A Appendices
1.1 A.1 Peer-Influence Under Homophily: Results and Inference Strategies
Binary peer-influence effect with normalized homophily: Consider now binary peer-influence effect with normalized homophily. For the untreated individuals, we have
$$\begin{aligned} Y_i(Z_i=0,(Z_j)_{j \in {\mathcal {N}_i}}) = \alpha + \beta _0 \mathbf 1 _{\sum _{j \in {\mathcal {N}_i}} Z_j > 0 } + h_0 \sum _{j \in {\mathcal {N}_i}} \frac{X_j}{ |\mathcal {N}_i | } + \epsilon _i(0, \sigma ^2_Y) \end{aligned}$$
(11)
where \(\epsilon _i(0, \sigma ^2_Y)\) are idependent and identically distributed with zero mean and \(\sigma ^2_Y\) variance.
As before, consider estimating the peer-influence parameter \(\beta _o\) using a difference in means estimator. Partition the set of untreated individuals into sets \(M^{(0)}_0 := \{ i: Z_i=0, \sum _{j \in \mathcal {N}_i}Z_j = 0 \}\) (the set of untreated individuals with no treated neighbors) and \(M^{(1)}_0:= \{ i: Z_i=0, \sum _{j \in \mathcal {N}_i}Z_j > 0 \}\) (the set of untreated individuals with at least one treated neighbors). Then, the difference in means estimator for \(\beta _0\) is given by:
$$\begin{aligned} \hat{\beta _0} = \underset{i \in M^{(1)}_0}{avg} Y_i - \underset{i \in M^{(0)}_0}{avg} Y_i \end{aligned}$$
(12)
Unlike in the case with unnormalized homophily, the difference of means estimator for peer-influence remains unbiased in the presence of normalized homophily. This is further highlighted in Theorem 2 below. Furthermore, for most sparse and dense models for the underlying graph, Theorem 2 can be used to show that \(\hat{\beta }_0\) is a consistent estimator of peer-influence under normalized homophily.
Theorem 2
Consider the difference in means estimator \(\hat{\beta }_0\) for binary peer-influence effect \(\beta _0\). Under the presence of normalized homophily in our model (11), the mean squared error of \(\hat{\beta }_0\) (conditional on the treatment \(\mathbf Z \)) is:
$$\begin{aligned} \begin{aligned} \mathbb {E}&[ (\hat{\beta }_0 - \beta _0)^2 | \mathbf Z ] = \\&h_0^2 \sigma _X^2 \Bigg ( \underset{i,j \in M^{(0)}_0}{avg} \frac{| \mathcal {N}_i \cap \mathcal {N}_j|}{|\mathcal {N}_i||\mathcal {N}_j|} + \underset{i,j \in M^{(1)}_0}{avg} \frac{| \mathcal {N}_i \cap \mathcal {N}_j|}{|\mathcal {N}_i||\mathcal {N}_j|} - \ 2 \underset{i \in M^{(0)}_0, j \in M^{(1)}_0}{avg} \frac{| \mathcal {N}_i \cap \mathcal {N}_j|}{|\mathcal {N}_i||\mathcal {N}_j|} \Bigg ) + \sigma _Y^2 \Bigg ( \frac{1}{|M^{(0)}_0|} + \frac{1}{|M^{(1)}_0|} \Bigg ) \end{aligned} \end{aligned}$$
(13)
Linear peer-influence effect with unnormalized homophily: We now consider modeling peer-influence as a linear function of the number of treated neighbors \(\mathbf{peer }( (Z_j)_{ j \in {\mathcal {N}_i} } ) = \sum _{j \in {\mathcal {N}_i}} Z_j\). For the untreated individuals under unnormalized homophily, this gives:
$$\begin{aligned} Y_i(Z_i=0,(Z_j)_{j \in {\mathcal {N}_i}}) = \alpha + \beta _0 \sum _{j \in {\mathcal {N}_i}} Z_j + h_0 \sum _{j \in {\mathcal {N}_i}} X_j + \epsilon _i(0, \sigma ^2_Y) \end{aligned}$$
(14)
where \(\epsilon _i(0, \sigma ^2_Y)\) are idependent and identically distributed with zero mean and \(\sigma ^2_Y\) variance.
Consider estimating the peer-influence parameter \(\beta _0\). Generalizing our methodology from the binary peer-influence case, we now develop a stratified estimator for \(\beta _0\). Let
$$M^{(k)}_0 := \{ i: Z_i=0, \sum _{j \in \mathcal {N}_i}Z_j = k \}$$
be the set of untreated individuals which have k treated neighbors. Then, an average of difference in means estimator for peer-influence is:
$$\begin{aligned} \hat{\beta }_0 \,{=}\, \frac{\sum _k \hat{\beta }_0^{(k)}}{\sum _k 1} \ for \ \hat{\beta }_0^{(k)} \,{=}\, \frac{1}{k} \Bigg ( \frac{\sum _{i \in M^{(k)}_0} Y_i}{| M^{(k)}_0 | } - \frac{\sum _{i \in M^{(0)}_0} Y_i}{| M^{(0)}_0 | } \Bigg ) = \frac{1}{k} \Bigg ( \underset{i \in M^{(k)}_0}{avg} Y_i - \underset{i \in M^{(0)}_0}{avg} Y_i \Bigg ). \end{aligned}$$
(15)
where we average over all k such that \(| M^{(k)}_0 | > 0 \) (so that \(\hat{\beta }_0^{(k)}\) is well-defined). Note that here we are averaging over the class of estimators \(\hat{\beta }_0^{(k)}\) under the assumption of linear peer-influence. In the case of nonlinearity, we can also consider each \(\hat{\beta }_0^{(k)}\) separately to understand the kth-level peer-influence effect in the network.
The presence of latent unnormalized homophily interferes and introduces bias to the estimation of linear peer-influence, as highlighted in Theorem 3 below.
Theorem 3
Consider the estimator \(\hat{\beta }_0\) for linear peer-influence effect \(\beta _0\). Under the presence of unnormalized homophily in our model (3), the mean squared error of \(\hat{\beta }_0\) (conditional on the treatment \(\mathbf Z \)) is:
$$\begin{aligned} \begin{aligned} \mathbb {E}&[ (\hat{\beta }_0 - \beta _0)^2 | \mathbf Z ] = \\&\Bigg ( \frac{h_0}{\sum _{k>0} 1} \sum _{k>0} \frac{1}{k} \Bigg ( \underset{i \in M^{(k)}_0}{avg} |\mathcal {N}_i| - \underset{i \in M^{(0)}_0}{avg} |\mathcal {N}_i| \Bigg ) \Bigg )^2 \\&+ \frac{1}{(\sum _{k>0} 1)^2} \sum _{k,l>0} \frac{1}{kl} \Bigg [ h_0^2 \sigma _X^2 \Bigg ( \underset{i \in M^{(k)}_0 ,j \in M^{(l)}_0}{avg} | \mathcal {N}_i \cap \mathcal {N}_j| + \underset{i,j \in M^{(0)}_0}{avg} | \mathcal {N}_i \cap \mathcal {N}_j| - \ 2 \underset{i \in M^{(0)}_0, j \in M^{(k)}_0}{avg} | \mathcal {N}_i \cap \mathcal {N}_j| \Bigg ) \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad + \sigma _Y^2 \Bigg ( \frac{1}{|M^{(0)}_0|} + \frac{ \mathbf 1 _{k=l} }{|M^{(k)}_0|} \Bigg ) \Bigg ] \end{aligned} \end{aligned}$$
(16)
Equation (16) highlights that unbiasedness estimation via optimal treatment allocation may be difficult computationally, as now we need to ensure balance across all the strata \( (M^{(k)}_0)_{k \ge 0} \). This motivates an alternative approach of unbiased estimation.
