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Tractable Identification of Strategic Network Formation Models with Unobserved Heterogeneity
Network formation is a central problem in network economics. From the econometric perspective, the goal is to identify and estimate structural parameters in a model where agents are heterogeneous and strategically interdependent when making linking decisions. Substantial progress has been made on models with either unobserved heterogeneity or strategic interdependence, but combining both features in a tractable framework has remained an open problem.
This paper develops a tractable identification approach for strategic network formation models with unobserved individual fixed effects. We consider a latent utility framework where the formation of a link between agents $i$ and $j$ depends on observed dyadic characteristics, individual fixed effects that capture unobserved degree heterogeneity (such as sociability or popularity), and idiosyncratic pairwise shocks. The link formation decision is allowed to depend on endogenous network statistics arising from the equilibrium of the network formation game, such as the number of common friends between $i$ and $j$. This accommodates strategic complementarities and other forms of link interdependence that are relevant in many economic applications.
The main methodological challenge is that the mapping from model primitives to equilibrium network structure is generally intractable. In strategic network formation games, the equilibrium network depends on the entire profile of agent characteristics, fixed effects, and shock realizations through complex equilibrium conditions. This gives rise to at least three difficulties. First, the space of possible network structures is a discrete space with a combinatorially overwhelming number of elements\footnote{For a standard illustrative example: there are $2^{435}$ possible (undirected and unweighted) network structures on a set of 30 agents.}, making it very hard to handle both theoretically and computationally. Second, solution concepts for network formation problems (such as pairwise stability) usually do not have uniqueness properties, and sometimes the number of equilibria can be very large de2020econometric. Third, iterative procedures are in general not guaranteed to converge to an equilibrium network even when such an equilibrium does exist jackson2002evolution. Consequently, characterizing the equilibrium mapping, let alone inverting it for identification purposes, is typically infeasible, except in very simple special cases.
This intractability has led the existing literature to either (i) focus on models without strategic interdependence, such as in the seminal work by graham2017econometric and follow-up work by gao2020 and \citet*{gao2023logical}, or (ii) study strategic models under restrictions that eliminate or simplify the role of unobserved heterogeneity \citep*{mele2017structural,de2018identifying,sheng2020structural}.
Our approach avoids the need for equilibrium characterization by exploiting monotonicity restrictions to obtain identifying information without knowing the form of the equilibrium mapping. The technique, which we call “bounding by $c$,” treats endogenous covariates as random variables and uses indicator function arguments to derive bounds on structural parameters that hold regardless of the equilibrium realization. This approach builds on and extends ideas from gao2026identification, who developed similar techniques for dynamic panel data models under a partial stationarity condition.
Based on our key technique, we derive a system of identifying restrictions based on different subnetwork configurations. Our primary restrictions exploit tetrad configurations, i.e., sets of four agents with a particular pattern of links, to difference out all individual fixed effects simultaneously. The resulting bounds depend only on the distribution of idiosyncratic shocks, which can be aggregated into identifying restrictions on the model parameters under a standard independence assumption on the idiosyncratic shocks. We also complement the tetrad restrictions with two additional types of restrictions: (1) “incomplete differencing” restrictions, such as triad-based restrictions, that do not eliminate all fixed effects but nevertheless provide identifying information, and (2) “cyclically differencing” restrictions that involve longer cycles of links than those in a tetrad.
Beyond partial identification, we establish conditions under which the structural parameters are point identified. When the pairwise shocks follow a logistic distribution and the endogenous covariates satisfy a comparison-pattern invariance condition---which holds for both the common-friends statistic and the standard Jaccard index---we show that a log-odds ratio of tetrad probabilities, conditioned on diagonal link absences that isolate the endogenous covariates from the tetrad shocks, identifies a linear index of the structural parameters. The fixed effects cancel algebraically through tetrad differencing, while the endogenous covariates decouple through the isolation conditioning; these two mechanisms operate separately. The resulting identification equation leads to a computationally simple conditional logit estimator that accommodates both endogenous network statistics and unobserved individual fixed effects, generalizing the tetrad logit of graham2017econometric to settings with strategic link interdependence.
We report a preliminary simulation exercise illustrating the finite-sample performance of the proposed identifying restrictions. We simulate networks under nested specifications that progressively add fixed effects and an endogenous covariate capturing local network structure, and compute the resulting identified set using the tetrad inequalities. The results show that the restrictions can deliver nontrivial bounds on the strategic-interaction parameter even in the full model with endogenous covariates and fixed effects. A larger-scale Monte Carlo study is left to future work.
This paper relates to several strands of the econometric literature on network formation; see de2020econometric for a recent survey. The first strand concerns dyadic models with unobserved heterogeneity but without strategic interdependence. graham2017econometric introduced the tetrad logit estimator, which differences out additive fixed effects by comparing link patterns across tetrads of agents. This approach achieves point identification of homophily parameters under a conditional logit specification. gao2020 extended these ideas to nonparametric identification of the homophily effect function, while \citet*{gao2023logical} developed logical differencing techniques for settings with non-transferable utilities where bilateral consent is required for link formation. Our paper builds on these ideas but generalizes the setting by allowing endogenous covariates arising from strategic interaction.
A complementary line of recent work develops general fixed-effects methods applicable to network data. bonhomme2023functional extend the functional differencing approach of bonhomme2012functional from panel data to network settings, deriving moment restrictions on model parameters that hold regardless of the form of heterogeneity and without requiring dense networks; their framework, however, treats the network as exogenous. \citet*{bonhomme2025moment} characterize all moment conditions for nonlinear panel data models that are robust to both unrestricted feedback and arbitrary heterogeneity, with applications to duration and count models. \citet*{dano2025binary} provide a systematic treatment of conditional likelihood and moment-based identification in binary logit models with general fixed effects, covering both panel and network (dyadic) data; their analysis subsumes the tetrad logit of graham2017econometric as a special case. Our paper differs from this body of work in that we address the additional complication of endogenous network covariates arising from strategic interaction, which requires a distinct identification strategy based on tetrad inequalities rather than conditional likelihood or moment equalities.
