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Who Matters to Whom? Identifying Peer Effects with Propagation Geometry
\noindentJEL classification: C31; C36; C57; D85.
\noindentKeywords: peer effects; social interactions; instrumental variables; weak identification; propagation geometry.
Peer effects are central to empirical work in education, health, crime, and technology adoption, where an individual's outcome responds to the outcomes or actions of peers connected through social, spatial, or organizational networks. The econometric challenge is that peer exposure is endogenous: it is mechanically correlated with unobservables (the “reflection problem”) and with shared shocks, so identification hinges on generating excluded variation that shifts peer exposure without directly shifting the outcome manski1993identification. A large empirical literature has therefore adopted the linear-in-means (LIM) model and its network generalizations, because LIM delivers transparent best-response foundations and a tractable class of instruments based on higher-order neighbor exposures such as $G^2X$ and $G^kX$ (see bramoulle2009identification,blume2011identification,goldsmith2013social). However, two facts are increasingly hard to ignore. First, the network topologies that arise in applications---high transitivity, community structure, and near-regular degree profiles---are precisely those in which adjacency-power instruments can become nearly collinear and weak. Second, LIM hard-codes a behavioral assumption: the relevant peer benchmark is the mean peer outcome, ruling out salience (“the best peer matters most”), downside comparison (“the weakest peer matters”), and rank-based norms (“keep up with the median/upper quartile”).
Recent work has begun to relax the mean benchmark by treating the peer aggregator (the “social norm”) as an estimand. boucher2024toward (BRUZ) develop a general peer-effects model in which the relevant peer benchmark is a CES aggregator with curvature parameter $\beta$, nesting LIM as $\beta=1$ and converging to max/min norms as $\beta\to\pm\infty$. In their framework, $\beta$ is economically meaningful---it governs who matters---and identifying it is part of the empirical task. They propose a one-step IV strategy based on scalar summaries of an exogenous predictor $\hat y=m(X)$ and derivatives of the aggregator with respect to $\beta$.
This paper provides a structural unification that nests (i) the LIM network model of bramoulle2009identification, (ii) CES peer-preference norms as in boucher2024toward, and (iii) “attention to salient peers” logic that underlies Brock--Durlauf-type social interactions brock2001discrete. Section (ref) formulates peer effects as a norm game: individuals choose actions, payoffs depend on own action and a scalar peer exposure index $N_i=\Phi_i(a_{-i};G,\theta)$, and equilibria are fixed points of best responses. Rather than impose a single functional form, we treat the peer exposure map $\Phi_i$ as a primitive and discipline it using economically interpretable axioms. Under mild regularity, the axioms yield the Kolmogorov--Nagumo class of quasi-arithmetic means, $\Phi_i(a)=\varphi^{-1}(\sum_j g_{ij}\varphi(a_j))$, providing a common lens on a large fraction of peer-exposure objects used in economics. Within this lens, LIM corresponds to the linear generator $\varphi(a)=a$, CES corresponds to the power generator $\varphi(a)=a^\beta$, and a smooth-max “attention” norm arises from log-sum-exp aggregation, which also has classic discrete-choice and entropy-regularization interpretations mcfadden1974conditional,luce1959individual,matejka2015rational,nesterov2005smooth. The appendix adds a complementary class of rank-based norms via loss minimization (weighted quantiles/medians), which are natural for robustness and for Manski-style nonparametric arguments koenker1978quantiles,manski1993identification. Section (ref) also provides equilibrium existence for this unified class (and uniqueness under transparent “weak interaction” conditions), so the peer-preference parameter $\theta$ is a genuine behavioral primitive rather than an ad hoc index.
A constructive identification toolkit follows naturally from the norm-game taxonomy. The central insight is that propagation is governed by a transport operator induced by the peer aggregator. In LIM the transport is constant and equals adjacency ($P=G$), so instrument design reduces to repeated linear propagation ($G^kX$). In CES and other smooth aggregators, the exposure map implies a state- and preference-dependent Jacobian transport operator evaluated at an exogenous predictor $\hat y$: \[ w_{ij}(\theta;\hat y)=\left.\frac{\partial \Phi_i(y;G,\theta)}{\partial y_j}\right|_{y=\hat y}, \qquad P_{ij}(\theta;\hat y)=\frac{w_{ij}(\theta;\hat y)}{\sum_{m\neq i}w_{im}(\theta;\hat y)}. \] This operator encodes “who matters to whom” under the maintained aggregator, and it is the appropriate object for multi-step propagation when the model is not linear-in-means. Section (ref) shows that once $P(\theta;\hat y)$ is taken seriously as the propagation primitive, instrument construction becomes a geometry problem rather than an algebra-of-$G$ problem: the analogue of $G^kX$ is $P^k(\theta;\hat y)X$, and the transport induces additional geometry objects---effective distances (shells/geodesics) and path nonredundancy (torsion)---that generate new excluded-variation directions. These directions are informative precisely in settings where one-step scalar moments (BRUZ-type summaries) or adjacency-power instruments become weak due to transitivity, regularity, and path redundancy.