Linear peer-influence effect with normalized homophily: For the peer-influence effect on untreated individuals under normalized homophily, we obtain:
$$\begin{aligned} Y_i(Z_i=0,(Z_j)_{j \in {\mathcal {N}_i}}) = \alpha + \beta _0 \sum _{j \in {\mathcal {N}_i}} Z_j + h_0 \sum _{j \in {\mathcal {N}_i}} \frac{X_j}{| \mathcal {N}_i |} + \epsilon _i(0, \sigma ^2_Y) \end{aligned}$$
(17)
where \(\epsilon _i(0, \sigma ^2_Y)\) are idependent and identically distributed with zero mean and \(\sigma ^2_Y\) variance.
To estimate the peer-influence parameter \(\beta _0\), the same stratified estimator as in the linear peer-influence with unnormalized homophily case can be applied:
$$\begin{aligned} \hat{\beta }_0 \,{=}\, \frac{\sum _k \hat{\beta }_0^{(k)}}{\sum _k 1} \ for \ \hat{\beta }_0^{(k)} \,{=}\, \frac{1}{k} \Bigg ( \frac{\sum _{i \in M^{(k)}_0} Y_i}{| M^{(k)}_0 | } \,{-}\, \frac{\sum _{i \in M^{(0)}_0} Y_i}{| M^{(0)}_0 | } \Bigg ) \,{=}\, \frac{1}{k} \Bigg ( \underset{i \in M^{(k)}_0}{avg} Y_i - \underset{i \in M^{(0)}_0}{avg} Y_i \Bigg ). \end{aligned}$$
(18)
where \(M^{(k)}_0 := \{ i: Z_i=0, \sum _{j \in \mathcal {N}_i}Z_j = k \}\) and we are averaging over all k such that \(| M^{(k)}_0 | > 0 \).
In the presence of normalized homophily, \(\hat{\beta }_0\) remains an unbiased estimator of peer-influence. This is highlighted in Theorem 4 below.
Theorem 4
Consider the estimator \(\hat{\beta }_0\) for linear peer-influence effect \(\beta _0\). Under the presence of normalized homophily in our model (11), \(\hat{\beta }_0\) is unbiased and the mean squared error of \(\hat{\beta }_0\) (conditional on the treatment \(\mathbf Z \)) is:
$$\begin{aligned} \begin{aligned} \mathbb {E}&[ (\hat{\beta }_0 - \beta _0)^2 | \mathbf Z ] = \\&\frac{1}{(\sum _{k>0} 1)^2} \sum _{k,l>0} \frac{1}{kl} \Bigg [ h_0^2 \sigma _X^2 \Bigg ( \underset{i \in M^{(k)}_0 ,j \in M^{(l)}_0}{avg} \frac{| \mathcal {N}_i \cap \mathcal {N}_j|}{|\mathcal {N}_i||\mathcal {N}_j|} + \underset{i,j \in M^{(0)}_0}{avg} \frac{| \mathcal {N}_i \cap \mathcal {N}_j|}{|\mathcal {N}_i||\mathcal {N}_j|} - \ 2 \underset{i \in M^{(0)}_0, j \in M^{(k)}_0}{avg} \frac{| \mathcal {N}_i \cap \mathcal {N}_j|}{|\mathcal {N}_i||\mathcal {N}_j|} \Bigg ) \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad + \sigma _Y^2 \Bigg ( \frac{1}{|M^{(0)}_0|} + \frac{ \mathbf 1 _{k=l} }{|M^{(k)}_0|} \Bigg ) \Bigg ] \end{aligned} \end{aligned}$$
(19)
The difference of means estimator for linear peer-influence remains unbiased in the presence of normalized homophily. Furthermore, for most sparse and dense models for the underlying graph, Theorem 2 can be used to show that \(\hat{\beta }_0\) is a consistent estimator of linear peer-influence under normalized homophily.
1.2 A.2 Disentangling Homophily from Estimation of Peer-Influence: Randomized Treatment Strategies
An algorithm for inference of linear peer-influence. We now use our general framework to design randomized treatments for the inference of linear peer-influence effects under homophily. We proceed to find the optimal treatment probabilities \(\theta _s\) for \(s = 1,\ldots ,r\) under a stochastic block model with r communities as before.
Let \( M_0^{(k)} \) denote the set of untreated individuals which have k neighbors (note that we are abusing notation here: Now, \( M_0^{(1)} \) represents untreated individuals which have exactly 1 neighbor, rather than at least 1 neighbor as before in the binary peer-influence case). First, we derive a proposition about \(M_0^{(k)}\) under our framework.
Proposition 2
Consider a stochastic block model (SBM) of N individuals in r communities. Denote the communities of the SBM by the sets \(B_1,\ldots ,B_r\), which are of respective sizes \(A_1,\ldots , A_r\) (where \(A_1+ \cdots +A_r = N\)). Let \(\mathbf P \) be the \(r \times r\) adjacency probability matrix between the r communities. We assign treatments independently to individuals such that individuals in \(B_s\) are treated with probability \(\theta _s\) for \(s=1,\ldots ,r\). Under such setup, let \( M_0^{(k)} \) denote the set of untreated individuals which have k treated neighbors. For ease of notation, let \(\{ s \in M_0^{(k)} \}\) denote the event that a fixed vertex in community s is in the set \(M_0^{(k)}\). Then,
$$\begin{aligned} \mathbb {P}( s \in M_0^{(k)} ) = (1 - \theta _s) \sum _{ \begin{array}{c} t_1,\ldots ,t_r: \\ \forall v =1,\ldots ,r \ 0 \le t_v \le A_v -\mathbf 1 _{ \{v=s \} }, \\ t_1+ \cdots +t_r = k \end{array}} \Bigg ( \prod _{v=1}^r \mathbf{Bin }( t_v; A_v -\mathbf 1 _{ \{v=s \} }, \theta _v P_{s,v} ) \Bigg ) \end{aligned}$$
(20)
where \(\mathbf{Bin }( t_v; A_v -\mathbf 1 _{ \{v=s \} }, \theta _v P_{s,v} ) = \left( {\begin{array}{c}A_v-\mathbf 1 _{ \{v=s \} }\\ t_v\end{array}}\right) \big ( \theta _v P_{s,v} \big )^{ t_v } \big ( 1 - \theta _v P_{s,v} \big )^{ A_v -\mathbf 1 _{ \{v=s \} } - t_v }\).
