**Authors:**

(1) Davide Viviano, Department of Economics, Harvard University;

(2) Lihua Lei, Graduate School of Business, Stanford University;

(3) Guido Imbens, Graduate School of Business and Department of Economics, Stanford University;

(4) Brian Karrer, FAIR, Meta;

(5) Okke Schrijvers, Meta Central Applied Science;

(6) Liang Shi, Meta Central Applied Science.

## Table of Links

Empirical illustration and numerical studies

## C Proofs

Throughout the proofs, expectations are conditional on the adjacency matrix A.

### C.1 Proof of Lemma 3.1

We have

### C.2 Proof of Lemma 3.2

### C.3 Proof of Lemma 3.3

We consider the case where two units are in the same or different clusters separately. We will refer to µi(Di , D−i) as µi(D) for notational convenience.

Following the same steps as for the case where i, j are in different clusters, accounting for Equation (27), the proof completes.

### C.4 Proof of Lemma 3.4

other units are not zero for individuals in the sets Bi , Gi defined in Lemma 3.2.

where the first inequality is due to Cauchy-Schwarz inequality and last equality follows from Assumption 5. The proof completes after collecting the terms.

### C.5 Proof of Theorem 3.5

### C.6 Proof of Theorem 3.6

### C.7 Proof of Theorem 4.1

The bias follows directly from Lemma 3.1. We now discuss the variance component. Under Lemmas 3.2, 3.3, and following Equations (28), (29), we can write

### C.8 Proof of Theorem 4.2

### C.9 Proof of Theorem 4.3

This paper is available on arxiv under CC 1.0 license.