This paper relates to the econometrics of social interactions and the identification of peer effects, beginning with the reflection problem manski1993identification and extending to network-based identification strategies such as bramoulle2009identification and subsequent work on network instruments and their limitations (see also blume2011identification,goldsmith2013social,boucherBramoulle2026binary). Relatedly, tchuente2019weak emphasizes that network-IV strategies may require many or highly correlated instruments, leading to weak identification or many-instruments bias, and proposes regularized 2SLS estimators as a remedy. It complements structural and reduced-form literatures that emphasize heterogeneity in “which peers matter,” including the CES peer-preference approach of boucher2024toward and recent results showing that allowing richer action structures can fundamentally change shock propagation in network games sadlerYeh2026. It also connects to the discrete-choice and information-theoretic foundations of attention/salience norms luce1959individual,mcfadden1974conditional,matejka2015rational and to quantile-based loss minimization koenker1978quantiles, as well as to recent work defining network structure behaviorally via equilibrium implications jacksonStorms2026. More broadly, it speaks to the econometric analysis of network data and strategic interaction models graham2020econometric,kline2020econometric,rootSadler2026, by providing a constructive route from a maintained aggregator to an instrument menu that matches the implied influence geometry.
Section (ref) develops the norm-game framework and the taxonomy of peer aggregators, and establishes equilibrium existence (and uniqueness under weak interactions). Section (ref) develops geometry-induced instrument families implied by the aggregator-induced transport. Section (ref) discusses identification and connects completeness-type conditions to the enlarged excluded information set generated by transport geometry. Section (ref) provides Monte Carlo evidence. Section (ref) applies the framework to NetHealth and compares CES, smooth-max, and quantile norms across steps, sleep, and GPA. Section (ref) concludes and discusses extensions.
This section builds a unifying structural framework that nests: (i) the linear-in-means network model of bramoulle2009identification, (ii) generalized peer norms with peer-preference parameters as in boucher2024toward, and (iii) the “attention to salient peers” logic that underlies Brock--Durlauf-type social interactions brock2001discrete. We isolate a small set of primitives, state economically interpretable aggregator classes, and provide equilibrium existence (and uniqueness under transparent “weak interaction” conditions). The appendix then adds an additional aggregator class (quantile/median norms) and additional technical proofs.
Individuals $i=1,\dots,n$ choose an action $a_i$ in a feasible set $A_i\subset \mathbb{R}$ (extensions to $\mathbb{R}^d$ are immediate). Let $a=(a_1,\dots,a_n)$ and $a_{-i}$ denote the vector without component $i$.
The network is summarized by a row-stochastic interaction matrix $G=(g_{ij})$ with $g_{ij}\ge 0$ and $\sum_{j\neq i} g_{ij}=1$ (when observations are grouped, e.g.\ schools $s$, all objects can be indexed by $s$; we suppress $s$ for readability.)
\paragraph{Peer aggregator (norm/exposure).} We model peer influence through a scalar peer exposure index
where $\Phi_i$ is an aggregator mapping peers' actions to a scalar, and $\theta$ indexes peer preference/salience (e.g.\ a CES parameter $\beta$ or an attention parameter $\kappa$).
\paragraph{Payoffs.} Preferences depend on own action and peer exposure:
where $x_i$ are observables (and unobservables can be added additively).
\paragraph{Nash equilibrium.} A pure-strategy Nash equilibrium is $a^\star\in \prod_i A_i$ such that, for each $i$,
Rather than fix a single functional form, we treat the peer exposure map as a primitive: for each $i$, let \[ N_i \;=\;\Phi_i(a_{-i};g_i), \qquad g_i=(g_{ij})_{j\neq i},\ \ g_{ij}\ge 0,\ \ \sum_{j\neq i} g_{ij}=1, \] where $a_j$ is a peer action/outcome and $g_{ij}$ are interaction weights. We then discipline $\Phi_i$ using economically transparent axioms that appear (explicitly or implicitly) across the peer-effects, aggregation, and discrete-choice literatures.
\paragraph{Axioms that deliver a “mean-like” class (Kolmogorov--Nagumo).} A large share of peer-exposure objects used in economics can be motivated as a notion of “weighted mean” satisfying four basic requirements:
Under mild regularity (continuity and strict monotonicity), (A1)--(A4) characterize the class of quasi-arithmetic means:
a classic result associated with Kolmogorov and Nagumo and presented in standard functional-equations treatments (e.g.\ aczel1966lectures).\footnote{Many modern summaries refer to this as the Kolmogorov--Nagumo theorem for (weighted) quasi-arithmetic means; see, e.g., singpurwalla2020mean for an accessible discussion.} Equation (ref) provides a unifying lens: different economic models correspond to different choices of the generator $\varphi$ (and sometimes to additional structure restricting $\varphi$).
\paragraph{From “mean-like” to economically interpretable families.} We now list four families that are both empirically useful and microfounded by simple strengthening of the axioms.
Why this taxonomy matters for identification and geometry. The point of Table (ref) is not only flexibility; it is structure. For quasi-arithmetic families (ref) (including LIM, CES, smooth-max), the model implies a marginal influence field (a Jacobian with respect to peers' actions), which is the primitive object behind our propagation/geometry-based instruments. Quantile norms (ref) are non-smooth but still admit subgradient-based influence notions (useful for robustness and for Manski-style partial identification), which we develop in the appendix.