The main idea behind the homophily disentangling strategy is to ensure that in every community \(B_s\) on our stochastic block model, there are equal (expected) numbers of individuals being affected by different levels of peer-influence. In the case of linear peer-influence, this means choosing treatment values such that inside every community s, each individual has an equal probability of being in sets \(M_0^{(k)}\) for different peer-influence levels k. Under a stochastic block model, values of k range from 0 to \(N-1\) (as one individual can have at most \(N-1\) treated neighbors). However, in practice, we can choose to consider \(k=0,1, \ldots , K\) where K is the maximum degree of the actual observed network. Therefore, through an optimal assignment of treatments, we wish to satisfy
$$ \forall s \,{=}\, 1,\ldots , r, \quad \mathbb {P}( s \in M_0^{(0)} ) \,{=}\, \mathbb {P}( s \in M_0^{(1)} ) = \cdots = \mathbb {P}( s \in M_0^{(K-1)} ) = \mathbb {P}( s \in M_0^{(K)} ), $$
where expressions for each \(\mathbb {P}( s \in M_0^{(k)} )\) as functions of \(\theta _s\) for \(s=1,\ldots , r\) are obtained from Proposition 2 above. This gives Kr conditions to satisfy for r variables \(\theta _s \in [0,1]\) (for \(s=1,\ldots , r\)), so we can approach this as a constrained optimization problem as considered in the binary peer-influence case before.
B Tables of Main Results
1.1 B.1 Analytical Results
1.2 B.2 Randomized Treatment Strategies to Disentangle Homophily
C Proofs
1.1 C.1 Proof of Theorem 1 (See p. xxx)
Theorem 1
Consider the difference in means estimator \(\hat{\beta }_0\) for binary peer-influence effect \(\beta _0\). Under the presence of unnormalized homophily in our model (3), the mean squared error of \(\hat{\beta }_0\) (conditional on the treatment \(\mathbf Z \)) is:
$$\begin{aligned} \begin{aligned} \mathbb {E}&[ (\hat{\beta }_0 - \beta _0)^2 | \mathbf Z ] = \Bigg ( h_0 \Bigg ( \underset{i \in M^{(1)}_0}{avg} |\mathcal {N}_i| - \underset{i \in M^{(0)}_0}{avg} |\mathcal {N}_i| \Bigg ) \Bigg )^2 \\&\quad + h_0^2 \sigma _X^2 \Bigg ( \underset{i,j \in M^{(0)}_0}{avg} | \mathcal {N}_i \cap \mathcal {N}_j| + \underset{i,j \in M^{(1)}_0}{avg} | \mathcal {N}_i \cap \mathcal {N}_j| - \ 2 \underset{i \in M^{(0)}_0, j \in M^{(1)}_0}{avg} | \mathcal {N}_i \cap \mathcal {N}_j| \Bigg ) + \sigma _Y^2 \Bigg ( \frac{1}{|M^{(0)}_0|} + \frac{1}{|M^{(1)}_0|} \Bigg ) \end{aligned} \end{aligned}$$
(5)
Proof
Recall the definition of the difference in means estimator for binary peer-influence (4).
$$\begin{aligned} \hat{\beta _0} = \underset{i \in M^{(1)}_0}{avg} Y_i - \underset{i \in M^{(0)}_0}{avg} Y_i \end{aligned}$$
where \(M^{(0)}_0 := \{ i: Z_i=0, \sum _{j \in \mathcal {N}_i}Z_j = 0 \}\) (the set of untreated individuals with no treated neighbors) and \(M^{(1)}_0:= \{ i: Z_i=0, \sum _{j \in \mathcal {N}_i}Z_j > 0 \}\) (the set of untreated individuals with at least one treated neighbors). The response variables \((Y_i)_{i=1,\ldots , N}\) are defined by:
$$ Y_i(Z_i=0,(Z_j)_{j \in {\mathcal {N}_i}}) = \alpha + \beta _0 \mathbf 1 _{\sum _{j \in {\mathcal {N}_i}} Z_j > 0 } + h_0 \sum _{j \in {\mathcal {N}_i}} X_j + \epsilon _i(0, \sigma ^2_Y)$$
$$ Y_i(Z_i=1,(Z_j)_{j \in {\mathcal {N}_i}},(X_j)_{j \in {\mathcal {N}_i}}) = \tau + Y_i(Z_i=0,(Z_j)_{j \in {\mathcal {N}_i}},(X_j)_{j \in {\mathcal {N}_i}}) + \beta _1 \mathbf 1 _{\sum _{j \in {\mathcal {N}_i}} Z_j > 0 } + h_1 \sum _{j \in {\mathcal {N}_i}} X_j $$
\(\epsilon _i(0, \sigma ^2_Y)\) for \(i=1,\ldots , N\) are the noise terms in the network, indepedent and identically distributed with zero mean and variance \(\sigma ^2_Y\). Note that the sets \(M^{(0)}_0\) and \(M^{(1)}_0\) are \(\mathbf Z \) measurable and that latent homophily variables \(\mathbf X = (X_j)_{j=1,\ldots , N}\) are independent of \(\mathbf Z = (Z_j)_{j=1,\ldots , N}\). Therefore,
$$\begin{aligned} \mathbb {E}[ \hat{\beta _0} | \mathbf Z ]&= \frac{\sum _{i \in M^{(1)}_0} \mathbb {E} \Big [ Y_i | \mathbf Z \Big ] }{|M^{(1)}_0|} - \frac{\sum _{i \in M^{(0)}_0} \mathbb {E} \Big [ Y_i | \mathbf Z \Big ] }{|M^{(0)}_0|} \\&= \frac{\sum _{i \in M^{(1)}_0} \mathbb {E} \Big [ \beta _0 + \sum _{j \in \mathcal {N}_i}X_j \Big ] }{|M^{(1)}_0|} - \frac{\sum _{i \in M^{(0)}_0} \mathbb {E} \Big [ \sum _{j \in \mathcal {N}_i}X_j \Big ] }{|M^{(0)}_0|} \\&= \beta _0 + \frac{\sum _{i \in M^{(1)}_0} h_0 |\mathcal {N}_i| }{|M^{(1)}_0|} - \frac{\sum _{i \in M^{(0)}_0} h_0 |\mathcal {N}_i| }{|M^{(0)}_0|} \\&= \beta _0 + h_0 \Bigg ( \underset{i \in M^{(1)}_0}{avg} |\mathcal {N}_i| - \underset{i \in M^{(1)}_0}{avg} |\mathcal {N}_i| \Bigg ). \end{aligned}$$