The second strand studies strategic network formation under various equilibrium concepts. mele2017structural analyzes exponential random graph models (ERGMs) as potential games with strategic complementarities, developing simulation-based estimation methods. However, ERGMs are known to suffer from degeneracy problems and computational challenges in large networks. \citet*{de2018identifying} and sheng2020structural study partial identification in strategic network formation models under simultaneity, using subnetwork restrictions to bound the identified set. sheng2020structural provides particularly elegant results using small subnetwork configurations, but her analysis does not accommodate agent-level fixed effects. menzel2026strategic develops a many-agent asymptotic approximation for pairwise stable networks with anonymous and non-anonymous interaction effects, characterizing the limiting distribution of link intensities through a fixed-point system. His identification of payoff parameters requires parametric specification of the utility function and solution of the equilibrium fixed point, and does not address individual fixed effects of the type we consider here. Our approach addresses the fixed-effects gap by combining the differencing logic from the dyadic literature with techniques that accommodate strategic interdependence, while avoiding the need for equilibrium computation.
On the inference side, leung2019treatment and leung2019normal develop laws of large numbers and central limit theorems for network moments in sparse strategic formation models embedded in a latent position space, using a branching-process subcriticality condition to control strategic dependencies. menzel2021clt provides an alternative central limit theorem based on the exchangeability of agents' potential values, which does not require positional homophily or $K$-locality and applies to general $D$-adic network moments. In Section (ref), we provide primitive conditions under which the conditional probabilities underlying our identifying restrictions can be consistently estimated from a single large network, drawing on these frameworks.
To our knowledge, this paper is the first in the econometric literature to provide identification results for network formation models that allow for both (i) unobserved individual fixed effects and (ii) endogenous network statistics arising from strategic link formation. Our contributions are as follows. First, we show that tetrad-based configurations can be used to difference out all individual fixed effects even when endogenous covariates are present, yielding bounds on structural parameters that are informative under reasonable support conditions. Second, we develop complementary “incomplete differencing” restrictions that deliver additional identification power by combining the two traditional “difference-out” and “aggregate-out” approaches for the handling of fixed effects in the econometric literature. Third, we demonstrate that the “bounding by $c$” technique from the panel data literature in gao2026identification can be productively adapted to network settings, suggesting broader applicability of the key idea.
Methodologically, our paper builds upon the recent work on semiparametric identification with non-separable unobservables. The “bounding by $c$” technique we employ shares conceptual foundations with the partial stationarity approach of gao2026identification for dynamic panel models, and the “multi-inequality aggregation” also relates to the logical operations underlying \citet*{gao2023logical}. The commonality across these papers is the use of arithmetic and logical operations based on monotonicity and inequality conditions. In particular, the “bounding by $c$” technique in gao2026identification and the present paper provides a way to “eliminate” the potentially complicated endogenous variables when establishing identifying restrictions based on homogeneity assumptions on the structural error terms.
Finally, our paper relates to the broader literature on games with incomplete information and heterogeneous agents. aguirregabiria2007sequential and \citet*{bajari2007estimating} develop estimation methods for dynamic games that avoid full solution of the game by using conditional choice probability representations. While our setting differs (we study a static network formation game rather than a dynamic game), the spirit of avoiding intractable equilibrium computations through the use of observable implications is similar. That said, the exact techniques we exploit to achieve this in our network formation context are naturally different from theirs.
The remainder of the paper is organized as follows. Section (ref) presents the model setup, introduces notation, and states our maintained assumptions. Section (ref) derives the main identifying restrictions, beginning with tetrad-based restrictions, then developing triad-based and weighted alternatives, and culminating in a general cycle-based differencing framework. Section (ref) provides primitive sufficient conditions under which the high-level identifying assumptions hold, embedding the model in the sparse network framework of leung2019treatment. Section (ref) establishes conditions for point identification under logistic errors and a comparison-pattern invariance condition on the endogenous covariates. Section (ref) provides simulation results based on tetrad-based identifying restrictions. Section (ref) concludes.
Consider a set of $n$ agents indexed by $i=1,\ldots,n$, and an unweighted network among them represented by an $n\times n$ adjacency matrix $Y$, where $Y_{ij}=1$ if a link exists between agents $i$ and $j$, and $Y_{ij}=0$ otherwise. We focus on undirected networks, i.e., $Y_{ij}=Y_{ji}$ for all $i,j$. Throughout, we index distinct unordered pairs as $(i,j)$ with $i<j$; all dyadic quantities ($Y_{ij}$, $Z_{ij}$, $X_{ij}$, $\varepsilon_{ij}$) are understood to be symmetric in their indices.
We consider the following network formation model with both link interdependence and fixed effects, where a link between agents $i$ and $j$ exists if and only if the latent surplus from the link is non-negative:
where:
The covariate function $w_n$ may depend on the network size $n$; this generality is essential for accommodating sparse network asymptotics (Section (ref)). Specifically, when agents are embedded in a latent space with positions $\Xi_i$ and the network is sparse with $O(1)$ expected degree, the sparsity scaling $r_n \to 0$ rescales positions to $\bar{\Xi}_i := r_n^{-1}\Xi_i$, and the covariate function takes the form $w_n(Z_i, Z_j) = \tilde{w}(\bar{\Xi}_i, \bar{\Xi}_j, W_i, W_j)$ for a fixed function $\tilde{w}$ that does not depend on $n$. The $n$-dependence of $w_n$ thus enters only through the position rescaling, while the structural parameters $\theta_0 = (\beta_0, \gamma_0)$ remain fixed---analogous to the spatial weight matrix $W_n$ in spatial autoregressive models. This structure ensures that the conditional link probability $\mathbb{P}(Y_{ij} = 1 \mid Z_{ij} = z)$ is an $O(1)$ quantity for any fixed $z$ in the support of $Z_{ij}$, even though the marginal link probability vanishes under sparsity. In many specifications $w_n$ does not actually depend on $n$, but we allow it for generality.