The main text uses the CES (power-mean) norm as the workhorse because it is (i) widely interpretable for economists and (ii) directly connected to the peer-preference estimand in boucher2024toward. For each $i$,
As $\beta\to 1$ this is close to the mean; as $\beta\to+\infty$ it approaches the max; as $\beta\to-\infty$ it approaches the min. Thus $\beta$ is a peer-salience parameter: it governs how concentrated influence is among neighbors.
\paragraph{Domain.} For general $\beta$, (ref) is naturally defined on $a_j>0$. This is not restrictive for many outcomes: one may model $a_j=\exp(s_j)$ (log-actions), or shift/scale the outcome to be positive.
We now state an equilibrium existence theorem that applies to all aggregators in Table (ref), including the CES norm and the smooth-max norm, under standard continuity and concavity conditions.
\paragraph{Interpretation.} Existence is not special to a particular peer norm. It is a consequence of standard economic regularity: bounded feasible actions, continuous peer exposure, and concavity in own action.
Multiple equilibria are possible when peer effects are strong. A sufficient uniqueness condition is a simple “weak interaction” restriction: agents do not respond too strongly to changes in the peer exposure, and the exposure does not change too strongly with peers' actions.
Assume strict concavity in $a_i$ so best responses are single-valued and can be written as
Define the equilibrium operator $T:\prod_i A_i\to\prod_i A_i$ by
Then equilibria are fixed points $a=T(a)$.
\paragraph{Economic meaning.} $L_\Phi$ measures how sensitive peer exposure is to peers' actions (“how much the peer environment moves”). $L_b$ measures how strongly an agent reacts to that exposure (“how much the agent moves”). Uniqueness holds when their product (peer amplification) is below one.
Take $\Phi_i(a_{-i};G)=\sum_{j\neq i} g_{ij}a_j=(Ga)_i$. If best responses are linear in exposure,
then the equilibrium satisfies $a = \alpha + \rho G a$, which is the standard linear-in-means / spatial-lag structure. A sufficient uniqueness condition is $|\rho|<1$ (since $G$ is row-stochastic, $\|Ga-Ga'\|_\infty\le \|a-a'\|_\infty$). This recovers the familiar “peer effects not too strong” uniqueness condition used in the linear peer-effects literature.
Take $\Phi_i=\Phi_i^{\text{CES}}(\cdot;\beta)$ from (ref). Then (ref) implies that equilibrium depends on $\beta$ through the peer-exposure function. The parameter $\beta$ is economically meaningful: it governs how much weight is placed on high- versus low-action peers. Equilibrium existence follows from Theorem (ref) under continuity and concavity; uniqueness follows from Theorem (ref) under a weak-interaction condition (Appendix (ref) provides sufficient bounds).
To connect to Brock--Durlauf logic, define a smooth-max peer exposure
This captures the idea that individuals respond more to salient (high-action) peers, with $\kappa$ indexing how sharp that salience is. When actions are binary, it is often more convenient to work with choice probabilities $p_i\in[0,1]$ and a logit-type equilibrium mapping of the form
which is a continuous fixed point on $[0,1]^n$. Existence follows from Brouwer; uniqueness follows under a weak-interaction condition because $\sup_t\Lambda'(t)\le 1/4$ (Appendix (ref)).
The norm-game framework delivers a sharp separation between (i) how peers enter behavior and (ii) the causal response of outcomes to that peer exposure. The first component is structural and can be disciplined by economic axioms through a peer aggregator \(\Phi\); the second component can be left nonparametric.
Let \(y_i\) denote an equilibrium outcome (or action) and define the endogenous peer exposure index
where \(G\) is the interaction matrix and \(\theta\) indexes preference parameters (e.g.\ \(\beta\) in power-mean norms). We then consider the general nonparametric structural form
where \(m(\cdot,\cdot)\) is an unknown response function.
Equation (ref) nests many familiar specifications: linear-in-means arises when \(m\) is linear and \(\Phi\) is the mean operator; BRUZ arises when \(m\) is linear and \(\Phi\) is the CES/power mean; attention/extremes-based models arise when \(\Phi\) is smooth-max and \(m\) is induced by a link function.
Manski's reflection problem states that if \(N_i\) is an equilibrium object (a function of peers' outcomes and unobservables), then \(m(\cdot,\cdot)\) is generally not nonparametrically identified without excluded variation or strong support restrictions manski1993identification. The aim of this paper is to show that once \(\Phi\) is disciplined by economic axioms, it also implies a model-consistent propagation structure that can generate excluded variation systematically. We develop that propagation structure in Sub-Sections (ref)--(ref), and return to Manski-style nonparametric identification in Section (ref).
Recent peer-effects theory emphasizes that the relevant peer “norm” need not be the average of neighbors. Following boucher2024toward, define the CES/power-mean peer norm in group \(s\):
where \(\beta\) indexes peer preference (salience): \(\beta=1\) is mean-like; \(\beta\to+\infty\) is max-like; \(\beta\to-\infty\) is min-like.
To connect (ref) to a structural peer-effects equation, adopt the BRUZ notation:
Here \(\lambda_1\) measures spillover intensity, \(\lambda_2\) conformity, and \(\beta\) determines which peers are salient.
