Adjusting for Network Size and Composition Effects in Exponential-Family Random Graph Models

1 Introduction

• ERGMs represent social structure through network statistics, including homophily and triad closure, in an interpretable and parsimonious statistical model.The model is an exponential family whose sufficient statistics capture departures from independent dyads.Source: §1 · p. 3
• Because an ERGM’s sample space is whole networks, a fit may not transfer directly to a different set of actors, especially when relevant actor attributes differ.Applying the original estimates to another actor set can yield unrealistic network structure; canonical ERGMs preserve density as size grows, so ties per actor increase proportionally.Source: §1 · pp. 3–4
• Egocentric data provide only a subset of actors, so inferring the network’s structural properties requires a theory of how those properties vary with network size.For personal relationships, full dyad censuses are difficult to collect in large, sparse networks and may raise confidentiality concerns.Source: §1 · pp. 4–5
• The paper proposes an offset for network size and develops an approach that fits ERGMs to egocentrically sampled data by constructing similarly structured networks at varying sizes.It tests parameter comparability by fitting models to those constructed networks.Source: §1 · p. 5

1.1 Notation

• The analysis treats networks as sets of binary ties, with no self-loops or other structural constraints; its reasoning also applies to directed ties.The stated setup excludes bipartite networks, though the paper says its approach can accommodate networks with much lower within-group than between-group density.Source: §1.1 · p. 5
• Actor attributes are treated as fixed exogenous information, and relevant dyadic attributes are assumed derivable from the attributes of the two actors.The notation also defines a tie-toggle operation and uses set-builder expressions to specify which network elements a statistic depends on.Source: §1.1 · pp. 5–6

2 Desirable properties of invariant models

• The paper evaluates cross-size and cross-composition model behavior through local network properties, since defining empirical structural similarity across different actor sets would require assuming what similarity means.It asks which probability models yield similarly structured networks for similar parameter values across different sizes and compositions.Source: §2 · p. 6

2.1 Locality

• Local social behavior should remain similar when an actor’s own network perspective is unchanged, even as global network structure changes.The paper frames global structure as a product of local behavior and constraints.Source: §2.1 · pp. 6–7

2.2 Degree distribution under scaling without composition changes

• Because actors have finite resources for relationships, adding actors should not substantially raise degree indefinitely, motivating declining marginal size effects on degree distributions.Source: §2.2 · p. 7

2.3 Mixing properties

• An actor’s expected degree should depend on how the actor’s affinities match the population’s attribute composition, which affects potential-partner availability.The paper describes tie counts by actor attributes as mixing matrices, modeled through population counts, group-specific activity, and selectivity.Source: §2.3 · p. 7

3 Structure of exponential-family random graph models

• A general curved ERGM combines network sufficient statistics with parameter mappings and a normalizing constant, which may be intractable depending on the statistics.Source: §3 · p. 8

3.1 Change statistics

• Change statistics describe how a network statistic changes when a dyad is toggled, and determine the conditional probability of a tie given the rest of the network.The paper interprets dyad selection as an opportunity for actors to make a tie decision, while cautioning that this is an analogy rather than a realistic temporal process.Source: §3.1 · pp. 8–9

3.2 Locality

• Locality concerns which network states and exogenous attributes a statistic’s change can depend on; a density statistic is nonlocal because its change depends on network size.The paper considers three locality notions based on dyadic dependence.Source: §3.2 · p. 9

3.2.1 Locality based on dyadic independence

• Dyadic independence requires each dyad’s change statistic to depend only on that dyad and its actors’ attributes, yielding independent dyads but a limited model class.Source: §3.2.1 · p. 10

3.2.2 Locality based on Markov dependence

• Under Markov dependence, a dyad’s change statistic may depend on dyads incident on either of its actors; useful attribute-based locality must also depend on the realized social neighborhood.The paper defines a Markov-graph-local statistic as depending on dyads incident on the focal actors and attributes of actors tied to either one.Source: §3.2.2 · pp. 10–11

