edges can be co-dependent. The
focus of this paper is causal inference based on the plausible representation
of complex social structure via ERNM.

We utilize a Bayesian framework for the causal quantities and the network
model. This allows for the incorporation of prior information, as well the
automatic accounting of uncertainty in a theoretically consistent manner. The
Bayesian approach does not require appeals to asymptotic arguments for its
validity. Indeed, asymptotics for causal quantities in our setting are
conceptually difficult as their is no single asymptotic framework that is
compelling. In particular, the number of nodes, $n$, is a fundamental
characteristic of the social process and not a sampling design characteristic,
as it is in most of Statistics. For example, the interactions of a class with
5 students will be quite different from a class of size 75. Hence asymptotic
approximations must identify credible invariant parametrisations Krivitsky et
al. (2011). Different values of $n$ change the fundamental structure of the
social network, as dependent edge behaviour is strongly related to the number
of nodes in a network. We develop a modification to the exchange algorithm
(Murray et al., 2006) to allow for sampling from ERNM posteriors, which we use
to infer the posterior potential outcomes, to estimate causal estimands of
interest.

This paper is structured as follows. Section 2 introduces our general network
setting and our notation. Section 3 considers the DAG of a network process
over time allowing for network uncertainty with a chain graph approximation.
Section 4 defines causal quantities of interest, based on the chain graph
structure in terms of equilibrium potential outcomes. Section 5 briefly
describes a simulation study, the details of which are contained in the
supplement. Section 6 considers a case-study of a network from the National
Longitudinal Study of Adolescent Health (Harris et al., 2007) and gives
estimates of unknown causal quantities. Section 7 provides general discussion
of the method, and its ability to generate credible causal inference.

##  2 Notation and Setting

We consider a known fixed set of $N$ nodes. Each node has a random nodal
outcome $Y_{i}$, thus the whole network outcome is $Y=(Y_{1},\ldots Y_{N})$.
Realizations of the random nodal covariates $Y_{i}$ and $Y$ are denoted with
lower case $y_{i}$ and $y$. For this paper we only consider binary outcomes
(although the ideas are easily extended to non-binary outcomes). Each node is
also permitted to have known fixed multivariate nodal covariates similarly
denoted $X=(X_{1},\ldots X_{N})$.

We also allow for inter-connections between nodes. We denote the random edges
as the random variable $A=\\{A_{i,j}\\}_{i=1,j=1}^{N}$, with realizations $a$.
$A$ can be considered a random adjacency matrix. We also restrict $A_{i,j}$ to
be binary with $1$ indicating a connection and $0$ representing the absence of
a connection. For this paper, we make the restriction that our networks are
undirected i.e. $A_{i,j}=A_{j,i}\quad\forall i,j$.

A network is defined to be a set $\\{Y,X,A\\}$. When considering the dynamic
network of the process over time we indicate the outcomes at time $t$ with
superscript, for example, the outcome random variable for node $i$ at time $t$
is $Y_{i}^{t}$ and the whole network at time t is $\\{Y^{t},X,A^{t}\\}$. If
the temporal dynamics have an equilibrium distribution, we denote it by
$\\{Y,X,A\\}$.

We represent the treatment of nodes via the treatment vector $Z=(z_{1},\ldots
z_{N})$ with realizations $z$. For the purposes of this paper we consider the
treatment to be applied prior to the evolution of the network process though
perhaps conditional on the fixed nodal covariates. We leave allowing for time
varying treatments assignments and outcome evolution to future research,
though we believe it to be compatible with our approach.

In the following section we will introduce a DAG to represent the dependence
structure of the social network. We emphasize that the nodes in the DAGs
represent random variables in the stochastic social process underlying the
above described social network setting (rather than actors in the social
network). That is, nodes in the DAG represent random variables in the social
process generating the network, they may either be treatments, random nodal
covariates, or random edges.