This gives the bias of \(\hat{\beta _0}\): \(\mathbb {E} \Big [ \hat{\beta _0} - \beta _0 | \mathbf Z \Big ] = h_0 \Bigg ( \underset{i \in M^{(1)}_0}{avg} |\mathcal {N}_i| - \underset{i \in M^{(1)}_0}{avg} |\mathcal {N}_i| \Bigg )\). Similarly,
$$\begin{aligned} var[ \hat{\beta _0} | \mathbf Z ]&= var \Bigg ( \frac{\sum _{i \in M^{(1)}_0} Y_i }{|M^{(1)}_0|} - \frac{\sum _{j \in M^{(0)}_0} Y_j }{|M^{(0)}_0|} \ \Bigg | \ \mathbf Z \Bigg ) \\&= \frac{ var(\sum _{i \in M^{(1)}_0} Y_i \ \big | \ \mathbf Z )}{|M^{(1)}_0|^2} + \frac{ var(\sum _{j \in M^{(0)}_0} Y_j \ \big | \ \mathbf Z )}{|M^{(0)}_0|^2} - \frac{ 2 { cov}( \sum _{i \in M^{(1)}_0} Y_i , \sum _{j \in M^{(0)}_0} Y_j \ \big | \ \mathbf Z )}{|M^{(0)}_0| |M^{(1)}_0|} \\&= \frac{ \sum _{i \in M^{(1)}_0} \sum _{k \in M^{(1)}_0}{} { cov}( Y_i , Y_k \ \big | \ \mathbf Z )}{|M^{(1)}_0|^2} + \frac{ \sum _{j \in M^{(0)}_0} \sum _{l \in M^{(0)}_0} { cov}( Y_j, Y_l \ \big | \ \mathbf Z )}{|M^{(0)}_0|^2} \\&\qquad - \frac{ 2 \sum _{i \in M^{(1)}_0} \sum _{j \in M^{(0)}_0} { cov}( Y_i , Y_j \ \big | \ \mathbf Z )}{|M^{(0)}_0| |M^{(1)}_0|} \\&= \underset{i, k \in M^{(1)}_0}{avg} { cov}( Y_i , Y_k \ \big | \ \mathbf Z ) + \underset{j,l \in M^{(0)}_0}{avg} { cov}( Y_i , Y_k \ \big | \ \mathbf Z ) - 2 \underset{i \in M^{(0)}_0, j \in M^{(1)}_0}{avg} { cov}( Y_i , Y_j \ \big | \ \mathbf Z ). \end{aligned}$$
For \(i \in M^{(1)}_0\) and \(k \in M^{(1)}_0\), by the law of total covariance and as \(\mathbf X \) are i.i.d.,
$$\begin{aligned} { cov}( Y_i , Y_k \ \big | \ \mathbf Z )&= \mathbb {E}[ { cov}( Y_i , Y_k \ \big | \ \mathbf X , \mathbf Z ) \ \Big | \ \mathbf Z ] + cov \Big ( \mathbb {E}[ Y_i \big | \mathbf X , \mathbf Z ], \mathbb {E}[ Y_k \big | \mathbf X , \mathbf Z ] \ \Big | \ \mathbf Z \Big ) \\&= \sigma _Y^2 \mathbbm {1}_{ \{i=k \} } \ + \ cov \Big ( \alpha + \beta _0 + h_0 \sum _{a \in \mathcal {N}_i} X_a, \alpha + \beta _0 + h_0 \sum _{b \in \mathcal {N}_k} X_b \Big ) \\&= \sigma _Y^2 \mathbbm {1}_{ \{i=k \} } \ + h_0^2 \ cov \Big ( \sum _{a \in \mathcal {N}_i} X_a, \sum _{b \in \mathcal {N}_k} X_b \Big ) \\&= \sigma _Y^2 \mathbbm {1}_{ \{i=k \} } \ + h_0^2 \sigma _X^2 |\mathcal {N}_i \cap \mathcal {N}_j | \end{aligned}$$
Similarly for \(j \in M^{(0)}_0\) and \(l \in M^{(0)}_0\),
$$\begin{aligned} { cov}( Y_j , Y_l \ \big | \ \mathbf Z )&= \sigma _Y^2 \mathbbm {1}_{ \{j=l \} } \ + \ cov \Big ( \alpha + h_0 \sum _{a \in \mathcal {N}_j} X_a, \alpha + h_0 \sum _{b \in \mathcal {N}_l} X_b \ \Big | \ \mathbf Z \Big ) \\&= \sigma _Y^2 \mathbbm {1}_{ \{j=l \} } \ + h_0^2 \sigma _X^2 |\mathcal {N}_j \cap \mathcal {N}_l | \end{aligned}$$
For \(i \in M^{(1)}_0\) and \(j \in M^{(0)}_0\), by the law of total covariance and as \(\mathbf X \) are i.i.d.,
$$\begin{aligned} { cov}( Y_i , Y_j \ \big | \ \mathbf Z )&= \mathbb {E}[ { cov}( Y_i , Y_j \ \big | \ \mathbf X , \mathbf Z ) \ \Big | \ \mathbf Z ] + cov \Big ( \mathbb {E}[ Y_i \big | \mathbf X , \mathbf Z ], \mathbb {E}[ Y_k \big | \mathbf X , \mathbf Z ] \ \Big | \ \mathbf Z \Big ) \\&= 0 + \ cov \Big ( \alpha + \beta _0 + h_0 \sum _{a \in \mathcal {N}_i} X_a, \alpha + h_0 \sum _{b \in \mathcal {N}_k} X_b \Big ) \\&= h_0^2 \ cov \Big ( \sum _{a \in \mathcal {N}_i} X_a, \sum _{b \in \mathcal {N}_k} X_b \Big ) \\&= h_0^2 \sigma _X^2 |\mathcal {N}_i \cap \mathcal {N}_j |. \end{aligned}$$
Therefore,
$$\begin{aligned} var[ \hat{\beta _0} | \mathbf Z ]&= \underset{i, k \in M^{(1)}_0}{avg} { cov}( Y_i , Y_k \ \big | \ \mathbf Z ) + \underset{j,l \in M^{(0)}_0}{avg} { cov}( Y_i , Y_k \ \big | \ \mathbf Z ) - 2 \underset{i \in M^{(0)}_0, j \in M^{(1)}_0}{avg} { cov}( Y_i , Y_j \ \big | \ \mathbf Z ) \\&= \underset{i, k \in M^{(1)}_0}{avg} \bigg ( \sigma _Y^2 \mathbbm {1}_{ \{i=k \} } \ + h_0^2 \sigma _X^2 |\mathcal {N}_i \cap \mathcal {N}_k | \bigg ) + \underset{j,l \in M^{(0)}_0}{avg} \bigg ( \sigma _Y^2 \mathbbm {1}_{ \{j=l \} } \ + h_0^2 \sigma _X^2 |\mathcal {N}_j \cap \mathcal {N}_l | \bigg ) \\&\qquad - 2 \underset{i \in M^{(0)}_0, j \in M^{(1)}_0}{avg} \bigg (h_0^2 \sigma _X^2 |\mathcal {N}_j \cap \mathcal {N}_l | \bigg ) \\&= h_0^2 \sigma _X^2 \Bigg ( \underset{i,j \in M^{(0)}_0}{avg} | \mathcal {N}_i \cap \mathcal {N}_j| + \underset{i,j \in M^{(1)}_0}{avg} | \mathcal {N}_i \cap \mathcal {N}_j| - \ 2 \underset{i \in M^{(0)}_0, j \in M^{(1)}_0}{avg} | \mathcal {N}_i \cap \mathcal {N}_j| \Bigg ) \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad + \sigma _Y^2 \Bigg ( \frac{1}{|M^{(0)}_0|} + \frac{1}{|M^{(1)}_0|} \Bigg ) \end{aligned}$$