The structural parameters of interest are $\theta_{0}:=\left(\beta_{0},\gamma_{0}\right)\in \Theta$. Throughout this paper, we use the shorthand notation:
so that the link formation equation becomes
A distinguishing feature of our model is the presence of the endogenous covariates $X_{ij}$ and the fixed effects $A_i,A_j$. In particular, the endogenous covariates $X_{ij}$ may arise as functions of the realized network. Specifically, we allow:
where $\phi_{ij}$ is a known function mapping the realized network $Y$ and exogenous characteristics $Z=(Z_{1},\ldots,Z_{n})$ to the dyadic covariate. For example, a component of $X_{ij}$ might be the number of common friends between $i$ and $j$:
which captures transitivity effects in network formation. Other examples include measures of local clustering, degree statistics, and their rescaled or normalized variants.
Since $X_{ij}$ depends on the entire network $Y$, and $Y$ is determined by equation (ref) simultaneously for all pairs, the model describes a strategic network formation game under transferable utilities with pairwise stability as the solution concept. In general, the game may have multiple equilibria for a given realization of primitives $(Z,A,\varepsilon)$, where $A=(A_{1},\ldots,A_{n})$ and $\varepsilon=(\varepsilon_{ij})_{i<j}$ collect the fixed effects and idiosyncratic shocks respectively. We represent the realized network abstractly as:
where $g$ incorporates both the equilibrium correspondence and an (arbitrary and possibly unknown) equilibrium selection mechanism that picks a single equilibrium for each realization of the primitives.
The equilibrium mapping $g$ is generally intractable to characterize. Even for simple specifications of $\phi_{ij}$, the fixed-point nature of the equilibrium creates a complex interdependence structure, which is exacerbated by the discrete combinatorial nature of the graph space. As will be shown below, our identification approach does not require characterization of $g$: we extract identifying information using monotonicity restrictions that hold regardless of the complexity of the equilibrium correspondence and any equilibrium selection mechanisms.
We maintain the following assumptions throughout the analysis.
Assumption (ref) is standard in the network formation literature. Note that arbitrary dependence between $Z_{i}$ and $A_{i}$ within an agent is allowed, as is arbitrary dependence between $A_i$ and $\varepsilon_{ij}$. The latter flexibility is inessential for the tetrad-based restrictions in Section (ref), since the tetrad differencing construction eliminates all fixed effects algebraically, but it avoids imposing unnecessary restrictions on the model.
Assumption (ref) is an exogeneity condition requiring that the observables $Z$ are independent of the unobserved pairwise shocks. Note that we do not assume that $X$ is independent of $\varepsilon$: since $X$ is a function of the equilibrium network, it is endogenous and generally correlated with the idiosyncratic shocks. The key distinction is between the exogenous covariates $Z$ (which satisfy independence) and the endogenous covariates $X$ (which do not).
Assumption (ref) is another standard assumption in the network formation literature on the idiosyncrasy of link-level surplus shocks.
Assumption (ref) is a high-level regularity condition asserting that conditional probabilities of tetrad events can be consistently estimated from a single large network. The conditioning is on the individual-level characteristics $(Z_i,Z_j,Z_h,Z_k)$, which is consistent with the general subnetwork formulation (Assumption (ref) below). Since the link formation equation (ref) depends on agents $i$ and $j$ only through the dyadic covariate $Z_{ij}=w_n(Z_i,Z_j)$, it is convenient to define the shorthand
for the vector of dyadic covariates determined by the tetrad. Because $\zeta_{ijhk}$ is a deterministic function of $Z_S$, any conditional probability $\mathbb{P}(E\mid \zeta_{ijhk}=\zeta)$ is well-defined under the finer conditioning of Assumption (ref), and we use the shorthand $\mathbb{E}[\cdot\mid\zeta]$ throughout Section (ref) for readability.
This assumption is satisfied when the network exhibits sufficiently weak dependence so that a Law of Large Numbers (LLN) applies to tetrad statistics. We view Assumption (ref) as a convenient high-level condition for presenting the identification arguments; Section (ref) provides primitive sufficient conditions under which it holds for the class of strategic formation games considered here, drawing on the network limit theory of leung2019treatment.
This section develops the main identifying restrictions for the structural parameters $\theta_{0}=(\beta_{0},\gamma_{0})$. We begin with tetrad-based restrictions that achieve complete elimination of all individual fixed effects, then present additional triad-based restrictions as well as a general class of cycle-based ones.
Our core idea is to combine the “tetrad differencing” technique with the “bounding-by-$c$” technique together to derive bounds free of potentially complicated endogenous variables and unobserved fixed effects.
Consider a tetrad of distinct agents $ijhk\equiv(i,j,h,k)$. We first explain the tetrad-differencing technique. Specifically, consider the following tetrad event
which by (ref) is equivalent to the event \[v_{ij}\leq\delta_{ij},\quad v_{hk}\leq\delta_{hk},\quad v_{ik}>\delta_{ik},\quad v_{jh}>\delta_{jh}. \] which implies the following
We observe that the fixed effects are all differenced out in $\Delta v$:
Since $\Delta v=\Delta\varepsilon$ by (ref), the inequality (ref) is equivalent to
Now, we combine the above with the “bounding-by-$c$” technique. Specifically, consider the intersection of the tetrad event (ref) with the event $\{\Delta\delta\leq c\}$, i.e.,
which by (ref) implies \[ \Delta\varepsilon<\Delta\delta \text{ and }\ \Delta\delta\leq c. \] The above further implies
an inequality on the exogenous $\Delta\varepsilon$ without fixed effects or endogeneity issues.