##  3 Causal Framework

DAGs provide a means for precisely specifying the structure of relationships
between random variables (see Pearl (2009) for an introduction). Pearl (1995)
developed strict criteria for identification of causal effects, and showed
equivalence with the potential outcome framework in Pearl (2010)

Nodes in the DAG represent random variables, with edges drawn between nodes
being strictly uni-directional, and cycles of edges prohibited, so that so
called “feedback” loops are not permitted. The interpretation of a directed
line from node $i$ to node $j$ is that $i$ is causally effects $j$. For a node
indexing set $V$, denote the corresponding random variables $\\{x_{v}\\}_{v\in
V}$. This graphical structure encodes the following factorization of their
joint probability distribution $p(x)=\prod_{v\in V}p(x_{v}|x_{pa(v)})$, where
$pa(v)$ are the parents of $v,$ that is the nodes in the DAG without edges
into $v$.

The causal effect of $X$ on $Y$ is represented by $p(y|do(X=x)),$ the
distribution of the random variable $Y$ when $X$ has been externally set to
$x$, for example by an external intervention. This fits an intuitive
understanding of a causal effect in terms of counter factual inference.

Chain graphs permit undirected as well as directed edges, which allow for
different Markov properties from DAGs. In fact chain graphs can represent
dependence structures that are not possible under a DAG. We omit subtleties of
their Markov properties discussed in detail in Frydenberg (1990) and utilized
for causal analysis in Lauritzen and Richardson (2002)111We consciously do not
give a complete description of chain graph machinery in this paper for reasons
of brevity. It is complex and subtle, and should not be avoided for full
understanding. We wish to avoid making the required definitions for concepts
such as the boundaries, closures, anterior sets, terminal sets, underlying
graph, moralization, and others which are required to fully specify chain
graphs and understand the variety of causal structures that they may
represent.. In their full generality, chain graphs can express complex
dependence structures. However, in our case, our example chain graph only has
one chain component which results in none of the outcomes being rendered
conditionally independent, thus we do not require an in-depth review for the
purposes of this paper. Practically, one possible interpretation of the
undirected edges in a chain graph is that the two variables interact in a
causal feedback sense over time.

DAGs representing the causal structure of outcomes of nodes in networks under
both interference and contagion are given a clear and detailed treatment in
Ogburn and VanderWeele (2014). We generalize the related conjecture in Ogburn
et al. (2020) for a chain graph approximation of causal structure of a social
network, which slowly evolves over time. This is based on an interpretation of
causality with feedback relationships over time which chain graphs can be used
to explain (Lauritzen and Richardson, 2002). Our generalization is to consider
a social network where the connections are not fixed and are motivated by the
empirical observation that social connections are often strongly dependent on
other nodes’ connections as well as other nodes’ covariates, treatments and
outcomes.

As an illustration for our chain graph approximation, Figure 1 represents the
structure of a network evolving over time for a three node network similar to
that of Ogburn et al. (2020) and Tchetgen Tchetgen et al. (2020). Note the
analogous connections between node 3 and 1 are omitted for clarity. Block
arrows denote multiple arrows into the variables described therein. The full
DAG with all arrows becomes quickly unwieldy. In the notation of Section 2,
the nodal treatments are $\\{Z_{1},Z_{2},Z_{3}\\}$, nodal outcome variables
$\\{Y_{1},Y_{2},Y_{3}\\}$, and we denote the undirected edges between nodes
$i$ and $j$ as $A_{i,j}$. Directed network edges as well as fixed nodal
covariates can also be built-in but are omitted here for clarity. Superscripts
denote the status of a variable at that time step e.g. $Y_{2}^{t}$ is the
outcome for node $2$ at time $t$.

Figure 1: Full DAG of temporal network formation feedback process, in the
three node network case. Treatment variables $Z_{j}$ are permitted to causally
effect all of the other variables. Outcomes $Y_{i}^{t}$ and $Y_{j}^{t}$ are
permitted to causally effect edges $A^{t}_{i,j}$, as well as outcomes
$Y_{i}^{t+1}$ and $Y_{j}^{t+1}$. Edges $A_{i,j}^{t}$ are permitted to causally
effect edges $A_{k,l}^{t+1}$ with no restrictions on $k$ and $l$. In practice
the functional form of models will usually restrict the causal impact of node
$i$’s outcome and treatment to only the set of neighbouring nodes $\\{j\\}$,
that is, where there is an edge present between node $i$ and $j$.

Under this DAG, each of the treatment variables $Z_{j}$ are permitted to
causally effect all of the other outcome variables, as well as edges involving
$j$. We suggest in practice the implicit assumption will be made that the
direct effect of $Z_{i}$ on outcome $Y^{t}_{j}$ only occurs if edge
$A^{t-1}_{i,j}$ is present in the specification of the model for the outcomes
at time $t$.

