- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Letter
  • Open Access

Directed percolation in temporal networks

Arash Badie-Modiri1, Abbas K. Rizi1, Márton Karsai2,3, and Mikko Kivelä1

  • 1Department of Computer Science, School of Science, Aalto University, FI-0007 Espoo, Finland
  • 2Department of Network and Data Science Central European University, 1100 Vienna, Austria
  • 3Alfréd Rényi Institute of Mathematics, 1053 Budapest, Hungary

PDF

Phys. Rev. Research 4, L022047 – Published 25 May, 2022

DOI: https://doi.org/10.1103/PhysRevResearch.4.L022047

Abstract

Connectivity and reachability on temporal networks, which can describe the spreading of a disease, the dissemination 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 yet been developed for temporal networks. Here, we 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 a 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 characterizing 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.

View figure in article

Physics Subject Headings (PhySH)

See Also

Directed percolation in random temporal network models with heterogeneities

Arash Badie-Modiri, Abbas K. Rizi, Márton Karsai, and Mikko Kivelä
Phys. Rev. E 105, 054313 (2022)

Article Text

Supplemental Material

References (80)

  1. P. Holme and J. Saramäki, Temporal Network Theory, Computational Social Sciences (Springer, New York, 2019).
  2. R. Lambiotte and N. Masuda, A Guide to Temporal Networks (World Scientific, Singapore, 2016), Vol. 4.
  3. P. Holme and J. Saramäki, Temporal networks, Phys. Rep. 519, 97 (2012).
  4. P. Holme, Modern temporal network theory: A colloquium, Eur. Phys. J. B 88, 234 (2015).
  5. D. J. Daley and D. G. Kendall, Epidemics and rumours, Nature (London) 204, 1118 (1964).
  6. C. Castellano, S. Fortunato, and V. Loreto, Statistical physics of social dynamics, Rev. Mod. Phys. 81, 591 (2009).
  7. C. Tripp-Barba, C. Alcaraz, and M. A. Igartua, Special issue on “Modeling and performance evaluation of wireless ad-hoc networks”, Ad Hoc Networks 52, 1 (2016).
  8. N. Nassir, M. Hickman, A. Malekzadeh, and E. Irannezhad, A utility-based travel impedance measure for public transit network accessibility, Transp. Res. Part A: Policy Pract. 88, 26 (2016).
  9. H. Hinrichsen, Non-equilibrium critical phenomena and phase transitions into absorbing states, Adv. Phys. 49, 815 (2000).
  10. G. Ódor, Universality classes in nonequilibrium lattice systems, Rev. Mod. Phys. 76, 663 (2004).
  11. H. Hinrichsen, Non-equilibrium phase transitions, Physica A (Amsterdam) 369, 1 (2006).
  12. M. Henkel, H. Hinrichsen, S. Lübeck, and M. Pleimling, Non-equilibrium Phase Transitions (Springer, New York, 2008), Vol. 1.
  13. S. R. Broadbent and J. M. Hammersley, Percolation processes: I. Crystals and mazes, Math. Proc. Cambridge Philos. Soc. 53, 629 (1957).
  14. J. Blease, Directed-bond percolation on hypercubic lattices, J. Phys. C: Solid State Phys. 10, 925 (1977).
  15. F. Schlögl, Chemical reaction models for non-equilibrium phase transitions, Z. Phys. 253, 147 (1972).
  16. H. Gerola, P. Seiden, and L. Schulman, Theory of dwarf galaxies, Astrophys. J. 242, 517 (1980).
  17. N. Van Lien and B. Shklovskii, Hopping conduction in strong electric fields and directed percolation, Solid State Commun. 38, 99 (1981).
  18. P. Bak and K. Sneppen, Punctuated Equilibrium and Criticality in a Simple Model of Evolution, Phys. Rev. Lett. 71, 4083 (1993).
  19. E. Domany and W. Kinzel, Equivalence of Cellular Automata to Ising Models and Directed Percolation, Phys. Rev. Lett. 53, 311 (1984).
  20. W. Kinzel, Phase transitions of cellular automata, Z. Phys. B: Condens. Matter 58, 229 (1985).
  21. T. E. Harris, Contact interactions on a lattice, Ann. Probab. 2, 969 (1974).
  22. I. Jensen, Critical Behavior of the Pair Contact Process, Phys. Rev. Lett. 70, 1465 (1993).
  23. J. Mendes, R. Dickman, M. Henkel, and M. C. Marques, Generalized scaling for models with multiple absorbing states, J. Phys. A: Math. Gen. 27, 3019 (1994).
  24. R. M. Ziff, E. Gulari, and Y. Barshad, Kinetic Phase Transitions in an Irreversible Surface-Reaction Model, Phys. Rev. Lett. 56, 2553 (1986).
