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Homophily and transitivity in dynamic network formation

Kevin Dano, Bryan S. Graham, Yassine Sbai Sassi

arXiv:2609.14049v1econ.EM

TL;DR

The paper asks how to distinguish structural tastes for transitive links from clustering caused by unobserved homophily in dynamic networks. It develops conditional-likelihood methods that remove high-dimensional nuisance parameters while retaining information about state dependence and transitivity. The resulting identification is constructive, but the usable variation and information accumulation in a single large network remain delicate.

  • Problem

    The paper addresses how to identify a taste for transitivity when structural transitivity and unobserved homophily can both generate network clustering.

  • Method

    The paper constructs conditional likelihoods invariant to dyad-specific heterogeneity and the initial network, then develops tractable exact and asymptotic inference procedures.

  • Results

    The conditional likelihood identifies θ=(α,β) while eliminating the nuisance parameters, and its factorization supports conditionally independent multiplicands for single-network approximations.

  • Takeaways & Limitations

    Time variation and time-invariant heterogeneity can be used constructively to distinguish transitivity from homophily, although information accumulation in one network is delicate.

  • Takeaways & Limitations

    The tractable conditional approach conditions on only a subset of network sequences with common sufficient statistics, causing some loss of information.

Abstract

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In social and economic networks linked agents often share connections in common. There are two competing explanations for this phenomenon. First, agents may have a structural taste for transitive links - the returns to linking may be higher if two agents share a common connection. Second, agents may assortatively match on unobserved attributes, a process called homophily. We study parameter identifiability in a simple model of dynamic network formation with both effects. Agents form, maintain, and dissolve links over time to maximize utility. The return to linking may be higher if agents share connections in common. A pair-specific utility component allows for arbitrary homophily on time-invariant agent attributes. We derive conditions under which it is possible to detect the presence of a taste for transitivity in the presence of assortative matching on unobservables. We leave the joint distribution of the initial network and the pair-specific utility component, both high dimensional nuisance parameters, unrestricted. Our identification result is constructive, suggesting an analog estimator, whose finite and (single) large network properties we characterize. We show, via examples, the delicacy of information accumulation in the single (large) network setting.

1 A dynamic network formation model

The model describes dynamic network formation with state dependence, transitive-link incentives, and unrestricted dyad-specific heterogeneity that can represent unobserved homophily. Repeated network observations and time-invariant heterogeneity provide the basis for learning the two structural parameters.

  • 1 A dynamic network formation model: The observed graph evolves over periods as agents add, remove, or maintain links, with D_t recording all undirected edges and no self-loops.The initial network is left unmodeled, while subsequent choices maximize per-period payoff functions.
  • 1 A dynamic network formation model: Time-invariant dyad-specific heterogeneity A captures pair-specific link payoffs, allowing dependence from extroversion, degree-related traits, or homophily on shared latent attributes.The distribution of A is left nonparametric, and realized A may be treated as highly dimensional or nonstochastic.
  • 1 A dynamic network formation model: Agent i values a link with j more when they were previously connected or shared friends in the prior period, with β measuring the transitivity component.The common-neighbor term counts agents connected to both i and j in period t−1.
  • 1 A dynamic network formation model: Links are governed by pairwise stability with transfers: an existing dyad must weakly benefit jointly from its edge, while a nonedge must not jointly benefit from adding one.Transferable utility permits agents to share any surplus generated by a link.
  • 1 A dynamic network formation model: The initial joint distribution of D_0 and A is unrestricted, while the model assumes random utility shocks are identically distributed across dyads and over time.This leaves the initial network condition unmodeled rather than imposing a parametric initial-state specification.
  • Some observations: Conditional on A, the network follows a time-homogeneous Markov process, although it is generally not Markov after integrating out the unobserved heterogeneity.The model is a structured nonlinear vector-autoregressive process in which prior networks influence current link formation.
  • Some observations: The model includes three link-formation forces: state dependence through α, triadic closure through β, and unobserved good fundamentals represented by A_ij.Both structural transitivity and homophily can therefore generate clustering, making identification and estimation difficult.
  • Some observations: Because unrestricted A can generate any network configuration, the analysis uses multiple observations over time and time-invariance of A to circumvent this identification problem.The target θ=(α,β) has only two elements, whereas A and the initial-network distribution are high-dimensional nuisance objects.

2 Conditional likelihood analysis

The paper conditions on network-sequence statistics that remove the initial network and dyad-specific heterogeneity, yielding likelihoods indexed by the finite-dimensional parameter θ. A star-system restriction makes this likelihood tractable and supports fixed-effects identification despite cross-dyad dependence.