Outcomes $Y_{i}^{t}$ and $Y_{j}^{t}$ are permitted to causally effect edges
$A^{t}_{i,j}$, as well as outcomes $Y_{i}^{t+1}$ and $Y_{j}^{t+1}$. Similarly
to the treatments’ causal effects, we will in practice implicitly assume that
the direct effect of outcome $i$ at time $t$ on outcome $j$ at time $t+1$ only
occurs if edge $A^{t}_{i,j}$ is present, which depends on the specification of
the model for the outcomes at time $t$.

Edges $A_{i,j}^{t}$ are permitted to causally effect edges $A_{k,l}^{t+1}$
with no restrictions on $k$ and $l$. Again, in practice, our models will
restrict this dependence to enable us to estimate causal quantities.

We note that there are key edges that are not present in this DAG. Edges and
outcomes, from time step $t$ can only influence edges and outcomes at time
step $t+1$, not others. We argue that this is plausible under slow evolution
of a network process over time. In general, it is exceedingly rare that enough
data are available to identify relationships as posited in Figure 1.

As noted in Lauritzen and Richardson (2002), the equilibrium distribution of a
so called infinite DAG can be represented as a chain graph. Similarly to
Ogburn et al. (2020), but modelling edges as random, the DAG in Figure 1 can
be approximated by the chain graph shown in Figure 2. The chain component
(undirected component) of this chain graph is close to being complete since we
allow every nodal outcome to influence every other nodal outcome, however we
only allow outcomes $Y_{i}$ and $Y_{j}$ to influence edge $A_{i,j}$.

This suggests that the complex structure of such a DAG can be approximated
with the chain graph factorization:

$\displaystyle p(Y,A,Z)=\left(\prod_{i=1}^{n}p(z_{i})\right)p(Y,A|Z)$ (1)

As an approximation to the true temporal DAG a chain graph model of causality
serves to render the causal effects tractable in practice. See Ogburn et al.
(2020) and Lauritzen and Richardson (2002) for a fuller explanation.

As an illustration of the generality of this chain graph’s dependence
structure we define a conditional independence property, that we refer to as
local conditional independence as follows:

$\displaystyle Y_{i}^{T}\perp\\!\\!\\!\perp
Y_{j}^{T},Z_{j}~{}~{}\Bigg{|}~{}~{}A_{i,j}^{T}=0,~{}~{}Z_{i},~{}~{}\\{Y_{l}^{T}:A_{i,l}^{T}=1\\}$
(2)

That is, nodal outcomes are conditionally independent given all neighbours,
and that they are not connected. Equation (2) does not hold a priori due to
the dependence induced by the random edge $A_{i,j}$ unlike in the fixed
network case where it does. That is, the chain graph retains dependencies
between outcomes of non-connected nodes, even when conditioning on neighbours.
In fact, due to the close-to-complete nature of the chain component, there are
few conditional independence assumptions that can be concretely made. Indeed
the complexity of such systems is the reason why social network modelling has
proven to be difficult.

Figure 2: Chain graph approximation of a three node temporal social network
process with treatment

Conditional independence properties have been considered for ERGMs (Snijders
et al., 2006). Common properties induced by model choice in ERGMs are the so-
called Markov assumption that alternatively, the so called social circuit
assumption requires the Markov assumption condition in addition to a four-
cycle condition (Snijders et al., 2006). Both of these commonly imposed
assumptions severely limit possible dependence structures. In practice the
social network modelling community has found the social circuit assumption to
yield well fitting models in a wide variety of situations (Goldenberg et al.,
2010).

We do not comment on the validity of the approximation of the DAG by the chain
graph. Ogburn et al. (2020) gave simulations supporting their version of this
approximation. We will only consider recovering causal estimands based on the
assumption that the chain graph approximation holds.

The remainder of this paper concerns modelling $P(A,Y|Z)$ in order to estimate
causal quantities. We specify our causal estimands using the equilibrium
distribution potential outcomes. The lack of conditional independence
assumptions that we are able to make in Section 4, is the direct consequence
of the nearly complete chain graph model specified in this section.

