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  • Open Access

Spatiotemporal activity-driven networks

Zsófia Simon* and Jari Saramäki

  • *Contact author: zsofia.simon@aalto.fi
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Phys. Rev. E 114, 014306 – Published 7 July, 2026

DOI: https://doi.org/10.1103/9wrc-j8m9

Abstract

Temporal-network models have provided key insights into how time-varying connectivity shapes dynamical processes such as spreading. Among them, the activity-driven model is a widely used, analytically tractable benchmark. Yet many temporal networks, such as those of physical proximity, are also embedded in space, and spatial constraints are known to affect dynamics unfolding on the networks strongly. Despite this, there is a lack of similar simple and solvable models for spatiotemporal contact structures. Here we introduce a spatial activity-driven model in which short-range contacts are more frequent. This model is analytically tractable and captures the joint effects of space and time. We show analytically and numerically that the model reproduces several characteristic features of social and contact networks, including strong and weak ties, clustering, and triangles having weights above the median. These traits can be attributed to space acting as a form of memory. Simulations of spreading dynamics on top of the model networks further illustrate the role of space, highlighting how localization slows down spreading. Furthermore, the framework is well suited for modeling social distancing in a principled way as an intervention measure aimed at reducing long-range links. We find that, unlike for nonspatial networks, even a small spatially targeted reduction in the total number of contacts can be very effective. More broadly, by offering a tractable framework, the model enables systematic exploration of dynamical processes on spatiotemporal networks.

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References (44)

