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Separable models for dynamic signed networks

Alberto Caimo, Isabella Gollini

TL;DR

A separable temporal generative framework based on multi-layer exponential random graph models, characterised by the assumption of conditional independence between the sign and interaction effects, is proposed, which preserves the flexibly and explanatory power inherent in the binary network specification while adhering to consistent balance theory assumptions.

Abstract

Signed networks capture the polarity of relationships between nodes, providing valuable insights into complex systems where both supportive and antagonistic interactions play a critical role in shaping the network dynamics. We propose a separable temporal generative framework based on multi-layer exponential random graph models, characterised by the assumption of conditional independence between the sign and interaction effects. This structure preserves the flexibly and explanatory power inherent in the binary network specification while adhering to consistent balance theory assumptions. Using a fully probabilistic Bayesian paradigm, we infer the doubly intractable posterior distribution of model parameters via an adaptive Metropolis-Hastings approximate exchange algorithm. We illustrate the interpretability of our model by analysing signed relations among U.S. Senators during Ronald Reagan's second term (1985-1989). Specifically, we aim to understand whether these relations are consistent and balanced or reflect patterns of supportive or antagonistic alliances.

Separable models for dynamic signed networks

TL;DR

A separable temporal generative framework based on multi-layer exponential random graph models, characterised by the assumption of conditional independence between the sign and interaction effects, is proposed, which preserves the flexibly and explanatory power inherent in the binary network specification while adhering to consistent balance theory assumptions.

Abstract

Signed networks capture the polarity of relationships between nodes, providing valuable insights into complex systems where both supportive and antagonistic interactions play a critical role in shaping the network dynamics. We propose a separable temporal generative framework based on multi-layer exponential random graph models, characterised by the assumption of conditional independence between the sign and interaction effects. This structure preserves the flexibly and explanatory power inherent in the binary network specification while adhering to consistent balance theory assumptions. Using a fully probabilistic Bayesian paradigm, we infer the doubly intractable posterior distribution of model parameters via an adaptive Metropolis-Hastings approximate exchange algorithm. We illustrate the interpretability of our model by analysing signed relations among U.S. Senators during Ronald Reagan's second term (1985-1989). Specifically, we aim to understand whether these relations are consistent and balanced or reflect patterns of supportive or antagonistic alliances.
Paper Structure (18 sections, 15 equations, 9 figures, 3 tables, 1 algorithm)

This paper contains 18 sections, 15 equations, 9 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Exponential random graph models
  3. Temporal exponential random graph models
  4. Signed network structures
  5. Multi-layer interpretation
  6. Separable modelling
  7. Layered separable temporal models
  8. Model terms
  9. Bayesian inference
  10. Software
  11. Comparative simulation study
  12. Application to US Congressional data
  13. Model specification
  14. Posterior inference
  15. Marginal estimation
  16. ...and 3 more sections

Figures (9)

  • Figure 1: Triadic configurations of a signed network. Solid lines represent positive interactions, dashed lines represent negative interactions.
  • Figure 2: Graphical representation of (a) 2-layer ERGMs defined in Equation \ref{['eq:ERGM_2']} and (b) separable 2-layer ERGMs defined in Equation \ref{['eq:sepERGM_2']}. Squares represent observed variables (network layers); circles represent parameters and arrows represent directed dependencies. In panel (a), both $\mathbf{x}$ and $\mathbf{z}$ are influenced by a common parameter $\boldsymbol{\vartheta}.$ In panel (b), $\mathbf{x}$ and $\mathbf{z}$ each have their own parameters, denoted by $\boldsymbol{\xi}$ and $\boldsymbol{\zeta}$, respectively. These parameters are distinct, meaning the two processes are conditionally independent given $\mathbf{x}.$
  • Figure 3: Graphical representation of 2-layer STERGMs for signed network data conditional on an initial network state (not included).
  • Figure 4: Endogenous edgewise shared partners configurations: (a) esf$^{+}$: positive edgewise shared friends; (b) ese$^{+}$: positive edgewise shared enemies; (c) ese$^{-}$: negative edgewise shared enemies; (d) esf$^{-}$: negative edgewise shared friends.
  • Figure 5: MLE estimates (black dots) with 95% confidence intervals are shown for the TERGM and STERGM based on fri:meh:thu:kau22, while the 2-layer STERGM displays posterior means (black dots) with 95% credible intervals. In the 2-layer STERGM plot, the white dots with black outline indicate the true parameter values used to generate the data.
  • ...and 4 more figures