Directed Percolation in Temporal Networks (CCS 2021)

Connectivity transitions in static networks are well described by percolation theory, yet the corresponding description is not developed for temporal networks. We map the connectivity problem of temporal networks to directed percolation and show that the reachability phase transition in random temporal network models, as induced by any limited-waiting-time process, appears with the mean-field exponents of directed percolation. Furthermore, we measure the central thermodynamic quantities adapted to large-scale real temporal networks to uncover reachability transitions in their global connectedness.

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1. Arash Badie-Modiri, Abbas K. Rizi, Márton Karsai, Mikko Kivelä https://doi.org/10.1103/PhysRevResearch.4.L022047 https://doi.org/10.1103/PhysRevE.105.054313 Directed Percolation in Temporal Networks
2. Saramäki, J. (2019). Temporal Networks: Past, present, future. Slides for keynote at Complex Networks 2019. Available from: https://jarisaramaki.com. Reachability on Temporal Networks
3. Badie-Modiri et al. (2019). Efficient limited-time reachability estimation in temporal networks. Physical Review E 101.5 (2020): 052303.
4. Saramäki, J. (2019). Temporal Networks: Past, present, future. Slides for keynote at Complex Networks 2019. Available from: https://jarisaramaki.com.
10. Event graphs Event Graphs Badie-Modiri et al. (2019). Efficient limited-time reachability estimation in temporal networks. Physical Review E 101.5 (2020): 052303.
11. Event graphs Event Graphs Badie-Modiri et al. (2019). Efficient limited-time reachability estimation in temporal networks. Physical Review E 101.5 (2020): 052303.
14. Saramäki, J. (2019). Temporal Networks: Past, present, future. Slides for keynote at Complex Networks 2019. Available from: https://jarisaramaki.com. Reachability in Temporal Networks out-component in-component
15. Isotropic bond Percolation Directed bond Percolation Hinrichsen, H. (2000). Non-equilibrium critical phenomena and phase transitions into absorbing states. Advances in physics 49.7 (2000): 815-958. Directed Percolation
16. Isotropic bond Percolation Directed bond Percolation Hinrichsen, H. (2000). Non-equilibrium critical phenomena and phase transitions into absorbing states. Advances in physics 49.7 (2000): 815-958. Directed Percolation Multiplication (Offspring production)
17. Isotropic bond Percolation Directed bond Percolation Hinrichsen, H. (2000). Non-equilibrium critical phenomena and phase transitions into absorbing states. Advances in physics 49.7 (2000): 815-958. Directed Percolation Coalescence Death
18. Hinrichsen, H. (2000). Non-equilibrium critical phenomena and phase transitions into absorbing states. Advances in physics 49.7 (2000): 815-958. Isotropic bond Percolation Directed bond Percolation Directed Percolation Coalescence Death Multiplication (Offspring production) Absorbing state (extinction) Active state (survival)
19. p < p c p > p c p = p c
20. p < p c p > p c p = p c V T M = Cluster mass: number of occupied sites V = Cluster volume: number of unique nodes T = Survival time: time to the last occupied site M
21. p > p c p = p c p < p c V T M = Cluster mass: number of occupied sites V = Cluster volume: number of unique nodes T = Survival time: time to the last occupied site M p = p c
22. ρ = Occupation density: fraction of occupied sites ꭕ = Susceptibility: changes in face of perturbation p > p c p = p c
23. Directed percolation in temporal networks Directed Percolation in Temporal Networks
24. Directed percolation in temporal networks Directed Percolation in Temporal Networks
25. Directed percolation in temporal networks Directed Percolation in Temporal Networks Spontaneous occupation by external field
26. DP in Temporal Networks: Analytical Solutions Out-component size from the self-consistency equations: Static density from the mean-field rate equations:
27. DP in Temporal Networks: Empirical Validation
28. DP in Temporal Networks: Empirical Validation • Erdős–Rényi • Random regular • 1-4D grid lattices • Poisson process • Renewal process with power-law inter-event times • Hawkes (self-exciting) processes Network topologies: Edge dynamics:
29. DP Measures in Real-world Temporal Networks US air transport Helsinki public transport Mobile phone calls Twitter mentions
30. Reticula $ pip install -U reticula https://reticula.network Supports: • Undirected/directed static networks • Undirected/directed static hypernetworks • Undirected/directed/delayed temporal networks • Undirected/directed/delayed temporal hypernetworks
31. Reticula You can: • Generate random networks • Load networks from file • Interoperate with NetworkX • Calculate various network properties • Reachability/connectivity analysis • Randomise temporal networks
32. Reticula def average_volume(g, time): adj = ret.temporal_adjacency.limited_waiting_time[g.edge_type()](time) clusters = ret.out_cluster_size_estimates( temporal_network=g, temporal_adjacency=adj, time_resolution=1.0, seed=0) return sum([c.volume_estimate() for _, c in clusters])/len(clusters) n = 5000 state = ret.mersenne_twister() g = ret.random_fully_mixed_temporal_network[ret.int64]( size=n, rate=0.2/n, max_time=2048, random_state=state) waiting_times = np.linspace(0, 10, num=50) with ThreadPoolExecutor(max_workers=8) as e: volumes = e.map(partial(average_volume, g), waiting_times) plt.plot(waiting_times, list(volumes)) More than a million events
33. Reticula Estimates limited waiting-time reachability from every possible starting point and every possible starting time on a temporal network with a million events for 50 different values of maximum waiting-time in less than four minutes.
34. • Connectivity in many temporal networks can be explained by directed percolation. • This finding is robust for various temporal and topological heterogeneities. • Event graphs map temporal network problems into DAG problems. Upshot Open Problems • What is the effect of introducing temporal motifs? • What are the generalisation limits? Additional parameters? What about other cutoffs other than “limited waiting-time”? • Real-world networks?
35. “Directed percolation in temporal networks” https://doi.org/10.1103/PhysRevResearch.4.L022047 “Directed percolation in random temporal network models with heterogeneities” https://doi.org/10.1103/PhysRevE.105.054313 Reticula: https://reticula.network https://arxiv.org/abs/2207.10771 Slides: https://speakerdeck.com/arashbm Upshot