Causal Inference from a Single Social Network
Suppose each observed outcome contains a disturbance shared with its neighbors. Adding more observations then adds some new information and some repeated information. The theory of (Ogburn et al. 2024) asks when an intervention mean can nevertheless be estimated with a normal limit from a single growing network.
1. The main result
Let \(K_{\max,n}\) be the largest network neighborhood size and let \(\psi_n\) be the intervention mean for the network of size \(n\). Under the paper’s causal structure, summary-map, nuisance-estimation, moment, and dependence conditions:
Theorem 1. If \(K_{\max,n}^2/n\to0\), the proposed estimator has a normal limit
for an appropriate variance normalization satisfying \(n/K_{\max,n}^2\le C_n\le n\).
Read the degree condition together with the model assumptions. It bounds how much local dependence can accumulate; identification, nuisance estimation, and moment control are separate requirements.
2. Why squared degree appears
Suppose each unit’s influence contribution depends on its own neighborhood. Two contributions can be dependent when those neighborhoods overlap, even if the two focal units are not directly linked. A neighborhood of size at most \(K\) can overlap with roughly \(K^2\) others.
The distinction is visible in the variance of an average. Expanding the square gives
There are at most order \(nK^2\) potentially nonzero covariance terms. If each is bounded, division by the squared sample size gives the upper scale \(K^2/n\). This count explains why degree enters the normalization. Establishing a normal limit requires control of higher-order dependence as well, which is supplied by the paper’s CLT.
3. Identification and dependence are separate
Locality of dependence also needs to be distinguished from causal identification. A latent variable can affect nearby units only and still confound treatment and outcome. Restricting the distance over which dependence travels does not remove that confounding.
The intervention formula uses measured covariates and specified treatment/outcome summaries. Correctly estimating these summaries does not compensate for a missing confounder or an incorrect exposure map. The network TMLE construction explains the targeting step once the identifying model is given.
4. The variance must follow the graph
An i.i.d. variance estimator retains only the terms \(i=j\) in the display. Under overlapping neighborhoods, the omitted off-diagonal terms can be of the same order or larger. A valid variance procedure must estimate or otherwise account for the relevant covariance structure under the paper’s assumptions.
Conditioning on observed baseline covariates can change the target and remove some baseline-distribution uncertainty. It does not erase treatment and outcome dependence. The distinction should be fixed before deriving the influence function or reporting a standard error.
A hub illustrates why average degree is insufficient here. Many units may each have few neighbors while all sharing one highly connected neighbor. Their influence contributions can then contain the same disturbance.
5. Building the dependence graph for influence contributions
Let \(N_i\) denote the set of primitive disturbances needed to construct \(D_i\). In a local model, disjoint disturbance sets imply independence. The relevant dependence graph joins \(i\) and \(j\) whenever \(N_i\cap N_j\ne\varnothing\). It can be denser than the original social graph.
If each closed neighborhood has at most \(K\) vertices and the graph is undirected, then
With \(\mathbb{E} D_i=0\) and \(\mathbb{E} D_i^2\leq M\), Cauchy–Schwarz gives \(|\operatorname{Cov}(D_i,D_j)|\leq M\). Hence
Overlapping neighborhoods identify the pairs that may contribute covariance. Some contributions may be small, zero, or negative, so the variance can be smaller than \(K^2/n\). Its actual scale is set by their sum, with a nondegeneracy condition needed for a nonzero limiting variance.
6. A small example showing the missing variance term
On a cycle, let \(\varepsilon_i\) be independent, centered, variance-one disturbances, and define \(D_i=\varepsilon_i+\varepsilon_{i+1}\) with cyclic indices. Each \(D_i\) has variance two and covariance one with each of its two neighbors. Therefore
The same answer follows from \(\sum_iD_i=2\sum_i\varepsilon_i\). Keeping only individual squared contributions would target \(2/n\), half the true variance. The network is sparse and the rate is still \(\sqrt n\); sparsity alone does not justify ignoring its covariance terms.
For another illustration, suppose contributions are identical within independent groups of size \(h\). The mean is an average of \(n/h\) independent group shocks, so its variance is proportional to \(h/n\). A bound based on \(K^2/n\) can be loose for this graph. The theorem supplies a sufficient regime and a permissible normalization range, rather than a universal formula for an effective sample size.
A linear disturbance model gives another way to calculate the covariance sum exactly. Let \(D_i=\sum_j b_{ij}\varepsilon_j\), where the primitive \(\varepsilon_j\) are independent with mean zero and variances \(\sigma_j^2\). Rearranging the average gives
The diagonal-only calculation would instead give \(n^{-2}\sum_j\sigma_j^2\sum_i b_{ij}^2\). Its error is the sum of the omitted products \(2b_{ij}b_{\ell j}\) for contributions sharing disturbance \(j\). This identifies the source of dependence more directly than the presence of an edge between the focal units.
If \(b_{ij}\) indicates membership in a closed neighborhood of an undirected graph and \(\sigma_j^2=1\), the exact variance is \(n^{-2}\sum_j(d_j+1)^2\). On a regular graph of degree \(d\), this is \((d+1)^2/n\), whereas keeping only the diagonal gives \((d+1)/n\). This example uses unnormalized neighborhood sums, whose individual variances grow with degree; it is not an example satisfying a uniform variance bound for arbitrarily growing \(d\). It illustrates why both the weights and the graph matter.
In this linear example, rearrangement reduces the problem to an independent weighted sum. A CLT can then be checked by controlling large weighted shocks relative to the total variance. Nonlinear network influence functions generally do not admit this rearrangement; the single-network theory must control their dependence directly and also bound the nuisance-estimation remainder.
7. Why a variance bound is not yet a CLT
For independent observations, controlling each large summand and the total variance can be enough for a Lindeberg argument. With dependence, products involving connected collections of indices also matter. A connected collection of \(r\) vertices can be built by choosing a root, then attaching each new vertex to any previously selected vertex.
If the dependence graph has maximum degree \(\Delta\), the number of such ordered constructions is at most
The factor \(j\) is essential: a new vertex may attach to any of the \(j\) previous vertices, not only to the last one. A path count would miss branching configurations. Moment bounds control the size of the corresponding terms, and the paper’s dependence assumptions and normalization make the relevant errors negligible.
Pair counting thus establishes a variance bound but leaves a further question: do higher-order terms permit normal approximation? The disturbance assumptions and moment bounds in the formal CLT answer that question. A sparse covariance pattern by itself does not.
8. A hub can violate a local-noise argument
Consider a star where every leaf contribution contains the same centered hub disturbance \(H\):
Their average converges to \(H\) rather than zero if \(\operatorname{Var}(H)>0\). Even though the original graph has average degree close to two, all leaf neighborhoods overlap at the hub. The maximum degree is of order \(n\), and the theorem’s degree condition fails.
Each leaf repeats the hub shock, so averaging over leaves cannot remove it. Measured baseline covariates can only remove the uncertainty they explain. If the unobserved shock also affects treatment, it can invalidate identification in addition to preventing the proposed CLT.
9. Matching nuisance rates and the target
An asymptotically linear expansion takes the form
The normal approximation requires \(\sqrt{C_n}R_n=o_p(1)\), not merely \(R_n=o_p(1)\). Product-rate conditions for outcome and assignment estimates are one way to control a second-order remainder; dependent empirical-process terms need their own bounds or a splitting scheme that supplies the needed separation.
One can instead define the intervention mean conditional on the realized baseline network. This removes the baseline-distribution contribution to that target and its gradient, while treatment and outcome randomness can remain dependent. The choice of conditional or unconditional target must therefore precede the variance calculation.