Let $F_{\Delta}$ denote the CDF of $\Delta\varepsilon$. Since the intersection event (ref) implies (ref), taking conditional probabilities given $\zeta_{ijhk}=\zeta$ yields
by the independence of $\varepsilon$ and $Z$ in Assumption (ref).\footnote{Although $\Delta\delta$ involves the endogenous covariates $X_{ij}$ and hence is not $\sigma(Z)$-measurable, the inequality (ref) is valid because the bound $\mathbf{1}\{\text{tetrad event}\}\cdot\mathbf{1}\{\Delta\delta\leq c\}\leq\mathbf{1}\{\Delta\varepsilon\leq c\}$ holds pointwise (i.e., for every realization of $(Y,X,Z,A,\varepsilon)$), and taking conditional expectations of both sides preserves the inequality. The right-hand side $\mathbb{P}(\Delta\varepsilon\leq c\mid \zeta_{ijhk}=\zeta)=F_\Delta(c)$ follows from $\varepsilon\perp Z$, since $\zeta_{ijhk}$ is a measurable function of $(Z_1,\ldots,Z_n)$.}
Similarly, we can obtain another bound by considering the “flipped” event \[ Y_{ij}=Y_{hk}=0\ \text{ and }\ Y_{ik}=Y_{jh}=1\ \text{ and } \ \Delta\delta>c, \] which implies $\Delta\varepsilon>\Delta\delta>c$ and hence \[(1-Y_{ij})(1-Y_{hk})Y_{ik}Y_{jh}\mathbf{1}\{\Delta\delta>c\}\leq\mathbf{1}\{\Delta\varepsilon>c\},\] yielding
or equivalently,
To state our main results, it is helpful to make explicit the dependence of the index difference on the model parameters and the tetrad $ijhk$. In the following we write the tetrad index difference
and two parametrized conditional probabilities (expectations)
We now exploit the above to build identifying restrictions, which have two variants depending whether researchers impose an additional parametric assumption on $\varepsilon$.
We first explain the case where no parametric assumption is made on $\varepsilon$ and $F_\Delta$ is left unknown. In this case, (ref) and (ref) are not identifying restrictions per se since they involve the unknown $F_\Delta(c)$. However, (ref) and (ref) together imply \[ p_{L}(\zeta,c;\theta_0)\ \leq F_\Delta(c)\ \leq p_{U}(\zeta,c;\theta_0), \] and then, since the middle term is constant across $\zeta$, \[ \sup_{\zeta}p_{L}(\zeta,c;\theta_0)\ \leq F_\Delta(c)\ \leq \inf_{\zeta}p_{U}(\zeta,c;\theta_0), \] which becomes an identifying restriction once we drop the unknown middle term $F_\Delta(c)$. We summarize our result in the following theorem.
Next, we consider the case where researchers impose an additional parametric assumption on $\varepsilon$, so that $F_\Delta(c;\theta_\varepsilon)$ is unknown up to a finite-dimensional parameter $\theta_\varepsilon$. Practically, if $\varepsilon$ is assumed to follow a distribution with location-scale parameters such as the logistic distribution graham2017econometric and the normal distribution dzemski2019empirical, then there is no need to explicitly introduce the parameter $\theta_\varepsilon$ given the scope of location and scale normalization in model (ref). That said, we keep the notation $\theta_\varepsilon$ for theoretical completeness.
With $F_\Delta(c;\theta_\varepsilon)$ known up to $\theta_\varepsilon$, the inequalities (ref) and (ref) become identifying restrictions themselves, which we summarize in the following theorem.
Several features of Theorems (ref) and (ref) are worth noting. First, the tetrad construction eliminates all individual fixed effects by purely algebraic differencing within a four-agent configuration, so identification does not rely on sufficient-statistic arguments (as in parametric logit) nor on restrictions such as $A_{i}\perp Z_{i}$. Second, the “bounding by $c$” step addresses the endogeneity of equilibrium network statistics by treating the endogenous index components as random variables and conditioning on observable events that imply inequalities in $\Delta\varepsilon$; in particular, we never need to model, estimate, or simulate the conditional distribution of $X_{ij}$. Third, because the restrictions are derived from monotonicity and equilibrium feasibility alone, they remain valid without computing the equilibrium mapping $g$ and are robust to equilibrium multiplicity and unknown equilibrium selection. We note that the $n$-dependence of the covariate construction $w_n$ (introduced in Section (ref)) plays no role in the identification arguments above: the bounding-by-$c$ restrictions hold for each fixed network size $n$ and are purely algebraic. The $n$-dependence becomes relevant only at the estimation stage, where one must verify that the conditional probabilities $p_L$ and $p_U$ can be consistently estimated; see Remark (ref) for a detailed discussion.
While tetrad restrictions achieve complete elimination of fixed effects, they are not the only source of identifying restrictions. We now develop complementary restrictions based on triads---subnetwork configurations involving three agents. Triad-based restrictions do not fully eliminate all fixed effects, but they have two practical advantages. First, triads are combinatorially more abundant than tetrads ($\binom{n}{3}$ versus $\binom{n}{4}$ configurations), yielding more observations per network. Second, they exploit different aspects of the latent variable structure and can tighten the identified set when combined with tetrad restrictions.
In each triad configuration, the differencing construction eliminates some agents' fixed effects but retains others. The agents whose fixed effects survive---i.e., those with nonzero incidence load $\sigma_i\neq 0$ in the notation of Section (ref)---are the retained agents. The residual latent composite depends on agent heterogeneity only through the retained agents, and by the i.i.d.\ structure of $(Z_i,A_i)$ (Assumption (ref)), its conditional law given the retained agents' exogenous characteristics does not depend on the characteristics of the differenced-out agents. This yields bounds that are sharper than unconditional versions, because they avoid averaging over the heterogeneity in $A_i\mid Z_i$ for the retained agents. We consider several variants below.
