[Paper Review] Tie-decay temporal networks in continuous time and eigenvector-based centralities

Walid Ahmad, Mason A. Porter+1 more|arXiv (Cornell University)|May 1, 2018

Complex Network Analysis Techniques47 references3 citations

TL;DR

This paper introduces a continuous-time network model that distinguishes between discrete interactions (contacts) and evolving relationship strengths (ties) using a tie-decay mechanism. It extends PageRank to this framework, enabling efficient, mathematically tractable centrality analysis in time-evolving networks, demonstrated on a UK NHS controversy retweet network and applicable to streaming data.

ABSTRACT

Network theory is a useful framework for studying interconnected systems of interacting agents. Many networked systems evolve continuously in time, but most existing methods for the analysis of time-dependent networks rely on discrete or discretized time. In this paper, we propose an approach for studying networks that evolve in continuous time by distinguishing between interactions, which we model as discrete contacts, and ties, which represent strengths of relationships as functions of time. To illustrate our tie-decay network formulation, we adapt the well-known PageRank centrality score to the tie-decay framework in a mathematically tractable and computationally efficient way. We demonstrate our framework on a synthetic example and then use it to study a network of retweets during the 2012 National Health Service controversy in the United Kingdom. Our work also provides guidance for similar generalizations of other tools from network theory to continuous-time networks with tie decay, including for applications to streaming data.

Motivation & Objective

• To address the limitation of discrete-time network analysis by modeling networks that evolve continuously in time.
• To distinguish between transient interactions (contacts) and persistent relationship strengths (ties) as time-varying functions.
• To develop a computationally efficient and mathematically tractable extension of PageRank for continuous-time networks with tie decay.
• To demonstrate the framework on real-world streaming data, such as social media interactions during a political controversy.
• To provide a foundation for generalizing other network analysis tools to continuous-time settings with decaying ties.

Proposed method

• Models interactions as discrete contacts and relationships as time-decaying ties, where tie strength decays exponentially over time without interaction.
• Defines a continuous-time adjacency matrix where edge weights are functions of time, based on the time since the last contact.
• Adapts the PageRank algorithm by replacing discrete-time transition matrices with continuous-time transition operators derived from the tie-decay model.
• Uses a system of ordinary differential equations (ODEs) to model the evolution of node influence over continuous time.
• Employs numerical integration to compute centrality scores efficiently, enabling real-time or near-real-time analysis.
• Validates the approach using a synthetic network to verify correctness and then applies it to a real retweet network from the 2012 UK NHS controversy.

Experimental results

Research questions

• RQ1 How can network centrality be meaningfully defined and computed in networks that evolve continuously over time?
• RQ2 Can the PageRank algorithm be generalized to continuous-time networks with decaying ties while preserving mathematical tractability and computational efficiency?
• RQ3 How do tie-decay dynamics affect the identification of influential nodes in time-evolving networks?
• RQ4 What are the practical implications of using continuous-time models for analyzing real-world streaming data, such as social media interactions?
• RQ5 To what extent can this framework be extended to other network analysis tools beyond centrality?

Key findings

• The proposed tie-decay framework successfully models continuous-time network evolution by capturing the decay of relationship strength over time without new interactions.
• The continuous-time PageRank extension maintains mathematical coherence and enables efficient computation through ODE-based dynamics.
• The method identifies influential nodes in the 2012 NHS controversy retweet network with greater temporal precision than discrete-time alternatives.
• The framework demonstrates scalability and suitability for streaming data applications due to its computational efficiency.
• The synthetic example confirms the correctness and stability of the proposed centrality computation under varying decay rates.
• The approach provides a generalizable pathway for adapting other network-theoretic tools to continuous-time settings with decaying ties.