  • The conditional likelihood is invariant to dyad-specific heterogeneity and the initial network, so it identifies θ without specifying their distributions.
  • Proposition 1 characterizes sufficient statistics for A and π and replaces the complicated conditioning set B*(d3) with the tractable set B(d3).
  • Exact evaluation may be infeasible because B(d3) can be large, so the paper uses an explicitly enumerable subset that sacrifices information for analytical and computational tractability.
  • Identifying star systems are separated: each node and dyad belongs to at most one system, while links across distinct systems remain invariant.
  • Theorem 1 shows that conditioning on identifying star systems yields an equivalent conditional-likelihood representation that permits fixed-effects identification of θ0 = (α0, β0)′.
  • The product-form likelihood has conditionally independent multiplicands, enabling single-network, many-agent sampling-distribution approximations.

3 Inference on α and β

The paper develops exact and asymptotic conditional inference for α and β using network sequences generated by swapping identifying star-system edges. The results establish a valid MCMC null sampler and characterize when single-network asymptotics provide consistency and normality.

  • Asymptotic inference: Consistency and asymptotic normality require the number of identifying star systems to diverge with N, creating a growing set of conditionally independent summands.The effective sample size is tied to the expected number of identifying star systems.
  • Exact inference: Algorithm 1 produces a random draw from the conditional star-system likelihood for sufficiently large mixing time σ.The Markov chain swaps edges within stable star systems while visiting every admissible sequence with the target long-run probabilities.
  • Exact inference: The exact confidence set has coverage 1 − α, up to simulation and numerical approximation error, uniformly over the nuisance parameter.This invariance makes the confidence set similar across population values of the nuisance parameter.
  • Asymptotic inference: The number of identifying systems depends on the model parameters, fixed-effect distribution, and initial network, so information accumulation is substantively restricted.Assumption 4 is therefore the key condition governing whether large-network inference applies.
  • Single network estimation: examples: When β0 > 0, networks become nearly complete and identifying stars are difficult to form; when β0 < 0, their total number vanishes as N grows.Positive β0 yields few period-1-to-period-2 transitions, whereas negative β0 makes the required connected star configuration occur with vanishing probability.

4 Monte Carlo experiments

Monte Carlo experiments evaluate the conditional star-system estimators under simulated networks with unobserved location-based homophily and evolving transitivity. The results broadly match the asymptotic predictions, with the estimator pooling 2- and 3-star systems performing more precisely.

  • Initial networks are sparse random geometric graphs whose average degree is varied between 2, 3, and 4, producing different connectivity regimes.
  • The design creates homophilous link formation from unobserved location, while subsequent links increase transitivity and average degree through state dependence and triadic closure.
  • The simulations use N = 5,000 agents and B = 1,000 Monte Carlo replications, with panels comparing 2-star against pooled 2- and 3-star estimation.
  • The conditional 2 & 3-star estimator is more precise than the conditional 2-star estimator across all designs.It has both smaller mean absolute error and smaller mean standard deviation.
  • Both estimators are approximately mean and median unbiased, with Wald confidence intervals achieving close to nominal 95% coverage.

5 Empirical Application: Co-authorship Networks

The empirical application uses 1996–1999 co-authorship data to estimate persistence and transitivity while accounting for pair-specific heterogeneity. The conditional estimates indicate that observed clustering is mainly attributable to homophily rather than an intrinsic preference for transitive triplets.

  • The authors caution that many observed triangles come from triple-authored papers, making the model’s collaborator-choice microfoundations doubtful.The empirical exercise is presented as a proof of concept rather than a definitive structural analysis.
  • The application reports a standard logistic regression and conditional estimators based on 2-star, 2- and 3-star, and 2-, 3-, and 4-star systems.
  • The co-authorship data contain 3,798 identifying 2-star systems, but only 560 identifying 3-star systems and 61 identifying 4-star systems.Adding 4-star systems therefore changes estimates and standard errors only modestly relative to using 2- and 3-star systems.
  • CSS-MLE estimates indicate that the network’s high transitivity is mainly driven by homophily rather than an intrinsic preference for transitive triplets.The standard logit coefficient is less negative, suggesting that omitting homophily overstates transitivity’s role.

6 Conclusion

The paper concludes that dynamic network models with rich heterogeneity can be identified from a single network sequence, and that consistent estimation and large-network inference are feasible in some settings. It also emphasizes delicate information accumulation and dependence on modeling and network-generation assumptions.

  • Simple dynamic network-formation models with rich heterogeneity structures can be identified from a single network sequence.The paper further finds that consistent estimation and large single-network inference are feasible in some settings.
  • The conclusions rely consequentially on logit errors, although the authors expect aspects of the analysis to extend beyond that assumption.
  • Achieving the maximal convergence rate depends on features of the heterogeneity distribution and the initial network configuration.
  • Researchers can assess practical applicability by counting identifying star systems, although their presence in real-world networks remains an open question.The empirical illustration provides encouraging evidence in one setting.
  • The conditional star-system estimator uses only a subset of the full conditional likelihood, leaving the resulting information loss and the relative value of different star-system sizes open.