Consider a triad $(i,j,k)$ and the configuration where agent $i$ is linked to both $j$ and $k$, but $j$ and $k$ are not linked. Given the event $\{Y_{ij}=Y_{ik}=1,\,Y_{jk}=0\}$, the link inequalities imply
The left-hand side equals
so intersecting with $\{\delta_{ij}+\delta_{ik}-\delta_{jk}\le c\}$ yields the indicator bound \[ Y_{ij}Y_{ik}(1-Y_{jk}) \mathbf{1}\{\delta_{ij}+\delta_{ik}-\delta_{jk}\le c\}\le\mathbf{1}\{u_{i,jk}\le c\}. \] Unlike the tetrad case, the fixed effect $A_i$ does not cancel: the incidence loads are $\sigma_i=+2$, $\sigma_j=\sigma_k=0$, so agent $i$ is the sole retained agent whose fixed effect survives. The residual $u_{i,jk}=2A_i+\varepsilon_{ij}+\varepsilon_{ik}-\varepsilon_{jk}$ depends on $(A_i,\varepsilon_{ij},\varepsilon_{ik},\varepsilon_{jk})$, which by Assumptions (ref)--(ref) are jointly independent of $(Z_j,Z_k)$ conditional on $Z_i$. Consequently, the conditional distribution of $u_{i,jk}$ given $Z_i=z_i$ does not depend on $(Z_j,Z_k)$. Let
denote this conditional CDF. Taking expectations conditional on $(Z_i=z_i,Z_j=z_j,Z_k=z_k)$ and then taking suprema over $(z_j,z_k)$ for each fixed $z_i$ yields the moment-inequality bounds stated below.
One may then incorporate the information in Proposition (ref) into the characterization of the identified set by arguments analogous to those in Theorems (ref) and (ref).
A complementary restriction compares two links originating from a common agent. Specifically, fix $i$ and consider the event that $i$ links to $j$ but not to $k$. Given the observable event $\{Y_{ij}=1,\,Y_{ik}=0\}$, the link inequalities imply \[ v_{ij}-v_{ik}<\delta_{ij}-\delta_{ik}. \] Since $v_{ij}-v_{ik}=(A_{i}+A_{j}+\varepsilon_{ij})-(A_{i}+A_{k}+\varepsilon_{ik})$, the fixed effect $A_i$ cancels (it has incidence load $\sigma_i=0$), leaving
Using the “bounding-by-$c$” technique and intersecting the above with $\{\delta_{ij}-\delta_{ik}\le c\}$, we obtain \[ Y_{ij}(1-Y_{ik}) \mathbf{1}\{\delta_{ij}-\delta_{ik}\le c\}\le\mathbf{1}\{u_{i,j,k}\le c\}. \] Here agents $j$ and $k$ are the retained agents ($\sigma_j=+1$, $\sigma_k=-1$), while agent $i$ is differenced out ($\sigma_i=0$). The residual $u_{i,j,k}=A_j+\varepsilon_{ij}-A_k-\varepsilon_{ik}$ depends on $(A_j,A_k,\varepsilon_{ij},\varepsilon_{ik})$, which is independent of $Z_i$ but may depend on $(Z_j,Z_k)$ through the correlation of $A_j$ with $Z_j$ and $A_k$ with $Z_k$. Let
denote the conditional CDF, which does not depend on $Z_i$. Replicating the “flipped” argument, taking expectations conditional on $(Z_i=z_i,Z_j=z_j,Z_k=z_k)$, and then taking the supremum over $z_i$ for each fixed $(z_j,z_k)$, we obtain the following.
The tetrad and triad restrictions developed above use equal weights on links. A natural extension is to consider weighted combinations of link indicators, which can generate additional identifying restrictions by exploiting different linear combinations of the latent inequalities. We illustrate the idea with a single example; the systematic development of weighted and more general differencing schemes is presented in Section (ref).
Within a tetrad $(i,j,k,\ell)$, consider the link configuration:
The event $\{Y_{ij}=Y_{ik}=1,\,Y_{i\ell}=0\}$ implies $v_{ij}\le \delta_{ij}$, $v_{ik}\le \delta_{ik}$, and $v_{i\ell}>\delta_{i\ell}$. Summing these inequalities with weights $(+1,+1,-2)$, we obtain \[ v_{ij}+v_{ik}-2v_{i\ell}<\delta_{ij}+\delta_{ik}-2\delta_{i\ell}. \] Since $v_{ij}+v_{ik}-2v_{i\ell}=(A_{j}+\varepsilon_{ij})+(A_{k}+\varepsilon_{ik})-2(A_{\ell}+\varepsilon_{i\ell})$, by intersecting with $\{\delta_{ij}+\delta_{ik}-2\delta_{i\ell}\le c\}$ we can derive the bound
The incidence loads are $\sigma_i=0$, $\sigma_j=+1$, $\sigma_k=+1$, $\sigma_\ell=-2$, so agent $i$ is differenced out while $j$, $k$, and $\ell$ are retained. The weighted latent index on the right-hand side depends on $(A_j,A_k,A_\ell,\varepsilon_{ij},\varepsilon_{ik},\varepsilon_{i\ell})$, which is independent of $Z_i$ but may depend on $(Z_j,Z_k,Z_\ell)$ through the correlation of each $A_m$ with $Z_m$. Let
denote the conditional CDF, which does not depend on $Z_i$. Taking expectations conditional on $(Z_i=z_i,Z_j=z_j,Z_k=z_k,Z_\ell=z_\ell)$ and then taking the supremum over $z_i$ for each fixed $(z_j,z_k,z_\ell)$ yields the following.
The potential advantage of weighted differencing is that it generates a richer family of moment inequalities, indexed by the weights, which may tighten the identified set beyond what is achievable with the equal-weight tetrad and triad restrictions alone.
It should now be clear that our core identification idea can be generalized further beyond the identifying restrictions developed in Sections (ref)--(ref), which are all instances of a unified principle: assign signed weights to links in a subgraph, and when the weighted sum of link indicators factors into a product of indicators, the “bounding by $c$” technique yields bounds involving a weighted sum of latent variables. Fixed effects cancel at any agent whenever the sum of weights on links incident to that agent equals zero. We now formalize this observation.
