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Likelihood Inference for Latent Network Models under Snowball Sampling

This paper addresses the systematic bias in network parameter estimates caused by ignoring snowball sampling mechanisms by deriving an exact likelihood for continuous latent space models and developing a stochastic Expectation-Maximization algorithm that significantly improves inference accuracy and model fit compared to naive approaches.

Original authors: Nurzhan Sapargali, Sergio Buttazzo, Göran Kauermann

Published 2026-06-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Nurzhan Sapargali, Sergio Buttazzo, Göran Kauermann

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Problem: The "Snowball" Effect

Imagine you want to understand the entire social circle of a massive city, but you can't talk to everyone. Instead, you start with one person (the "seed") and ask, "Who are your friends?" Then you ask those friends, "Who are your friends?" and so on.

This is called Snowball Sampling. It's like rolling a snowball down a hill; it starts small but picks up more and more snow (people) as it goes.

The Trap:
The problem is that this method is biased. People with lots of friends (popular hubs) are much more likely to get picked up by the snowball than people with only one or two friends. If you just look at the people you collected and try to guess how the whole city is connected, you will get it wrong. You will think the city is much more tightly connected and "clumpy" than it actually is, because you've accidentally gathered mostly the popular crowd.

The Solution: A New Mathematical Lens

The authors of this paper developed a new way to do the math so that researchers can fix this bias. They created a "corrected lens" that accounts for the fact that the snowball sampling method naturally misses the quiet, isolated people.

Here is how they did it, broken down into three parts:

1. The "Invisible Wall" Analogy

Imagine the people you didn't find are standing behind an invisible wall.

The Naive Mistake: If you ignore the sampling method, you assume everyone you didn't find is just like the people you did find. You assume they are all hanging out in the middle of the party.
The Correction: The authors realized that if a person wasn't found by the snowball, it means they didn't have a connection to the people you did find in the earlier rounds. They are "invisible" because they are too far away from the main group.
The Math Magic: The paper proves that for a specific class of models (called Continuous Latent Space models), you can calculate the probability of this "invisibility" without needing to see the missing people. It's like knowing that if you didn't hear a noise, the person making it must be very far away, even if you can't see them. This allows them to write a perfect mathematical formula (a likelihood) that includes the missing people without actually having to interview them.

2. The "Hidden Map" (Latent Space)

To understand the network, the authors imagine every person has a secret coordinate on a hidden map.

If two people are close together on this hidden map, they are likely to be friends.
If they are far apart, they probably aren't.

The Naive Approach: When researchers ignore the sampling bias, they draw this hidden map too small. They squish everyone together into a tiny, crowded room. Because everyone is so close on this tiny map, the math predicts that everyone should be friends with everyone else. This leads to a prediction that the network has way more connections than it actually does.

The Corrected Approach: The new method stretches the map out. It realizes, "Ah, the people we missed are actually very far away on the map, which is why the snowball didn't roll over to them." This creates a much larger, more accurate map where the distances between people reflect reality.

3. The Real-World Test: The Patent Inventors

To prove their method works, the authors tested it on a real, massive network: German semiconductor inventors (about 6,000 people who work together on patents).

The Experiment: They took 500 different "snowballs" (samples) from this network and tried to reconstruct the whole picture using two methods: the old "naive" way and their new "corrected" way.
The Results:
- The Naive Way: It drew a map where everyone was squished together. It predicted the network had nearly twice as many connections as it actually did. It thought the world was much smaller and more connected than it is.
- The Corrected Way: It drew a map that was 8 times wider (more spread out). This matched the real data much better. It correctly predicted the number of connections and the "shape" of the network.

Why This Matters

The paper shows that if you use the old, naive method, you might draw the wrong conclusions.

Example: If you are studying how geography affects collaboration, the naive method might say, "Oh, distance doesn't matter much because everyone is so close on our tiny map."
The Truth: The corrected method shows that distance does matter a lot. The further apart two inventors are, the less likely they are to work together. The naive method hides this reality by squishing the map.

The Bottom Line

The authors didn't just say, "Snowball sampling is biased." They built a specific mathematical tool that fixes that bias for a wide range of network models. They showed that by accounting for the "invisibility" of the people you missed, you can get a true picture of the network, even when you can't talk to everyone.

In short: Don't just look at the snowball you rolled; use their new math to figure out who was left behind in the snow, so you can draw the map of the whole world correctly.

Technical Summary: Likelihood Inference for Latent Network Models under Snowball Sampling

Problem Statement
Large-scale network data is rarely fully observable due to logistical, budgetary, or ethical constraints. While snowball sampling is a standard method for collecting data from hard-to-reach populations, naive inference that ignores the sampling mechanism yields systematically biased parameter estimates. Specifically, because snowball sampling preferentially recruits high-degree vertices, the resulting subgraph appears more clustered and homogeneous than the underlying population. When practitioners fit network models to these samples as if they were complete populations, they systematically underestimate population heterogeneity and variance. Existing literature offers design-based estimators for specific statistics or model-based approaches that require computationally intractable marginalization over unobserved network configurations (e.g., summing over exponentially many graphs). There is a lack of an exact, closed-form likelihood for snowball samples that is valid across a broad class of network models and computationally efficient.

