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Generalized network autoregressive modelling of longitudinal networks with application to presidential elections in the USA

Guy Nason, Daniel Salnikov, Mario Cortina-Borja

TL;DR

This work extends the GNAR framework to incorporate known community structure and asymmetric inter‑community interactions in longitudinal networks, enabling parsimonious yet expressive modelling of high‑dimensional time series. It introduces the community‑α GNAR with variable lag/r‑stage orders and interaction terms, together with finite‑sample error bounds for conditional LS estimation and model‑selection tools based on NACF/PNACF (Corbit/R‑Corbit). The approach is illustrated on 1976–2020 US presidential election data, revealing distinct dynamics among Red, Blue, and Swing states and evidence of cross‑community influences, especially involving Swing states. The combination of stationarity conditions, LS estimation, nonasymptotic bounds, and practical model selection provides a robust toolkit for analysing dynamic network time series in political science, finance, and beyond.

Abstract

Longitudinal networks are becoming increasingly relevant in the study of dynamic processes characterised by known or inferred community structure. Generalised Network Autoregressive (GNAR) models provide a parsimonious framework for exploiting the underlying network and multivariate time series. We introduce the community-$α$ GNAR model with interactions that exploits prior knowledge or exogenous variables for analysing interactions within and between communities, and can describe serial correlation in longitudinal networks. We derive new explicit finite-sample error bounds that validate analysing high-dimensional longitudinal network data with GNAR models, and provide insights into their attractive properties. We further illustrate our approach by analysing the dynamics of $\textit{Red, Blue}$ and $\textit{Swing}$ states throughout presidential elections in the USA from 1976 to 2020, that is, a time series of length twelve on 51 time series (US states and Washington DC). Our analysis connects network autocorrelation to eight-year long terms, highlights a possible change in the system after the 2016 election, and a difference in behaviour between $\textit{Red}$ and $\textit{Blue}$ states.

Generalized network autoregressive modelling of longitudinal networks with application to presidential elections in the USA

TL;DR

This work extends the GNAR framework to incorporate known community structure and asymmetric inter‑community interactions in longitudinal networks, enabling parsimonious yet expressive modelling of high‑dimensional time series. It introduces the community‑α GNAR with variable lag/r‑stage orders and interaction terms, together with finite‑sample error bounds for conditional LS estimation and model‑selection tools based on NACF/PNACF (Corbit/R‑Corbit). The approach is illustrated on 1976–2020 US presidential election data, revealing distinct dynamics among Red, Blue, and Swing states and evidence of cross‑community influences, especially involving Swing states. The combination of stationarity conditions, LS estimation, nonasymptotic bounds, and practical model selection provides a robust toolkit for analysing dynamic network time series in political science, finance, and beyond.

Abstract

Longitudinal networks are becoming increasingly relevant in the study of dynamic processes characterised by known or inferred community structure. Generalised Network Autoregressive (GNAR) models provide a parsimonious framework for exploiting the underlying network and multivariate time series. We introduce the community- GNAR model with interactions that exploits prior knowledge or exogenous variables for analysing interactions within and between communities, and can describe serial correlation in longitudinal networks. We derive new explicit finite-sample error bounds that validate analysing high-dimensional longitudinal network data with GNAR models, and provide insights into their attractive properties. We further illustrate our approach by analysing the dynamics of and states throughout presidential elections in the USA from 1976 to 2020, that is, a time series of length twelve on 51 time series (US states and Washington DC). Our analysis connects network autocorrelation to eight-year long terms, highlights a possible change in the system after the 2016 election, and a difference in behaviour between and states.

Paper Structure

This paper contains 37 sections, 14 theorems, 135 equations, 7 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contribution and paper structure
  3. Review of GNAR models
  4. The GNAR model
  5. Introducing community structure into GNAR
  6. Different model order between communities
  7. Introducing interactions between communities
  8. Comparison with other models
  9. The parsimonious GNAR family
  10. Conditions for stationarity
  11. Conditional least-squares estimation for GNAR models
  12. Statistical efficiency for GNAR model specifications
  13. Model selection
  14. Modelling presidential elections in the USA
  15. Analysis of network autocorrelation and model selection
  16. ...and 22 more sections

Key Result

Theorem 1

[gnar_paper] Let $\boldsymbol{X}_t$ be a local-$\alpha$$\operatorname{GNAR} (p, [s_k] )$ process linked to a static network. If the parameters in $X_{i, t} = \sum_{k = 1}^{p} ( \alpha_{i, k} X_{i, t - k} + \sum_{r = 1}^{s_k} \sum_{c = 1}^C \beta_{k r c} Z_{i, t - k}^{r, c} ) + u_{i, t},$ where $u_{i

Figures (7)

  • Figure 1: USA state-wise network, blue nodes are Blue states (Democrat nominee won at least 75$\%$ of elections), orange nodes are Red states (Republican nominee won at least $75\%$ of elections), and grey nodes are Swing states (neither party won at least $75\%$ of elections).
  • Figure 2: Illustration of a longitudinal network. The first plot is the fiveNet network (included in the GNAR package); see gnar_package. The latter are nodal time series realisations of length 100 generated by a $\hbox{community-$\alpha$}$$\operatorname{GNAR} \left ( [1, 2], \{ [1], [1, 1] \}, 2 \right )$ model. Blue nodes belong to community $K_1 = \{2, 3, 4 \}$ and orange ones to community $K_2 = \{ 1, 5 \}$.
  • Figure 3: R-Corbit plot for 1000 long realisation generated by a stationary $\hbox{community-$\alpha$}$$\operatorname{GNAR} \left ( [1, 2], \{ [1], [1, 1] \}, 2 \right )$, where the underlying network is fiveNet, $K_1 = \{2, 3, 4\}$ and $K_2 = \{1, 5 \}$; see Figure \ref{['fig: 2-communal fiveNet']}. The maximum lag is equal to six and maximum $\hbox{$r$-stage}$ depth is equal to three. The PNACF cut-offs are $(1, [1])$ for $K_1$ and $(2, [1, 1])$ for $K_2$.
  • Figure 4: Cross-correlation heat-map plots for a 1000 long realisation generated by a stationary $\hbox{community-$\alpha$}$$\operatorname{GNAR} \left ( [1, 2], \{ [1], [1, 1] \}, 2 \right )$, where the underlying network is fiveNet, $K_1 = \{2, 3, 4\}$ and $K_2 = \{1, 5 \}$; see Figure \ref{['fig: 2-communal fiveNet']}. Figure \ref{['fig: one-lag cross-corr']} shows the one-lag cross-correlation matrix, and \ref{['fig: four-lag cross-corr']} the four-lag one. The plots highlights the two communities and that the sample cross-correlation between nodes in different communities is close to zero. The diagonal corresponds to node-wise autocorrelations, which are coloured grey so as not to detract from the main object of study: the cross-correlations.
  • Figure 5: PNACF R-Corbit plot of the series $\boldsymbol{X}_{t}$. The underlying network is the USA state-wise network; see Figure \ref{['fig: usa net']}, and Section \ref{['sec: model selection']} for plot description.
  • ...and 2 more figures

Theorems & Definitions (34)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Remark 3
  • Theorem 2
  • Remark 4
  • Corollary 3
  • Remark 5
  • ...and 24 more