Clearly, a weighted link configuration achieves complete fixed-effect elimination if $\sigma_{i}=0$ for every $i\in S$, i.e., if the weights incident to each agent sum to zero. The identifying restrictions from earlier sections correspond to specific weighted link configurations:
We are now ready to present an umbrella result for general weighted link configurations. Fix a weighted link configuration $(E_S,\omega)$ on $S$, and let
Given the event $\{\prod_{e\in E^{+}}Y_{e}\prod_{e\in E^{-}}(1-Y_{e})=1\}$, each $e\in E^{+}$ implies $v_{e}\le\delta_{e}$ and each $e\in E^{-}$ implies $v_{e}>\delta_{e}$. Multiplying the corresponding inequalities by $\omega_{e}$ and summing yields \[ \sum_{e\in E_S}\omega_{e}v_{e}<\sum_{e\in E_S}\omega_{e}\delta_{e}. \] Then \[ \sum_{e\in E_S}\omega_{e}v_{e}=\sum_{i\in S}\sigma_{i}A_{i}+\sum_{e\in E_S}\omega_{e}\varepsilon_{e}=:U_S. \] Hence, the fixed effect of any agent with $\sigma_i=0$ is differenced out, while agents with $\sigma_i\neq 0$ contribute residual fixed-effect terms. As in the triad and weighted-differencing cases, we can sharpen the resulting bounds by conditioning on the exogenous characteristics of the retained agents. Since $U_S=\sum_{i\in S_R}\sigma_i A_i+\sum_{e\in E_S}\omega_e\varepsilon_e$, where $S_R:=\{i\in S:\sigma_i\neq 0\}$, the conditional law of $U_S$ given $Z_{S_R}=z_R$ does not depend on $Z_{S_0}$: the retained agents' fixed effects are integrated out through the conditional distribution of $A_i\mid Z_i=z_i$ for $i\in S_R$, while the differenced-out agents' characteristics serve as profiling variables. Adding the “bounding-by-$c$” event $\{\sum_{e\in E_S}\omega_{e}\delta_{e}\leq c\}$ yields a bound in terms of $U_S$, which translates to the following.
To apply the cycle-based restrictions in Proposition (ref) and aggregate them over a class $\mathcal{W}$ that involve subnetwork structures larger than tetrads, we need an analogue of Assumption (ref) that guarantees identification of the required subnetwork conditional probabilities.
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Define the joint surplus from a link between $i$ and $j$, viewed as a function of the endogenous covariate value $x$, as
Because the endogenous covariate $X_{ij} = \phi_{ij}(Y,Z)$ depends on the realized network, the value of $x$ at which the surplus is evaluated is itself an equilibrium object. Each potential link $(i,j)$ falls into exactly one of three categories, determined entirely by the exogenous primitives $(Z,A,\varepsilon)$:
The connected components $C_i$ of the non-robustness graph $\bm{D}$ partition the agent set $\mathcal{N}_n$, but the strategic neighborhoods $C_i^+$ can overlap: an agent who is a robust neighbor of two distinct non-robustness components belongs to both of their strategic neighborhoods. Nevertheless, strategic neighborhoods localize the equilibrium: the link outcomes within any strategic neighborhood $C^+$ depend only on the primitives of agents in $C^+$. To see why, consider a non-robust link $(a,b)$ with $a,b\in C$ for some component $C$. The endogenous covariate $X_{ab}=\phi_{ab}(Y,Z)$ depends on other link outcomes, but only links incident to $a$ or $b$ contribute. Any such link $(a,k)$ is either robustly absent ($Y_{ak}=0$, no contribution), robustly present ($k\in\mathcal{N}_{\bm\Pi}(a,1)\subset C^+$, value determined by primitives of $\{a,k\}\subset C^+$), or non-robust ($k\in C_a=C$, status determined by the equilibrium within $C^+$). Hence the equilibrium on $C^+$ forms a self-contained fixed-point system in the primitives of agents in $C^+$ leung2019treatment. We formalize this via the following assumptions.
Assumption (ref) is satisfied by common friends $\text{CF}_{ij}=\sum_k Y_{ik}Y_{jk}$, the Jaccard index, degree statistics, and other standard endogenous covariates used in the network formation literature.
Unscaled common-friends counts $\mathrm{CF}_{ij}=\sum_k Y_{ik}Y_{jk}$ can grow with network size in dense networks. In the sparse regime of Assumption (ref) below, however, expected degree is $O(1)$, so the expected number of common friends is also $O(1)$. The Jaccard index takes values in $[0,1]$ by construction, and normalized common friends $\mathrm{CF}_{ij}/n$ are bounded as well. For unnormalized count statistics in denser networks, explicit trimming or normalization would be needed to satisfy Assumption (ref).
Assumption (ref) requires that equilibrium play within each strategic neighborhood is determined locally. As discussed in leung2019treatment, this is satisfied by myopic best-response dynamics, which are widely used in the theoretical and econometric literature on network formation.
Assumption (ref) ensures that chains of strategic dependencies do not propagate through the network. It follows from the branching-process domination argument in Theorem 1 of leung2019treatment under sparsity and subcriticality. Intuitively, exploring $C_i^+$ via breadth-first search is akin to growing a subcritical branching process that almost surely terminates.
The proof is given in Appendix (ref).
The identified sets in Theorems (ref) and (ref) are defined by conditional moment inequalities and are therefore generally set-valued. It is natural to ask when these restrictions are sharp enough to yield point identification. In this section we give conditions under which $\theta_0=(\beta_0,\gamma_0)$ is point identified, exploiting the algebraic structure of the logistic distribution.
When $\gamma_0\neq 0$, two complications arise. First, the endogenous covariates $X_{ij}=\phi_{ij}(Y,Z)$ create dependence among the four tetrad link outcomes, so the product formula for $\mathbb{P}(\text{tetrad}\mid\zeta,A)$ breaks down. Second, the tetrad index difference $\Delta\delta=\Delta Z'\beta_0+\Delta X'\gamma_0$ depends on $A$ through the equilibrium, so the conditional-on-$A$ odds ratio varies with $A$ and cannot be factored out when integrating over $A$. We handle both problems by conditioning on the realized endogenous covariate values and restricting attention to isolated tetrads in which the endogenous covariates for the four tetrad links are invariant across the tetrad and flipped patterns.
