Polyads Reduce Bias in Sparse Multi-Way Networks

High-dimensional fixed effects are useful in trade, health, and other network data because they absorb unobserved heterogeneity along several axes. They also create a familiar problem for nonlinear count models: as the network grows, the number of nuisance parameters can grow with it, and maximum likelihood can inherit incidental-parameter bias. This paper introduces the Polyads estimator for weighted multi-way networks, aimed at estimating structural parameters while conditioning away arbitrary fixed-effect structures.

The setting is a $D$-partite count network with Poisson intensities of the form $\lambda_i = \exp(\beta_*^\top X_i + \sum_{g\in G}\theta^g_{g(i)})$. The target is $\beta_*$, while the fixed effects may span rows, columns, time, products, physician groups, or other network dimensions. The authors' answer is to replace global conditional maximum likelihood with many small conditional classification problems built from polyads.

Core Contribution

The key construction is a polyad: a $2\times D$ block of network indices whose $2^D$ induced edges can be assigned positive and negative signs. Applying a polyad transformation shifts count mass across those edges while preserving the sufficient statistics for the fixed effects. That preservation is the mechanism that removes the nuisance parameters from the conditional likelihood.

Rather than conditioning on the full set of all networks with the same generalized degrees, the estimator conditions on the orbit generated by one polyad at a time. For each active polyad, the resulting likelihood depends on a generalized difference-in-differences feature $\tilde X_\xi = \sum_i s_\xi(i)X_i$, not on the fixed effects. Summing these convex polyad losses gives the estimator $\hat\beta_\Xi = \arg\min_\beta \hat L_\Xi(Y\mid X,\beta)$.

Technical Approach

The paper gives both an inference theory and an implementation path. Lemma 1 derives the gradient and Hessian of each polyad loss, with the Hessian equal to a conditional variance times $\tilde X_\xi\tilde X_\xi^\top$. Lemma 2 shows strict convexity when the polyad difference features vary enough. Theorem 1 proves consistency under assumptions controlling dependence among active polyads and curvature around $\beta_*$. Theorem 2 proves asymptotic normality without requiring all network dimensions to grow; short panels are allowed as long as the total graph size grows.

Results and Analysis

The real-data application uses French health insurance claims from 2016 to 2018 to study a May 2017 GP fee increase of 8.7%. Because the full grouped network has 69,265 patient groups, 16,941 doctors, 34 months, about 40 billion dyads, and 56,034,015 positive edges, the authors use subsampling plus random-effects meta-analysis. Across 2%, 3%, and 4% subsamples, Polyads estimates a distance effect around $1.23$ to $1.73$ after scaling by $10^4$, and a same-city effect around $-7.60$ to $-3.91$ after scaling by $10^2$. The interpretation is that the reform weakened distance and same-city frictions, consistent with patients traveling farther after the fee increase. The computational table also supports the sparsity claim: at the 2% subsample, Polyads takes 5 minutes versus 19 minutes for PPML and 53 minutes for debiased PPML; at 4%, it takes 111 minutes versus 203 and 315 minutes.

Caveats

The estimator pays for local conditioning with some statistical-efficiency loss relative to full conditional maximum likelihood; Appendix A bounds this loss using feature concentration and local overlap. The method is also designed for count data and fixed-effect dependence, not arbitrary dependence between observations. Dense networks remain the main computational weak spot: the paper says PPML is still competitive when most potential edges are realized.