Closed-form likelihoods for link-tracing designs beyond special cases

Derive closed-form expressions for the likelihood sum over all network completions induced by link-tracing sampling designs beyond the special cases currently known, thereby making design-aware network inference directly tractable.

Background

The paper situates its contribution within the broader problem of likelihood-based inference from network samples obtained through link-tracing designs. Under such designs, the likelihood generally requires summing over every possible completion of the unsampled portion of the network consistent with the observed sample. This completion space is combinatorially large, and existing approaches commonly rely on Markov chain Monte Carlo approximations.

The authors identify the Erdős–Rényi model with full-neighbourhood recruitment as a tractable special case and derive its exact likelihood. The broader derivation of closed-form likelihoods for other link-tracing settings remains unresolved and is included as an open problem because it would extend exact, sampling-aware inference beyond the independent-edge case.

References

Beyond a handful of special cases, deriving this sum in closed form remains an open problem.

— Exact Likelihood Inference for Snowball-Sampled Erdős-Rényi Networks  (2608.14129 - Sapargali et al., 14 Aug 2026) in Section 1, Introduction

It therefore cannot be evaluated from the sample alone, and we leave a full treatment of endogenous ego selection to future work.

— Exact Likelihood Inference for Snowball-Sampled Erdős-Rényi Networks  (2608.14129 - Sapargali et al., 14 Aug 2026) in Section 6.1, Endogenous Ego Selection