Fix a tetrad of distinct agents $(i,j,h,k)$. Recall that the four tetrad links are $E_t:=\{ij,hk,ik,jh\}$; the remaining two links within the quadruplet are the diagonal links $\{ih,jk\}$. Write $Y_{-E_t}$ for the collection of all link outcomes outside $E_t$ and $\varepsilon_{E_t}:=(\varepsilon_{ij},\varepsilon_{hk},\varepsilon_{ik},\varepsilon_{jh})$ for the tetrad shocks. Define the tetrad endogenous covariate vector
and the tetrad exogenous covariate vector $\zeta_{ijhk}$ as in (ref).
We introduce three structural conditions on admissible tetrads. Together they guarantee that the log-odds of the tetrad pattern versus the flipped pattern is a linear function of the covariates, allowing the fixed effects to cancel.
For an admissible tetrad $(i,j,h,k)$ with $Y_{ih}=Y_{jk}=0$, define the tetrad pattern and flipped pattern
Assumption (ref) requires that the endogenous covariates of the four tetrad links are invariant across the two comparison patterns. It is weaker than requiring that $X_e$ not depend on any tetrad link outcome globally (as would follow from the bilinear condition $X_{ij}=\psi(\{Y_{il}Y_{jl}\}_{l\neq i,j},Z)$): bilinearity implies (ref) globally, whereas Assumption (ref) only requires it on the comparison sample $T_t\cup F_t$. The following examples verify the condition for the two leading specifications.
Assumption (ref) requires that the tetrad-specific shocks $\varepsilon_{E_t}$ do not affect any link outcome outside the tetrad. Under the strategic-neighborhood framework of Section (ref), a sufficient condition is that no agent outside the tetrad belongs to a non-robustness component containing a tetrad agent: formally, $C_a\cap\{i,j,h,k\}^c=\emptyset$ for each $a\in\{i,j,h,k\}$, where $C_a$ is agent $a$'s connected component in the non-robustness graph $\bm{D}$. This ensures that all non-robust links involving tetrad agents are links among the four tetrad agents themselves, so perturbing $\varepsilon_{E_t}$ cannot propagate outside the tetrad. Note that this condition does not require the four agents' strategic neighborhoods to be mutually disjoint---agents $i$ and $j$ may share robust neighbors, and the link $Y_{ij}$ itself may be non-robust, provided the non-robustness component containing $i$ and $j$ lies entirely within $\{i,j,h,k\}$. Under the bounded-strategic-neighborhood condition (Assumption (ref)), this isolation condition holds for a positive fraction of quadruplets.
Assumption (ref) rules out equilibrium multiplicity within the tetrad on the comparison sample $T_t\cup F_t$. Note that $T_t$ and $F_t$ themselves can never coexist as equilibria under Assumption (ref): their best-response conditions require the shock $\varepsilon_e$ to fall on opposite sides of the same threshold $\tilde\delta_e$ for every $e\in E_t$, so the two events are mutually exclusive. What Assumption (ref) additionally rules out is coexistence of $T_t$ (or $F_t$) with a third tetrad pattern.
In a sparse network with expected degree $O(1)$, each link exists with probability $O(n^{-1})$, so the diagonal-absence condition holds with probability approaching $1$ for any given quadruplet. Consequently, the number of admissible tetrads is $\Theta(n^4)$, and after the greedy packing argument of Proposition (ref) one retains $\Theta(n)$ independent tetrads---the same asymptotic order as in the partial-identification setting.
The point identification result requires a strengthened version of Assumption (ref) that conditions on the realized endogenous covariates and the diagonal link absences.
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The tetrad differencing eliminates all individual fixed effects algebraically---exactly as in the $\gamma_0=0$ case---while the endogenous covariates are handled by the isolation conditioning. The two mechanisms do not interfere with each other: fixed effects cancel regardless of the endogenous covariates, and endogenous covariates decouple regardless of the fixed effects.
The proof is given in Appendix (ref).
The proof of Theorem (ref) is constructive and yields a computationally simple estimator.
We examine the finite-sample behavior of the tetrad-based partial identification approach through simulation, focusing on how the network size $n$, unobserved heterogeneity (fixed effects) $A$, the support size of the discrete exogenous covariate $Z$, and the endogenous covariate $X$ affect the sharpness of the identified set. For simplicity, we use only the baseline tetrad restrictions (Theorems (ref)--(ref)), leaving the longer-cycle restrictions and aggregation over $\mathcal{W}$ for future work.
We conduct simulation studies under three nested specifications, which differ by whether individual fixed effects and/or endogenous covariates are present.
\paragraph{Baseline Model.}
The individual-level exogenous characteristics $Z_{i}$ is a two-dimensional vector $(Z_{1i},Z_{2i})\in[-10,10]^{2}$, and the vector of exogenous dyadic covariates is constructed as $Z_{ij}=|Z_{i}-Z_{j}|$. The true link formation equation is
Under all three specifications, the shocks $\varepsilon_{ij}$ are i.i.d. following a $\text{Logistic}(0,1)$ distribution.
\paragraph{FE-only Model.}
We add individual-level unobserved fixed effects $A_{i}\sim\mathcal{N}(\rho Z_{1i},\sigma_{A}^{2})$ to the baseline model, where $\rho\in[0,1]$ captures the correlation between unobserved heterogeneity and observed characteristics. The true link formation equation becomes
Note that this is equivalent to the main model (ref) with $\tilde A_i:=-A_i$ playing the role of the fixed effect (and $-\varepsilon_{ij}$ for the shock, which has the same logistic distribution by symmetry).\footnote{Under this reparametrization, $\tilde A_i\sim\mathcal{N}(-\rho Z_{1i},\sigma_A^2)$ and equation (ref) becomes $Y_{ij}=\mathbf{1}\{Z_{ij,1}+\gamma_0 Z_{ij,2}+\varepsilon_{ij}\ge\tilde A_i+\tilde A_j\}$, matching the sign convention in (ref). The tetrad restrictions are invariant to this reparametrization since $A_i$ cancels in all cases.}
\paragraph{Full Model.}
We further add an endogenous dyadic covariate $X_{ij}$ to the model while reducing the dimension of $Z_{ij}$ to 1, i.e., $Z_{i}\in\mathcal{Z}=[-10,10]$ and $Z_{ij}=|Z_{i}-Z_{j}|$. We set the coefficient for $Z_{ij}$ to be $\beta_{0}=1$, fixed and known. The endogenous covariate is the Jaccard index of common friends: \[ X_{ij} = \frac{|N(i)\cap N(j)|}{|N(i)\cup N(j)|}, \] where $N(i)=\{k:\,Y_{ik}=1\}$ denotes the neighbor set of $i$. The true link formation equation becomes
Under this model, the probabilities involved in the identifying restriction (ref) no longer have a closed form. Therefore, we need to approximate the probabilities using a realized network.
