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Inference for dynamic Erdős-Rényi random graphs under regime switching

Michel Mandjes, Jiesen Wang

TL;DR

The paper tackles the problem of inferring edge-on/off-time distributions and the hidden regime in a pair of regime-switching dynamic Erdős-Rényi graphs, using only aggregated subgraph counts from the observed graph. It develops a method-of-moments estimator based on single-snapshot and lag-1 cross moments for edges, complete subgraphs, and stars, deriving closed-form moment expressions under an alternating renewal background. A two-step procedure is proposed: first identify equilibrium quantities $( abla, ho)$ from single-snapshot moments, then recover the six means $(E X_1,E X_2,E Z_1)$ and related quantities from cross-moments, with extensions to higher lag moments. Numerical experiments demonstrate accurate parameter recovery and suggest asymptotic normality, validating the approach for moderate network sizes and different parametric families. This framework enables learning hidden dynamics and regime-driven observations in domains where only coarse network statistics are accessible, with potential applications in economics, communications, and biology.

Abstract

This paper examines a model involving two dynamic Erdős-Rényi random graphs that evolve in parallel, with edges in each graph alternating between being present and absent according to specified on- and off-time distributions. A key feature of our setup is regime switching: the graph that is observed at any given moment depends on the state of an underlying background process, which is modeled as an alternating renewal process. This modeling framework captures a common situation in various real-world applications, where the observed network is influenced by a (typically unobservable) background process. Such scenarios arise, for example, in economics, communication networks, and biological systems. In our setup we only have access to aggregate quantities such as the number of active edges or the counts of specific subgraphs (such as stars or complete graphs) in the observed graph; importantly, we do not observe the mode. The objective is to estimate the on- and off-time distributions of the edges in each of the two dynamic Erdős-Rényi random graphs, as well as the distribution of time spent in each of the two modes. By employing parametric models for the on- and off-times and the background process, we develop a method of moments approach to estimate the relevant parameters. Experimental evaluations are conducted to demonstrate the effectiveness of the proposed method in recovering these parameters.

Inference for dynamic Erdős-Rényi random graphs under regime switching

TL;DR

The paper tackles the problem of inferring edge-on/off-time distributions and the hidden regime in a pair of regime-switching dynamic Erdős-Rényi graphs, using only aggregated subgraph counts from the observed graph. It develops a method-of-moments estimator based on single-snapshot and lag-1 cross moments for edges, complete subgraphs, and stars, deriving closed-form moment expressions under an alternating renewal background. A two-step procedure is proposed: first identify equilibrium quantities from single-snapshot moments, then recover the six means and related quantities from cross-moments, with extensions to higher lag moments. Numerical experiments demonstrate accurate parameter recovery and suggest asymptotic normality, validating the approach for moderate network sizes and different parametric families. This framework enables learning hidden dynamics and regime-driven observations in domains where only coarse network statistics are accessible, with potential applications in economics, communications, and biology.

Abstract

This paper examines a model involving two dynamic Erdős-Rényi random graphs that evolve in parallel, with edges in each graph alternating between being present and absent according to specified on- and off-time distributions. A key feature of our setup is regime switching: the graph that is observed at any given moment depends on the state of an underlying background process, which is modeled as an alternating renewal process. This modeling framework captures a common situation in various real-world applications, where the observed network is influenced by a (typically unobservable) background process. Such scenarios arise, for example, in economics, communication networks, and biological systems. In our setup we only have access to aggregate quantities such as the number of active edges or the counts of specific subgraphs (such as stars or complete graphs) in the observed graph; importantly, we do not observe the mode. The objective is to estimate the on- and off-time distributions of the edges in each of the two dynamic Erdős-Rényi random graphs, as well as the distribution of time spent in each of the two modes. By employing parametric models for the on- and off-times and the background process, we develop a method of moments approach to estimate the relevant parameters. Experimental evaluations are conducted to demonstrate the effectiveness of the proposed method in recovering these parameters.
Paper Structure (9 sections, 1 theorem, 26 equations, 4 figures, 2 tables)

This paper contains 9 sections, 1 theorem, 26 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Literature review
  3. Method of moments
  4. Edges
  5. Complete graphs
  6. Stars
  7. Estimator
  8. Numerical examples
  9. Conclusion

Key Result

Proposition 1

Let the two stars $S_{\ell}$ and $\Tilde{S}_{\ell}$ share $m$ vertices. Then Table tab:1 presents, for the scenarios ${\mathscr C}_1$, ${\mathscr C}_2$, and ${\mathscr C}_3$, the number of common edges and the number of occurrences.

Figures (4)

  • Figure 1: Illustration of common edges for $S_{4}$ and $\Tilde{S}_{4}$ when $m = 2$ (left panels) and $m=3$ (right panels), where the grey edges and nodes represent ${S}_4$, and the blue edges and nodes represent $\Tilde{S}_4$.
  • Figure 2: Histograms of parameters in Case I.
  • Figure 3: Histograms of parameters in Case II.
  • Figure 4: Histograms of parameters in Case III

Theorems & Definitions (3)

  • Proposition 1
  • proof
  • Remark 1