##  4 Neighbourhood Potential Outcomes

Hudgens and Halloran (2008), Toulis and Kao (2013) and Aronow and Samii (2017)
all considered potential outcome-based frameworks of assumptions and
definitions as a basis for causal inference for nodal covariates. As per the
previous section, our estimands are based on equilibrium distributions, and
relate to the chain graph causal model.

Our strategy is to pursue model-based Bayesian imputation of potential
outcomes Imbens and Rubin (2015). The principle assumption that we make is
that the data generating process is the one which we specify. In reality,
network processes are very complex and simple models are unlikely to be an
exact representation. Thus the plausibility of our causal claims should be
criticized by the reader through that lens. In addition, we state assumptions
restricting the dependence structure of the process that will enable us to
feasibly impute missing potential outcomes from network simulations.

We let $Y_{i}(Z,A,Y_{-i})$ be the equilibrium potential outcome of node $i$,
given the treatment vector of all nodes Z, the network A, and the outcomes of
all the other nodes $Y_{-i}$. We omit any nodal covariates $X$ for notational
clarity, as they are fixed and, assumed for this paper, not related to the
treatment assignment mechanism.

Typically we consider one-step neighbourhoods, that is we allow only nodes
connected to the ego to effect the nodal outcome. Thus we dramatically reduce
the number of unique potential outcomes for any given node, by requiring that
only the treatment assignment, edges involving a node and outcomes of the
neighbours of the nodes matter, not the whole network. We note that this is,
in fact, a highly restrictive assumption. We are vastly compressing the space
of possible data generating processes for such a social network, though
intuitively we believe this to be a plausible assumptions in real data cases.
In relation to the DAGs in the previous section, we now state the practical
restriction we make on the functional form of the joint distribution:

###### Assumption 1

One Step Neighbourhoods

$\displaystyle\forall i\qquad
Y_{i}(Z,A,Y_{-i})=Y_{i}(Z_{i},Z_{\mathscr{N}_{i}(A)},\mathscr{N}_{i}(A),Y_{\mathscr{N}_{i}(A)})$
(3)

where $\mathscr{N}_{i}(A)$ is the neighbourhood of node i in the network A.
That is, the individual outcomes only depend on treatments, edges and outcomes
in their own one-step neighbourhood.

We also require the unconfounded treatment assignment in which we explicitly
include nodal covariates:

###### Assumption 2

Unconfounded Treatment Assignment Assumption Under Network Assignment

$\displaystyle P(Z|X,A,\mathbb{Y})=P(Z|X,A,\mathbb{Y}^{\prime})\quad\forall
Z,X,A,\mathbb{Y}\quad and\quad\mathbb{Y}^{\prime}$

where $A$ is a network, X is a set of nodal covariates on the nodes, and
$\mathbb{Y}$ are the potential outcomes.

Should we want to account for the uncertainty in the network it is possible to
make the assumption, as in Kao (2017), that the causal link between the
network and outcomes can be broken by the inclusion of nodal covariates.
Specifically:

###### Assumption 3

Unconfounded Network Assumption under Network Interference

$\displaystyle P(A|X,\mathbb{Y})=P(A|X,\mathbb{Y}^{\prime})\quad\forall
A,X,\mathbb{Y}\quad and\quad\mathbb{Y}^{\prime}$

We note that Assumption 2 also follows as a consequence of the chain graph
approximation, as in the three node example in Figure 2 (See Lauritzen and
Richardson, 2002, for discussion on this).

Assumption 3 essentially assumes away the main problem of network causal
inference, that the network structure and the potential outcomes are related.
The idea is that after including enough nodal covariates, the association
between the network and the nodal potential outcomes breaks down. Note that
this assumption trivially holds if the network is considered fixed.

Denote the missing potential outcomes as $\mathbb{Y}_{miss}$ and the observed
as $\mathbb{Y}_{obs}$. Using Assumptions 1 and 2, Kao (2017) show that :

$\displaystyle
P(\mathbb{Y}_{miss}|X,Z,A,\mathbb{Y}_{obs})=P(\mathbb{Y}_{miss}|X,\mathbb{Y}_{obs})$
(4)

Kao (2017) propose choosing covariates $X$ to include so that Assumption 3 is
met, and suggest that the addition of covariates derived from modelling the
network $A$ may be included to achieve this.