In order to generate a realized network, we first discretize $\mathcal{Z}$ and generate $Z_{i}$ from $\mathcal{Z}$ by randomly sampling with equal probabilities. Since $X_{ij}$ depends on the realized network $Y$, it requires solving for a network consistent with (ref) given the shocks and fixed effects. We set $A_{i}\overset{\text{i.i.d}}{\sim}\mathcal{N}(0,1)$ and generate data by iterating a link-update procedure (starting from a zero $Y$ matrix and recomputing $X_{ij}$ after each update) until convergence. The parameter to be identified is $\gamma_{0}$.
We use $\gamma$ to denote the parameter to be identified. Following Theorem (ref), we evaluate the (sample analogue of the) tetrad criterion $Q^{\mathrm{tetrad}}(\gamma)$ defined in (ref), where the relevant conditional probabilities $p_{L}$ and $p_{U}$ are computed (i) in closed form under the baseline model, (ii) by Monte Carlo integration over fixed effects under the FE-only model, and (iii) by sample proportions under the full model. The identified set is the set of values for which the criterion is non-positive, i.e. $Q^{\mathrm{tetrad}}(\gamma) \leq 0.$
Under the full model, we also implement the stronger criterion that enforces both sides of the tetrad restriction under the parametric assumption on the logistic distribution of $\varepsilon_{ij}$, as in Theorem (ref); the resulting identified set is denoted by $\Gamma_{\mathrm{strict}}$.
We evaluate the criterion functions on a finite grid of $\gamma$ values. The supremum is computed by the GenSA algorithm under the baseline model and the FE-only model, while it is computed by grid-search under the full model.
Figure (ref) shows how the value of the main criterion function $Q(\gamma)$ changes with $\gamma$ between $-10$ and $10$ when the true parameter is $\gamma_{0}=1$. In the figure, $Q$ is plotted after shifting by $+1$, so the identified set $\{\gamma:Q(\gamma)\le0\}$ corresponds to the region where the plotted curve attains its minimum value of $1$. The identified set for $\gamma$ is $[1,5]$.
Table (ref) reports the identifying results for $\gamma$ under the FE-only model (ref) with fixed-effect designs that vary (i) the dispersion of individual fixed effects and (ii) the strength of correlation between individual fixed effects and observed individual characteristics $Z_{i}$.
Under the FE-only model, introducing unobserved heterogeneity substantially weakens identification of $\gamma$ relative to the baseline model without fixed effects. In particular, while the baseline model yields a finite identified set $[1,5]$ when the true value is $\gamma_{0}=1$, the FE-only model produces noticeably wider sets that become sensitive to both the dispersion and endogeneity of fixed effects. When fixed effects are uncorrelated with individual characteristics and have small dispersion, the identified set remains bounded above but expands a little compared to the baseline; increasing heterogeneity makes the upper bound disappear, yielding one-sided identification. Holding dispersion fixed, correlation between the fixed effects and individual characteristics also weakens identification when the correlation is strong. Overall, unobserved heterogeneity, especially when strongly correlated with observables or with large dispersion, reduces the informativeness of the identifying criterion and tends to eliminate finite upper bounds on $\gamma$, but the sign of $\gamma$ can still be identified in all cases.
Table (ref) reports the identified sets obtained from both the main criterion $Q(\gamma)\le0$ and the strict criterion (Theorem (ref)), using the Jaccard index as the endogenous covariate with $n=100$ and a single network draw.
The identified set tightens as the support size $|\mathcal{Z}|$ grows, since more exogenous variation enables more informative tetrad comparisons. At $|\mathcal{Z}|=21$, the strict identified set is $[4,11]$, a bounded interval of width $7$ containing the true value $\gamma_{0}=4$. This shows that the tetrad restrictions can produce nontrivial bounds on the strategic-interaction parameter even with endogenous covariates and unobserved individual fixed effects.
These results are preliminary: they are based on a single network draw at $n=100$, and the current design fixes $\beta_0$ at its true value and searches only over $\gamma$, which does not address the challenges of joint identification over the full parameter vector $(\beta,\gamma)$. A more systematic Monte Carlo study---varying network sizes, averaging across repeated draws, searching over the full parameter space, and constructing formal confidence sets for the identified set (e.g., along the lines of andrews2013inference and \citealt*{chernozhukov2013intersection})---is under investigation and will be reported in a subsequent version of this paper.
This paper develops a tractable identification approach for strategic network formation models with endogenous network statistics and unobserved individual fixed effects. The main idea is a “bounding-by-$c$” construction applied to subnetwork configurations (tetrads, triads, and more general weighted cycles), which produces moment-inequality restrictions without requiring characterization of the equilibrium mapping. Section (ref) gives primitive conditions under which the high-level Assumption (ref) holds, embedding the model in the sparse network framework of leung2019treatment and establishing consistency of the tetrad conditional probability estimator via a greedy packing argument.
Several directions remain for future work. First, the central limit theorems of leung2019normal and menzel2021clt can be used to develop formal inference procedures for the identified sets, building on the econometric theory of inference based on conditional moment inequalities \citep*{andrews2013inference,chernozhukov2013intersection}. Second, a more systematic set of Monte Carlo simulations would provide a fuller picture of the finite-sample performance of both the set-identification approach and the point identification result, together with the associated estimator. Third, an empirical application using real-world network data is a natural next step.