# Directed Percolation in Temporal Networks

arXiv: https://arxiv.org/abs/2107.01510 · HTML: https://mlsanity.com/paper/2107.01510/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Jul 2021.

## Built on by (most cited first, 5 of 5)

- [De Bruijn goes Neural: Causality-Aware Graph Neural Networks for Time Series Data on Dynamic Graphs](https://mlsanity.com/paper/2209.08311/) (2022)
- [A Survey of Link Prediction in Temporal Networks](https://mlsanity.com/paper/2502.21185/) (2025)
- [Initialisation and Network Effects in Decentralised Federated Learning](https://mlsanity.com/paper/2403.15855/) (2024)
- [Weisfeiler and Leman Follow the Arrow of Time: Expressive Power of Message Passing in Temporal Event Graphs](https://mlsanity.com/paper/2505.24438/) (2025)
- [Decentralised Federated Learning over Temporal Networks: The Role of Heterogeneities](https://mlsanity.com/paper/2607.03171/) (2026)

## Abstract (arXiv)

Connectivity and reachability on temporal networks, which can describe the spreading of a disease, decimation of information or the accessibility of a public transport system over time, have been among the main contemporary areas of study in complex systems for the last decade. However, while isotropic percolation theory successfully describes connectivity in static networks, a similar description has not been yet developed for temporal networks. Here address this problem and formalize a mapping of the concept of temporal network reachability to percolation theory. We show that the limited-waiting-time reachability, a generic notion of constrained connectivity in temporal networks, displays directed percolation phase transition in connectivity. Consequently, the critical percolation properties of spreading processes on temporal networks can be estimated by a set of known exponents characterising the directed percolation universality class. This result is robust across a diverse set of temporal network models with different temporal and topological heterogeneities, while by using our methodology we uncover similar reachability phase transitions in real temporal networks too. These findings open up an avenue to apply theory, concepts and methodology from the well-developed directed percolation literature to temporal networks.