  1. P. Holme and J. Saramäki, Temporal networks, Phys. Rep. 519, 97 (2012).
  2. N. Masuda and P. Holme, eds., Temporal Network Epidemiology (Springer, Singapore, 2017).
  3. J. Saramäki and P. Holme, eds., Temporal Network Theory, 2nd ed. (Springer, Cham, 2023).
  4. A. Vazquez, B. Rácz, A. Lukács, and A.-L. Barabási, Impact of non-Poissonian activity patterns on spreading processes, Phys. Rev. Lett. 98, 158702 (2007).
  5. M. Karsai, M. Kivelä, R. Pan, K. Kaski, J. Kertész, A. Barabási, and J. Saramäki, Small but slow world: How network topology and burstiness slow down spreading, Phys. Rev. E 83, 025102(R) (2011).
  6. S. Unicomb, G. Iñiguez, J. P. Gleeson, and M. Karsai, Dynamics of cascades on burstiness-controlled temporal networks, Nat. Commun. 12, 133 (2021).
  7. N. Perra, B. Gonçalves, R. Pastor-Satorras, and A. Vespignani, Activity driven modeling of time varying networks, Sci. Rep. 2, 469 (2012).
  8. I. Pozzana, K. Sun, and N. Perra, Epidemic spreading on activity-driven networks with attractiveness, Phys. Rev. E 96, 042310 (2017).
  9. S. Liu, N. Perra, M. Karsai, and A. Vespignani, Controlling contagion processes in activity driven networks, Phys. Rev. Lett. 112, 118702 (2014).
  10. G. Laurent, J. Saramäki, and M. Karsai, From calls to communities: A model for time-varying social networks, Eur. Phys. J. B 88, 301 (2015).
  11. M. Barthélemy, Spatial networks, Phys. Rep. 499, 1 (2011).
  12. C. Hens, U. Harush, S. Haber, R. Cohen, and B. Barzel, Spatiotemporal signal propagation in complex networks, Nat. Phys. 15, 403 (2019).
  13. D. Balcan, V. Colizza, B. Gonçalves, H. Hu, J. J. Ramasco, and A. Vespignani, Multiscale mobility networks and the spatial spreading of infectious diseases, Proc. Natl. Acad. Sci. USA 106, 21484 (2009).
  14. V. Belik, T. Geisel, and D. Brockmann, Natural human mobility patterns and spatial spread of infectious diseases, Phys. Rev. X 1, 011001 (2011).
  15. D. Brockmann and D. Helbing, The hidden geometry of complex, network-driven contagion phenomena, Science 342, 1337 (2013).
  16. J. Stehlé et al., Simulation of an SEIR infectious disease model on the dynamic contact network of conference attendees, BMC Med. 9, 87 (2011).
  17. P. Sapiezynski, A. Stopczynski, D. D. Lassen, and S. Lehmann, Interaction data from the Copenhagen networks study, Sci. Data 6, 315 (2019).
  18. F. Schlosser, B. F. Maier, O. Jack, D. Hinrichs, A. Zachariae, and D. Brockmann, COVID-19 lockdown induces disease-mitigating structural changes in mobility networks, Proc. Natl. Acad. Sci. USA 117, 32883 (2020).
  19. M. Ajelli, B. Gonçalves, D. Balcan, V. Colizza, H. Hu, J. Ramasco, S. Merler, and A. Vespignani, Comparing large-scale computational approaches to epidemic modeling: Agent-based versus structured metapopulation models, BMC Infect. Dis. 10, 190 (2010).
  20. M. Chinazzi et al., The effect of travel restrictions on the spread of the 2019 novel coronavirus (COVID-19) outbreak, Science 368, 395 (2020).
  21. A. Arenas, W. Cota, J. Gómez-Gardeñes, S. Gómez, C. Granell, J. Matamalas, D. Soriano-Paños, and B. Steinegger, Modeling the spatiotemporal epidemic spreading of COVID-19 and the impact of mobility and social distancing interventions, Phys. Rev. X 10, 041055 (2020).
  22. J. Jia, X. Lu, Y. Yuan, G. Xu, J. Jia, and N. Christakis, Population flow drives spatio-temporal distribution of COVID-19 in China, Nature (London) 582, 389 (2020).
  23. D. Soriano-Paños, W. Cota, S. Ferreira, G. Ghoshal, A. Arenas, and J. Gómez-Gardeñes, Modeling communicable diseases, human mobility, and epidemics: A review, Ann. Phys. 534, 2100482 (2022).
  24. 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, rsif.2020.1000 (2021).
  25. M. P. Rast, Contact statistics in populations of noninteracting random walkers in two dimensions, Phys. Rev. E 105, 014103 (2022).
  26. R. Masoumi, J. Gambaudo, and M. Génois, Simple crowd dynamics to generate complex temporal contact networks, arXiv:2405.06508.
  27. J.-P. Onnela, J. Saramäki, J. Hyvönen, G. Szabó, M. A. de Menezes, K. Kaski, A.-L. Barabási, and J. Kertész, Analysis of a large-scale weighted network of one-to-one human communication, New J. Phys. 9, 179 (2007).
  28. M. Penrose, Random Geometric Graphs, Oxford Studies in Probability Vol. 5 (Oxford University Press, Oxford, 2003).
  29. J. Powell, P. A. Lewis, N. Roberts, M. Garcia-Finana, and R. I. M. Dunbar, Orbital prefrontal cortex volume predicts social network size: An imaging study of individual differences in humans, Proc. R. Soc. B 279, 2157 (2012).
  30. G. Miritello, E. Moro, R. Lara, R. Martínez-López, S. G. B. Roberts, and R. I. M. Dunbar, Time as a limited resource: Communication strategy in mobile phone networks, Social Netw. 35, 89 (2013).
  31. J. Saramäki, E. A. Leicht, E. López, S. G. Roberts, F. Reed-Tsochas, and R. I. Dunbar, Persistence of social signatures in human communication, Proc. Natl. Acad. Sci. USA 111, 942 (2014).
  32. J. Fournet and A. Barrat, Contact patterns among high school students, PLoS ONE 9, e107878 (2014).
  33. A. Barrat, C. Cattuto, A. Tozzi, P. Vanhems, and N. Voirin, Measuring contact patterns with wearable sensors: Methods, data characteristics and applications to data-driven simulations of infectious diseases, Clin. Microbiol. Infect. 20, 10 (2014).
  34. R. Mastrandrea, J. Fournet, and A. Barrat, Contact patterns in a high school: A comparison between data collected using wearable sensors, contact diaries and friendship surveys, PLoS One 10, e0136497 (2015).
  35. G. Krings, M. Karsai, S. Bernharsson, V. D. Blondel, and J. Saramäki, Effects of time window size and placement on the structure of aggregated networks, EPJ Data Sci. 1, 4 (2012).
  36. S. Wasserman, Social Network Analysis: Methods and Applications (Cambridge University Press, Cambridge, 1994).
  37. M. Barthélemy, Crossover from scale-free to spatial networks, Europhys. Lett. 63, 915 (2003).
  38. J. Dall and M. Christensen, Random geometric graphs, Phys. Rev. E 66, 016121 (2002).
  39. H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev. 42, 599 (2000).
  40. J.-P. Onnela, J. Saramäki, J. Hyvönen, G. Szabó, D. Lazer, K. Kaski, J. Kertész, and A.-L. Barabási, Structure and tie strengths in mobile communication networks, Proc. Natl. Acad. Sci. USA 104, 7332 (2007).
  41. R. Lambiotte, V. D. Blondel, C. de Kerchove, E. Huens, C. Prieur, Z. Smoreda, and P. V. Dooren, Geographical dispersal of mobile communication networks, Physica A 387, 5317 (2008).
  42. J.-P. Onnela, S. Arbesman, M. C. González, A.-L. Barabási, and N. A. Christakis, Geographic constraints on social network groups, PLoS ONE 6, e16939 (2011).
  43. D. Krioukov, F. Papadopoulos, M. Kitsak, A. Vahdat, and M. Boguñá, Hyperbolic geometry of complex networks, Phys. Rev. E 82, 036106 (2010).
  44. M. Boguñá, I. Bonamassa, M. De Domenico, S. Havlin, D. Krioukov, and M. Á. Serrano, Network geometry, Nat. Rev. Phys. 3, 114 (2021).

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Phys. Rev. E 114, 014306– Published 7 July, 2026
  • Received 28 November 2025
  • Accepted 9 June 2026
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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.

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