We require only Assumptions 1 and 2, and modelling the potential outcomes
jointly with the network. We argue that this is more realistic in most
situations, and forces the researches into more realistic modelling choices to
arrive at causal inference for nodal outcomes.

The relaxation of Assumption 3 breaks down the proof of Equation (4), with the
analogous result under the relaxed assumption being:

$\displaystyle
P(\mathbb{Y}_{miss}|X,Z,A,\mathbb{Y}_{obs})=P(\mathbb{Y}_{miss}|X,A,Y_{obs})$
(5)

The retention of the network $A$ in the conditioning variables, requires that
we model its posterior distribution in the imputation of the missing potential
outcomes. We next state the causal estimands that we will infer under these
relaxed assumptions.

###### Definition 1

Primary Effects

$\displaystyle\xi_{i}$
$\displaystyle=\frac{1}{|(Z,A,Y)_{\mathscr{N}_{i}}|}\sum_{(z,a,y)\in(Z,A,Y)_{\mathscr{N}_{i}}}Y_{i}(z_{i}=1,z,a,y)-Y_{i}(z_{i}=0,z,a,y)$
(6) $\displaystyle\xi^{ave}$ $\displaystyle=\frac{1}{N}\sum_{i=1}^{N}\xi_{i}$
(7)

where $(Z,A,Y)_{\mathscr{N}_{i}}$ are the set of possible unique treatment and
outcome neighbourhoods for node $i$. Noting that neighbourhood treatments and
outcomes are induced by the network $A$.

###### Definition 2

$k$-peer Treatment Effect

$\displaystyle\delta^{k}_{i}$
$\displaystyle=\frac{1}{|(Z,A)_{\mathscr{N}_{i}}|}\sum_{(z,a,y)\in(Z,A)^{k}_{\mathscr{N}_{i}}}Y_{i}(z_{i}=0,z,a,y)-Y_{i}(z_{i}=0,(z,a)\in(Z,A)^{0}_{\mathscr{N}_{i}},y)$
(8) $\displaystyle\delta_{k}$
$\displaystyle=\frac{1}{N}\sum_{i=1}^{N}\delta^{k}_{i}$ (9)

where $(Z,A)^{k}_{\mathscr{N}_{i}}$ is the set of combinations of node-i-local
edge realizations and treatment assignments such that node $i$ is exposed to
$k$ other treated nodes.

###### Definition 3

$k$-peer Outcome Effect For Binary Outcomes

$\displaystyle\delta^{k}_{i}$
$\displaystyle=\frac{1}{|(A,Y)_{\mathscr{N}_{i}}|}\sum_{(z,a,y)\in(A,Y)^{k}_{\mathscr{N}_{i}}}Y_{i}(z_{i}=0,z,a,y)-Y_{i}(z_{i}=0,z,(a,y)\in(A,Y)^{0}_{\mathscr{N}_{i}})$
(10) $\displaystyle\delta_{k}$
$\displaystyle=\frac{1}{N}\sum_{i=1}^{N}\delta^{k}_{i}$ (11)

where $(A,Y)^{k}_{\mathscr{N}_{i}}$ is the set of combinations of node-i-local
edge and outcome realizations such that node $i$ is exposed to $k$ other nodes
with outcome $1$.

We note that the $k$-peer outcome effect could be considered as a special case
of a $k$-peer treatment effect where the treatment results in outcome $1$,
almost surely. However we find it convenient to express the estimand
separately as it is often of substantive interest in networks where some other
treatment is administered, but the $k$-peer outcome effect is important.

Tchetgen Tchetgen et al. (2020) and Ogburn et al. (2020) utilized Markov
random field models with coding or pseudo likelihood estimators to power a
Gibbs sampling procedure, from which they estimated causal effects of
treatment in situations where the observed data treatment was permitted to be
affected by covariates. Toulis and Kao (2013) allowed for edge uncertainty
utilizing a Poisson edge model, and a linear outcome model. We note that such
a model does not account for the dependent nature of tie formation usually
observed in real social processes.