Because \(\tilde y_{-is}(\beta)\) is an equilibrium object, it is generally endogenous. BRUZ construct moments using an exogenous predictor \(\hat y_{is}=m(x_{is})\) (e.g.\ reduced-form OLS/ML) and exploit isolated individuals (no friends) to separately identify private components. Their non-isolate moments use the instrument vector
The BRUZ moments deliberately compress network information into one-step scalar summaries: \(\tilde y_{-is}(\hat y,\beta)\) and its \(\beta\)-slope. Our contribution is to show that the same structural primitive \(\Phi\) implies a full who-matters-to-whom influence structure that generates richer excluded variation.
The central object is the Jacobian (marginal influence field) of the peer exposure map. For a general aggregator \(N_i=\Phi_i(y;G,\theta)\), define
\paragraph{CES case.} For \(\Phi_i=\tilde y_{-i}(\beta)\) in (ref),
\paragraph{Anchoring to observables.} As in BRUZ, evaluate influence objects at an exogenous predictor \(\hat y=m(X)\):
Row-normalize to obtain the influence-propagation kernel
Econometric interpretation. \(P_{ij,s}(\beta;\hat y)\) is the model-implied influence share of \(j\) in \(i\)'s peer exposure, computed from observables via \(\hat y\). When \(\beta\neq 1\), \(P\) generally differs from \(G\) even in group-like graphs.
Let \(X_s\) stack \(x_{is}\). For \(k\ge 2\), define
This is the direct analogue of BDF's \(G^kX\), but with propagation dictated by the structural norm through marginal influence.
To define “distance” in a way that is transparent to econometricians, interpret distance as how easily influence can travel along strong links. Convert influence shares into frictions
and define the effective influence distance as the shortest-path friction:
Define shells \(\mathcal S_{is}(h):=\{j: d_{\beta,s}(i,j)\in(h-1,h]\}\) and shell instruments
A key reason higher-order network IV can be weak is path redundancy (transitivity). Measure local non-redundancy by comparing direct influence to two-step influence:
A torsion-weighted two-step instrument is
The instrument sets (ref)--(ref) are all functions of observables \((X,G,\hat y)\) and \(\beta\). They extend classical \(G^2X\) by (i) reweighting links by model-implied salience, (ii) using effective influence distance rather than hop distance, and (iii) concentrating identifying power on non-redundant paths.
Two nesting facts guide interpretation. First, in linear-in-means the marginal influence field is constant, so the model-implied influence operator coincides with the adjacency operator and the resulting propagation instruments are exactly the familiar $G^kX$. Second, BRUZ corresponds to using only one-step scalar summaries of the peer aggregator evaluated at an exogenous predictor (the predicted peer norm and its $\beta$-derivative), while our framework additionally permits instruments built from multi-step propagation and path structure.
This section explains how our geometry-based instrument construction addresses Manski's reflection problem in a nonparametric way, and how familiar network-IV strategies appear as special cases.
Manski's reflection problem can be summarized as follows: when the peer environment is an equilibrium object, it is generally endogenous, so without excluded variation it is not possible to identify the causal response to that peer environment nonparametrically manski1993identification.
We formalize this by writing outcomes for individual $i$ in group $s$ (e.g.\ a school) as
where $x_{is}$ are observed covariates, $G_s$ is the interaction matrix, $S_{is}$ is an endogenous peer exposure index, $g(\cdot)$ is the (unknown) private component, and $h(\cdot)$ is the (unknown) peer-response function of interest. Endogeneity arises because $S_{is}$ depends on peers' outcomes/actions and therefore inherits peers' unobservables (reflection), and may also be correlated with $\varepsilon_{is}$ through sorting or common shocks.
In our framework, the peer exposure index is disciplined by a structural aggregator:
where $y_s$ stacks outcomes in group $s$ and $\beta$ indexes peer preference / salience (e.g.\ a CES parameter). The key point is that the same primitive $\Phi$ that defines exposure also implies a propagation law (“who matters to whom”) that can be exploited to generate excluded variation.
Define the marginal-influence (Jacobian) matrix implied by $\Phi$:
To anchor this object in observables, we evaluate at an exogenous predictor $\hat y_{is}=m(x_{is})$ constructed from $X_s$ (e.g.\ OLS/ML), with $\hat y_s$ measurable w.r.t.\ $(X_s,G_s)$:
Let $w_{ij,s}(\beta;\hat y):=W_{ij,s}(\hat y_s;\beta)$ and row-normalize to obtain the normalized influence operator
whenever the denominator is positive (for isolates set the row to zero). Economically, $P_{ij,s}(\beta;\hat y)$ is the model-implied influence share: it measures how strongly $j$ contributes at the margin to $i$'s peer exposure, under the structural aggregator and evaluated at the observable field $\hat y$.
From $P_s(\beta;\hat y)$ we generate excluded variables in three complementary ways:
\paragraph{Geometry-augmented instrument signature.} Fix $K\ge 2$ and $H\ge 2$. Collect the geometry-induced instruments into the single signature
where:
with $P_s(\beta;\hat y)$ the normalized influence operator defined in (ref), $\hat y_s=m(X_s)$ an exogenous predictor, and $\mathcal{S}_{is}(h)$ the $h$-th strong-influence shell induced by the effective-distance metric (equations defining frictions and shortest-path distance appear above in this section). The wedge non-redundancy (torsion) term is
$Z_{is}(\beta)$ enlarges the excluded $\sigma$-field beyond one-step summaries by using model-consistent propagation structure: multi-step influence ($P_s^kX_s$), localization along strong influence chains (shells), and explicit emphasis on non-redundant higher-order paths (torsion).