Now we can recall the bias–variance decomposition of the MSE to obtain
$$\begin{aligned} \mathbb {E}&[ (\hat{\beta }_0 - \beta _0)^2 | \mathbf Z ] = \Big ( \mathbb {E}[ \hat{\beta _0} - \beta _0 | \mathbf Z ] \Big )^2 + var[ \hat{\beta _0} | \mathbf Z ] \\&= \Bigg ( h_0 \Bigg ( \underset{i \in M^{(1)}_0}{avg} |\mathcal {N}_i| - \underset{i \in M^{(0)}_0}{avg} |\mathcal {N}_i| \Bigg ) \Bigg )^2 \\&\quad + h_0^2 \sigma _X^2 \Bigg ( \underset{i,j \in M^{(0)}_0}{avg} | \mathcal {N}_i \cap \mathcal {N}_j| + \underset{i,j \in M^{(1)}_0}{avg} | \mathcal {N}_i \cap \mathcal {N}_j| - \ 2 \underset{i \in M^{(0)}_0, j \in M^{(1)}_0}{avg} | \mathcal {N}_i \cap \mathcal {N}_j| \Bigg )\\&\quad + \sigma _Y^2 \Bigg ( \frac{1}{|M^{(0)}_0|} + \frac{1}{|M^{(1)}_0|} \Bigg ) \end{aligned}$$
as required. \(\square \)
1.2 C.2 Proof of Theorem 3 (See p. xxx)
Theorem 3
Consider the difference in means estimator \(\hat{\beta }_0\) for binary peer-influence effect \(\beta _0\). Under the presence of unnormalized homophily in our model (3), the mean squared error of \(\hat{\beta }_0\) (conditional on the treatment \(\mathbf Z \)) is:
$$\begin{aligned} \mathbb {E} [ (\hat{\beta }_0 - \beta _0)^2 | \mathbf Z ] =&\, \Bigg ( h_0 \Bigg ( \underset{i \in M^{(1)}_0}{avg} |\mathcal {N}_i| - \underset{i \in M^{(0)}_0}{avg} |\mathcal {N}_i| \Bigg ) \Bigg )^2 \nonumber \\&+ h_0^2 \sigma _X^2 \Bigg ( \underset{i,j \in M^{(0)}_0}{avg} | \mathcal {N}_i \cap \mathcal {N}_j| + \underset{i,j \in M^{(1)}_0}{avg} | \mathcal {N}_i \cap \mathcal {N}_j| - \ 2 \underset{i \in M^{(0)}_0, j \in M^{(1)}_0}{avg} | \mathcal {N}_i \cap \mathcal {N}_j| \Bigg )\nonumber \\&+ \sigma _Y^2 \Bigg ( \frac{1}{|M^{(0)}_0|} + \frac{1}{|M^{(1)}_0|} \Bigg ) \end{aligned}$$
(16)
Proof
We proceed as similar to the binary peer-influence estimator case. Recall the definition of the estimator for linear peer-influence (21):
$$\begin{aligned} \hat{\beta }_0 \,{=}\, \frac{\sum _k \hat{\beta }_0^{(k)}}{\sum _k 1} \ for \ \hat{\beta }_0^{(k)} \,{=}\, \frac{1}{k} \Bigg ( \frac{\sum _{i \in M^{(k)}_0} Y_i}{| M^{(k)}_0 | } - \frac{\sum _{i \in M^{(0)}_0} Y_i}{| M^{(0)}_0 | } \Bigg ) = \frac{1}{k} \Bigg ( \underset{i \in M^{(k)}_0}{avg} Y_i - \underset{i \in M^{(0)}_0}{avg} Y_i \Bigg ), \end{aligned}$$
(21)
where \(M^{(k)}_0 := \{ i: Z_i=0, \sum _{j \in \mathcal {N}_i}Z_j = k \}\) (the set of untreated individuals with k treated neighbors). The response variables \((Y_i)_{i=1,\ldots , N}\) are defined by:
$$ Y_i(Z_i\,{=}\,0,(Z_j)_{j \in {\mathcal {N}_i}}) \,{=}\, \alpha + \beta _0 \sum _{j \in {\mathcal {N}_i}} Z_j + h_0 \sum _{j \in {\mathcal {N}_i}} X_j \,{+}\, \epsilon _i(0, \sigma ^2_Y)$$
$$Y_i(Z_i=1,(Z_j)_{j \in {\mathcal {N}_i}},(X_j)_{j \in {\mathcal {N}_i}}) = \tau + Y_i(Z_i=0,(Z_j)_{j \in {\mathcal {N}_i}},(X_j)_{j \in {\mathcal {N}_i}}) + \beta _1 \sum _{j \in {\mathcal {N}_i}} Z_j + h_1 \sum _{j \in {\mathcal {N}_i}} X_j $$
\(\epsilon _i(0, \sigma ^2_Y)\) for \(i=1,\ldots , N\) are the noise terms in the network, indepedent and identically distributed with zero mean and variance \(\sigma ^2_Y\). Note that sets \(M^{(k)}_0\) are \(\mathbf Z \) measurable and that latent homophily variables \(\mathbf X = (X_j)_{j=1,\ldots , N}\) are independent of \(\mathbf Z = (Z_j)_{j=1,\ldots , N}\). Therefore,
$$\begin{aligned} \mathbb {E}[ \hat{\beta }^{(k)}_0 | \mathbf Z ]&= \frac{1}{k} \Bigg ( \frac{\sum _{i \in M^{(k)}_0} \mathbb {E} \Big [ Y_i | \mathbf Z \Big ] }{|M^{(k)}_0|} - \frac{\sum _{i \in M^{(0)}_0} \mathbb {E} \Big [ Y_i | \mathbf Z \Big ] }{|M^{(0)}_0|} \Bigg ) \\&= \frac{1}{k} \Bigg ( \frac{\sum _{i \in M^{(k)}_0} \mathbb {E} \Big [ k \beta _0 + \sum _{j \in \mathcal {N}_i}X_j | \mathbf Z \Big ] }{|M^{(k)}_0|} - \frac{\sum _{i \in M^{(0)}_0} \mathbb {E} \Big [ \sum _{j \in \mathcal {N}_i}X_j | \mathbf Z \Big ] }{|M^{(0)}_0|} \Bigg ) \\&= \beta _0 + \frac{1}{k} \Bigg ( \frac{\sum _{i \in M^{(k)}_0} \mathbb {E} \Big [ \sum _{j \in \mathcal {N}_i}X_j | \mathbf Z \Big ] }{|M^{(k)}_0|} - \frac{\sum _{i \in M^{(0)}_0} \mathbb {E} \Big [ \sum _{j \in \mathcal {N}_i}X_j | \mathbf Z \Big ] }{|M^{(0)}_0|} \Bigg ) \\&= \beta _0 + \frac{1}{k} \Bigg ( \frac{\sum _{i \in M^{(k)}_0} h_0 |\mathcal {N}_i| }{|M^{(k)}_0|} - \frac{\sum _{i \in M^{(0)}_0} h_0 |\mathcal {N}_i| }{|M^{(0)}_0|} \Bigg ) \\&= \beta _0 + \frac{h_0}{k} \Bigg ( \underset{i \in M^{(k)}_0}{avg} |\mathcal {N}_i| - \underset{i \in M^{(1)}_0}{avg} |\mathcal {N}_i| \Bigg ), \end{aligned}$$
