Source-linked AI summary
Modeling social networks from sampled data
Mark S. Handcock, Krista J. Gile
TL;DR
The paper addresses network inference when sampling leaves actors and ties unobserved. It develops likelihood-based inference for adaptive network designs and finds that such samples can yield effective, practical estimates, while design-based inference faces unobservable dyadic sampling probabilities.
Problem
Network inference commonly assumes complete, reliable observation of all links, although surveys and censuses miss actors and ties.
Method
The paper develops a theoretical and computational likelihood framework for inference from partially observed networks generated by adaptive sampling designs.
Results
Adaptive samples produced effective parameter estimates with modest MLE bias and errors increasing slowly as unobserved dyads increased.
Takeaways & Limitations
Sampled networks can be representative without simple random dyad sampling, making inference from adaptive network samples tractable.
Takeaways & Limitations
Design-based inference requires dyadic sampling probabilities that are unobservable even for single-wave link-tracing samples.
Abstract
from arXiv · showhide
Network models are widely used to represent relational information among interacting units and the structural implications of these relations. Recently, social network studies have focused a great deal of attention on random graph models of networks whose nodes represent individual social actors and whose edges represent a specified relationship between the actors. Most inference for social network models assumes that the presence or absence of all possible links is observed, that the information is completely reliable, and that there are no measurement (e.g., recording) errors. This is clearly not true in practice, as much network data is collected though sample surveys. In addition even if a census of a population is attempted, individuals and links between individuals are missed (i.e., do not appear in the recorded data). In this paper we develop the conceptual and computational theory for inference based on sampled network information. We first review forms of network sampling designs used in practice. We consider inference from the likelihood framework, and develop a typology of network data that reflects their treatment within this frame. We then develop inference for social network models based on information from adaptive network designs. We motivate and illustrate these ideas by analyzing the effect of link-tracing sampling designs on a collaboration network.
1. Introduction.
The paper addresses inference for social networks when actors and ties are only partially observed through sampling. It develops a framework for sampled-network inference and illustrates it with adaptive link-tracing designs in a law-firm collaboration network.
- Network models represent relational data and stochastic mechanisms governing ties among social actors and other entities.
- The application analyzes collaborative ties among 36 New England law-firm partners using nodal and relational attributes.
- Survey sampling is necessary because complete information on actors and relational ties is difficult to obtain, especially in large or hard-to-find populations.
- The paper develops a theoretical framework for inference from network data partially observed because of sampling.
- The framework considers network sampling concepts, sampled-data inference, design-based limitations, likelihood-based inference, ERGMs, and a collaboration-network study.
2. Network sampling design.
The paper formalizes sampled network data by distinguishing observed ties, unobserved ties, and the sampling design. It defines conventional and adaptive designs according to whether observed survey information directs subsequent sampling.
- The sampling design is a probability distribution for the matrix indicating which network elements were sampled.
- When sampled dyads are determined by sampled nodes, D records dyads incident on at least one selected node.
- Observed network data consist of Yobs and D, while complete data additionally include the unobserved portion Ymis.
- Conventional designs do not use survey information to direct later sampling, whereas adaptive designs use observed data for that purpose.
- Adaptive designs satisfy a missing-at-random condition in which observation propensity can depend on observed network data.
2.1. Some adaptive designs for undirected networks.
For undirected networks, the paper describes ego-centric and link-tracing designs that observe dyads through sampled nodes and may expand samples across waves.
- An ego-centric design randomly selects individuals and observes all dyads involving at least one selected individual.
- The ego-centric design is conventional because its sampling distribution does not depend on the network Y.
- A link-tracing design enrolls subsequent nodes based on observed relations with previously sampled nodes.
- One-wave link-tracing design: In one-wave link-tracing, the initial sample leads to enrolling all individuals reported to have at least one relation with it, followed by observing their dyads.
- Multi-wave link-tracing design: In k-wave link-tracing, partners are enrolled repeatedly for k waves; saturated link-tracing continues until a wave adds no individuals.
2.2. Some adaptive designs for directed networks.
For directed networks, the paper adapts ego-centric and one-wave link-tracing designs by sampling outgoing dyads and enrolling recipients of observed arcs.
- A directed ego-centric design randomly selects individuals and observes all directed dyads originating from them.
- Because directed dyads are observed only when their tail node is sampled, this ego-centric design is conventional.
- A directed one-wave link-tracing design enrolls individuals receiving an arc from a member of the initial sample.
- After new recipients are selected, the design observes all directed dyads originating from those newly selected individuals.
- Multi-wave link-tracing design: For multi-wave directed link-tracing, successive samples depend on the observed graph rather than unobserved portions.
3. Inferential frameworks.
The section contrasts design-based and likelihood-based inference for partially observed networks. Design-based methods depend on sampling probabilities that are often unobservable for link-tracing designs, whereas amenable adaptive designs permit likelihood inference based on the observed network information.
- Design-based inference: Design-based inference treats the network as fixed and attributes uncertainty entirely to the sampling design, without requiring a model for the data.Its estimators nevertheless require the sampling procedure to be fully known.
- Design-based inference: The Horvitz–Thompson approach estimates network quantities by inverse-probability weighting observed dyads.The relevant probability is the probability that dyad (i,j) is observed.
