Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Estimating Graph Dynamics from Population Observations

Peter Braunsteins, Michel Mandjes, Florian Montalescot

Abstract

In this paper we consider a population process evolving on a dynamic random graph. The dynamic random graph is an Erdős--Rényi graph that is resampled every time unit, independently of the previous ones, with `edge existence probability' $p$. The population process consists of $M$ individuals which reside at the vertices of the dynamic graph. At each point in time any of the $M$ individuals, supposing it resides at a vertex with $k$ neighbors, jumps to an adjacent vertex with probability $k/(k+1)$ (where this adjacent vertex is picked uniformly at random), and with probability $1/(k+1)$ it stays where it is. We suppose we observe the numbers of individuals at each of the vertices, but not the evolving random graph itself. We propose two estimators for $p$, and establish their consistency and asymptotic normality.

Estimating Graph Dynamics from Population Observations

Abstract

In this paper we consider a population process evolving on a dynamic random graph. The dynamic random graph is an Erdős--Rényi graph that is resampled every time unit, independently of the previous ones, with `edge existence probability' . The population process consists of individuals which reside at the vertices of the dynamic graph. At each point in time any of the individuals, supposing it resides at a vertex with neighbors, jumps to an adjacent vertex with probability (where this adjacent vertex is picked uniformly at random), and with probability it stays where it is. We suppose we observe the numbers of individuals at each of the vertices, but not the evolving random graph itself. We propose two estimators for , and establish their consistency and asymptotic normality.
Paper Structure (12 sections, 5 theorems, 62 equations, 5 figures)

This paper contains 12 sections, 5 theorems, 62 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Model and objective
  3. Method-of-moments based estimator
  4. Auxiliary results
  5. Second moment
  6. Estimator
  7. Asymptotic normality
  8. Asymptotic normality of the sample-mean vector
  9. Asymptotic normality of the covariance estimator
  10. Asymptotic normality of the parameter estimator
  11. Least-squares based estimator
  12. Numerical experiments

Key Result

Lemma 1

For $i\in\{1,\ldots,n\}$ and $t\in{\mathbb N}$,

Figures (5)

  • Figure 1: Our dynamic network, together with the individuals moving on it, shown at three successive time points ($n=6$, $M=10$).
  • Figure 2: QQ-plot of the empirical distribution of $\widehat{p}_T$ (vertical axis; 'empirical quantiles') against the normal distribution (horizontal axis; 'theoretical quantiles'), based on $R=2000$ replications, with $n=7$, $M=14$, and $T=4000$. Left panel: $p=0.25$, middle panel: $p=0.50$, right panel: $p=0.75$.
  • Figure 3: QQ-plot of the empirical distribution of $\bar{p}_T$ (vertical axis; 'empirical quantiles') against the normal distribution (horizontal axis; 'theoretical quantiles'), based on $R=2000$ replications, with $n=7$, $M=14$, and $T=4000$. Left panel: $p=0.25$, middle panel: $p=0.50$, right panel: $p=0.75$.
  • Figure 4: Left panel: $\lambda(p)$ as a function of $p$. Right panel: $\mu(p)$ as a function of $p$. In the setting considered, $n=7$ and $M=14$. The estimates underlying the right panel are based on runs of length $T=4000$ and $R=2000$ replications.
  • Figure 5: Left panel: the standard deviations of the estimators $\widehat{p}_T$ and $\bar{p}_T$; orange curve corresponds to $\sqrt{\newline\,} \widehat{p}_T$ and blue curve to $\sqrt{\newline\,} \bar{p}_T$. In the setting considered, $n=7$ and $M=14$. Right panel: $\nu(p)$ as a function of $p$. The estimates are based on runs of length $T=4000$ and $R=2000$ replications.

Theorems & Definitions (12)

  • Remark 1
  • Lemma 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Proposition 1
  • Theorem 1
  • Theorem 2
  • Remark 5
  • Theorem 3
  • ...and 2 more