which gives the bias of the estimator \(\hat{\beta }_0 = \frac{\sum _k \hat{\beta }_0^{(k)}}{\sum _k 1}\) to be:
$$ \mathbb {E}[ \hat{\beta }_0 - \beta _0 | \mathbf Z ] = \frac{h_0}{\sum _{k>0} 1} \sum _{k>0} \frac{1}{k} \Bigg ( \underset{i \in M^{(k)}_0}{avg} |\mathcal {N}_i| - \underset{i \in M^{(0)}_0}{avg} |\mathcal {N}_i| \Bigg ).$$
Similarly, \( var[ \hat{\beta _0} | \mathbf Z ] = \frac{1}{(\sum _k 1)^2} \sum _{k> 0}\sum _{l > 0} { cov}( \hat{\beta }^{(k)}_0, \hat{\beta }^{(l)}_0 )\), where
$$\begin{aligned} { cov}( \hat{\beta }^{(k)}_0, \hat{\beta }^{(l)}_0 \Big | \mathbf Z )&= \frac{1}{kl} cov \Bigg ( \frac{\sum _{i \in M^{(k)}_0} Y_i }{|M^{(k)}_0|} - \frac{\sum _{j \in M^{(0)}_0} Y_j }{|M^{(0)}_0|} , \frac{\sum _{i \in M^{(l)}_0} Y_i }{|M^{(l)}_0|} - \frac{\sum _{j \in M^{(0)}_0} Y_j }{|M^{(0)}_0|} \ \Bigg | \ \mathbf Z \Bigg ) \\&= \frac{1}{kl} \Bigg ( \frac{\sum _{i \in M^{(k)}_0, j \in M^{(l)}_0}{} { cov}( Y_i, Y_j | \mathbf Z ) }{|M^{(k)}_0||M^{(l)}_0|} + \frac{\sum _{i \in M^{(0)}_0, j \in M^{(0)}_0}{} { cov}( Y_i, Y_j | \mathbf Z ) }{|M^{(0)}_0|^2} \\&\qquad \qquad - \frac{\sum _{i \in M^{(k)}_0, j \in M^{(0)}_0}{} { cov}( Y_i, Y_j | \mathbf Z )}{|M^{(k)}_0||M^{(0)}_0|} - \frac{\sum _{i \in M^{(0)}_0, j \in M^{(l)}_0}{} { cov}( Y_i, Y_j | \mathbf Z ) }{|M^{(0)}_0||M^{(l)}_0|} \Bigg ). \end{aligned}$$
For \(i \in M^{(k)}_0\) and \(j \in M^{(l)}_0\), by the law of total covariance and as \(\mathbf X \) are i.i.d.,
$$\begin{aligned} { cov}( Y_i , Y_j \ \big | \ \mathbf Z )&= \mathbb {E}[ { cov}( Y_i , Y_j \ \big | \ \mathbf X , \mathbf Z ) \ \Big | \ \mathbf Z ] + cov \Big ( \mathbb {E}[ Y_i \big | \mathbf X , \mathbf Z ], \mathbb {E}[ Y_j \big | \mathbf X , \mathbf Z ] \ \Big | \ \mathbf Z \Big ) \\&= \sigma _Y^2 \mathbbm {1}_{ \{i=j \} } \ + \ cov \Big ( \alpha + k \beta _0 + h_0 \sum _{a \in \mathcal {N}_i} X_a, \alpha + l\beta _0 + h_0 \sum _{b \in \mathcal {N}_j} X_b \Big ) \\&= \sigma _Y^2 \mathbbm {1}_{ \{i=j \} } \ + h_0^2 \ cov \Big ( \sum _{a \in \mathcal {N}_i} X_a, \sum _{b \in \mathcal {N}_j} X_b \Big ) \\&= \sigma _Y^2 \mathbbm {1}_{ \{i=j \} } \ + h_0^2 \sigma _X^2 |\mathcal {N}_i \cap \mathcal {N}_j |. \end{aligned}$$
This gives
$$\begin{aligned} { cov}( \hat{\beta }^{(k)}_0, \hat{\beta }^{(l)}_0 \Big | \mathbf Z )&= \frac{1}{kl} \Bigg ( \frac{\sum _{i \in M^{(k)}_0, j \in M^{(l)}_0}{} { cov}( Y_i, Y_j | \mathbf Z ) }{|M^{(k)}_0||M^{(l)}_0|} + \frac{\sum _{i \in M^{(0)}_0, j \in M^{(0)}_0}{} { cov}( Y_i, Y_j | \mathbf Z ) }{|M^{(0)}_0|^2} \\&\qquad \qquad - \frac{\sum _{i \in M^{(k)}_0, j \in M^{(0)}_0}{} { cov}( Y_i, Y_j | \mathbf Z )}{|M^{(k)}_0||M^{(0)}_0|} - \frac{\sum _{i \in M^{(0)}_0, j \in M^{(l)}_0}{} { cov}( Y_i, Y_j | \mathbf Z ) }{|M^{(0)}_0||M^{(l)}_0|} \Bigg ) \\&= \frac{1}{kl} \Bigg ( \frac{\sum _{i \in M^{(k)}_0, j \in M^{(l)}_0}\sigma _Y^2 \mathbbm {1}_{ \{i=j \} } \ + h_0^2 \sigma _X^2 |\mathcal {N}_i \cap \mathcal {N}_j | }{|M^{(k)}_0||M^{(l)}_0|} \\&\qquad \qquad + \frac{\sum _{i \in M^{(0)}_0, j \in M^{(0)}_0}\sigma _Y^2 \mathbbm {1}_{ \{i=j \} } \ + h_0^2 \sigma _X^2 |\mathcal {N}_i \cap \mathcal {N}_j | }{|M^{(0)}_0|^2} \\&\qquad \qquad - \frac{\sum _{i \in M^{(k)}_0, j \in M^{(0)}_0} h_0^2 \sigma _X^2 |\mathcal {N}_i \cap \mathcal {N}_j |}{|M^{(k)}_0||M^{(0)}_0|} - \frac{\sum _{i \in M^{(0)}_0, j \in M^{(l)}_0} h_0^2 \sigma _X^2 |\mathcal {N}_i \cap \mathcal {N}_j | }{|M^{(0)}_0||M^{(l)}_0|} \Bigg ) \\&= \frac{1}{kl} \Bigg ( \sigma _Y^2 \frac{ \mathbbm {1}_{ \{l = k \} }}{|M^{(k)}|} + h_0^2 \sigma _X^2 \frac{\sum _{i \in M^{(k)}_0, j \in M^{(l)}_0} |\mathcal {N}_i \cap \mathcal {N}_j | }{|M^{(k)}_0||M^{(l)}_0|} \\&\qquad \qquad + \sigma _Y^2 \frac{ 1 }{|M^{(0)}|} + h_0^2 \sigma _X^2 \frac{\sum _{i \in M^{(0)}_0, j \in M^{(0)}_0} |\mathcal {N}_i \cap \mathcal {N}_j | }{|M^{(0)}_0||M^{(0)}_0|} \\&\qquad \qquad - h_0^2 \sigma _X^2 \frac{\sum _{i \in M^{(k)}_0, j \in M^{(0)}_0} |\mathcal {N}_i \cap \mathcal {N}_j | }{|M^{(k)}_0||M^{(0)}_0|} - h_0^2 \sigma _X^2 \frac{\sum _{i \in M^{(0)}_0, j \in M^{(l)}_0} |\mathcal {N}_i \cap \mathcal {N}_j | }{|M^{(0)}_0||M^{(l)}_0|} \Bigg ) \\&= \frac{1}{kl} \Bigg [ h_0^2 \sigma _X^2 \Bigg ( \underset{i \in M^{(k)}_0 ,j \in M^{(l)}_0}{avg} | \mathcal {N}_i \cap \mathcal {N}_j| + \underset{i,j \in M^{(0)}_0}{avg} | \mathcal {N}_i \cap \mathcal {N}_j| - \underset{i \in M^{(k)}_0, j \in M^{(0)}_0}{avg} | \mathcal {N}_i \cap \mathcal {N}_j| \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad - \underset{i \in M^{(0)}_0, j \in M^{(l)}_0}{avg} | \mathcal {N}_i \cap \mathcal {N}_j| \Bigg ) + \sigma _Y^2 \Bigg ( \frac{1}{|M^{(0)}_0|} + \frac{ \mathbf 1 _{k=l} }{|M^{(k)}_0|} \Bigg ) \Bigg ]. \end{aligned}$$