With a suitably simple MRF model with a fixed network or a separable network
and outcome model, the causal effects can usually be computed directly from
the realized parameter values. For an ERNM this is not the case. The missing
potential outcomes can be derived from complete network simulations as
follows, yielding the causal estimands:

$\displaystyle P(\mathbb{Y}_{miss}|X,A_{obs},Y_{obs})$
$\displaystyle=\int_{\Theta}P(\mathbb{Y}_{miss}|\theta,X,A_{obs},Y_{obs})p(\theta|X,A_{obs},Y_{obs})d\theta$
(12)
$\displaystyle\approx\frac{1}{MN}\sum_{j=1}^{M}\sum_{i=1}^{N}Y_{miss,i}(A_{j},Y_{j})$
(13)

where $Y_{miss,i}(A_{j},Y_{j})$ is the unobserved potential outcome for the
$i^{th}$ node realized in $j^{th}$ of $M$ networks which are simulated from
the posterior distribution. Note that when a network simulation does not
result in the neighbourhood treatment of interest for node $i$ the potential
outcome is omitted in the sum and the averaging denominator reduced
correspondingly.

##  5 Simulation Study

In this section we describe a simulation study to verify our method’s ability
to recover causal effects.

We considered four possible data generating social processes for 100 node
networks where half of the nodes receive the treatment before the social
process evolves. The network model for the social process included
transitivity, homophily and contagion. There is mild spillover through a peer
treatment effect and contagion through a peer outcome effect, where in
addition to homophily, peer outcomes and treatments also increase the
propensity of a positive nodal outcome.

The study demonstrates that, in the presence of treatment spillover and
outcome interference, the distribution of the ERNM posterior causal estimands
are consistent with the expected posterior distributions across all estimands
of interest. A full-development is given in the supplement.

##  6 Case-Study of Smoking Behavior within a High School

In this section we give a real data case-study utilizing ERNM to relax the
fixed network assumption as well as conditional unconfoundedness of the edges
and nodal potential outcomes. In this case we do not know the ground truth, so
the purpose of this section is to demonstrate that our method produces
plausible posterior estimand distributions and to highlight the differences
between methods for these data. We also performed a simulation study with
known data generating processes which is contained in the supplement.

We consider one of the school social networks from the well known Adolescent
Health study (Harris et al., 2007). The network we used for this example has
$869$ nodes of which $462$ were male and $407$ were female, with $344$ having
reported trying a cigarette at least once. For consistency with Fellows and
Handcock (2012), gender was coded as $1$ for male and $0$ for females, hence
the gender coefficients reported correspond to males.

We consider the following estimands and estimate them under different
frameworks:

  1. 1.

Average main treatment effect of gender on smoker status

  2. 2.

$k$-peer outcome effect of peer smoker status on smoker status of the ego.

We consider the following models for imputing the required potential outcomes
to claim causal inference:

  1. 1.

Logistic regression.

  2. 2.

Markov random field model.

  3. 3.

Separable nodal and network model, ERGM model for network, logistic regression
for the potential outcomes.

  4. 4.

Full ERNM model with potential outcome as a random nodal covariates.

We used a version of the exchange algorithm Murray et al. (2006), with an
extension which allows for efficient sampling from the ERNM posterior. The
development is given in the supplement. While informative priors are
compatible with the computational framework, here we report based on a uniform
prior over all parameters. The results do not appear to be sensitive to the
choice of prior (see the supplement for a discussion).

Table 1:  Summary of posterior distributions of network models, posterior means are shown with posterior standard errors in parentheses | Logistic Regression | ERGM+Logistic | ERNM
---|---|---|---
Edges | NA | -5.77(0.05) | -5.86(0.04)
GWESP | NA | 1.54(0.03) | 1.55(0.03)
GWDEG | NA | 1.62(0.43) | 2.11(0.4)
Grade Homophily | NA | 1.64(0.1) | 1.57(0.08)
Sex Homophily | NA | 0.5(0.07) | 0.55(0.08)
Smoke Homophily | NA | 0.6(0.09) | 0.34(0.09)
Intercept | -1.7(0.16) | -1.7(0.16) | 0.82(0.08)
Gender | 0.53(0.15) | 0.53(0.15) | -0.11(0.02)
Smoker neighbours | 0.37(0.04) | 0.37(0.04) | -0.36(0.14)
Stochastic Edges | No | Yes | Yes
Stochastic Covariates | Smoker Status | Smoker Status | Smoker Status
Separable Likelihood | NA | Yes | No

Table 1 gives a summary of the posterior distributions each of the models,
showing the posterior means with the posterior standard errors in parentheses.
We note that parameters should not be compared across models, as the
functional forms are different, we show this table to summarize the
posteriors, but to also highlight the differences between the models. We were
not able to fit the corresponding MRF model to these data as, as expected, it
displayed degeneracy (Handcock, 2003; Fellows and Handcock, 2012).

