Spline tie-decay temporal networks

TL;DR

This paper introduces a spline-based extension to tie-decay temporal networks, allowing for more flexible modeling of interaction dynamics with minimal additional computational cost.

Contribution

It develops a spline kernel for tie-decay networks, enabling delayed and smooth interaction rates, and demonstrates its mathematical properties and applications.

Findings

Spline tie-decay networks support $C^1$-continuous interaction rates.

The method is computationally efficient with marginal memory increase.

Applications include network embedding, opinion dynamics, and epidemic spreading.

Abstract

Increasing amounts of data are available on temporal, or time-varying, networks. There have been various representations of temporal network data each of which has different advantages for downstream tasks such as mathematical analysis, visualizations, agent-based and other dynamical simulations on the temporal network, and discovery of useful structure. The tie-decay network is a representation of temporal networks whose advantages include the capability of generating continuous-time networks from discrete time-stamped contact event data with mathematical tractability and a low computational cost. However, the current framework of tie-decay networks is limited in terms of how each discrete contact event can affect the time-dependent tie strength (which we call the kernel). Here we extend the tie-decay network model in terms of the kernel. Specifically, we use a cubic spline function for modeling short-term behavior of the kernel and an exponential decay function for long-term behavior, and graft them together. This spline version of tie-decay network enables delayed and $C^1$-continuous interaction rates between two nodes while it only marginally increases the computational and memory burden relative to the conventional tie-decay network. We show mathematical properties of the spline tie-decay network and numerically showcase it with three tasks: network embedding, a deterministic opinion dynamics model, and a stochastic epidemic spreading model.