3.2.3 Locality based on partial conditional independence

• Partial conditional dependence permits dependence between dyads sharing an actor or connected through the specified cross-edges, and constrains relevant statistics to that neighborhood.Source: §3.2.3 · p. 11

3.3 Scaling without composition changes

• For any linear ERGM, a parameter setting that yields mean degree μ at one network size will yield a different expected mean degree at another size.With θ = 0, the distribution is an Erdős–Rényi graph with density 1/2, whose expected dyad count grows quadratically while ties needed for constant mean degree grow linearly.Source: §3.3 · p. 12

4.1 Model statement

• The proposed offset is a fixed component of the ERGM, not a freely estimated parameter, and is chosen so mean degree converges asymptotically absent other terms.It multiplies conditional dyad odds by 1/n; the paper also states that local statistic effects then do not depend on network size and composition.Source: §4 · p. 12, §4.1 · pp. 12–13

4.2 Erdős–Rényi model

• As network size grows, the offset ERGM’s degree distribution converges to Poisson(exp(θ)), making θ determine mean degree rather than density.For sufficiently large networks with similar mean degree, the maximum-likelihood estimates of θ would be similar.Source: §4.2 · p. 14

4.3 Selective mixing model

• With proportional group composition, expected ties from an actor to a group are asymptotically proportional to that group’s population fraction and the actor’s affinity for it.Expected overall degree therefore depends on how group availability matches actor affinities; under a linear η mapping, the MLEs are similar across sizes and compositions when proportional mixing holds.Source: §4.3 · p. 15

5 Inference from egocentrically sampled data

• Egocentric data identify only ties incident on sampled egos, not whether alters overlap or egos refer to one another.The method therefore considers hypothetical full networks that could have produced the observed reports.Source: §5 · pp. 15–16

5.1.1 Dyad census statistics

• For undirected dyad statistics, each reported tie is counted twice; same-category and cross-category ties require different counting expressions.The cross-category expression accounts for both possible ego–alter reporting directions.Source: §5.1.1 · pp. 16–17

5.1.2 Actor census statistics

• Local actor statistics, including degree counts, can be computed by summing each ego’s function of their own attributes and nominated alters.For example, the number of actors with degree d is the sum of indicators that each ego has d nominations.Source: §5.1.2 · p. 17

5.2 Sampling and consistency

• The inference assumes a representative ego sample and a local underlying process; then sampled degree distributions and selective mixing represent those in the full network.Representative samples are simple random samples or properly weighted stratified samples.Source: §5.2 · p. 18
• When sampled reports conflict or imply fractional ties, the authors average conflicting reports and treat fractional statistics as mean-value parameters.The inferred network’s composition and attribute-specific average relations are approximately population-like, yielding approximately similar structure.Source: §5.2 · pp. 18–19

6 Application to National Health and Social Life Survey data; 6.2 Missing data and weighting

• For each target size, weighted bootstrap resampling of NHSLS egos and nominations produces pseudo-populations and network statistics for fitting the same ERGM.The approach constructs networks of increasing size with similar structure to compare model results across sizes.Source: §6 · p. 19, §6.2 · p. 20

6.1 NHSLS Data; 6.2 Missing data and weighting

• The analysis uses active sexual partnerships reported by NHSLS respondents and restricts the hypothetical network to people aged 18–59.The survey was a stratified random sample of 3,432 American men and women; post-stratification weights adjust for sample design and nonresponse, and cases with missing or invalid attributes were excluded.Source: §6.1 · p. 19, §6.2 · p. 20

6.3.1 Sex

• The model includes sex-specific partner propensity and monogamy, same-sex partnership prevalence, and older-male-younger-female age asymmetry.The data report an average 1.8-year age difference in heterosexual partnerships, with female partners younger.Source: §6.3.1 · pp. 21–22