The classic BDF intuition is: the endogenous regressor is the peer term (e.g.\ $Gy$), but peers' outcomes respond to peers-of-peers covariates, so $G^2X$ can shift the peer term without directly entering $i$'s outcome.
Our instruments apply the same equilibrium-feedback logic under general peer norms. The difference is that what counts as a “peer” and how influence propagates is dictated by the aggregator through $P_s(\beta;\hat y)$:
With the structural equation (ref), identification of the unknown peer-response function $h(\cdot)$ is a nonparametric IV problem: $S_{is}$ is endogenous, so we require excluded variables that shift $S_{is}$ but are excluded from the outcome equation. Our construction provides such excluded variables by enlarging the instrument menu to the geometry-induced signature $Z_{is}(\beta)$.
\paragraph{Interpretation.} Manski's negative message is fundamentally about insufficient independent variation: when the only available shifters are low-rank (one-step) summaries such as group means or $G_sX_s$, the endogenous peer regressor moves almost one-for-one with the outcome, so there is too little excluded variation to separate endogenous social effects from correlated effects and contextual forces (the “reflection” problem). In our setting, the identifying assumption is still a standard NPIV completeness/injectivity condition, but the key distinction is that geometry changes its plausibility. Geometry enlarges the excluded $\sigma$-field from one-step summaries to a rich, within-network collection of structured multi-step and path-based objects generated by the model-implied influence operator $P_s(\beta;\hat y)$ (e.g., $P_s^kX_s$, shell and torsion components). Because these instruments vary across nodes even within the same group and exploit non-redundant propagation paths, they can generate high-rank excluded variation in the peer exposure precisely in network topologies where one-step moments collapse, making completeness (and hence point identification) more plausible in empirically relevant networks.
The same objects also unify well-known identification strategies:
This example shows a setting where the one-step BRUZ instruments based on $\tilde y_{-i}(\hat y,\beta)$ and $\partial_\beta \tilde y_{-i}(\hat y,\beta)$ provide essentially no excluded variation for a subset of nodes, while two-step/geodesic instruments remain relevant.
\paragraph{Environment: two disconnected stars (one school).} There are two hubs $h\in\{a,b\}$ and disjoint peripheral sets $\mathcal P_a$ and $\mathcal P_b$. Each peripheral $i\in\mathcal P_h$ has exactly one friend (hub $h$), hubs are not linked. Row-normalized weights satisfy \[ g_{ih}=1\ (i\in\mathcal P_h),\qquad g_{hi}=1/|\mathcal P_h|\ (i\in\mathcal P_h). \]
\paragraph{Peer-preference model (set conformity to zero for transparency).}
For peripherals $i\in\mathcal P_h$, $\tilde y_{-i}(\beta)=y_h$ (single neighbor). For hubs, \[ \tilde y_{-h}(\beta) = \left(\frac{1}{|\mathcal P_h|}\sum_{j\in\mathcal P_h} y_j^\beta\right)^{1/\beta}. \]
\paragraph{One-step BRUZ objects collapse for peripherals.} Let $\hat y_i=m(x_i)$. For a peripheral $i\in\mathcal P_h$, \[ \tilde y_{-i}(\hat y,\beta) = \left(\hat y_h^\beta\right)^{1/\beta} = \hat y_h \quad\Rightarrow\quad \partial_\beta \tilde y_{-i}(\hat y,\beta)=0. \] If hubs have identical covariates $x_a=x_b$, then $\hat y_a=\hat y_b$ and the one-step predicted peer norm is constant across peripherals, with a zero $\beta$-derivative. Hence one-step BRUZ moments provide no excluded first-stage variation for $\tilde y_{-i}(\beta)=y_h$ for peripherals.
\paragraph{Two-step / geodesic instruments remain relevant.} Although peripherals only “see” the hub directly, the hub outcome depends on the entire peripheral set through $\tilde y_{-h}(\beta)$. Therefore, covariates of distance-2 nodes shift $y_h$ and hence $\tilde y_{-i}(\beta)$. The distance-2 shell instrument is \[ Z^{\mathrm{geo}}_{i,2} = \sum_{j:\ d(i,j)=2} x_j = \sum_{j\in\mathcal P_h\setminus\{i\}} x_j, \] which is excluded from (ref) for peripheral $i$ (not direct friends), but relevant via the hub equilibrium. Likewise, multi-step influence instruments $(P^2(\beta;\hat y)X)_i$ aggregate the hub's neighborhood covariates with $\beta$-dependent weights.
This section provides simulation evidence on when and why geometry-based instruments strengthen identification of the peer-effect intensity relative to standard one-step network IV and scalar-moment instruments. The Monte Carlo is designed to isolate the mechanism emphasized in the theory: multi-step propagation through non-redundant paths can generate excluded variation even when one-step summaries collapse.
Each experiment is built to answer three practical questions:
We simulate $S$ groups (“schools”) indexed by $s$, each with $n_s$ individuals. Within each group we generate an interaction matrix $G_s=(g_{ij,s})$ (row-normalized for non-isolates) and covariates $X_s=(x_{1s},\dots,x_{n_ss})'$.