  25. D. Dhar, The collapse of directed animals, J. Phys. A: Math. Gen. 20, L847 (1987).
  26. H. Hinrichsen, On possible experimental realizations of directed percolation, Braz. J. Phys. 30, 69 (2000).
  27. K. A. Takeuchi, M. Kuroda, H. Chaté, and M. Sano, Directed Percolation Criticality in Turbulent Liquid Crystals, Phys. Rev. Lett. 99, 234503 (2007).
  28. G. Lemoult, L. Shi, K. Avila, S. V. Jalikop, M. Avila, and B. Hof, Directed percolation phase transition to sustained turbulence in Couette flow, Nat. Phys. 12, 254 (2016).
  29. M. Sano and K. Tamai, A universal transition to turbulence in channel flow, Nat. Phys. 12, 249 (2016).
  30. A. Barrat, M. Barthelemy, and A. Vespignani, Dynamical Processes on Complex Networks (Cambridge University Press, Cambridge, 2008).
  31. R. Pastor-Satorras, C. Castellano, P. Van Mieghem, and A. Vespignani, Epidemic processes in complex networks, Rev. Mod. Phys. 87, 925 (2015).
  32. D. Kempe, J. Kleinberg, and A. Kumar, Connectivity and inference problems for temporal networks, J. Comput. Syst. Sci. 64, 820 (2002).
  33. J. Moody, The importance of relationship timing for diffusion, Social Forces 81, 25 (2002).
  34. P. Holme, Network reachability of real-world contact sequences, Phys. Rev. E 71, 046119 (2005).
  35. R. K. Pan and J. Saramäki, Path lengths, correlations, and centrality in temporal networks, Phys. Rev. E 84, 016105 (2011).
  36. I. Scholtes, N. Wider, R. Pfitzner, A. Garas, C. J. Tessone, and F. Schweitzer, Causality-driven slow-down and speed-up of diffusion in non-Markovian temporal networks, Nat. Commun. 5, 5024 (2014).
  37. J. Stehlé, N. Voirin, A. Barrat, C. Cattuto, L. Isella, J.-F. Pinton, M. Quaggiotto, W. Van den Broeck, C. Régis, B. Lina, and P. Vanhems, High-resolution measurements of face-to-face contact patterns in a primary school, PLoS One 6, e23176 (2011).
  38. S. Dai, H. Bouchet, A. Nardy, E. Fleury, J.-P. Chevrot, and M. Karsai, Temporal social network reconstruction using wireless proximity sensors: model selection and consequences, EPJ Data Sci. 9, 19 (2020).
  39. A. Aleta, G. F. de Arruda, and Y. Moreno, Data-driven contact structures: from homogeneous mixing to multilayer networks, PLoS Comput. Biol. 16, e1008035 (2020).
  40. R. Parshani, M. Dickison, R. Cohen, H. E. Stanley, and S. Havlin, Dynamic networks and directed percolation, Europhys. Lett. 90, 38004 (2010).
  41. M. E. J. Newman, Spread of epidemic disease on networks, Phys. Rev. E 66, 016128 (2002).
  42. E. Kenah and J. M. Robins, Second look at the spread of epidemics on networks, Phys. Rev. E 76, 036113 (2007).
  43. E. Kenah and J. C. Miller, Epidemic percolation networks, epidemic outcomes, and interventions, Interdiscip. Perspect. Infect. Dis. 2011, 543520 (2011).
  44. A. K. Rizi, A. Faqeeh, A. Badie-Modiri, and M. Kivelä, Epidemic spreading and digital contact tracing: Effects of heterogeneous mixing and quarantine failures, Phys. Rev. E 105, 044313 (2022).
  45. T. Hiraoka, A. K. Rizi, M. Kivelä, and J. Saramäki, Herd immunity and epidemic size in networks with vaccination homophily, arXiv:2112.07538.
  46. P. Crescenzi, C. Magnien, and A. Marino, Approximating the temporal neighbourhood function of large temporal graphs, Algorithms 12, 211 (2019).
  47. A. Badie-Modiri, M. Karsai, and M. Kivelä, Efficient limited-time reachability estimation in temporal networks, Phys. Rev. E 101, 052303 (2020).
  48. A. Casteigts, A.-S. Himmel, H. Molter, and P. Zschoche, Finding Temporal Paths Under Waiting Time Constraints, Algorithmica 83, 2754 (2021).
  49. S. Thejaswi, J. Lauri, and A. Gionis, Restless reachability problems in temporal graphs, arXiv:2010.08423.
  50. A.-S. Himmel, M. Bentert, A. Nichterlein, and R. Niedermeier, Efficient computation of optimal temporal walks under waiting-time constraints, in International Conference on Complex Networks and Their Applications (Springer, New York, 2019), pp. 494–506.
  51. D. R. De Souza and T. Tomé, Stochastic lattice gas model describing the dynamics of the SIRS epidemic process, Physica A (Amsterdam) 389, 1142 (2010).
  52. M. Kivelä, J. Cambe, J. Saramäki, and M. Karsai, Mapping temporal-network percolation to weighted, static event graphs, Sci. Rep. 8, 12357 (2018).
