Source-linked AI summary
Exponential-Family Random Graph Models for Valued Networks
Pavel N. Krivitsky
TL;DR
Binary ERGMs cannot directly represent valued relations, so count and other valued networks have often been dichotomized, losing information and potentially introducing bias. This paper generalizes ERGMs to valued networks, particularly unbounded counts, develops corresponding network statistics, and applies them to interaction data. The applications examine how transitivity relates to faction allegiance and actor heterogeneity, while revealing stability–power trade-offs for subtle effects.
Problem
Complex ERGMs have largely been limited to binary ties, requiring valued network data to be dichotomized and thereby losing information and potentially introducing bias.
Method
The paper generalizes ERGMs to valued networks, especially unbounded count dyads, and develops statistics for common social-network features.
Results
In the karate-club analysis, faction similarity is strongly positive, while transitivity loses potential significance when both effects are included.
Takeaways & Limitations
The applications show that valued ERGMs can examine friend-of-a-friend effects alongside homophily and individual heterogeneity in interaction networks.
Takeaways & Limitations
Count ERGMs can have complex parameter spaces and degeneracy-like bimodality, creating computational and inferential difficulties; subtle-effect detection trades off stability against power.
Abstract
from arXiv · showhide
Exponential-family random graph models (ERGMs) provide a principled and flexible way to model and simulate features common in social networks, such as propensities for homophily, mutuality, and friend-of-a-friend triad closure, through choice of model terms (sufficient statistics). However, those ERGMs modeling the more complex features have, to date, been limited to binary data: presence or absence of ties. Thus, analysis of valued networks, such as those where counts, measurements, or ranks are observed, has necessitated dichotomizing them, losing information and introducing biases. In this work, we generalize ERGMs to valued networks. Focusing on modeling counts, we formulate an ERGM for networks whose ties are counts and discuss issues that arise when moving beyond the binary case. We introduce model terms that generalize and model common social network features for such data and apply these methods to a network dataset whose values are counts of interactions.
1 Introduction
ERGMs encode social-network features through sufficient statistics, but complex ERGMs have largely been restricted to binary ties. This forces valued relations to be dichotomized, losing information and potentially introducing bias; the paper extends ERGMs to valued networks, especially counts.
- Valued relations include counts, continuous measurements, and ordered or weighted relationships across domains such as interaction, trade, and advice networks.
- ERGMs are generative network models whose sufficient statistics represent features such as degree distribution, homophily, and triad closure.
- Most ERGM applications have treated each relationship as present or absent rather than valued.
- Dichotomizing valued data loses information and may introduce biases in ERGM analysis.
- The paper generalizes ERGMs to valued networks, focusing on count dyad values while retaining much of binary ERGMs’ flexibility and interpretability.
2 ERGMs for binary data
Binary ERGMs define distributions over network spaces using parameters, sufficient statistics, and a normalizing constant. Change statistics provide local tie interpretations and connect dyad-independent models to logistic regression and GLMs.
- The network sample space is a set of possible tie configurations, potentially restricted by structural or survey-induced constraints.
- A binary ERGM specifies a network distribution using a parameter vector, sufficient statistics, a parameter mapping, and a normalizing constant.
- Change statistics measure how sufficient statistics change when a dyad is added or removed, enabling local interpretations of conditional tie probabilities.
- For edge count, the change statistic is 1, while triangle-related change statistics count common neighbors and connect positive coefficients to higher tie odds.
- When change statistics are constant in the rest of the network, dyadic-independent ERGMs reduce to logistic regression, with MLE and MPLE equivalent.
3 ERGM for counts
Count ERGMs represent each dyad by an unbounded nonnegative count and add a reference measure h that shapes baseline dyad distributions. Unlike binary models, count specifications require nontrivial parameter-space constraints.
- Count ERGMs define networks as mappings that assign each dyad a value in the nonnegative integers.
- A count ERGM’s probability mass function includes sufficient statistics, parameters, a reference measure h, and a normalizing constant.
- For count networks, the normalizing-condition constraint may be complex because the sample space is infinite, unlike the binary case.