\paragraph{Structural equation with a fixed CES peer exposure.} The Monte Carlo targets identification of the peer-effect parameter $\lambda_0$ holding the curvature parameter $\beta$ fixed. This reflects the empirical use case where the researcher specifies an exposure mapping and seeks robust identification of the peer effect. Specifically, we define the endogenous peer exposure \[ w_{is}(\beta_{\mathrm{fix}})\;=\;\Phi_{is}^{\mathrm{CES}}(y_s;G_s,\beta_{\mathrm{fix}}), \qquad \Phi_{is}^{\mathrm{CES}}(y_s;G_s,\beta) = \Big(\sum_{j\neq i} g_{ij,s} y_{js}^{\beta}\Big)^{1/\beta}, \] and simulate outcomes from
We ensure positivity of the CES mapping by working with shifted outcomes $y_{is,+}=y_{is}+c$ (with $c>0$) inside $\Phi^{\mathrm{CES}}$.
\paragraph{Shocks.} Baseline shocks are i.i.d.\ $\varepsilon_{is}\sim \mathcal{N}(0,\sigma_\varepsilon^2)$. (We also consider correlated-shock variants $\varepsilon_{is}=u_s+\nu_{is}$ in additional experiments; all inference is clustered by group.)
\paragraph{Equilibrium computation.} In each replication we generate outcomes by solving the network norm game to equilibrium, since the DGP is a simultaneous system; solving the fixed point ensures the simulated data satisfy the maintained structural peer-effects equation and its implied propagation operator.
Given $(X_s,G_s)$ and $(\lambda_0,\beta_{\mathrm{fix}})$, we solve (ref) by fixed-point iteration: \[ y_s^{(t+1)} = X_s\gamma_0 + \lambda_0 \Phi^{\mathrm{CES}}(y_{s,+}^{(t)};G_s,\beta_{\mathrm{fix}}) + \zeta_s \mathbf{1} + \varepsilon_s, \] initialized at $y_s^{(0)}=X_s\gamma_0$. We monitor convergence by $\|y_s^{(t+1)}-y_s^{(t)}\|/\|y_s^{(t)}\|$ below a tolerance and cap iterations; non-convergent draws are recorded.
To highlight identification gains, we use a dispersion-bridge design within each group. Nodes are partitioned into two blocks with different outcome dispersion, and a small number of bridge links connect the blocks. This topology creates substantial multi-step propagation while making one-step scalar summaries comparatively fragile.
For transparency, we report simple diagnostics that characterize the exposure and the influence field in each design: (i) dispersion of the endogenous exposure $w(\beta_{\mathrm{fix}})$ (e.g.\ $\mathrm{sd}(w)/\mathrm{mean}(w)$), (ii) concentration of Jacobian weight shares (mean and upper-tail of $\max_j P_{ij}$), and (iii) dispersion of the Jacobian row-sum intensity $s_i=\sum_j g_{ij}\hat y_{js}^{\beta_{\mathrm{fix}}-1}$.
We compare two instrument menus (holding the DGP fixed):
\paragraph{(A) One-step scalar instruments (BRUZ-style).} Let $\hat y_{is}$ be an exogenous predictor of $y_{is}$ (constructed below). The BRUZ menu uses \[ Z^{\mathrm{BRUZ}}_{is}(\beta_{\mathrm{fix}}) = \Big[x_{is},\ \Phi_{is}^{\mathrm{CES}}(\hat y_s;G_s,\beta_{\mathrm{fix}}),\ \partial_\beta \Phi_{is}^{\mathrm{CES}}(\hat y_s;G_s,\beta)\big|_{\beta=\beta_{\mathrm{fix}}}\Big], \] where $\partial_\beta \Phi$ is computed numerically by finite differences.\footnote{Including $\partial_\beta\Phi$ improves robustness of the BRUZ first stage even when $\beta$ is treated as fixed, and we keep this specification for comparability with the profile-based implementations.}
\paragraph{(B) Geometry-augmented instruments (this paper).} Let $P_s(\beta_{\mathrm{fix}};\hat y)$ denote the row-normalized Jacobian-weight operator evaluated at $\hat y$. The geometry menu augments BRUZ with multi-step and shell-based excluded variation: \[ Z^{\mathrm{GEO}}_{is}(\beta_{\mathrm{fix}}) = \Big[ Z^{\mathrm{BRUZ}}_{is}(\beta_{\mathrm{fix}}),\ (P_s^2X_s)_{is},\ \partial_\beta(P_s^2X_s)\big|_{\beta=\beta_{\mathrm{fix}}},\ (\mathrm{Shell}_2(G_s)X_s)_{is} \Big], \] where $\mathrm{Shell}_2(G_s)$ is the exact distance-2 adjacency (row-normalized). This “poster-boy” menu mirrors the implementation used in the dominance simulations.
\paragraph{Estimation of $(\gamma,\lambda)$ with fixed $\beta$.} For each replication we treat $\beta_{\mathrm{fix}}$ as given and estimate $(\gamma,\lambda)$ by 2SLS/GMM using the chosen menu. We report first-stage diagnostics for the endogenous exposure $w(\beta_{\mathrm{fix}})$ (partial $R^2$ and F-statistics), and second-stage performance for $\hat\lambda$.
