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Generating Temporal Contact Graphs Using Random Walkers

Anton-David Almasan, Sergey Shvydun, Ingo Scholtes, Piet Van Mieghem

TL;DR

RWIG is shown to be a realistic model for temporal human contact graphs, which may place RWIG on a same footing as the Erdos–Renyi (ER) and Barabasi–Albert (BA) models for fixed graphs.

Abstract

We study human mobility networks through timeseries of contacts between individuals. Our proposed Random Walkers Induced temporal Graph (RWIG) model generates temporal graph sequences based on independent random walkers that traverse an underlying graph in discrete time steps. Co-location of walkers at a given node and time defines an individual-level contact. RWIG is shown to be a realistic model for temporal human contact graphs, which may place RWIG on a same footing as the Erdos-Renyi (ER) and Barabasi-Albert (BA) models for fixed graphs. Moreover, RWIG is analytically feasible: we derive closed form solutions for the probability distribution of contact graphs.

Generating Temporal Contact Graphs Using Random Walkers

TL;DR

RWIG is shown to be a realistic model for temporal human contact graphs, which may place RWIG on a same footing as the Erdos–Renyi (ER) and Barabasi–Albert (BA) models for fixed graphs.

Abstract

We study human mobility networks through timeseries of contacts between individuals. Our proposed Random Walkers Induced temporal Graph (RWIG) model generates temporal graph sequences based on independent random walkers that traverse an underlying graph in discrete time steps. Co-location of walkers at a given node and time defines an individual-level contact. RWIG is shown to be a realistic model for temporal human contact graphs, which may place RWIG on a same footing as the Erdos-Renyi (ER) and Barabasi-Albert (BA) models for fixed graphs. Moreover, RWIG is analytically feasible: we derive closed form solutions for the probability distribution of contact graphs.
Paper Structure (31 sections, 8 theorems, 69 equations, 8 figures, 4 tables, 2 algorithms)

This paper contains 31 sections, 8 theorems, 69 equations, 8 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Random Walkers Induced temporal Graph (RWIG)
  3. Formulation of RWIG
  4. State space of RWIG
  5. Contact graph probability: Examples
  6. Introductory example
  7. 2-clique contact graph
  8. 3-clique contact graph
  9. Contact graph probability distribution
  10. Amassed clique contact graphs
  11. Main theorem
  12. Steady-state contact graphs
  13. Unlabelled contact graphs
  14. A combinatorial computation of the steady-state graph
  15. Empirical analysis
  16. ...and 16 more sections

Key Result

Theorem 1

The probability of an $m$-clique contact graph $g = \{\mathcal{A}_1, \mathcal{A}_2, ..., \mathcal{A}_m\}$ at discrete time $k$ is

Figures (8)

  • Figure 1: RWIG. (a) Contact graphs. (b) Random walkers traversing the Markov graph.
  • Figure 2: Cliques (a) as partitions on the walker set (b).
  • Figure 3: Process of creating amassed clique contact graphs.
  • Figure 4: Most probable 4 realisations of the contact graph $G_\infty$ formed by 4 walkers.
  • Figure 5: Probability density of the unlabelled contact graph $G^u_\infty$ formed by 4 walkers.
  • ...and 3 more figures

Theorems & Definitions (16)

  • Theorem 1
  • proof
  • Example 1
  • Theorem 2
  • Lemma 1
  • Lemma 2
  • Example 2
  • Lemma 3
  • Example 3
  • Example 4
  • ...and 6 more