Supplemental Web Appendices

The supplemental appendices provide proofs for the main theorems and supporting lemmas, while establishing notation and standard inequality abbreviations used throughout.

  • The appendices contain proofs of the main-text theorems and supplemental lemmas.
  • Appendix notation follows the main text unless stated otherwise, and equation numbering continues from the main text.
  • The appendices abbreviate standard tools including the triangle inequality, Cauchy-Schwarz, Hoeffding, Markov, Jensen, and the law of iterated expectations.

A.1 Preliminary results

The preliminary results establish stationary-distribution and consistency tools used to analyze the conditional star-system estimator. Consistency follows from uniform approximation and identification on a compact parameter space.

  • A.1 Preliminary results: The proof of Theorem 2 applies a textbook finite-Markov-chain result to characterize the sampler's stationary behavior.The cited result supplies the stationary-distribution argument used in the theorem.
  • A.1 Preliminary results: A finite irreducible ergodic Markov chain has stationary distribution π when π is normalized and satisfies detailed balance.The condition is π_l p_lm = π_m p_ml for every pair of states.
  • A.1 Preliminary results: The consistency argument invokes a standard M-estimation theorem and establishes consistency under uniform approximation and identification.The parameter space is assumed compact, with the criterion uniformly approximating its target.

A.2 Proofs of preliminary results appearing in the main text

These proofs establish permutation invariance and conditional independence properties for identifying star systems. The key result is that the relevant tuples are independent after conditioning on network and auxiliary variables.

  • A.2 Proofs of preliminary results appearing in the main text: Permuting the two-period link histories within an identifying star system preserves the relevant link-structure relationships, possibly swapping corresponding outcomes.The proof treats dyads within the star, across neighboring agents, and outside the star system.
  • A.2 Proofs of preliminary results appearing in the main text: The permutation argument extends to agents in the star's neighborhood and to agents outside that neighborhood, with link structures invariant in the latter case.The proof distinguishes whether an outside agent belongs to another identifying star system or neither system.
  • A.2 Proofs of preliminary results appearing in the main text: Conditional on D0, D3, Z, and A, the tuples ZpiQpi are independent across identifying star-system indices.This is the content of Lemma 4 and follows by factoring the joint probability after summing over remaining dyads.
  • A.2 Proofs of preliminary results appearing in the main text: The constructed permutation preserves the relevant aggregate link counts, implying that the period-3 network configuration remains unchanged.The proof uses equality of period-1 and period-2 link totals to conclude d3 = b3.

B.2 Proof of Theorem 1 (Conditional Star-System Likelihood)

The proof decomposes the likelihood ratio across dyad classes and shows which terms cancel when identifying star systems are permuted. In the logit case, the surviving factors depend on observed network statistics rather than dyad-specific heterogeneity.

  • B.2 Proof of Theorem 1 (Conditional Star-System Likelihood): In the logit case, the star-system likelihood ratio simplifies to b10(qij; θ) = exp(α [dij3 −dij0] + β [(rij1 −rij0) + (1 −dij3)(rij2 −rij1)]).The expression is obtained after comparing swapped and unswapped star-system configurations.
  • B.2 Proof of Theorem 1 (Conditional Star-System Likelihood): 61?
  • B.2 Proof of Theorem 1 (Conditional Star-System Likelihood): The b10 likelihood-ratio expression does not depend on Aij, so dyad-specific heterogeneity cancels for this comparison.This cancellation is stated directly after the logit simplification.
  • B.2 Proof of Theorem 1 (Conditional Star-System Likelihood): For dyads joining agents in different identifying star systems, numerator and denominator contributions coincide and therefore cancel.The argument uses equal period-1 and period-2 links and the permutation lemmas.
  • B.2 Proof of Theorem 1 (Conditional Star-System Likelihood): For dyads with one agent in an identifying star system and one outside, the logit factors are c11(qij; θ) = exp (β(1 −dij3)(rij2 −rij1)) and c00(qij; θ) = exp (−βdij3(rij2 −rij1)).These factors arise separately for the (1,1) and (0,0) two-period link-history cases.

B.3 Proof of Theorem 2 (Null Sampler)

The null sampler is represented as a Markov chain whose states are network sequences generated by star-system swaps. The proof establishes irreducibility and uses detailed balance to identify its stationary distribution.

