Exposure probabilities$\mathbb{P}_\pi(D_i(\boldsymbol{Z}) = d)$ for each degree class. Under Bernoulli($\pi$) assignment, $D_i$ counts the number of unit $i$'s treated neighbors. Units with different degrees have different exposure distributions. by degree
Average focal expected outcomes (AFEOs)Each exposure level $e$ defines a focal map $f_e(\boldsymbol{z}) = \mathbf{1}\{d_i(\boldsymbol{z}) = e\}$, selecting units whose exposure equals $e$. The AFEO for exposure $e$ is: $$\text{AFEO}(\pi, f_e, y) = \langle \mathbb{E}_\pi[y_i(\boldsymbol{Z}) \mid d_i(\boldsymbol{Z})=e] \rangle_{i \in \text{supp}_{f_e}(\pi)}$$ First average over treatment assignments within each unit (assignment-averaging), then average across units (unit-averaging). With a correctly specified exposure map, the inner expectation collapses (all assignments giving the same exposure yield the same outcome), so only unit-averaging remains. by degree
Exposure mapA function $d_i: \{0,1\}^n \to \mathcal{D}$ mapping a treatment vector to a low-dimensional summary of unit $i$'s exposure. Here: $d_i(\boldsymbol{z}) = \min(\text{\#treated neighbors of } i,\, 2)$. Each exposure level $e \in \{0, 1, 2+\}$ induces a focal map $f_e$ that selects units with that exposure, which in turn defines an AFEO.: d ∈ {0, 1, 2+} treated neighbors.
Correctly specifiedAn exposure map is correctly specified if it gives level sets of the potential outcome function: whenever $d_i(\boldsymbol{z}) = d_i(\boldsymbol{z}')$, then $y_i(\boldsymbol{z}) = y_i(\boldsymbol{z}')$. This means the exposure map captures all relevant variation in how treatments affect outcomes, so the AFEO reduces to a simple average of $y_i(e)$ over units..
(Unlike panel (b) of Figure 2 in the paper, which shows AFEOs varying with π under a misspecified exposure map, here the exposure map is correctly specified throughout, so the AFEOs do not depend on π.)
Deg 2 (3 units)
0.10
0.10
0.80
Deg 3 (2 units)
0.70
0.70
0.10
Overall AFEOsThe overall AFEO for exposure $e$ averages across all units: $$\text{AFEO}(f_e, y) = \langle y_i(e) \rangle_{i \in [n]}$$ On this graph, units have two different degrees, so the overall AFEO is a weighted average of degree-specific AFEOs: $(n_A \cdot \langle y_i(e)\rangle_A + n_B \cdot \langle y_i(e)\rangle_B)/n$. These are what a researcher observes without knowing degree-specific values.
Expected Average Outcome (EAO)$\text{EAO}(\pi, y) = \mathbb{E}_\pi\!\big[\langle y_i(\boldsymbol{Z}) \rangle_{i \in [n]}\big]$: the expected population-average outcome under policy $\pi$. For the EAO, the two orders of averaging (over units and over assignments) coincide under any policy (an instance of Proposition 3 in the paper), and it is sufficient for policy choice with utilitarian welfare.
Expected Focal Average Outcomes (EFAOs)For exposure level $e$, the EFAO is: $$\text{EFAO}(\pi, f_e, y) = \mathbb{E}_\pi\!\big[\langle y_i(\boldsymbol{Z}) \rangle_{d_i(\boldsymbol{Z})=e} \,\big|\, \exists\, i: d_i(\boldsymbol{Z})=e\big]$$ First average over the (random) set of units with exposure $e$ in a given assignment (unit-averaging), then average over assignments (assignment-averaging). The opposite order from AFEOs. These two orders generally differ because the set of focal units varies across assignments. Contrasts of EFAOs for different policies (or different exposures) typically lack an interpretation as summaries of unit-level causal effects.
Key insight: The 6 degree-specific AFEOs fully determine the EAO curve (solid line).
But the 3 overall AFEOs (averaging across degree classes) do not identify the EAO —
they only give partial identification boundsThe range of $\text{EAO}(\pi)$ values consistent with the 3 overall AFEOs alone, maximizing and minimizing over all degree-specific AFEOs $\bar{y}_A(d), \bar{y}_B(d) \in [0,1]$ that produce those overall values. (shaded region).
This is because the exposure distribution is not unit-homogeneousAn exposure distribution is unit-homogeneous under policy $\pi$ if $\mathbb{P}_\pi(D_i(\boldsymbol{Z}) = d)$ is the same for all units $i$. This fails on irregular graphs where units have different degrees. on an irregular graph.
The naive plug-inEstimates $\text{EAO}(\pi)$ as $\sum_d \bar{P}(D=d \mid \pi) \cdot \bar{y}(d)$, using overall AFEOs with average exposure probabilities, as if the exposure distribution were unit-homogeneous. This is what you would compute if you ignored degree heterogeneity. (dashed line) uses overall AFEOs as if exposure were homogeneous —
it can be misleading because it ignores that units with different degrees have different
exposure probabilities.