Now, we can recall the bias–variance decomposition of the MSE to obtain
$$\begin{aligned} \mathbb {E} [ (\hat{\beta }_0 - \beta _0)^2 | \mathbf Z ]&= \Big ( \mathbb {E}[ \hat{\beta _0} - \beta _0 | \mathbf Z ] \Big )^2 + var[ \hat{\beta _0} | \mathbf Z ] \\&= \Big ( \mathbb {E}[ \hat{\beta _0} - \beta _0 | \mathbf Z ] \Big )^2 + \frac{1}{(\sum _k 1)^2} \sum _{k> 0}\sum _{l> 0} { cov}( \hat{\beta }^{(k)}_0, \hat{\beta }^{(l)}_0 ) \\&= \Bigg ( \frac{h_0}{\sum _{k>0} 1} \sum _{k>0} \frac{1}{k} \Bigg ( \underset{i \in M^{(k)}_0}{avg} |\mathcal {N}_i| - \underset{i \in M^{(0)}_0}{avg} |\mathcal {N}_i| \Bigg ) \Bigg )^2 \\&\qquad + \frac{1}{(\sum _{k>0} 1)^2} \sum _{k,l>0} \frac{1}{kl} \Bigg [ h_0^2 \sigma _X^2 \Bigg ( \underset{i \in M^{(k)}_0 ,j \in M^{(l)}_0}{avg} | \mathcal {N}_i \cap \mathcal {N}_j| + \underset{i,j \in M^{(0)}_0}{avg} | \mathcal {N}_i \cap \mathcal {N}_j| \\&\qquad - \underset{i \in M^{(0)}_0, j \in M^{(k)}_0}{avg} | \mathcal {N}_i \cap \mathcal {N}_j| \Bigg ) - \underset{i \in M^{(l)}_0, j \in M^{(0)}_0}{avg} | \mathcal {N}_i \cap \mathcal {N}_j| + \sigma _Y^2 \Bigg ( \frac{1}{|M^{(0)}_0|} + \frac{ \mathbf 1 _{k=l} }{|M^{(k)}_0|} \Bigg ) \Bigg ] \\&= \Bigg ( \frac{h_0}{\sum _{k>0} 1} \sum _{k>0} \frac{1}{k} \Bigg ( \underset{i \in M^{(k)}_0}{avg} |\mathcal {N}_i| - \underset{i \in M^{(0)}_0}{avg} |\mathcal {N}_i| \Bigg ) \Bigg )^2 \\&\qquad + \frac{1}{(\sum _{k>0} 1)^2} \sum _{k,l>0} \frac{1}{kl} \Bigg [ h_0^2 \sigma _X^2 \Bigg ( \underset{i \in M^{(k)}_0 ,j \in M^{(l)}_0}{avg} | \mathcal {N}_i \cap \mathcal {N}_j| + \underset{i,j \in M^{(0)}_0}{avg} | \mathcal {N}_i \cap \mathcal {N}_j| \\&\qquad \qquad \qquad \qquad \qquad \qquad - 2 \underset{i \in M^{(0)}_0, j \in M^{(k)}_0}{avg} | \mathcal {N}_i \cap \mathcal {N}_j| \Bigg ) + \sigma _Y^2 \Bigg ( \frac{1}{|M^{(0)}_0|} + \frac{ \mathbf 1 _{k=l} }{|M^{(k)}_0|} \Bigg ) \Bigg ] \end{aligned}$$
as required. \(\square \)
1.3 C.3 Proof of Theorem 1 (See p. xxx)
Proposition 1
Consider a stochastic block model (SBM) of N individuals in r communities. Denote the communities of the SBM by the sets \(B_1,\ldots ,B_r\), which are of respective sizes \(A_1,\ldots ,A_r\) (where \(A_1+\cdots +A_r = N\)). Let \(\mathbf P \) be the \(r \times r\) adjacency probability matrix between the r communities. We assign treatments independently to individuals such that individuals in \(B_s\) are treated with probability \(\theta _s\) for \(s=1,\ldots ,r\). Under such setup, let \( M_0^{(0)} \) denote the set of untreated individuals which have no treated neighbors and let \( M_0^{(1)} \) denote the set of untreated individuals which have at least one treated neighbor. For ease of notation, let \(\{ s \in M_0^{(0)} \}\), \(\{ s \in M_0^{(1)} \}\) denote the event that a fixed vertex in community s is in the sets \(M_0^{(0)}\), \(M_0^{(1)}\) respectively. Then,
$$\begin{aligned} \mathbb {P}( s \in M_0^{(0)} ) = (1-\theta _s) \prod _{v=1}^{r}(1-P_{s,v}\theta _v)^{A_v - \mathbf 1 _{v=s}} \text {, and} \end{aligned}$$
(7)
$$\begin{aligned} \mathbb {P}( s \in M_0^{(1)} ) = (1-\theta _s) \bigg ( 1- \prod _{v=1}^{r}(1-P_{s,v}\theta _v)^{A_v - \mathbf 1 _{v=s}} \bigg ). \end{aligned}$$
(8)
Proof
Note that each vertex in the graph is assigned treatment independently and that under the stochastic block model the events of any pair of vertices being adjacent are independent. Therefore,
$$\mathbb {P}(s \text { has 0 treated neighbors} \ | \ s \text { is untreated} ) =\mathbb {P}(s \text { has 0 treated neighbors} ) $$
for all k and \(s=1,\ldots ,r\). This gives
$$\begin{aligned} \mathbb {P}( s \in M_0^{(0)} )&= \mathbb {P}(s \text { is untreated} )\mathbb {P}(s \text { has 0 treated neighbors} ) \\&= (1 - \theta _s ) \mathbb {P}(s \text { has 0 treated neighbors}) \\&= (1 - \theta _s) \mathbb {P} \Big ( \bigcap _{v=1}^r \{ s \text { has 0 treated neighbors in}\; B_v \} \Big ) \\&= (1 - \theta _s) \prod _{v=1}^r \mathbb {P}( s \text { has 0 treated neighbors in}\; B_v)\\&= (1 - \theta _s) \prod _{v=1}^r ( 1 - P_{s,v} \theta _v )^{ A_v - \mathbf 1 _{v=s} }. \end{aligned}$$