We consider three constructions to separate “oracle” from “realistic” performance:
Across $1000$ replications we report:
For each replication $r=1,\dots,R$:
\paragraph{Monte Carlo summary.} Tables (ref)--(ref) show that geometry-based instruments deliver large and robust identification gains for the peer-effect parameter $\lambda$ across both sample sizes and exposure curvatures. In the “dispersion-bridge” design, the one-step BRUZ menu is systematically weak: its first-stage partial $R^2$ is near zero (roughly $0.002$--$0.014$ at $n=600$ and $0.004$--$0.010$ at $n=2400$), with corresponding first-stage $F$ statistics in the weak-instrument range. This weak identification translates into severe finite-sample instability for $\hat\lambda$, with RMSEs that can be extremely large and highly sensitive to $\beta$ (e.g., RMSEs ranging from about $6$ to $360$ at $n=600$ and from about $25$ to $301$ at $n=2400$).
By contrast, the geometry-augmented menu produces a strong first stage in every cell, with partial $R^2$ around $0.58$ at $n=600$ and around $0.41$ at $n=2400$, and $F$ statistics well above conventional thresholds (about $120$ at $n=600$ and about $250$ at $n=2400$). Consistent with this, GEO yields accurate and stable estimation of $\lambda$ across all reported $\beta$ values: bias remains small (about $0.03$--$0.06$) and RMSE remains low (about $0.09$--$0.14$), with only modest variation across $\beta$ and $n$. Overall, the results confirm the mechanism emphasized in the theory: multi-step, geometry-based excluded variation can restore identification of peer effects in network topologies where one-step scalar moments effectively collapse.
This section illustrates how the geometry-based instrument construction can be implemented in a longitudinal social network setting and how conclusions about peer effects depend on the peer-exposure aggregator that disciplines what counts as a salient peer. The goal is not to claim a single “true” social norm, but to show how the same dataset can support very different peer-effect estimates depending on whether exposure is (i) mean-like (CES with small curvature), (ii) attention-to-extremes (smooth-max), or (iii) rank-based (quantile/median norms).
We use NetHealth, which combines (i) repeated network measurement and (ii) objective outcomes from wearable devices and administrative records. We focus on a core panel of participants with valid identifiers across the Fitbit, network, and baseline survey files. The analysis proceeds in “waves” indexed by $t=1,\dots,8$. For each wave we construct a short outcome window around the median survey date (a fixed-length window in days) and compute wave-specific outcomes and networks; then we stack the resulting wave samples to estimate a single (peer-effect, preference) parameter vector with cluster-robust inference clustered by individual (egoid).
\paragraph{Outcomes.} We consider three outcomes: (i) physical activity (Fitbit steps), (ii) sleep duration (Fitbit minsasleep), and (iii) academic performance (term GPA from course records). Fitbit activity reports daily steps (steps) and a compliance measure (complypercent, percent minutes wearing/using the device). Fitbit sleep reports minutes asleep (minsasleep). Wave-level outcomes are aggregated over the window (subject to a minimum number of valid days), and we include mean Fitbit compliance over the same window as a control (for GPA, compliance is set to 100 by construction).
\paragraph{Controls.} The baseline control vector $X$ includes an intercept, a male indicator (from the BasicSurvey variable gender_1), and mean Fitbit compliance in the analysis window (average complypercent); for GPA, compliance is set to 100 for all observations.
For each wave $t$, we construct an interaction matrix $G_t$ from the observed friendship links. We treat ties as undirected in the baseline construction and row-normalize $G_t$ for non-isolates. Because both one-step and geometry-based instruments rely on propagation through the network, isolates contain no peer-exposure information; in the baseline empirical pipeline we drop isolates within wave before stacking. (We report sensitivity to alternative conventions such as retaining isolates with zero exposure, using reciprocal ties only, or alternative weighting schemes.)
Let $y_{it}$ denote the outcome for individual $i$ in wave $t$ and $x_{it}$ the controls. The empirical specification is a linear peer-effects equation with an endogenous peer exposure index $S_{it}(\theta)$ disciplined by a structural aggregator:
where $\lambda$ is the peer-effect coefficient of interest and $\theta$ is the peer-preference / salience parameter governing the exposure aggregator.
\paragraph{Aggregator menu (peer preferences).} We compare three families of exposure indices $S_{it}(\theta)$:
Endogeneity arises because $S_{it}(\theta)$ is constructed from peers' outcomes and therefore inherits peers' unobservables (reflection) and potentially sorting/common shocks. We implement two instrument menus:
\paragraph{BRUZ (one-step scalar moments).} Let $\hat y_{it}=m(x_{it})$ denote an exogenous predictor of $y_{it}$ constructed from observables. BRUZ uses excluded variation from one-step aggregator summaries evaluated at $\hat y$ and numerical derivatives w.r.t.\ the preference parameter: \[ Z^{\mathrm{BRUZ}}_{it}(\theta) = \big[x_{it},\ \Phi(\hat y_t;G_t,\theta),\ \partial_\theta \Phi(\hat y_t;G_t,\theta)\big]. \] This is the natural analogue of BDF/BRUZ instruments for general peer norms.