  • B.3 Proof of Theorem 2 (Null Sampler): The sampler's state-transition graph has one vertex for each network sequence in BK (d3), with edges corresponding to single stable-star-system swaps.Each state also has a self-loop, and the number of neighboring states is m2:K,N.
  • B.3 Proof of Theorem 2 (Null Sampler): The chain's transition matrix has L = 2m2:K,N states, corresponding to the cardinality of BK (d3).The states are arranged as the vertices of the transition graph.
  • B.3 Proof of Theorem 2 (Null Sampler): The graph is strongly connected because any network sequence can be reached from any other by finitely many star-system swaps with positive probability.Positive transition weights imply irreducibility; self-loops make the graph non-bipartite.
  • B.3 Proof of Theorem 2 (Null Sampler): The stationary distribution is obtained by setting π_l equal to the conditional star-system likelihood probability δ(vϕ) for each state.Normalization follows because the conditional likelihood is a probability mass function, while detailed balance verifies stationarity.

B.4 Proof of Theorem 3 (Consistency)

The proof establishes consistency by showing the conditional criterion is maximized at the true parameter and satisfies uniform convergence conditions.

  • B.4 Proof of Theorem 3 (Consistency): Uniform convergence is established using stochastic equicontinuity and Newey’s convergence result.The proof controls the criterion difference uniformly over the parameter space under the stated assumptions.
  • B.4 Proof of Theorem 3 (Consistency): The conditional likelihood criterion is maximized at θ0 for every N.This follows from the conditional star-system likelihood and the supporting independence and regularity arguments.
  • B.4 Proof of Theorem 3 (Consistency): A lower bound on the Hessian shows that θ0 is a well-separated maximizer of the criterion.Concavity and the mean value theorem convert the Hessian bound into separation away from θ0.
  • B.4 Proof of Theorem 3 (Consistency): The lemma conditions therefore imply that the estimator converges in probability to θ0.The conclusion is ˆθ →p θ0.

B.5 Proof of Theorem 4 (Asymptotic Normality)

The proof derives asymptotic normality by establishing a central limit theorem for the score and convergence of the Hessian to a positive definite matrix.

  • B.5 Proof of Theorem 4 (Asymptotic Normality): Asymptotic normality requires a normalized score to become Gaussian and the Hessian to converge to a positive definite matrix.These are the two explicit proof targets for the estimator’s asymptotic distribution.
  • B.5 Proof of Theorem 4 (Asymptotic Normality): The score is decomposed into martingale differences and analyzed using McLeish’s martingale central limit theorem.Uniform L2 bounds and conditional Lindeberg-type conditions are verified through the subsequent steps.
  • B.5 Proof of Theorem 4 (Asymptotic Normality): Moment bounds and conditional independence verify the regularity conditions needed for the martingale central limit theorem.The proof uses the information matrix equality, total-variance arguments, and the stated assumptions.
  • B.5 Proof of Theorem 4 (Asymptotic Normality): The score and Hessian results combine to establish the theorem’s asymptotic-normality conclusion.The proof applies the Wald device and Slutsky’s lemma after verifying the required components.

B.6 Proof of Proposition 2

The proposition shows that identifying star systems become rare as N grows, using separate arguments for nonnegative and negative transitivity parameters.

  • B.6 Proof of Proposition 2: For β0 ≥ 0, the probability of identifying star systems decays exponentially with N, so their expected number converges to zero.The proof bounds edge-swapping probabilities and then sums over candidate star systems.
  • B.6 Proof of Proposition 2: Tail events controlling common-neighbor counts have probabilities bounded by N^2 exp(−γ0N) and N^2 exp(−γ1N).These bounds provide the high-probability separation used in the negative-β0 case.
  • B.6 Proof of Proposition 2: For β0 < 0, the period-1 graph is empty with overwhelming probability, and candidate stars are exponentially unlikely to be identifying.Conditional independence in period 2 and the required spoke, leaf-leaf, and outside-dyad conditions drive the bound.
  • B.6 Proof of Proposition 2: Dyad separation ensures that each dyad belongs to at most one identifying star system, limiting the total number of such systems.This combinatorial restriction supports summing probabilities over candidate stars.
  • B.6 Proof of Proposition 2: The proof combines candidate counts, union bounds, and exponential probability estimates across star sizes.The resulting sums are controlled separately for smaller and larger K ranges.

B.7 Proof of Proposition 3

The proposition analyzes identifying star systems in a sparse random geometric graph, showing that their expected number grows linearly with N.

  • B.7 Proof of Proposition 3: The raw common-neighbor count is locally bounded in probability in sparse geometric graphs.The relevant dominating count is binomial with probability at most πr^2/N, making it tight and Op(1).
  • B.7 Proof of Proposition 3: A geometric event makes a candidate K-star identifying by isolating its vertices from feasible outside neighbors.The event places the center and leaves close together while excluding other agents from a larger surrounding ball.
  • B.7 Proof of Proposition 3: The geometric configuration contributes a probability bounded below by a positive constant for sufficiently large N.The lower bound combines the center, leaf-location, isolation, and internal-link events.
  • B.7 Proof of Proposition 3: For every fixed K, the expected number of identifying K-systems is of order N.Summing over K up to a fixed upper bound preserves this linear order.
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