- The reference measure h sets the baseline shape of dyad distributions and constrains the parameter space.
- With h(y)=1, a simple dyad-sum model yields an i.i.d. geometric distribution, whereas a factorial reference yields i.i.d. Poisson dyads with mean exp(θ).
4 Inference and implementation
Inference for valued ERGMs inherits exponential-family methods but faces intractable normalizing constants, complex parameter spaces, and degeneracy-like behavior. The implementation uses simulation-based fitting, constrained model choices, and diagnostics.
- Valued ERGMs inherit standard errors and deviance analysis, but Wald tests rely on asymptotics that are questionable under complex dependence.
- Likelihood inference commonly requires simulation because the normalizing constant is intractable and exact evaluation integrates over the network sample space.
- Geyer–Thompson and Robbins–Monro methods provide simulation-based routes for fitting valued ERGMs.
- Count models may have parameter spaces formed by intersections of up to |Y| linear half-spaces, making boundary detection difficult.
- The implementation adapts binary ergm software with a Metropolis–Hastings sampler for the count-network sample space and reference measure.
- The paper limits dependence or uses counterweights to reduce avalanche-like degeneracy and relies on MCMC diagnostics because existing formal diagnostics are not directly applicable.
5 Statistics and interpretation for count data
This section develops sufficient statistics for count-valued network data, focusing on a Poisson-reference ERGM without complex constraints.
- The section develops sufficient statistics for count data that represent network features of interest.
- The default framework is a Poisson-reference ERGM for count-valued dyads.
- Unless otherwise noted, the models are specified without complex constraints.
5.1 Interpretation of model parameters
The section extends binary ERGM interpretation tools to valued networks by relating parameters and local dyad changes to network statistics and conditional probabilities.
- Sufficient statistics encode the structural network properties that binary and valued ERGMs aim to model.
- Increasing a parameter θ_k strictly increases the expected value of its statistic g_k when other parameters remain fixed.This follows from a general exponential-family property for linear ERGMs.
- Valued ERGMs use discrete change statistics to interpret how changing one dyad value affects conditional probabilities while other dyads remain fixed.
5.2 Model specification statistics
The section proposes count-valued ERGM statistics for dyad distributions, sparsity, dispersion, mutuality, actor heterogeneity, and triad-related structure, while addressing dependence and estimation constraints.
- Dyad-independent statistics: Dyad-independent count statistics yield Poisson-regression-type models, with a unit increase in θ_k multiplying a dyad’s expected count by exp(θ_k).
- Zero modification: Zero-modified Poisson terms separately shape zero probability and the conditional distribution of nonzero counts, although θ1 also affects the probability of zero.When θ2 = 0, the distribution reduces to Poisson.
- Dispersion modeling: CMP terms represent under- and overdispersion, but fitting often fails for data dispersed as much as or more than the geometric distribution.The CMP coefficient is constrained to θCMP ≤1, with θCMP = 0 retaining Poisson dispersion.
- Dispersion modeling: Square-root-based statistics model dispersion using variance-stabilized counts, while retaining a parameter space Θ = R^p through a highest-order term of order y_i,j.
- Mutuality: Mutuality statistics alter the conditional distribution of one count given its reciprocal, and alternative formulations can be reparametrizations when an edge-count term is present.
- Actor heterogeneity and triad closure: A pooled within-actor covariance of variance-stabilized dyad values represents actor heterogeneity, while conservative triadic statistics reduce dependence and bimodality at the cost of sensitivity.
6 Examples
Two valued-network examples apply Poisson-reference ERGMs to interaction counts, comparing social forces and modeling dispersion and heterogeneity. In the karate club, faction cohesion better explains structure than transitivity, while fraternity transitivity disappears after accounting for actor heterogeneity.
- 6.1 Example 1: Social relations in a karate club: The karate-club analysis compares faction allegiance with friend-of-a-friend transitivity using counts of interactions across eight social contexts.The network includes 34 actors, faction alignments, and dyadic counts of contexts in which actors interacted.
- 6.1 Example 1: Social relations in a karate club: The karate model uses a Poisson-reference ERGM with terms for leader interaction intensity, faction similarity, and transitivity of interaction intensities.Faction membership is coded from strongly aligned with Hi through strongly aligned with John.
