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Longitudinal Network Models and Permutation-Uniform Markov Chains

William K. Schwartz, Sonja Petrović, Hemanshu Kaul

TL;DR

This framework facilitates introducing a new network model, simplifies analysis of some network and autoregressive models from the literature, including by permitting closed‐form expressions for maximum likelihood estimates for some models, and facilitates applying standard tools to longitudinal‐network Markov chains from either asymptotics or single‐observation exponential random graph models.

Abstract

Consider longitudinal networks whose edges turn on and off according to a discrete-time Markov chain with exponential-family transition probabilities. We characterize when their joint distributions are also exponential families with the same parameter, improving data reduction. Further we show that the permutation-uniform subclass of these chains permit interpretation as an independent, identically distributed sequence on the same state space. We then apply these ideas to temporal exponential random graph models, for which permutation uniformity is well suited, and discuss mean-parameter convergence, dyadic independence, and exchangeability. Our framework facilitates our introducing a new network model; simplifies analysis of some network and autoregressive models from the literature, including by permitting closed-form expressions for maximum likelihood estimates for some models; and facilitates applying standard tools to longitudinal-network Markov chains from either asymptotics or single-observation exponential random graph models.

Longitudinal Network Models and Permutation-Uniform Markov Chains

TL;DR

This framework facilitates introducing a new network model, simplifies analysis of some network and autoregressive models from the literature, including by permitting closed‐form expressions for maximum likelihood estimates for some models, and facilitates applying standard tools to longitudinal‐network Markov chains from either asymptotics or single‐observation exponential random graph models.

Abstract

Consider longitudinal networks whose edges turn on and off according to a discrete-time Markov chain with exponential-family transition probabilities. We characterize when their joint distributions are also exponential families with the same parameter, improving data reduction. Further we show that the permutation-uniform subclass of these chains permit interpretation as an independent, identically distributed sequence on the same state space. We then apply these ideas to temporal exponential random graph models, for which permutation uniformity is well suited, and discuss mean-parameter convergence, dyadic independence, and exchangeability. Our framework facilitates our introducing a new network model; simplifies analysis of some network and autoregressive models from the literature, including by permitting closed-form expressions for maximum likelihood estimates for some models; and facilitates applying standard tools to longitudinal-network Markov chains from either asymptotics or single-observation exponential random graph models.

Paper Structure

This paper contains 24 sections, 16 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Notation and Terminology
  3. Markov Chains with Low-Dimension Sufficient Statistics
  4. Permutation-Uniform Markov Chains
  5. Independence
  6. Convergence
  7. Symmetry
  8. Permutation Uniformity and CEFs
  9. Markov Chains of Graphs
  10. Finite Exchangeability
  11. Dyadic Independence
  12. Exponential Random Multigraph Models
  13. Independent Sequences and Multigraphs
  14. Examples from the Literature
  15. A Model of Loyalty
  16. ...and 9 more sections

Key Result

Theorem 1.2

$\mathscr{P}$ is the MEF from eq:def-mef with $\kappa(a, b) = P_{\bm{\theta}_0}(a, b)$ for some $\bm{\theta}_0 \in \Theta$ and all $a, b \in \mathscr{S}$ if and only if $\kappa(a, b) = L^1_{\bm{\theta}_0, a}(b)$ for some $\bm{\theta}_0 \in \Theta$ and all $a, b \in \mathscr{S}$, and $\{L^t_{\bm{\the for all $x_0, x_1, \dotsc, x_t \in \mathscr{S}$, $t \in \mathbb{N}_{>0}$, and $\bm{\theta} \in \The

Theorems & Definitions (43)

  • Example 1.1: [Example 4.3$\cdot$3, 358--359]Gani:1955gp
  • Theorem 1.2
  • Proposition 1.3: [357--358]Gani:1955gp
  • proof : Proof sketch.
  • Theorem 1.4
  • proof
  • Definition 2.1
  • Theorem 2.2
  • proof : Proof of \ref{['thm:uniform-chain']}
  • Example 2.3: Modular Autoregressive Model [Example 6.2, 66]diaconis-freedman-1999
  • ...and 33 more