- Design-based inference: Link-tracing designs complicate design-based inference because dyadic sampling probabilities differ from nodal probabilities and are not observable even for single-wave samples.Estimating them can require inferring unobserved relations from observed relations, reintroducing reliance on a model.
- Likelihood-based inference: Likelihood-based inference offers an alternative framework for network inference from link-tracing samples.The paper develops this approach after identifying the limitations of design-based estimators.
- Likelihood-based inference: A sampling design is amenable when it is adaptive and its sampling parameter is distinct from the stochastic network model parameter.Under amenability, the sampling design is ignorable for likelihood inference about the network parameter.
- Likelihood-based inference: For amenable standard designs, likelihood inference for the network parameter can use the face-value likelihood based only on the observed network information.The result applies to ego-centric, single-wave, and multi-wave sampling designs.
4. Exponential family models for networks.
The paper formulates ERGMs for network data and develops likelihood-based inference when only part of the network is observed. Conditional modeling of missing ties enables MCMC-based multiple imputation on networks consistent with the observed data.
- Exponential family random graph models: ERGMs represent network distributions through statistics g(y), which may include structural relations and node or edge covariates.The model family includes both relation-derived statistics and covariate information.
- Likelihood-based inference: Likelihood inference is extended to settings where the complete network Y is decomposed into observed data Yobs and missing data Ymis.The target is inference for η with Y = Yobs + Ymis possibly only partially observed.
- Likelihood-based inference: MCMC can approximate the likelihood by sampling missing-data configurations or by simulating from the conditional distribution given Yobs.The conditional simulation produces multiple imputations of the full network using a variant of standard ERGM MCMC.
- Likelihood-based inference: The conditional distribution of missing ties given observed ties is an ERGM on the constrained space of networks concordant with the observations.This restriction preserves the observed network information while varying only compatible missing portions.
- Likelihood-based inference: The sampled-data likelihood is only slightly more difficult than the complete-data case because separate MCMC chains estimate complete-data and conditional terms.The two terms are estimated using a chain on complete data and a chain conditional on yobs.
5. Two-wave link-tracing samples from a collaboration network.
The paper evaluates two-wave link-tracing samples from a 36-partner law-firm collaboration network using an ERGM with seniority, practice, homophily, density, and transitivity statistics. Sample-based estimates are generally close to complete-data estimates, although information varies substantially across samples of equal size.
- Model and data: The ERGM includes seniority and practice effects, homophily by practice, gender, and office, plus density and GWESP transitivity statistics.GWESP measures transitivity structure, while the homophily statistics indicate attribute matches between partners.
- Model and data: The analysis treats the fitted complete-network parameters as truth and assesses sampling effects in a 36-partner New England law-firm collaboration network.The network records an undirected tie when both partners indicate that they collaborate.
- Sampling design: Two-wave link-tracing samples begin from two randomly chosen seeds, producing datasets with between 2 and 36 actors and observed dyads involving at least one sampled node.The design constructs all possible datasets from the two seeds and partitions the sociomatrix into observed and unobserved blocks.
- Sampling design: The two isolated-seed sample is excluded because it observes only 69 of 630 dyads, no edges, and yields MLEs with negative infinite values.This sample lies on the boundary of the convex hull and is considered practically unreasonable for link-tracing research.
- Evaluation: Kullback–Leibler divergence compares each sampled-data MLE with the complete-data MLE, while bias, RMSE, and efficiency loss assess parameter-estimation accuracy.The divergence is computed using MCMC, and efficiency loss standardizes sampled-estimate error against full-data sampling variance.
- Results: As sampled dyads approach the full network, information approaches that of complete data, but samples of equal size can differ greatly in information content.The two smallest samples contain five nodes and 165 dyads, with divergences of about 14.
- Results: 2%–3% efficiency loss occurs on average, with very small bias and modest RMSE, although these averages conceal variation across individual samples.The mean-value parameterization likewise shows approximately unbiased estimates with small RMSE and efficiency losses.
- Conclusion: The paper presents a systematic framework for inference from partially observed designed samples, including adaptive network sampling designs.The framework includes a model-amenable network design and a likelihood-based inference result.
6. Discussion.
The paper argues that adaptive network samples can support principled likelihood-based inference when analyzed with complete-network models. Its application finds generally effective parameter estimates, modest bias, and information that depends on which nodes and dyads are observed.
- Sampled networks can be representative rather than biased when analyzed correctly; simple random dyad sampling is unnecessary for valid inference.Common sampling designs can be incorporated into the framework.
- Adaptive network samples can be analyzed with complete-network likelihood models, making inference principled and practical.
- Adaptive collaboration-network samples produce effective parameter estimates in the vast majority of cases, with only modest bias relative to complete-data estimates.Estimation error increases slowly as the number of unobserved dyads grows.
- Fisher information depends on the specific sampled nodes and dyads rather than only on sample size when network dependence terms are present.Information for GWESP increases when more sampled node pairs are connected by edges, and disappears for that term if no such dyads are observed.
- The framework treats faithfully unobserved dyads as missing by design, while distinguishing the controlled sampling design from out-of-design observation mechanisms.
Supplement: Software used in the simulation study
The simulation study uses R code based on the statnet open-source suite for network modeling, with code and documentation provided online.
- The study’s simulation code is written in the R statistical language.
- The code is based on statnet, an open-source software suite for network modeling.
- Code and documentation are provided through links to the statnet website.