  53. A. Badie-Modiri, A. K. Rizi, M. Karsai, and M. Kivelä, Directed percolation in random temporal network models with heterogeneities, Phys. Rev. E 105, 054313 (2022).
  54. H. H. K. Lentz, T. Selhorst, and I. M. Sokolov, Unfolding Accessibility Provides a Macroscopic Approach to Temporal Networks, Phys. Rev. Lett. 110, 118701 (2013).
  55. A. Mellor, Event graphs: Advances and applications of second-order time-unfolded temporal network models, Adv. Complex Syst. 22, 1950006 (2019).
  56. J. Saramäki, M. Kivelä, and M. Karsai, Weighted temporal event graphs, in Temporal Network Theory (Springer, New York, 2019), pp. 107–128.
  57. L. Kovanen, M. Karsai, K. Kaski, J. Kertész, and J. Saramäki, Temporal motifs in time-dependent networks, J. Stat. Mech. (2011) P11005.
  58. A. Mellor, Analysing collective behaviour in temporal networks using event graphs and temporal motifs, arXiv:1801.10527.
  59. M. Torricelli, M. Karsai, and L. Gauvin, weg2vec: Event embedding for temporal networks, Sci. Rep. 10, 7164 (2020).
  60. A. Mellor, The temporal event graph, J. Complex Networks 6, 639 (2018).
  61. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.4.L022047 for details on proofs and description of datasets.
  62. P. Grassberger, On phase transitions in Schlögl's second model, Z. Phys. B 47, 365 (1982).
  63. H. K. Janssen, On the nonequilibrium phase transition in reaction-diffusion systems with an absorbing stationary state, Z. Phys. B 42, 151 (1981).
  64. M. E. J. Newman, S. H. Strogatz, and D. J. Watts, Random graphs with arbitrary degree distributions and their applications, Phys. Rev. E 64, 026118 (2001).
  65. P. Grassberger and A. De La Torre, Reggeon field theory (Schlögl's first model) on a lattice: Monte Carlo calculations of critical behaviour, Ann. Phys. (Amsterdam) 122, 373 (1979).
  66. Bureau of Transportation Statistics, Bureau of Transportation Statistics website (2017), https://www.bts.gov/.
  67. R. Kujala, C. Weckström, R. K. Darst, M. N. Mladenović, and J. Saramäki, A collection of public transport network data sets for 25 cities, Sci. Data 5, 180089 (2018).
  68. J. Yang and J. Leskovec, Patterns of temporal variation in online media, in Proceedings of the Fourth ACM International Conference on Web Search and Data Mining (ACM, New York, 2011), pp. 177–186.
  69. M. Karsai, M. Kivelä, R. K. Pan, K. Kaski, J. Kertész, A.-L. Barabási, and J. Saramäki, Small but slow world: How network topology and burstiness slow down spreading, Phys. Rev. E 83, 025102(R) (2011).
  70. S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, Critical phenomena in complex networks, Rev. Mod. Phys. 80, 1275 (2008).
  71. D. R. Chialvo, Emergent complex neural dynamics, Nat. Phys. 6, 744 (2010).
  72. J. Hesse and T. Gross, Self-organized criticality as a fundamental property of neural systems, Front. Syst. Neurosci. 8, 166 (2014).
  73. C.-W. Shin and S. Kim, Self-organized criticality and scale-free properties in emergent functional neural networks, Phys. Rev. E 74, 045101(R) (2006).
  74. P. Bak and C. Tang, Earthquakes as a self-organized critical phenomenon, J. Geophys. Res.: Solid Earth 94, 15635 (1989).
  75. Y. Chen and Y. Zhou, Scaling laws and indications of self-organized criticality in urban systems, Chaos, Solitons Fractals 35, 85 (2008).
  76. P. Bak, C. Tang, and K. Wiesenfeld, Self-Organized Criticality: An Explanation of the 1/f Noise, Phys. Rev. Lett. 59, 381 (1987).
  77. P. Bak, C. Tang, and K. Wiesenfeld, Self-organized criticality, Phys. Rev. A 38, 364 (1988).
  78. B. Blasius, Power-law distribution in the number of confirmed COVID-19 cases, Chaos 30, 093123 (2020).
  79. J. Leitch, K. A. Alexander, and S. Sengupta, Toward epidemic thresholds on temporal networks: a review and open questions, Appl. Network Sci. 4, 105 (2019).
  80. A. Barrat, C. Cattuto, M. Kivelä, S. Lehmann, and J. Saramäki, Effect of manual and digital contact tracing on COVID-19 outbreaks: a study on empirical contact data, J. R. Soc. Interface 18, 20201000 (2021).

Outline

Information

Phys. Rev. Research 4, L022047– Published 25 May, 2022
  • Received 18 October 2021
  • Accepted 6 April 2022
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation