Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Time-dependent Personalized PageRank for temporal networks: discrete and continuous scales

David Aleja, Julio Flores, Eva Primo, Miguel Romance

TL;DR

This work advances the study of PageRank on temporal networks by formulating a time-dependent PageRank for both discrete and continuous time scales with time-varying personalization vectors and damping. It establishes a clear connection between discrete snapshots and a continuous-time model, and proves localization bounds that quantify how personalization changes can influence node rankings. The approach is demonstrated with theoretical results and real-world/synthetic examples, including Wikipedia and other temporal networks, highlighting the impact of time-varying personalization on centrality. The framework provides a robust tool for understanding dynamic importance in evolving networks and guides principled interpolation between discrete and continuous-time analyses.

Abstract

In this paper we explore the PageRank of temporal networks on both discrete and continuous time scales in the presence of personalization vectors that vary over time. Also the underlying interplay between the discrete and continuous settings arising from discretization is highlighted. Additionally, localization results that set bounds to the estimated influence of the personalization vector on the ranking of a particular node are given. The theoretical results are illustrated by means of some real and synthetic examples.

Time-dependent Personalized PageRank for temporal networks: discrete and continuous scales

TL;DR

This work advances the study of PageRank on temporal networks by formulating a time-dependent PageRank for both discrete and continuous time scales with time-varying personalization vectors and damping. It establishes a clear connection between discrete snapshots and a continuous-time model, and proves localization bounds that quantify how personalization changes can influence node rankings. The approach is demonstrated with theoretical results and real-world/synthetic examples, including Wikipedia and other temporal networks, highlighting the impact of time-varying personalization on centrality. The framework provides a robust tool for understanding dynamic importance in evolving networks and guides principled interpolation between discrete and continuous-time analyses.

Abstract

In this paper we explore the PageRank of temporal networks on both discrete and continuous time scales in the presence of personalization vectors that vary over time. Also the underlying interplay between the discrete and continuous settings arising from discretization is highlighted. Additionally, localization results that set bounds to the estimated influence of the personalization vector on the ranking of a particular node are given. The theoretical results are illustrated by means of some real and synthetic examples.
Paper Structure (4 sections, 4 theorems, 26 equations, 3 figures)

This paper contains 4 sections, 4 theorems, 26 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation and some preliminary definitions
  3. Discrete time scale case
  4. Continuous time scale case

Key Result

Theorem 3.3

If $(V,E(t))_{t\in I}$ is a temporal network of $n$ nodes with discrete time-scale $I=\{t_1,\dots,t_N\}$, then for every $k\in\{1,\dots, N\}$ and every node $i\in\{1,\dots, n\}$ where $X(t_k)=(1-\lambda(t_k))\left(Id-\lambda(t_k) P_B(t_k)\right)^{-1}.$

Figures (3)

  • Figure 1: Effects by changing the personalization vector in Definition \ref{['def:local']}. PageRank values of five nodes (a color for each node) of Italian Wikipedia temporal network (see Refs. [Jerome] and [Preusse]) corresponding to an uniform (left panel), Input type (middle panel) and Inverse-Input type (right panel) personalization vector.
  • Figure 2: A comparison by changing the personalization vector or the teleportation parameter in Definition \ref{['def:local']}. Panel a): Kendall Tau coefficients over time are shown between the three pairs of temporal PageRanks considered, with Uniform, Input and Inverse Input personalization vector. Panel b): Kendall Tau coefficients over time are shown between the three pairs of temporal PageRanks when the teleportation parameter varies.
  • Figure 3: The discrete PageRank (Definition \ref{['def:local']}) approximates to the continuous PageRank (Definition \ref{['def:cont-local']}). PageRank values of five nodes (a color for each node) of a synthetic graph \ref{['synthetic']} given by Definition \ref{['def:cont-local']} (continuous line) and by Definition \ref{['def:local']} with $N=5$ (dotted line) and with $N=9$ (dashed line) for the weights $\omega(s,t)=e^{-\alpha (t-s)}$ with $\alpha=-4$ (left panel), $\alpha=1$ (middle panel) and $\alpha=6$ (right panel).

Theorems & Definitions (11)

  • Remark 3.1
  • Definition 3.2
  • Theorem 3.3
  • proof
  • Remark 4.1
  • Definition 4.2
  • Proposition 4.3
  • proof
  • Definition 4.4
  • Theorem 4.5
  • ...and 1 more