For GWESP and GWDEG terms decay parameters were fixed at $0.5$. The use of
these geometrically weighted terms is in part required to avoid degeneracy
issues (Handcock, 2003; Snijders et al., 2006), but also implicitly induces
the social circuit dependence assumption for the ERGM, rather than the more
restrictive Markov assumption. However we used the ERNM style homophily terms
(Fellows and Handcock, 2012) for both the ERNM and ERGM model, which in fact
induce non local dependencies. Thus the dependence structures of the edges in
these models are unknown and best described as “complex”. In addition we
enforce homogeneity in school grade for the edgewise shared partners in the
GWESP term as there is a very strong grade structure to transitivity in this
network.

We do not interpret the posteriors directly, rather make comparison through
the main effect of gender and the smoker peer effect on the smoker status of a
node.

Table 2 shows the estimated main effect of gender, the interpretation of the
posteriors in this case is not qualitatively different. Table 3 and Figure 3
show the the $k$-peer effect estimates This is estimated as the additional
chance of smoking that having $k$ smoker friends has over having no smoker
friends. We believe these results suggest the true power of using the ERNM for
the estimation of causal estimands. The one and two peer effect for the ERNM
is much higher than the estimates using a linear potential outcome model with
either of the two other methods. This is much more in line with social
theories of the influence of friendship on behaviour; the first few marginal
exposures, have a much greater effect influence than subsequent exposures.

In addition we show Bayesian posterior prediction goodness of fit graphics in
the style of Hunter et al. (2008) in the supplement. These demonstrate that
networks simulated from the ERNM model posterior, correspond closely to the
observed data. The separable ERGM logistic regression model produces similar
networks in terms of edge structure, but not in terms of the network placement
of smoker nodes.

Table 2:  Main Gender effect ATEs with various methods Method | Posterior Mean | Posterior SD
---|---|---
Logistic Regression | 0.1105 | 0.0301
ERGM+Logistic | 0.0893 | 0.0336
ERNM | 0.0943 | 0.0375
Table 3:  k-peer smoker outcome effect ATEs, with various methods Peers | Logistic Regression | ERGM+Logistic Regression | ERNM
---|---|---|---
1 | 0.0650 | 0.0691 | 0.1196
2 | 0.1428 | 0.1507 | 0.2255
3 | 0.2292 | 0.2399 | 0.3253
4 | 0.3209 | 0.3317 | 0.4158
5 | 0.4110 | 0.4173 | 0.4971
6 | 0.4932 | 0.4936 | 0.5680
7 | 0.5651 | 0.5576 | 0.6259
Figure 3: Estimated k-peer effects for various methods

##  7 Discussion

In this paper we model causality when the underlying population is networked
and that network is unobserved. Our approach jointly stochastically models the
links and covariates in the network, better representing our state of
knowledge and their codependency.

Considering a DAG, we suggest that the estimating true causal effects in this
setting, is almost always intractable due to the usual lack of fine grained
temporal data, as well as the highly complex causal structure. We present a
chain graph approximation to the DAG, which allows for a dependence structure
that we believe to be compatible with such problems. We then frame the
approximate causal inference in terms of network equilibrium potential
outcomes, that is, potential outcomes that are free to depend on nodes in the
neighbourhood of the node in question. We propose the use of ERNMs to jointly
model both random edges and nodal covariates. We also develop a simple
modification to the exchange algorithm allowing for a feasible sampling from
the posterior distribution. We use the posterior distribution, through
simulation, to impute the distribution of the missing potential outcomes,
allowing the consideration of the distribution of the causal estimands. We
showed, using an AddHealth school dataset, that failing to account for the
network structure of the problem could lead to misleading qualitative
conclusions, in particular when considering the one- and two- peer outcome
effect.