- 6.1 Example 1: Social relations in a karate club: Faction similarity is highly significant and positive, whereas transitivity loses potential significance when both effects are included.The estimated correlation between their parameter estimates is −0.34, and faction allegiance is the stronger explanation at observation time.
- 6.2 Example 2: Interactions in a fraternity: The fraternity counts are strongly overdispersed relative to Poisson and geometric distributions, with mean 2.0 and standard deviation 3.4.The square root of within-actor count variance is 3.1, and removing values over 30 does not change the qualitative pattern.
- 6.2 Example 2: Interactions in a fraternity: In the fraternity network, apparent transitivity vanishes after modeling actor heterogeneity, indicating that highly social individuals better explain the excess transitive ties.Using a less conservative statistic raises significance to one-sided P-val. = 0.07, but fitting that effect produces degeneracy-like bimodality.
7 Discussion
The discussion extends valued ERGMs beyond count networks by allowing more general dyad-value spaces and reference measures, while identifying scope boundaries and future extensions.
- The paper generalizes ERGMs to unbounded-count networks and proposes terms for common network features, demonstrated on two contrasting networks.The analyses examine friend-of-a-friend effects alongside homophily and individual heterogeneity.
- Valued ERGMs can be formulated over a general dyad-value set S and a σ-finite measure space with reference measure Ph.Counts are the paper’s primary case, but the formulation is broader.
- For continuous data, the reference measure may be defined relative to Lebesgue measure, but its density-like function still requires specification.Binary and count data can instead use a function relative to counting measure.
- Mixed-zero continuous measurements, such as trade volumes, may require more complex reference-measure specifications.The cited example models log-transformed trade volume and handles zero observations separately.
- The framework assumes unconstrained sample spaces, whereas rank networks require permutation constraints and remain future work.Each actor’s unique ranking of alters induces a constrained sample space.
- The paper focuses on cross-sectional networks; longitudinal valued ERGMs and inference for partially observed valued networks are identified as natural extensions.Existing binary approaches provide precedents for both directions.
- Some model-fit procedures transfer readily, but valued-network goodness-of-fit characteristics and stability criteria require further development.MCMC diagnostics may need little modification, unlike some goodness-of-fit methods.
A A sampling algorithm for a Poisson-reference ERGM
The sampling algorithm uses local Poisson proposals for Poisson-reference ERGMs and adds occasional direct jumps to zero to improve mixing in zero-inflated networks.
- The algorithm samples Poisson-reference ERGMs with a Metropolis-Hastings procedure using a Poisson kernel centered at the current dyad value.The proposal mechanism can be adapted for highly overdispersed distributions.
- A direct proposal to zero, taken with probability π0, targets the zero inflation often observed in interaction-count networks.Setting π0 > 0 can speed up mixing.
- The dyad is selected uniformly, which may be inefficient for sparse networks because many selected dyads have zero values.The authors suggest adapting the tie-no-tie proposal to focus on nonzero dyads.
B Non-steepness of the Conway–Maxwell– Poisson family
The Conway–Maxwell–Poisson family used in valued ERGMs is neither regular nor steep, creating boundary-related and estimation concerns for highly dispersed data.
- The displayed CMP expressions define its probability model and normalizing constant, which underlie the regularity and steepness analysis.The supplied passages identify the pmf and normalizing constant but do not provide a complete readable expression.
- The Conway–Maxwell–Poisson family is not regular because its natural parameter space has a boundary at θ2 = 0.The natural parameter space is not open.
- The appendix presents a Poisson-reference ERGM sampler based on random dyad selection, Poisson proposals, optional jumps to zero, and Metropolis-Hastings acceptance.The algorithm returns a draw after T iterations from the specified model.
- The CMP family is not steep because finite first and second moments occur at its non-steep boundary.The proof uses the geometric distribution at that boundary.
- The non-steep boundary represents the most dispersed distribution CMP can model, so maximum-likelihood properties are not guaranteed for highly overdispersed data.This is a limitation of using CMP for such observations.