Methodology
The authors derive the exact likelihood for multi-wave snowball samples within the class of Continuous Latent Space (CLS) models. In CLS models, edges form independently across vertex pairs conditional on latent vertex-level quantities (e.g., positions in a geometric space).

Likelihood Derivation: The core theoretical contribution is the demonstration that conditional edge independence allows the marginalization over unobserved network configurations to reduce to a closed-form expression.
- Starting with the Erdős–Rényi (ER) model, the authors derive the joint probability of the observed adjacency matrix and the wave sets. They show that the likelihood factorizes into terms involving observed edges/non-edges within and between adjacent waves, and a product term representing the probability that unobserved vertices have no edges to the sampled waves $V(0)$ through $V(r-1)$ .
- This result is extended to the general CLS class. The likelihood of the snowball sample is shown to be equivalent to the likelihood of a CLS network where unsampled vertices are constrained to have no edges to the core sample ( $V(0) \cup \dots \cup V(r-1)$ ), while connections within the unsampled set and between the unsampled set and the final wave $V(r)$ are treated as missing.
- The authors address the identifiability of latent positions for unsampled vertices, showing they are identifiable only up to the equivalence class of configurations yielding the same joint exclusion probability.
Estimation Algorithm: To implement this likelihood for the Euclidean latent distance model (Hoff et al., 2002), the authors develop a Stochastic Expectation-Maximization (sEM) algorithm.
- E-step: The latent positions of sampled vertices are treated as missing data and sampled via a Markov Chain Monte Carlo (MCMC) procedure. The authors utilize an Elliptical Slice Sampler (ESS) for tuning-free updates. Crucially, the contribution of the vast number of unsampled vertices is not modeled individually but is integrated out as a single low-dimensional expected exclusion probability.
- M-step: Parameters are updated by maximizing the complete data log-likelihood at the sampled latent positions.
- Computational Efficiency: The integration over unsampled vertices is approximated using Quasi-Monte Carlo (QMC) methods (Sobol sequences), keeping computational overhead modest and avoiding the quadratic scaling typical of full-population MCMC.

Key Contributions

Exact Closed-Form Likelihood: The paper provides the first exact, closed-form likelihood for multi-wave snowball samples applicable to the broad class of CLS models, resolving the open problem of computing such distributions without enumerating unobserved networks.
Algorithmic Implementation: A stochastic EM algorithm is developed that integrates the sampling correction naturally with existing MCMC samplers for latent position inference, adding only a low-dimensional integral.
Theoretical Insight: The work clarifies that the non-ignorability of ego selection in CLS models arises when the selection mechanism depends on latent positions (e.g., degree-proportional selection), a scenario the authors note is not currently addressed by general model-based approaches.

Empirical Results
The methodology is applied to a large-scale co-inventor network of German semiconductor patent applicants ( $N \approx 6,000$ ) and validated through extensive simulations.

Simulation Studies: In simulations of Erdős–Rényi and distance models, the naive estimator (ignoring sampling) severely overestimates edge density and underestimates latent space variance. The snowball-corrected estimator remains approximately unbiased even with sparse samples.
Semiconductor Patent Application:
- Latent Space Variance: The naive procedure severely underestimates the spread of the latent space (estimating $\hat{\psi} \approx 25$ vs. $\hat{\psi} \approx 215$ for the corrected model).
- Edge Counts: The naive model produces networks with nearly twice the observed edge count (mean 41,154 vs. observed 21,073), whereas the corrected model closely reproduces the observed count (mean 24,093).
- Goodness-of-Fit: The corrected model achieves a spectral goodness-of-fit (SGOF) nine times better than the naive model (0.258 vs. 0.027).
- Covariate Effects: While both models qualitatively agree on patterns of gender homophily and geographic proximity, the naive model systematically overestimates baseline edge probabilities and attenuates the decay of collaboration probability with geographic distance. The corrected model reveals a steeper distance decay and higher variability in collaboration propensities.

Significance and Claims
The paper claims that ignoring the snowball sampling mechanism leads to quantitatively dramatic errors in network inference, altering conclusions about population heterogeneity and covariate effects. The proposed framework enables the first empirical stress test of latent distance models on large-scale networks exhibiting genuine small-world properties, a scale previously computationally prohibitive for likelihood-based inference.

The authors modestly note that while the framework significantly improves upon naive inference, the Euclidean distance model still struggles to perfectly fit small-world networks with heavy-tailed degree distributions (a known theoretical limitation). However, they argue that the corrected procedure exposes this model limitation more honestly than the naive approach, which artificially contracts the latent space to mask the misfit. The framework is presented as a foundation for future extensions to hyperbolic latent spaces, graphon models, and models with covariates, provided the conditional edge independence assumption holds.