6.3.2 Race

• Race terms represent each category’s overall partner propensity and within-category ties as race homophily.Black is the arbitrary baseline for the overall propensity terms.Source: §6.3.2 · pp. 22–23

6.3.3 Age

• Age terms model overall partner propensity and partner-age similarity, with both effects stronger at younger ages.The model uses quadratic functions of age and age differences, including square-root transformations.Source: §6.3.3 · pp. 23–25

6.4 Simulation study design

• Two simulation studies compare parameter estimates across 400 resample sizes from 600 to 12,000 and across 100 resamples each at sizes 1,000, 6,000, and 11,000.The authors treat bootstrap variability as a baseline for judging changes associated with network size.Source: §6.4 · p. 25

6.5 Results

• Across three resample sizes, estimates show negative same-sex partnership coefficients, positive race homophily, monogamy propensity for both sexes, and older-male-younger-female asymmetry.These coefficient patterns are reported as consistent with the authors’ expectations.Source: §6.5 · p. 26
• From 6,000 to 11,000 egos, parameter-estimate differences are smaller than one simulated standard error; estimates also asymptote across sizes 600–12,000.The authors suggest asymptotic size invariance, while noting age-difference effects may converge more slowly, possibly because of age normalization.Source: §6.5 · p. 26

7 Discussion

• With local network statistics, the offset yields size-appropriate mean statistics for group mixing and degree distributions, mapping similar distributions to similar parameter values.Source: §7 · pp. 26, 29
• A fairly complex ERGM demonstrated this behavior on networks of different sizes constructed to have similar structure.Source: §7 · p. 29
• The egocentric-data approach, combined with the adjustment, may allow sample-based parameter estimates to generalize to the population from which the sample was drawn.Source: §7 · p. 29
• The offset-based parametrization supports comparing networks despite different sizes or compositions and simulating networks with the same underlying structure across those differences.Source: §7 · p. 29
• The analysis covers dyadic and degree-distribution effects, but not triad closure, because the available egocentric data lack information about third-order effects.Source: §7 · p. 29
• A different offset might speed convergence, but statistics containing the edge count can absorb offset-coefficient changes without affecting convergence of other parameters.Source: §7 · pp. 29–30
• The authors lack an appropriate uncertainty measure for these estimates and report that standard errors for larger networks are too small.Source: §7 · p. 30
• Network-size effects on structure beyond density remain unexplored.Source: §7 · p. 30

• The probability is formed from the relative probabilities of networks with and without the dyad’s tie.Source: §A.1 · p. 34
• The conditional probability of a tie is the inverse-logit of the natural-parameter-weighted change statistics for dyad (i, j).Source: §A.1 · p. 34
• For each statistic, the change statistic is its value with the tie minus its value without the tie.Source: §A.1 · p. 34

• The algorithm starts from an arbitrary network and runs for a specified number of iterations.Source: §A.2 · p. 34
• At each iteration, it computes the selected dyad’s conditional tie probability and updates the network to include or omit that tie accordingly.Source: §A.2 · p. 35
• RandomChoose selects a member of a set at random, supporting the algorithm’s selection of a dyad to update.Source: §A.2 · p. 35

B.1 Size-invariant statistics of linear ERGMs

• The argument assumes network composition converges and considers statistics whose expectations remain constant or approach a finite limit as size increases.Source: §B.1 · pp. 35–36
• If a target statistic’s expectation does not already converge under a Bernoulli model, no linear ERGM can make its expectation size-invariant.Source: §B.1 · p. 36
• Under Bernoulli(1/2), expected mean degree diverges as network size grows, whereas density and subnetwork densities can be unaffected by size.Source: §B.1 · p. 36

B.2 Asymptotic degree distribution of a simple offset model

• The asymptotic degree distribution of the simple offset model is Poisson with mean exp(θ).Source: §B.2 · p. 37