Our primary contributions lie in considering the consequences of relaxing of
the commonly made fixed network assumption and proposing the use of suitable
social network models to estimate causal effects. This relaxation complicates
the causal structure considerably and necessitates the use of complex models
to derive causal estimands. Clearly the relaxation of this assumption also
allows for greater generalizability. Our inferences hold for the given node
set and social process, whereas assuming the fixed network narrows the scope
to that observed network only. Our method is only applicable to the given node
set. However we suggest that the posteriors derived from the given network can
serve as strong priors for “similar” networks. In general, qualitative
features of the posteriors from the given network can useful inform other
analyses. It is not possible to make further statements than this for networks
of different sizes.

We believe that in the context of social network analysis, where individual
attributes are heavily influenced by social context, such covariation of edges
and nodal covariates is overwhelmingly more representative of many social
processes. That is, in many social networks we believe it highly likely that
individual characteristics and the connections that form between individuals
are strongly dependent. This is especially true if the network evolves over
time. We also note that avoiding such issues by letting the size of the
network become large and invoking asymptotic arguments, fundamentally mis-
interprets the problem. The phenomenon of interest; the inter dependencies of
actors, is a result of the small size of the network, for example dependence
assumptions implicitly made in modelling a $30$ node network usually do not
make sense to apply to say a network of $3000$ nodes. Thus arguments that rely
on the number of nodes being large, are incoherent, as they rely on changing
the structure of the problem itself, to understand uncertainty. As our
approach is fully Bayesian on the fixed node set, we do not require asymptotic
arguments on the number of nodes in a network.

Notable by its absence is discussion of suitable prior distributions for
ERNMs. We note there has been some work developing conjugate prior
distributions for ERGMs (Wang, 2011) and strongly suspect that such an
approach may also be applicable in our setting. In practice flat Gaussian
prior are often used for ERGMs (Caimo and Friel, 2011). For the purpose of
demonstrating our approach, we used uniform priors, which make no account for
the geometry of exponential families, but allow us defer careful consideration
of possible priors to future work, while demonstrating the utility of our
method. We note that we performed similar posterior fits with flat Gaussian
priors, which did not effect the posteriors substantially.

The cost of our approach is the strong assumption that such a complex process,
can be adequately modelled by an ERNM. This is in general the main criticism
of model based causal inference approaches, that models are mis-specified with
unknown consequences. In a network setting this mis-specification is often
acute e.g. constant marginal effect of additional smoker friends in the linear
potential outcomes model. Our central argument is that a complex model is much
less mis-specified than current approaches. We have sought to justify this
with real data and simulations, though propose this as a future area of
research. For example, how dependent do outcomes in networks need to be to
invalidate conclusions made with unrealistic models?

The mis-specification may seem to be cause for pessimism, however we emphasize
that network settings are indeed the extreme case of small data, as we usually
only have one observations of a network on a fixed set of nodes. Thus
intuitively we should expect strict functional form assumptions to be required
to generate any meaningful statistical, and especially causal inference. In
fact we argue that approaching network problems with simpler assumptions is
problematic, whilst potentially less prone to mis-specification in the sense
that simple models can be used, this easily glosses over the inherent
difficulty of dealing with network data where nodes and edges are strongly
dependent on other nodes and edges.

We believe that meaningful steps can be made towards causal inference on
networks, through careful consideration of the complex causal structure of
such problems. Whilst we make strict assumption on the function form of this,
if the researcher is unwilling to make such assumptions, we opine casual
inference is out of reach. We suggest it is better to acknowledge the
complexity of the situation, and therefore claim that causal inference is not
possible, than employing highly restrictive assumptions on the dependence
structure of the data generating process, to allow simpler models to be
employed.

##  8 Acknowledgements

This article is based upon work supported by the National Science
Foundation(NSF, MMS-0851555, SES-1357619, IIS-1546259) and National Institute
of Child Health and Human Development (NICHD, R21HD063000, R21HD075714 and
R24-HD041022). The content is solely the responsibility of the authors and do
not necessarily represent the official views of the National Institutes of
Health or the National Science Foundation.

##  9 Supplementary Materials

Supplement

The supplement contains an additional chain graph approximation diagram, a
review of ERNM and Bayesian computation for them, a simulation study showing
that the method is able to accurately infer the causal effects and a goodness
of fit analysis for the modelling of the adolescent health network, including
a MCMC convergence analysis. (pdf)

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