where the \(A_v-\mathbf 1 _{v=s}\) arises from noting that s can have at most \(A_s - 1\) neighbors in \(B_s\) (it cannot connect to itself). Note that sets \(M_0^{(0)}\) and \(M_0^{(1)}\) partition the set of untreated individuals. Therefore,
$$\begin{aligned} \mathbb {P}( s \in M_0^{(1)} )&= \mathbb {P}( Z_s = 0 ) - \mathbb {P}( s \in M_0^{(0)} ) \\&= (1-\theta _s) - (1 - \theta _s) \prod _{v=1}^r ( 1 - P_{s,v} \theta _v )^{ A_v - \mathbf 1 _{v=s} } \\&= (1-\theta _s) \Big (1 - \prod _{v=1}^r ( 1 - P_{s,v} \theta _v )^{ A_v - \mathbf 1 _{v=s} } \Big ). \end{aligned}$$
\(\square \)
1.4 C.4 Proof of Proposition 2 (See p. xxx)
Proposition 2
Consider a stochastic block model (SBM) of N individuals in r communities. Denote the communities of the SBM by the sets \(B_1,\ldots , B_r\), which are of respective sizes \(A_1,\ldots , A_r\) (where \(A_1+\cdots +A_r = N\)). Let \(\mathbf P \) be the \(r \times r\) adjacency probability matrix between the r communities. We assign treatments independently to individuals such that individuals in \(B_s\) are treated with probability \(\theta _s\) for \(s=1,\ldots ,r\). Under such setup, let \( M_0^{(k)} \) denote the set of untreated individuals which have k treated neighbors. For ease of notation, let \(\{ s \in M_0^{(k)} \}\) denote the event that a fixed vertex in community s is in the set \(M_0^{(k)}\). Then,
$$\begin{aligned} \mathbb {P}( s \in M_0^{(k)} ) = (1 - \theta _s) \sum _{ \begin{array}{c} t_1,\ldots ,t_r: \\ \forall v =1,\ldots ,r \ 0 \le t_v \le A_v -\mathbf 1 _{ \{v=s \} }, \\ t_1+\cdots +t_r = k \end{array}} \Bigg ( \prod _{v=1}^r \mathbf{Bin }( t_v; A_v -\mathbf 1 _{ \{v=s \} }, \theta _v P_{s,v} ) \Bigg ) \end{aligned}$$
(20)
where \(\mathbf{Bin }( t_v; A_v -\mathbf 1 _{ \{v=s \} }, \theta _v P_{s,v} ) = \left( {\begin{array}{c}A_v-\mathbf 1 _{ \{v=s \} }\\ t_v\end{array}}\right) \big ( \theta _v P_{s,v} \big )^{ t_v } \big ( 1 - \theta _v P_{s,v} \big )^{ A_v -\mathbf 1 _{ \{v=s \} } - t_v }\).
Proof
Note that each vertex in the graph is assigned treatment indpendently. Therefore,
$$\mathbb {P}(s \text { has k treated neighbors} \ | \ s \text { is untreated} ) =\mathbb {P}(s \text { has k treated neighbors} ) $$
for all k and \(s=1,\ldots ,r\). This gives
$$\begin{aligned} \mathbb {P}( s \in M_0^{(k)} )&= \mathbb {P}(s \text { is untreated} )\mathbb {P}(s \text { has k treated neighbors} ) \\&= (1 - \theta _s ) \mathbb {P}(s \text { has k treated neighbors}) \\&= (1 - \theta _s) \sum _{ \begin{array}{c} t_1,\ldots ,t_r: \\ \forall v =1,\ldots ,r \ 0 \le t_v \le A_v -\mathbf 1 _{ \{v=s \} }, \\ t_1+\cdots +t_r = k \end{array}} \mathbb {P} \Big ( \bigcap _{v=1}^r \{ s \text { has}\; t_v\;{\text {treated neighbors in}}\; B_v \} \Big ) \\&= (1 - \theta _s) \sum _{ \begin{array}{c} t_1,\ldots ,t_r: \\ \forall v =1,\ldots ,r \ 0 \le t_v \le A_v -\mathbf 1 _{ \{v=s \} }, \\ t_1+\cdots +t_r = k \end{array}} \Big ( \prod _{v=1}^r \mathbb {P}( s \text { has}\; t_v\;\text {treated neighbors in}\; B_v) \Big ). \end{aligned}$$
We now wish to evaluate \(\mathbb {P}( s \;\text {has}\; t_k\;\text {treated neighbors in}\; B_v)\). Let \(n_v\) be the number of neighbors s (denoting a fixed individual in community \(B_s\)) has in \(B_v\). Under a stochastic block model setup,
$$ n_v \sim Bin(A_v-\mathbf 1 _{v=s}, P_{s,v}) $$
$$ t_v | n_v \ \sim Bin(n_v, \theta _v) $$
where the \(A_v-\mathbf 1 _{v=s}\) arises from noting that s can have at most \(A_s - 1\) neighbors in \(B_s\) (it cannot connect to itself). We want the unconditional distribution of \(t_v\). Recall that moment generating function of \(X \sim Bin(N,p)\) is \(\mathbb {E}( z^X ) = ( (1-p) + pz )^N\). Therefore,
$$\begin{aligned} \mathbb {E} [ z^{t_v} ] = \mathbb {E}[ \mathbb {E}[ z^{t_v} | n_v]] = \mathbb {E} \Big [ \Big ( (1-\theta _s) + \theta _s z \Big )^{n_v} \Big ]&= \bigg ( (1-P_{s,v}) + P_{s,v}\Big ( (1-\theta _s) + \theta _s z \Big ) \bigg )^{A_v-\mathbf 1 _{v=s}} \\&= \bigg ( (1- \theta _s P_{s,v}) + P_{s,v}\theta _s z \bigg )^{A_v}, \end{aligned}$$
giving \(t_v \sim Bin(A_v-\mathbf 1 _{v=s}, \theta _s P_{s,v})\). This gives
$$ \mathbb {P}( s \text { has}\; t_v \;\text {treated neighbors in}\; B_v) = Bin ( t_v; A_v -\mathbf 1 _{ \{v=s \} }, \theta _v P_{s,v} ) $$
from which (20) directly follows. \(\square \)