\paragraph{GEO (multi-step geometry-induced instruments).} Geometry augments BRUZ by adding excluded variation generated from the model-implied influence operator $P_t(\theta;\hat y)$ (the normalized Jacobian weights of the exposure mapping evaluated at $\hat y$), and then propagating covariates along multi-step influence paths (and optionally shells/torsion): \[ Z^{\mathrm{GEO}}_{it}(\theta) = \big[Z^{\mathrm{BRUZ}}_{it}(\theta),\ (P_t^2(\theta;\hat y)X_t)_{it},\ \ldots \big]. \] Intuitively, $P^kX$ replaces “friends-of-friends” with “influencers-of-influencers” under the aggregator-consistent influence geometry, expanding the excluded $\sigma$-field beyond one-step summaries.
\paragraph{Exogenous predictor and cross-fitting.} Following best practice for generated instruments, we construct $\hat y=m(X)$ using a pooled cross-fitted linear predictor (K-fold sample splitting), so that an observation's own realized outcome does not leak into its own instruments through estimation of $m(\cdot)$.
For each aggregator family, we estimate $(\gamma,\lambda,\theta)$ by a profile-IV procedure over a grid of the preference parameter (e.g.\ $\beta$ for CES, $\kappa$ for smooth-max, and fixed $q$ for quantiles), and we report cluster-robust standard errors clustered by egoid. In practice, BRUZ and GEO can select different profile minimizers, so we report $\hat\theta^{\mathrm{BRUZ}}$ and $\hat\theta^{\mathrm{GEO}}$ separately along with $\hat\lambda$ and its standard error.
Table (ref) reports stacked IV estimates for each outcome (steps, sleep, GPA) under each aggregator (CES, smooth-max, quantile norms), comparing BRUZ and GEO.
\paragraph{Steps.} Under CES and smooth-max exposure, both menus deliver negative peer-effect estimates, with GEO generally producing smaller standard errors; the preferred preference parameter differs across menus (e.g.\ CES curvature differs between BRUZ and GEO). Quantile norms yield noisier and sign-sensitive estimates, consistent with rank-based exposure being a different behavioral object than mean-like or attention-to-extremes exposure.
\paragraph{Sleep.} CES exposure yields near-zero peer effects. Under attention and upper-quantile norms, GEO can select substantially different preference parameters than BRUZ and can deliver positive point estimates, though precision varies by aggregator and outcome. This pattern is consistent with the idea that sleep-related peer influence may operate through salient or aspirational peers rather than average peers, but it also highlights that identification of preferences can be fragile in sparse networks and motivates the geometry diagnostics discussed in the implementation notes.
\paragraph{GPA.} GPA effects are small under CES, while smooth-max and quantile norms can deliver different point estimates and different preferred preference parameters under BRUZ vs GEO. Given the thinner usable sample for GPA (relative to Fitbit outcomes), we treat GPA primarily as a mechanism and robustness outcome rather than a headline estimate.
\paragraph{Takeaway for identification.} The central empirical lesson is that “peer effects” are not a single estimand independent of modeling choices: the mapping from peers' outcomes to exposure (mean-like vs attention vs rank norms) is itself a behavioral primitive, and the geometry-based instrument menu provides a disciplined way to generate excluded variation that is consistent with that primitive. The Monte Carlo evidence suggests that GEO can materially strengthen first-stage diagnostics in settings where one-step moments are weak, and this application provides a practical template for applying the same logic to real network data.
This section provides a replicable recipe for taking the model to data. The goal is not to prescribe a single “correct” pipeline, but to lay out default steps that mirror standard applied practice while making the construction of geometry-induced instruments transparent.
\paragraph{Objects.} Fix a group index $s$ (e.g.\ school) and let $G_s$ denote the observed network (or any baseline connectivity object). The empirical outcome equation features an endogenous peer-exposure index $E_{is}(\beta;\cdot)$ (e.g.\ a CES norm, smooth-max, quantile norm), with peer-preference parameter(s) $\beta$. Instruments are generated from the induced transport (influence) matrix $P_s(\beta;\hat y)$, evaluated at a predetermined proxy $\hat y=m(X)$, and then transformed into multi-step, shell-based, and torsion-weighted objects. Let $Z_{is}(\beta)$ collect all such instruments. \paragraph{Algorithm 1 (Replication checklist).}
This paper offers a unifying perspective on peer-effects identification: propagation is governed by a transport operator induced by the peer aggregator. The linear-in-means model is the constant-transport benchmark ($P=G$), which recovers the familiar $G^kX$ instrument families and clarifies their well-known fragilities under transitivity, regularity, and path redundancy. By contrast, peer-preference aggregators---such as CES norms and their extensions---generate a state-dependent Jacobian transport operator.
Its powers $P^k$, along with geometry-induced objects (shells/geodesics and torsion), produce new excluded-variation families tailored to how influence actually flows under the maintained aggregator.
The main implication is practical: identification can be strengthened by moving from one-step summaries to multi-step, influence-consistent propagation. Because geometry-based instruments exploit marginal influence heterogeneity and non-redundant paths, they can remain informative precisely in settings where one-step scalar moments (e.g., BRUZ-type summaries) or algebraic powers of $G$ become nearly collinear and weak. More broadly, the framework makes explicit that “peer effects” depend on the behavioral primitive used to map peer outcomes into exposure; different aggregators induce different transports and therefore different estimands. This transport view provides a constructive route to diagnosis (via profile curvature and first-stage strength) and to design (via instrument menus that match the implied influence geometry), and it suggests natural extensions to alternative aggregators such as attention (smooth-max) and rank norms (quantiles).